Finite Group Databasebeta

← #373587

Subgroups — #373587

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21058C211yesmaximal, characteristic
2#21052C211yesmaximal, characteristic
3#20431C211yesmaximal, characteristic, complemented
4#11376C211yesmaximal, characteristic, complemented
5#11269C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11053C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10934C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10876C211yesmaximal, characteristic, complemented
13#10809C211yesmaximal, characteristic, complemented
14#10808C211yesmaximal, characteristic, complemented
15#10706C211yesmaximal, characteristic, complemented
16C2×#427C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1928C411yescharacteristic, complemented
32#1928C411yescharacteristic, complemented
33#1927C411yescharacteristic, complemented
34#1927C411yescharacteristic, complemented
35#1926C411yescharacteristic, complemented
36#1926C411yescharacteristic, complemented
37#1925C411yescharacteristic, complemented
38#1925C411yescharacteristic, complemented
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1768—21no—
44#1768—21no—
45#1767—21no—
46C4×C4⋊C8—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#1728C2211yescharacteristic
62C2×#336C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C4×#95C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#427—41no—
83#427—41no—
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88#423—22no—
89#423—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×#104—22no—
99C2×#104—22no—
100C2×#104—22no—
101C2×#104—41no—
102C2×#104—41no—
103C2×#104—41no—
104#416—22no—
105#416—22no—
106#416—41no—
107#416—41no—
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116#408—22no—
117#408—22no—
118#408—22no—
119#408—41no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—22no—
125#405—22no—
126#405—22no—
127#405—41no—
128#405—41no—
129#405—41no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8—22no—
137C2×C4×C8—41no—
138C2×C4×C8—41no—
139#391C2×C411yescharacteristic
140#391C2×C411yescharacteristic
141#390C2×C411yescharacteristic
142#390C2×C411yescharacteristic
143#386—21no—
144#386—21no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C22⋊C4—21no—
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153#376—21no—
154C2×#95—21no—
155C2×#95—21no—
156C43—21no—
157#341—22no—
158#341—22no—
159#341—41no—
160#341—41no—
161#340C2×C411yescharacteristic
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#339—22no—
166#339—22no—
167#339—22no—
168#339—22no—
169#339—22no—
170#339—41no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—22no—
178#336—21no—
179#336—21no—
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—21no—
197C2×#41—21no—
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
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
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#4112yes—
229C2×M16#4112yes—
230C2×M16#4112yes—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
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×C8C22×C411yescharacteristic
258C22×C8C22×C411yescharacteristic
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8#4112yes—
266C22×C8#4112yes—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117#4112yes—
278#117#4112yes—
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—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×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4112yes—
319#107—41no—
320#107—41no—
321#106—41no—
322#106—41no—
323C4⋊C8—24no—
324C4⋊C8—24no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—22no—
328C4⋊C8—22no—
329#104—44no—
330#104—44no—
331#103—22no—
332#103—22no—
333#103—41no—
334#103—41no—
335#102—22no—
336#102—22no—
337#102—41no—
338#102—41no—
339C22⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343#97—42no—
344#97—42no—
345#97—42no—
346C4×C8—42no—
347C4×C8—42no—
348C4×C8—42no—
349C4×C8—44no—
350#95—24no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—42no—
360C24—21no—
361C24C2×#4111yescharacteristic
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41—41no—
374#41—41no—
375#41—41no—
376#41—41no—
377#41—41no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
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×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
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×C42−1+411yescharacteristic
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—22no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—22no—
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—22no—
488C2×C8—22no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—41no—
495C2×C8—41no—
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#12312yes—
516C2×C8#12312yes—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4#11712yes—
528C4⋊C4#11712yes—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—42no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—81no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42—42no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—81no—
564C42—81no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23C2×C4×D411yescharacteristic
577C23#52911yescharacteristic
578C23#52911yescharacteristic
579C23#55211yescharacteristic
580C23#55311yescharacteristic
581C23#55311yescharacteristic
582C23#55511yescharacteristic
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—81no—
592D4—81no—
593D4—81no—
594D4—81no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
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—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4#44312yes—
650C2×C4C2×#11711yescharacteristic
651C2×C4C4×#4111yescharacteristic
652C2×C4C4×#4111yescharacteristic
653C2×C4C4×#4111yescharacteristic
654C2×C4#51811yescharacteristic
655C2×C4#52011yescharacteristic
656C2×C4C2×#12311yescharacteristic
657C2×C4#52511yescharacteristic
658C2×C4#52911yescharacteristic
659C2×C4#52911yescharacteristic
660C2×C4#52911yescharacteristic
661C2×C4#52911yescharacteristic
662C2×C4#53811yescharacteristic
663C2×C4#53811yescharacteristic
664C2×C4#54211yescharacteristic
665C2×C4#54211yescharacteristic
666C2×C4#54611yescharacteristic
667C2×C4#54611yescharacteristic
668C2×C4#54811yescharacteristic
669C2×C4#55011yescharacteristic
670C2×C4#55111yescharacteristic
671C2×C4#55211yescharacteristic
672C2×C4#55311yescharacteristic
673C2×C4#55311yescharacteristic
674C2×C4#55311yescharacteristic
675C2×C4#55611yescharacteristic
676C2×C4#55911yescharacteristic
677C2×C4#55911yescharacteristic
678C8—22no—
679C8—22no—
680C8—41no—
681C8—41no—
682C8—41no—
683C8—41no—
684C8—41no—
685C8—41no—
686C8—42no—
687C8—42no—
688C8—42no—
689C8—81no—
690C22—21no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—21no—
700C22—21no—
701C22—21no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22—81no—
709C22C4×#12311yescharacteristic
710C22#235211yescharacteristic
711C22#236311yescharacteristic
712C22#252511yescharacteristic
713C22#252911yescharacteristic
714C22#259311yescharacteristic
715C22#259511yescharacteristic
716C22C2×#44311yescharacteristic
717C22#295511yescharacteristic
718C22#295611yescharacteristic
719C22#295711yescharacteristic
720C4—22no—
721C4—44no—
722C4—42no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—42no—
727C4—82no—
728C4—81no—
729C4—81no—
730C4—81no—
731C4—81no—
732C4—81no—
733C4#229811yescharacteristic
734C4#233711yescharacteristic
735C4#235711yescharacteristic
736C4#237311yescharacteristic
737C4#244411yescharacteristic
738C4#244811yescharacteristic
739C4#245811yescharacteristic
740C4#246711yescharacteristic
741C2—21no—
742C2—21no—
743C2—21no—
744C2—21no—
745C2—161no—
746C2#1412511yescharacteristic
747C2#2159211yescharacteristic
748C2#2159411yescharacteristic
749C2#2278311yescharacteristic
750C2#2278411yescharacteristic
751C2#2311311yescharacteristic
752C2#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. #20431C2semidirect
4. #11376C2semidirect
5. #11269C2semidirect
11. #10909C2semidirect
12. #10876C2semidirect
13. #10809C2semidirect
14. #10808C2semidirect
15. #10706C2semidirect
31. #1928C4semidirect
32. #1928C4semidirect
33. #1927C4semidirect
34. #1927C4semidirect
35. #1926C4semidirect
36. #1926C4semidirect
37. #1925C4semidirect
38. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#210522
#204312
#113762
#112692
#111912
#110532
#109342
#109292
#109092
#108762
#108092
#108082
#107062