Finite Group Databasebeta

← #378075

Subgroups — #378075

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2C2×#1938C211yesmaximal, characteristic
3#21051C211yesmaximal, characteristic
4C2×#1922C211yesmaximal, characteristic, complemented
5#20439C211yesmaximal, characteristic, complemented
6#11317C211yesmaximal, characteristic, complemented
7#11209C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10934C211yesmaximal, characteristic
11#10933C211yesmaximal, characteristic
12#10909C211yesmaximal, characteristic, complemented
13#10877C211yesmaximal, characteristic, complemented
14#10853C211yesmaximal, characteristic, complemented
15#10707C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2297C2211yescharacteristic
32#2296C2211yescharacteristic
33#2068C411yescharacteristic, complemented
34#2068C411yescharacteristic, complemented
35#2067C411yescharacteristic, complemented
36#2067C411yescharacteristic, complemented
37#2066C411yescharacteristic, complemented
38#2066C411yescharacteristic, complemented
39#2065C411yescharacteristic, complemented
40#2065C411yescharacteristic, complemented
41#1938—24no—
42#1924—21no—
43#1924—21no—
44#1923—21no—
45#1923—21no—
46#1922—24no—
47#1771—21no—
48#1771—21no—
49#1770—21no—
50C8×C4⋊C4—21no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1735C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62#1729C2211yescharacteristic
63#1729C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C4×#95C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#427—22no—
85#427—22no—
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90#424—22no—
91#424D414yes—
92#423—22no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—41no—
99C2×C4⋊C8—41no—
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8C2×C411yescharacteristic
103C2×C4⋊C8C2×C411yescharacteristic
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104—41no—
110#416—22no—
111#416—22no—
112#416—41no—
113#416—41no—
114C2×#103C2×C411yescharacteristic
115C2×#103C2×C411yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122#408—22no—
123#408—22no—
124#408—22no—
125#408—41no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130#405—22no—
131#405—22no—
132#405—22no—
133#405—22no—
134#405—22no—
135#405—41no—
136C4×M16—22no—
137C4×M16—41no—
138C2×#97—21no—
139C2×#97—22no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—22no—
144C2×#97—41no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149#391C2×C411yescharacteristic
150#391C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#390C2×C411yescharacteristic
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#386—21no—
158#386—21no—
159#385—21no—
160#385—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C2×#95—21no—
166C2×#95—21no—
167#341—22no—
168#341—22no—
169#341—22no—
170#341—22no—
171#340C2×C411yescharacteristic
172#340C2×C411yescharacteristic
173#340C2×C411yescharacteristic
174#340C2×C411yescharacteristic
175#339—22no—
176#339—22no—
177#339—22no—
178#339—22no—
179#339—22no—
180#339—41no—
181#336—22no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—22no—
187#336—41no—
188#336D411yescharacteristic
189#336D411yescharacteristic
190#336D411yescharacteristic
191#336D411yescharacteristic
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×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C22×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4C22×C411yescharacteristic
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4#4111yescharacteristic
211C23×C4#4111yescharacteristic
212C23×C4C2411yescharacteristic
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
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×M16C2×D412yes—
233C2×M16#4112yes—
234C2×M16#4112yes—
235C2×M16#4112yes—
236C2×M16#4112yes—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
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×C8C22×C411yescharacteristic
264C22×C8C22×C411yescharacteristic
265C22×C8C22×C411yescharacteristic
266C22×C8C22×C411yescharacteristic
267C22×C8C2×D411yescharacteristic
268C22×C8C2×D411yescharacteristic
269C22×C8C2×D412yes—
270C22×C8C2×D411yescharacteristic
271C22×C8C2×D411yescharacteristic
272C22×C8#4112yes—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117C2×D412yes—
284#117#4112yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—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×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4112yes—
325#107—22no—
326#107—22no—
327#106—22no—
328#106—22no—
329C4⋊C8—24no—
330C4⋊C8—24no—
331C4⋊C8—41no—
332C4⋊C8—41no—
333C4⋊C8—41no—
334C4⋊C8—41no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337#104—44no—
338#104—44no—
339#103—22no—
340#103—22no—
341#103—41no—
342#103—41no—
343#102—22no—
344#102—22no—
345#102—41no—
