Finite Group Databasebeta

← #378601

Subgroups — #378601

Order 512.

957 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21027C211yesmaximal, characteristic, complemented
3#20387C211yesmaximal, characteristic
4#11376C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11053C211yesmaximal, characteristic
7#10934C211yesmaximal, characteristic
8#10929C211yesmaximal, characteristic
9#10909C211yesmaximal, characteristic, complemented
10#10886C211yesmaximal, characteristic, complemented
11#10876C211yesmaximal, characteristic, complemented
12#10853C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic
14#10791C211yesmaximal, characteristic
15#10706C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30#2296C2211yescharacteristic
31#2068C411yescharacteristic, complemented
32#2068C411yescharacteristic, complemented
33#2067C411yescharacteristic, complemented
34#2067C411yescharacteristic, complemented
35#2066C411yescharacteristic, complemented
36#2066C411yescharacteristic, complemented
37#2065C411yescharacteristic, complemented
38#2065C411yescharacteristic, complemented
39#1917—21no—
40#1917—21no—
41#1916—21no—
42#1916—21no—
43#1753—21no—
44#1753—21no—
45#1752—21no—
46C8×C22⋊C4—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131—41no—
73C2×#131—41no—
74C22×M16—21no—
75C22×M16C2311yescharacteristic
76C22×M16C2311yescharacteristic
77C22×M16C2311yescharacteristic
78C22×M16C2311yescharacteristic
79C23×C8—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#427—41no—
91#427—41no—
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96#423—22no—
97#423—41no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8—41no—
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×C4⋊C8C2×C411yescharacteristic
105C2×C4⋊C8C2×C411yescharacteristic
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—41no—
111C2×#104—41no—
112#416—41no—
113#416—41no—
114#416—41no—
115#416—41no—
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#103C2×C411yescharacteristic
119C2×#103C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124#408—22no—
125#408Q814yes—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128#405—22no—
129#405—22no—
130#405—41no—
131#405—41no—
132C4×M16—22no—
133C4×M16—41no—
134C2×#97—21no—
135C2×#97—21no—
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8—22no—
140C2×C4×C8—21no—
141C2×C4×C8—41no—
142C2×C4×C8—41no—
143#391C2×C411yescharacteristic
144#391C2×C411yescharacteristic
145#390C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#386—21no—
152#386—21no—
153#385—21no—
154#385—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C2×#95—21no—
160C2×#95—21no—
161#341—22no—
162#341—22no—
163#341—41no—
164#341—41no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#339—22no—
170#339—22no—
171#339—22no—
172#339—22no—
173#339—41no—
174#339—41no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336—21no—
183#336—22no—
184#336—41no—
185#336—41no—
186#336D411yescharacteristic
187#336D411yescharacteristic
188#336D411yescharacteristic
189#336D411yescharacteristic
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41C2×Q812yes—
201C2×#41#4112yes—
202C22×Q8C22×C411yescharacteristic
203C22×D4—21no—
204C22×D4C22×C411yescharacteristic
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4#4111yescharacteristic
209C23×C4#4111yescharacteristic
210C23×C4C2411yescharacteristic
211#131—44no—
212#131—44no—
213#131—44no—
214#131—44no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—24no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—42no—
236C2×M16—42no—
237C2×M16C2×Q812yes—
238C2×M16#4112yes—
239C2×M16#4112yes—
240C2×M16#4112yes—
241C2×M16#4112yes—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—24no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—42no—
270C22×C8—42no—
271C22×C8C22×C411yescharacteristic
272C22×C8C22×C411yescharacteristic
273C22×C8C22×C411yescharacteristic
274C22×C8C22×C411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×Q812yes—
280C22×C8#4112yes—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290#117—21no—
291#117—21no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42#4112yes—
333#107—41no—
334#107—41no—
335#106—41no—
336#106—41no—
337C4⋊C8—41no—
338C4⋊C8—41no—
339C4⋊C8—41no—
340C4⋊C8—41no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343#104—44no—
344#104—44no—
345#103—41no—
346#103—41no—
347#103—41no—
348#103—41no—
349#102—41no—
350#102—41no—
351#102—41no—
352#102—41no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355#97—42no—
356#97—42no—
357#97—42no—
358C4×C8—42no—
359C4×C8—42no—
360C4×C8—42no—
361C4×C8—44no—
362#95—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—42no—
372C24—21no—
373C24C2×#4111yescharacteristic
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—22no—
