Finite Group Databasebeta

← #7624501

Subgroups — #7624501

Order 512.

1245 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62364C211yesmaximal, characteristic, complemented
2C2×#3393C211yesmaximal, characteristic, complemented
3#33969C214yesmaximal, complemented
4#33965C212yesmaximal, complemented
5#33960C212yesmaximal, complemented
6#33944C214yesmaximal, complemented
7#33934C212yesmaximal, complemented
8#33931C212yesmaximal, complemented
9#33784C211yesmaximal, characteristic, complemented
10#33781C212yesmaximal, complemented
11#33777C212yesmaximal
12#33775C211yesmaximal, characteristic, complemented
13#33677C212yesmaximal, complemented
14#33670C211yesmaximal, characteristic, complemented
15#33665C211yesmaximal, characteristic
16#33664C212yesmaximal, complemented
17#33498C211yesmaximal, characteristic, complemented
18#3523—21no—
19C2×#556—21no—
20#3395C2211yescharacteristic, complemented
21#3393—21no—
22#3393—21no—
23#3393—22no—
24#3393C2212yes—
25#3393C2212yes—
26#3393C2214yes—
27#3393C2211yescharacteristic, complemented
28#3393C2212yescomplemented
29#3383C2211yescharacteristic, complemented
30#3383C2212yescomplemented
31Q8×D8C2211yescharacteristic, complemented
32#3364—21no—
33#3364—22no—
34#3359—21no—
35#3248—21no—
36#3248—22no—
37#3170C2212yes—
38#3170C2211yescharacteristic
39#3168C2211yescharacteristic
40#3168C2212yes—
41C2×#495C2212yes—
42C2×#494C2211yescharacteristic
43C2×#464—22no—
44C2×#461—21no—
45C2×C8×Q8C2211yescharacteristic
46C2×C4×Q16C2212yes—
47C2×C4×D8C2211yescharacteristic
48#2251C2214yes—
49#2251C2212yescomplemented
50#2249C2214yes—
51#2247C2212yescomplemented
52#2245C2212yescomplemented
53#2243C2214yes—
54#2242C2214yes—
55#2241C2212yescomplemented
56#2236C2211yescharacteristic
57#2236C2212yes—
58#2233C2211yescharacteristic
59#2229C2214yescomplemented
60#2228C2212yes—
61#2227C2214yescomplemented
62#2224C2212yes—
63#2211C2214yes—
64#2211C2214yes—
65#2211C2214yes—
66#2211C2214yescomplemented
67#2210C2212yes—
68#2210C2214yes—
69#2209C2214yescomplemented
70#2209C2212yescomplemented
71#2209C2214yescomplemented
72#2208C2212yes—
73#2208C2212yescomplemented
74#2188C2212yes—
75#2188C2212yes—
76#2188C2212yes—
77#2187C2211yescharacteristic
78#2187C2212yes—
79#2185C2212yes—
80#2184C2211yescharacteristic
81#2183C2212yescomplemented
82#2183C2214yescomplemented
83Q8×C16C2212yescomplemented
84#2176C2214yescomplemented
85#2175C2214yes—
86#2175C2214yescomplemented
87C4×Q32C2214yescomplemented
88C4×Q32C2214yescomplemented
89C4×SD32C2212yes—
90C4×D16C2212yes—
91#2163C2211yescharacteristic
92#2163C2212yes—
93#2162C2212yes—
94#2162C2211yescharacteristic
95C22×Q16—21no—
96C22×Q16C2312yes—
97C22×SD16—22no—
98C22×D8C2311yescharacteristic
99#557—21no—
100#557—22no—
101#557—21no—
102#556—21no—
103#556—22no—
104#556—22no—
105#556—22no—
106#556—24no—
107#556—41no—
108#556—41no—
109#556—42no—
