Finite Group Databasebeta

← #378677

Subgroups — #378677

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23451C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic, complemented
3#21304C211yesmaximal, characteristic, complemented
4#20839C211yesmaximal, characteristic
5C2×#1765C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11068C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10924C211yesmaximal, characteristic
11#10887C211yesmaximal, characteristic, complemented
12#10883C211yesmaximal, characteristic, complemented
13#10882C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#2026C412yescomplemented
32#2025C412yescomplemented
33#2024C412yescomplemented
34#2023C412yescomplemented
35#1995—24no—
36#1985—22no—
37#1984—22no—
38#1851—22no—
39#1850—22no—
40#1765—24no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1735C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—41no—
67C2×#131—41no—
68C22×M16—21no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8—21no—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78#533—22no—
79#533—22no—
80#533—41no—
81#533—41no—
82C2×C4⋊Q8C2×C412yes—
83C2×#125C2×C412yes—
84C2×C22⋊Q8—22no—
85C2×C4×Q8—22no—
86C2×#117C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#424—22no—
93#424—22no—
94#424—22no—
95#424—41no—
96#423—22no—
97#423—41no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104—41no—
107C2×#104Q814yescomplemented
108#416—22no—
109#416—22no—
110#416—41no—
111#416—41no—
112C2×#103C2×C412yes—
113C2×#103C2×C412yes—
114C2×#102C2×C412yes—
115C2×#102C2×C412yes—
116#408—22no—
117#408—41no—
118C2×C22⋊C8—22no—
119#405—22no—
120#405—41no—
121C2×#97—21no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—41no—
125C2×C4×C8—21no—
126C2×C4×C8—22no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#386—22no—
134C4×C22⋊C4—22no—
135#376—22no—
136#376—41no—
137C2×#95—22no—
138#341—22no—
139#341—41no—
140#341—41no—
141#341—41no—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339D414yes—
152#336—22no—
153#336—21no—
154#336—22no—
155#336—21no—
156#336—22no—
157#336—22no—
158#336—22no—
159#336C2×C412yes—
160#336C2×C412yes—
161#336D412yes—
162#336D412yes—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41C2×Q812yes—
174C2×#41#4112yes—
175C22×Q8—22no—
176C22×Q8C22×C411yescharacteristic
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184#131—28no—
185#131—44no—
186#131—44no—
187#131—44no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—42no—
213C2×M16C2×D412yes—
214C2×M16C2×Q812yes—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
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—42no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8C22×C412yes—
245C22×C8C22×C412yes—
246C22×C8C2×D412yes—
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8#4112yes—
250C22×C8#4112yes—
251C4⋊Q8—24no—
252C4⋊Q8—42no—
253#125—24no—
254#125—42no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257C4×Q8—44no—
258C4×Q8—44no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117C2×D412yes—
270#117#4112yes—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×Q812yes—
302C2×C42#4112yes—
303C2×C42#4112yes—
304C4⋊C8—44no—
305C4⋊C8—44no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308#104—28no—
309#104—44no—
310#103—24no—
311#103—42no—
312#102—24no—
313#102—42no—
314C22⋊C8—44no—
315#97—44no—
316#97—42no—
317C4×C8—42no—
318C4×C8—44no—
319#95—22no—
320#95—22no—
321#95—44no—
322C24—21no—
323C24C2×#4111yescharacteristic
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—22no—
331#41—22no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—41no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—82no—
348C2×Q8C4×Q812yes—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4C4×Q812yes—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)11yescharacteristic
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C42−1+411yescharacteristic
402M16—22no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—48no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—48no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8#12412yes—
462C2×C8#12412yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4C4×D414yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—44no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—82no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42C22⋊Q814yes—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23C2×C4×D411yescharacteristic
509C23#52911yescharacteristic
510C23#52911yescharacteristic
511C23#54611yescharacteristic
512C23#55111yescharacteristic
513C23#55211yescharacteristic
514C23#55411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—41no—
518Q8—41no—
519Q8—88no—
520D4—22no—
521D4—22no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—41no—
527D4—41no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×Q811yescharacteristic
565C2×C4C4×#4111yescharacteristic
566C2×C4C4×#4111yescharacteristic
567C2×C4#51811yescharacteristic
568C2×C4#52011yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4C2×#12411yescharacteristic
572C2×C4#52911yescharacteristic
573C2×C4#53211yescharacteristic
574C2×C4#53311yescharacteristic
575C2×C4#53811yescharacteristic
576C2×C4#53811yescharacteristic
577C2×C4#54211yescharacteristic
578C2×C4#54211yescharacteristic
579C2×C4D4211yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54611yescharacteristic
582C2×C4#54611yescharacteristic
583C2×C4#54811yescharacteristic
584C2×C4D4×Q811yescharacteristic
585C2×C4#55111yescharacteristic
586C2×C4#55111yescharacteristic
587C2×C4#55411yescharacteristic
588C2×C4#55411yescharacteristic
589C2×C4#55611yescharacteristic
590C2×C4#55911yescharacteristic
591C8—22no—
592C8—22no—
593C8—42no—
594C8—42no—
595C8—42no—
596C8—42no—
597C8—42no—
598C8—81no—
599C8—81no—
600C22—22no—
601C22—22no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—42no—
609C22—44no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22C4×#12411yescharacteristic
614C22#235811yescharacteristic
615C22#236011yescharacteristic
616C22#245011yescharacteristic
617C22#247711yescharacteristic
618C22#252711yescharacteristic
619C22#258911yescharacteristic
620C22#295512yes—
621C22#295712yes—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—81no—
630C4—81no—
631C4C4×C22⋊Q811yescharacteristic
632C4#233711yescharacteristic
633C4#236711yescharacteristic
634C4#236811yescharacteristic
635C4#245211yescharacteristic
636C4#247211yescharacteristic
637C4#249211yescharacteristic
638C4#255711yescharacteristic
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—82no—
644C2#1422011yescharacteristic
645C2#2165911yescharacteristic
646C2#2166311yescharacteristic
647C2#2288712yes—
648C2#2298412yes—
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. C2×#1995C2semidirect
3. #21304C2semidirect
6. #11317C2semidirect
7. #11269C2semidirect
11. #10887C2semidirect
12. #10883C2semidirect
13. #10882C2semidirect
15. #10694C2semidirect
31. #2026C4semidirect
32. #2025C4semidirect
33. #2024C4semidirect
34. #2023C4semidirect
107. C2×#104Q8semidirect

Maximal subgroups

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

subgroupindex
#234512
C2×#19952
#213042
#208392
C2×#17652
#113172
#112692
#110682
#109292
#109242
#108872
#108832
#108822
#107912
#106942