346#102—41no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—24no—
352#97—44no—
353#97—42no—
354#97—42no—
355#97—42no—
356C4×C8—24no—
357C4×C8—42no—
358C4×C8—42no—
359C4×C8—42no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—42no—
369#95—42no—
370C24—21no—
371C24C2×#4111yescharacteristic
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387#41—41no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
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×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4C4×D411yescharacteristic
444C22×C4C4×D411yescharacteristic
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C4H(5,𝔽2)11yescharacteristic
457C22×C42−1+411yescharacteristic
458C22×C42−1+411yescharacteristic
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16#12614yes—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—21no—
482C2×C8—22no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
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—22no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
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#12112yes—
526C2×C8#12112yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4C4×D412yes—
542C4⋊C4C4×D412yes—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—42no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—81no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—22no—
558C42—22no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—22no—
564C42—22no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23C2×C4×D411yescharacteristic
587C23#52911yescharacteristic
588C23#53211yescharacteristic
589C23#55211yescharacteristic
590C23#55311yescharacteristic
591C23#55311yescharacteristic
592C23#55611yescharacteristic
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597D4—41no—
598D4—41no—
599D4—41no—
600D4—41no—
601D4—81no—
602D4—81no—
603D4—81no—
604D4—81no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4—81no—
658C2×C4—81no—
659C2×C4#44412yes—
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C4×#4111yescharacteristic
662C2×C4C4×#4111yescharacteristic
663C2×C4C4×#4111yescharacteristic
664C2×C4#51811yescharacteristic
665C2×C4#51911yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4#52511yescharacteristic
668C2×C4C2×#12611yescharacteristic
669C2×C4#52911yescharacteristic
670C2×C4#52911yescharacteristic
671C2×C4#52911yescharacteristic
672C2×C4#53911yescharacteristic
673C2×C4#53911yescharacteristic
674C2×C4#54211yescharacteristic
675C2×C4#54211yescharacteristic
676C2×C4#54611yescharacteristic
677C2×C4#54611yescharacteristic
678C2×C4#54611yescharacteristic
679C2×C4#54711yescharacteristic
680C2×C4#54711yescharacteristic
681C2×C4#54811yescharacteristic
682C2×C4#54811yescharacteristic
683C2×C4#55011yescharacteristic
684C2×C4#55211yescharacteristic
685C2×C4#55311yescharacteristic
686C2×C4#55511yescharacteristic
687C2×C4#55611yescharacteristic
688C8—22no—
689C8—22no—
690C8—24no—
691C8—41no—
692C8—41no—
693C8—41no—
694C8—41no—
695C8—42no—
696C8—42no—
697C8—41no—
698C8—41no—
699C8—81no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—21no—
709C22—21no—
710C22—21no—
711C22—21no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22C4×#12111yescharacteristic
720C22#236211yescharacteristic
721C22#236411yescharacteristic
722C22#251211yescharacteristic
723C22#252911yescharacteristic
724C22#255811yescharacteristic
725C22#259411yescharacteristic
726C22C2×#44411yescharacteristic
727C22#295511yescharacteristic
728C22#295611yescharacteristic
729C22#295711yescharacteristic
730C4—22no—
731C4—42no—
732C4—42no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—82no—
737C4—82no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4—81no—
742C4—81no—
743C4C4×#12611yescharacteristic
744C4#233211yescharacteristic
745C4#236211yescharacteristic
746C4#236311yescharacteristic
747C4#242511yescharacteristic
748C4#242611yescharacteristic
749C4#247411yescharacteristic
750C4#248111yescharacteristic
751C2—21no—
752C2—21no—
753C2—21no—
754C2—21no—
755C2—161no—
756C2#1420611yescharacteristic
757C2#2151811yescharacteristic
758C2#2151911yescharacteristic
759C2#2308111yescharacteristic
760C2#2308211yescharacteristic
761C2#2314711yescharacteristic
762C2#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
4. C2×#1922C2semidirect
5. #20439C2semidirect
6. #11317C2semidirect
7. #11209C2semidirect
12. #10909C2semidirect
13. #10877C2semidirect
14. #10853C2semidirect
15. #10707C2semidirect
33. #2068C4semidirect
34. #2068C4semidirect
35. #2067C4semidirect
36. #2067C4semidirect
37. #2066C4semidirect
38. #2066C4semidirect
39. #2065C4semidirect
40. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
C2×#19382
#210512
C2×#19222
#204392
#113172
#112092
#111912
#109342
#109332
#109092
#108772
#108532
#107072