Finite Group Databasebeta

← #359203

Subgroups — #359203

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2960C211yesmaximal, characteristic, complemented
2C2×#2031C211yesmaximal, characteristic
3#20420C211yesmaximal, characteristic
4C2×#1764C211yesmaximal, characteristic
5C2×#1757C211yesmaximal, characteristic
6#11387C211yesmaximal, characteristic, complemented
7#11281C211yesmaximal, characteristic, complemented
8#11264C211yesmaximal, characteristic, complemented
9#11261C211yesmaximal, characteristic, complemented
10#11092C211yesmaximal, characteristic
11#11086C211yesmaximal, characteristic
12#11082C211yesmaximal, characteristic
13#10711C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15#10699C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#2960—24no—
19#2960—24no—
20C2×#446—22no—
21C2×C8×Q8—22no—
22C2×#424C2211yescharacteristic
23C2×#424C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C22×#104C2211yescharacteristic
27#2896C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#413C2211yescharacteristic
30C2×#410C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×#405C2211yescharacteristic
34C2×#405C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2031C418yescomplemented
38#1765—22no—
39#1764—24no—
40#1764—22no—
41#1757—24no—
42C2×#344C2211yescharacteristic
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46#1742C2211yescharacteristic
47#1740C2211yescharacteristic
48#1738C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1733C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#338C2211yescharacteristic
57C2×H(3,ℤ/4)C2211yescharacteristic
58C2×H(3,ℤ/4)C2211yescharacteristic
59C2×H(3,ℤ/4)C2211yescharacteristic
60C22×#41C2311yescharacteristic
61C2×#131—22no—
62C2×#131—22no—
63C2×#131—21no—
64C2×#131—21no—
65C2×#131—21no—
66C2×#131—22no—
67C2×#131—21no—
68C2×#131Q814yescomplemented
69C22×M16—21no—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C22×M16C2311yescharacteristic
76C23×C8—21no—
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79#533—22no—
80#533—22no—
81#533—22no—
82#533C2×C414yes—
83C2×C4⋊Q8—21no—
84C2×#125—21no—
85C2×C22⋊Q8—22no—
86C2×C4×Q8—22no—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C2×#117C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#446—216no—
94C8×Q8—216no—
95#424—22no—
96#424—22no—
97#424—22no—
98#424—22no—
99#423—22no—
100#423—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104#421—22no—
105#421C2×C414yes—
106C2×#104—22no—
107C2×#104—22no—
108C2×#104—21no—
109C2×#104—21no—
110C2×#104C2×C414yes—
111#419—21no—
112#418—21no—
113#417—22no—
114#416—22no—
115#416—22no—
116#416—22no—
117C2×#103—21no—
118C2×#102—21no—
119#413—22no—
120#413—22no—
121C2×#101—21no—
122C2×#100—21no—
123#410—22no—
124#410C2×C414yes—
125C2×#99—22no—
126#405—22no—
127#405—22no—
128#405—22no—
129#405—22no—
130#405—22no—
131#405—21no—
132#405—21no—
133#405—22no—
134#405—21no—
135#405—22no—
136#405—21no—
137#405D414yes—
138C2×#97—21no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97D412yes—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—22no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8D414yes—
151C2×C4×C8D412yes—
152#344—22no—
153#344—22no—
154#344—22no—
155#344C2×C414yes—
156#343—22no—
157#342—22no—
158#341—22no—
159#341—22no—
160#341—22no—
161#340—22no—
162#339—22no—
163#339—22no—
164#339—21no—
165#339—21no—
166#339C2×C414yes—
167#338—22no—
168H(3,ℤ/4)—22no—
169H(3,ℤ/4)—22no—
170H(3,ℤ/4)C2×C414yes—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41C22×C412yes—
182C2×#41C2×Q812yes—
183C22×Q8—22no—
184C22×Q8#4111yescharacteristic
185C22×D4—21no—
186C22×D4#4111yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C2411yescharacteristic
190#131—24no—
191#131—24no—
192#131—24no—
193#131—24no—
194#131—42no—
195#131—42no—
196#131—42no—
197#131—42no—
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—24no—
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—21no—
229C2×M16—21no—
230C2×M16—41no—
231C2×M16—42no—
232C2×M16—41no—
233C2×M16C22×C412yes—
234C2×M16C22×C412yes—
235C2×M16C2×D412yes—
236C2×M16C2×Q812yes—
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—24no—
250C22×C8—42no—
251C22×C8—41no—
252C22×C8—41no—
253C22×C8C2×D412yes—
254C22×C8C2×D412yes—
255C22×C8C2×D411yescharacteristic
256C22×C8C2×D412yes—
257C22×C8C2×D412yes—
258C22×C8C2×D411yescharacteristic
259C22×C8C2×Q812yes—
260C22×C8#4111yescharacteristic
261C22×C8#4111yescharacteristic
262C22×C8#4112yes—
263C22×C8#4111yescharacteristic
264C22×C8#4111yescharacteristic
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267#125—24no—
268#125—24no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C4×Q8—28no—
272C4×Q8—28no—
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#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117C22×C412yes—
