Finite Group Databasebeta

← #384923

Subgroups — #384923

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#20838C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic, complemented
4#11376C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11269C211yesmaximal, characteristic, complemented
7#11221C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11053C211yesmaximal, characteristic
10#10930C211yesmaximal, characteristic
11#10883C211yesmaximal, characteristic, complemented
12#10877C211yesmaximal, characteristic, complemented
13#10808C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2297C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1928C411yescharacteristic, complemented
30#1928C411yescharacteristic, complemented
31#1927C411yescharacteristic, complemented
32#1927C411yescharacteristic, complemented
33#1926C411yescharacteristic, complemented
34#1926C411yescharacteristic, complemented
35#1925C411yescharacteristic, complemented
36#1925C411yescharacteristic, complemented
37#1851—21no—
38#1851—21no—
39#1850—21no—
40#1850—21no—
41#1832—21no—
42#1832—21no—
43#1831—21no—
44#1831—21no—
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49#1735C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59#1729C2211yescharacteristic
60#1729C2211yescharacteristic
61#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—41no—
71C2×#131—41no—
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#424—22no—
97#424—41no—
98#423—41no—
99#423—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8D411yescharacteristic
108C2×C4⋊C8D411yescharacteristic
109C2×C4⋊C8D411yescharacteristic
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—22no—
113C2×#104—41no—
114C2×#104—41no—
115C2×#104—41no—
116#416—22no—
117#416—22no—
118#416—41no—
119#416—41no—
120C2×#103C2×C411yescharacteristic
121C2×#103C2×C411yescharacteristic
122C2×#103C2×C411yescharacteristic
123C2×#103C2×C411yescharacteristic
124C2×#102C2×C411yescharacteristic
125C2×#102C2×C411yescharacteristic
126C2×#102C2×C411yescharacteristic
127C2×#102C2×C411yescharacteristic
128#408—22no—
129#408—41no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×#97—41no—
133C2×#97—41no—
134C2×#97—41no—
135#391C2×C411yescharacteristic
136#391C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#386—21no—
140#386—21no—
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2×C411yescharacteristic
143C4×C4⋊C4C2×C411yescharacteristic
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—21no—
150C2×#95—21no—
151C2×#95—21no—
152C43—21no—
153#341—41no—
154#341—41no—
155#341—41no—
156#341—41no—
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#339—22no—
162#339—22no—
163#339—22no—
164#339—41no—
165#339—41no—
166#339—41no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—41no—
180#336—41no—
181#336—41no—
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184#336C2×C411yescharacteristic
185#336C2×C411yescharacteristic
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41#4112yes—
197C2×#41#4112yes—
198C22×Q8C22×C411yescharacteristic
199C22×D4—21no—
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4#4111yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4C2411yescharacteristic
207#131—44no—
208#131—44no—
209#131—44no—
210#131—44no—
211C2×M16—21no—
212C2×M16—21no—
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×M16—21no—
233C2×M16—21no—
234C2×M16—42no—
235C2×M16—42no—
236C2×M16—42no—
237C2×M16#4112yes—
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×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8C22×C411yescharacteristic
270C22×C8C22×C411yescharacteristic
271C22×C8C22×C411yescharacteristic
272C22×C8C22×C411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C22×C8C2×D411yescharacteristic
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117—21no—
288#117—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×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42#4112yes—
329#107—41no—
330#107—41no—
331#106—41no—
332#106—41no—
333C4⋊C8—41no—
334C4⋊C8—41no—
335C4⋊C8—41no—
336C4⋊C8—41no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#104—44no—
342#104—44no—
343#103—22no—
344#103—22no—
345#103—41no—
346#103—41no—
347#102—22no—
348#102—22no—
349#102—41no—
350#102—41no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353#97—44no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—42no—
363#95—42no—