380#41—22no—
381#41—22no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C4×Q812yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4C4×Q812yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C4×D411yescharacteristic
446C22×C4C4×D411yescharacteristic
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×Q811yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4H(5,𝔽2)11yescharacteristic
458C22×C4H(5,𝔽2)11yescharacteristic
459C22×C4H(5,𝔽2)11yescharacteristic
460C22×C4H(5,𝔽2)11yescharacteristic
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—48no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16#12614yes—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—22no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—48no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8—41no—
518C2×C8—41no—
519C2×C8—41no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—41no—
523C2×C8—41no—
524C2×C8—41no—
525C2×C8—41no—
526C2×C8—41no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8C22⋊Q812yes—
530C2×C8C22⋊Q812yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C22⋊C4—42no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—42no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42—41no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—22no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23C2×C4×D411yescharacteristic
591C23#52911yescharacteristic
592C23#53311yescharacteristic
593C23#55311yescharacteristic
594C23#55311yescharacteristic
595C23#55311yescharacteristic
596C23#55511yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—41no—
600Q8—41no—
601D4—22no—
602D4—22no—
603D4—42no—
604D4—42no—
605D4—42no—
606D4—42no—
607D4—41no—
608D4—41no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—24no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—81no—
661C2×C4—81no—
662C2×C4—81no—
663C2×C4#44312yes—
664C2×C4C2×C4×Q811yescharacteristic
665C2×C4C4×#4111yescharacteristic
666C2×C4C4×#4111yescharacteristic
667C2×C4C4×#4111yescharacteristic
668C2×C4#51811yescharacteristic
669C2×C4#51811yescharacteristic
670C2×C4C2×C22⋊Q811yescharacteristic
671C2×C4#52511yescharacteristic
672C2×C4C2×#12611yescharacteristic
673C2×C4#52911yescharacteristic
674C2×C4#52911yescharacteristic
675C2×C4#52911yescharacteristic
676C2×C4#53811yescharacteristic
677C2×C4#53811yescharacteristic
678C2×C4#54011yescharacteristic
679C2×C4#54011yescharacteristic
680C2×C4#54611yescharacteristic
681C2×C4#54611yescharacteristic
682C2×C4#54611yescharacteristic
683C2×C4#55011yescharacteristic
684C2×C4#55111yescharacteristic
685C2×C4#55111yescharacteristic
686C2×C4#55111yescharacteristic
687C2×C4#55111yescharacteristic
688C2×C4#55411yescharacteristic
689C2×C4#55411yescharacteristic
690C2×C4#55411yescharacteristic
691C2×C4#55411yescharacteristic
692C8—22no—
693C8—22no—
694C8—24no—
695C8—41no—
696C8—41no—
697C8—41no—
698C8—41no—
699C8—42no—
700C8—42no—
701C8—41no—
702C8—41no—
703C8—81no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—21no—
713C22—21no—
714C22—21no—
715C22—21no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—81no—
720C22—81no—
721C22—81no—
722C22—81no—
723C22C4×C22⋊Q811yescharacteristic
724C22#236311yescharacteristic
725C22#236411yescharacteristic
726C22#249711yescharacteristic
727C22#252611yescharacteristic
728C22#257611yescharacteristic
729C22#258711yescharacteristic
730C22C2×#44311yescharacteristic
731C22#295511yescharacteristic
732C22#295611yescharacteristic
733C22#295711yescharacteristic
734C4—22no—
735C4—44no—
736C4—41no—
737C4—41no—
738C4—41no—
739C4—42no—
740C4—82no—
741C4—81no—
742C4—81no—
743C4—81no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4C4×#12611yescharacteristic
748C4#233311yescharacteristic
749C4#236811yescharacteristic
750C4#236811yescharacteristic
751C4#245011yescharacteristic
752C4#245411yescharacteristic
753C4#247111yescharacteristic
754C4#248011yescharacteristic
755C2—21no—
756C2—21no—
757C2—21no—
758C2—21no—
759C2—82no—
760C2#1421811yescharacteristic
761C2#2162711yescharacteristic
762C2#2162911yescharacteristic
763C2#2311311yescharacteristic
764C2#2311511yescharacteristic
765C2#2314611yescharacteristic
766C2#2314811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21027C2semidirect
4. #11376C2semidirect
5. #11317C2semidirect
9. #10909C2semidirect
10. #10886C2semidirect
11. #10876C2semidirect
12. #10853C2semidirect
15. #10706C2semidirect
31. #2068C4semidirect
32. #2068C4semidirect
33. #2067C4semidirect
34. #2067C4semidirect
35. #2066C4semidirect
36. #2066C4semidirect
37. #2065C4semidirect
38. #2065C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#236092
#210272
#203872
#113762
#113172
#110532
#109342
#109292
#109092
#108862
#108762
#108532
#107912
#107062