288#117C22×C412yes—
289#117C22×C412yes—
290#117C2×D412yes—
291#117#4112yes—
292#117#4112yes—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4#4112yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4#4112yes—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42C22×C412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42#4111yescharacteristic
316C2×C42#4111yescharacteristic
317C4⋊C8—28no—
318C4⋊C8—28no—
319C4⋊C8—28no—
320#104—24no—
321#104—24no—
322#104—28no—
323#103—24no—
324#103—24no—
325#102—24no—
326#102—24no—
327#101—24no—
328#100—24no—
329#99—28no—
330#97—24no—
331#97—24no—
332#97—41no—
333#97—42no—
334#97—41no—
335#97—42no—
336C4×C8—24no—
337C4×C8—24no—
338C4×C8—24no—
339C4×C8—24no—
340C4×C8—42no—
341C4×C8—41no—
342C4×C8—42no—
343C4×C8—41no—
344C24—21no—
345C24C2×#4111yescharacteristic
346#41—22no—
347#41—22no—
348#41—22no—
349#41—22no—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—41no—
355#41—41no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—41no—
360#41—41no—
361#41—41no—
362C2×Q8—28no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—48no—
369C2×Q8C4×Q812yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4C4×Q812yes—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—44no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436M16—41no—
437M16—44no—
438M16—44no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
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—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8#12012yes—
467C2×C8#12012yes—
468C2×C8C22⋊Q812yes—
469C2×C8C22⋊Q812yes—
470C2×C8C22⋊Q812yes—
471C2×C8C22⋊Q812yes—
472C2×C8C22⋊Q812yes—
473C2×C8C22⋊Q812yes—
474C2×C8#12412yes—
475C2×C8#12411yescharacteristic
476C2×C8#12412yes—
477C2×C8#12411yescharacteristic
478C2×C8#12412yes—
479C2×C8#12411yescharacteristic
480C2×C8#12412yes—
481C2×C8#12411yescharacteristic
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4C4×D414yes—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42C4×D412yes—
518C42C4×D412yes—
519C42C4×D414yes—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—82no—
525C23C2×C4×D411yescharacteristic
526C23#52511yescharacteristic
527C23#52511yescharacteristic
528C23#52911yescharacteristic
529C23#52911yescharacteristic
530C23#52911yescharacteristic
531C23#53311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—416no—
535Q8—41no—
536Q8—41no—
537D4—22no—
538D4—22no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—41no—
544D4—41no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×Q811yescharacteristic
571C2×C4C4×#4111yescharacteristic
572C2×C4C4×#4111yescharacteristic
573C2×C4C4×#4111yescharacteristic
574C2×C4C2×#12011yescharacteristic
575C2×C4C2×C22⋊Q811yescharacteristic
576C2×C4C2×C22⋊Q811yescharacteristic
577C2×C4C2×C22⋊Q811yescharacteristic
578C2×C4#52511yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#52511yescharacteristic
581C2×C4#52511yescharacteristic
582C2×C4C2×#12411yescharacteristic
583C2×C4C2×#12411yescharacteristic
584C2×C4C2×#12411yescharacteristic
585C2×C4#52911yescharacteristic
586C2×C4#52911yescharacteristic
587C2×C4#52911yescharacteristic
588C2×C4#52911yescharacteristic
589C2×C4#52911yescharacteristic
590C2×C4#52911yescharacteristic
591C2×C4#53211yescharacteristic
592C2×C4#53211yescharacteristic
593C2×C4#53211yescharacteristic
594C2×C4#53311yescharacteristic
595C2×C4#53311yescharacteristic
596C8—42no—
597C8—42no—
598C8—42no—
599C8—42no—
600C8—42no—
601C8—42no—
602C8—42no—
603C8#39312yes—
604C8#39312yes—
605C8#39312yes—
606C8#39312yes—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22C4×#12011yescharacteristic
619C22C4×C22⋊Q811yescharacteristic
620C22C4×#12411yescharacteristic
621C22#239611yescharacteristic
622C22#239611yescharacteristic
623C22#239811yescharacteristic
624C22#239811yescharacteristic
625C4—24no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4C4×C22⋊Q811yescharacteristic
634C4C4×C22⋊Q811yescharacteristic
635C4C4×#12411yescharacteristic
636C4C4×#12411yescharacteristic
637C4C2×#39311yescharacteristic
638C4#238811yescharacteristic
639C4#239611yescharacteristic
640C4#239811yescharacteristic
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—21no—
646C2—21no—
647C2—82no—
648C2C4×#39311yescharacteristic
649C2#2383312yes—
subgroupG/H
centerC2×C8#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2398
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2960C2semidirect
6. #11387C2semidirect
7. #11281C2semidirect
8. #11264C2semidirect
9. #11261C2semidirect
13. #10711C2semidirect
14. #10706C2semidirect
15. #10699C2semidirect
37. #2031C4semidirect
68. C2×#131Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29602
C2×#20312
#204202
C2×#17642
C2×#17572
#113872
#112812
#112642
#112612
#110922
#110862
#110822
#107112
#107062
#106992