364C24—21no—
365C24C2×#4111yescharacteristic
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8#11712yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4#11712yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
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×C4C4×D411yescharacteristic
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C42−1+411yescharacteristic
451C22×C42−1+411yescharacteristic
452C22×C42−1+411yescharacteristic
453M16—22no—
454M16—22no—
455M16—48no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—48no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
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—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C22⋊C4—42no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—41no—
557C42—41no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—42no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—81no—
570C42—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C2×C4×D411yescharacteristic
583C23#53911yescharacteristic
584C23#53911yescharacteristic
585C23#55211yescharacteristic
586C23#55311yescharacteristic
587C23#55311yescharacteristic
588C23#55511yescharacteristic
589Q8—22no—
590Q8—22no—
591Q8—41no—
592Q8—41no—
593D4—22no—
594D4—22no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—42no—
599D4—41no—
600D4—41no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
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#44412yes—
656C2×C4C2×#11711yescharacteristic
657C2×C4C4×#4111yescharacteristic
658C2×C4#51811yescharacteristic
659C2×C4#52011yescharacteristic
660C2×C4#52011yescharacteristic
661C2×C4#52011yescharacteristic
662C2×C4#53411yescharacteristic
663C2×C4#53611yescharacteristic
664C2×C4#53811yescharacteristic
665C2×C4#53811yescharacteristic
666C2×C4#53811yescharacteristic
667C2×C4#53911yescharacteristic
668C2×C4#53911yescharacteristic
669C2×C4#53911yescharacteristic
670C2×C4#54111yescharacteristic
671C2×C4#54211yescharacteristic
672C2×C4#54611yescharacteristic
673C2×C4#54611yescharacteristic
674C2×C4#54811yescharacteristic
675C2×C4#55011yescharacteristic
676C2×C4#55111yescharacteristic
677C2×C4#55111yescharacteristic
678C2×C4#55211yescharacteristic
679C2×C4#55211yescharacteristic
680C2×C4H(3,𝔽4)11yescharacteristic
681C2×C4#56211yescharacteristic
682C2×C4#56211yescharacteristic
683C2×C4#56311yescharacteristic
684C8—41no—
685C8—41no—
686C8—41no—
687C8—41no—
688C8—41no—
689C8—41no—
690C8—41no—
691C8—41no—
692C8—42no—
693C8—81no—
694C8—81no—
695C8—81no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—21no—
703C22—21no—
704C22—21no—
705C22—21no—
706C22—21no—
707C22—21no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—81no—
712C22—81no—
713C22—81no—
714C22—81no—
715C22#232211yescharacteristic
716C22#235211yescharacteristic
717C22#236311yescharacteristic
718C22#274411yescharacteristic
719C22#276511yescharacteristic
720C22#278111yescharacteristic
721C22#278211yescharacteristic
722C22C2×#44411yescharacteristic
723C22#295511yescharacteristic
724C22#295611yescharacteristic
725C22#295711yescharacteristic
726C4—22no—
727C4—42no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—82no—
732C4—82no—
733C4—81no—
734C4—81no—
735C4—81no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4#231811yescharacteristic
740C4#233411yescharacteristic
741C4#234311yescharacteristic
742C4#237911yescharacteristic
743C4#268411yescharacteristic
744C4#269011yescharacteristic
745C4#269411yescharacteristic
746C4#273011yescharacteristic
747C2—21no—
748C2—21no—
749C2—21no—
750C2—21no—
751C2—82no—
752C2#1439311yescharacteristic
753C2#2209611yescharacteristic
754C2#2209711yescharacteristic
755C2#2278211yescharacteristic
756C2#2278411yescharacteristic
757C2#2311411yescharacteristic
758C2#2311511yescharacteristic
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 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20755C2semidirect
4. #11376C2semidirect
5. #11317C2semidirect
6. #11269C2semidirect
7. #11221C2semidirect
11. #10883C2semidirect
12. #10877C2semidirect
13. #10808C2semidirect
29. #1928C4semidirect
30. #1928C4semidirect
31. #1927C4semidirect
32. #1927C4semidirect
33. #1926C4semidirect
34. #1926C4semidirect
35. #1925C4semidirect
36. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#208382
#207552
#113762
#113172
#112692
#112212
#111912
#110532
#109302
#108832
#108772
#108082
#107912