110#552—41no—
111#552—42no—
112D4×Q8—21no—
113D4×Q8—41no—
114#541—41no—
115#541—42no—
116#533—21no—
117#533—22no—
118C2×C4⋊Q8—21no—
119#529—21no—
120#529—22no—
121C2×#126—42no—
122C2×#124—22no—
123C2×C22⋊Q8—41no—
124#519—41no—
125C2×C4×Q8—22no—
126C2×C4×Q8D411yescharacteristic
127C2×C4×D4—21no—
128C2×#117—41no—
129C2×Q32C2314yes—
130C2×Q32C2314yes—
131C2×Q32C2314yes—
132C2×SD32C2312yes—
133C2×SD32C2314yes—
134C2×SD32C2314yes—
135C2×D16C2312yes—
136#500C2311yescharacteristic
137#500C2312yes—
138#499C2311yescharacteristic
139#499C2312yes—
140#495—22no—
141#495—22no—
142#495C2314yes—
143#495C2314yes—
144#495C2311yescharacteristic
145#495C2312yes—
146#494—21no—
147#494—21no—
148#494C2312yes—
149#494C2312yes—
150#494C2311yescharacteristic
151#494C2312yes—
152#488—21no—
153#488—22no—
154#487—21no—
155#487—22no—
156#479—22no—
157#479—22no—
158#479—22no—
159#475—21no—
160#475—22no—
161#474—21no—
162#474—22no—
163#467—22no—
164#467—21no—
165#467C2311yescharacteristic
166#467C2312yes—
167#466C2311yescharacteristic
168#465—22no—
169#465—21no—
170#464—22no—
171#464—22no—
172#464—22no—
173#464—24no—
174#464—24no—
175#464—24no—
176#464—24no—
177#462—21no—
178#462—22no—
179#461—21no—
180#461—22no—
181#461—22no—
182#461—22no—
183#461—24no—
184C8×Q8D411yescharacteristic
185C8×Q8D411yescharacteristic
186C8×Q8D412yes—
187C8×Q8C2311yescharacteristic
188C8×Q8C2311yescharacteristic
189C8×Q8C2312yes—
190C4×Q16—21no—
191C4×Q16—22no—
192C4×Q16—22no—
193C4×Q16—22no—
194C4×Q16C2312yes—
195C4×Q16C2312yes—
196C4×Q16C2312yes—
197C4×Q16C2311yescharacteristic
198C4×Q16C2312yes—
199C4×Q16C2314yes—
200C4×Q16C2314yes—
201C4×SD16—21no—
202C4×SD16—22no—
203C4×D8—21no—
204C4×D8—21no—
205C4×D8C2311yescharacteristic
206C4×D8C2312yes—
207C4×D8C2312yes—
208C4×D8C2312yes—
209C2×#107C2311yescharacteristic
210C2×C4⋊C8D411yescharacteristic
211C2×C4⋊C8D412yes—
212C2×#103—22no—
213C2×#102—21no—
214C2×C4×C8C2311yescharacteristic
215C2×C4×C8C2312yes—
216#367C2312yes—
217#366C2312yes—
218#366C2314yes—
219C4⋊C16C2312yes—
220C4⋊C16C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2314yes—
223#358C2312yes—
224#358C2314yes—
225#358C2314yes—
226#358C2314yes—
227#358C2314yes—
228#358C2314yes—
229#357C2312yes—
230#357C2312yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2311yescharacteristic
236#346C2312yes—
237#346C2314yes—
238C4×C16C2312yes—
239C4×C16C2314yes—
240C22×Q8—22no—
241C22×Q8—41no—
242C22×Q8D811yescharacteristic
243C22×D4—21no—
244C23×C4—41no—
245C2×Q16—22no—
246C2×Q16—22no—
247C2×Q16—22no—
248C2×Q16—21no—
249C2×Q16—22no—
250C2×Q16—44no—
251C2×Q16—82no—
252C2×Q16C2414yes—
253C2×Q16C2411yescharacteristic
254C2×Q16C2412yes—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—21no—
258C2×SD16—22no—
259C2×SD16—84no—
260C2×D8—21no—
261C2×D8—42no—
262C2×D8C2412yes—
263C2×D8C2411yescharacteristic
264C22×C8C2×D412yes—
265C22×C8C2×D411yescharacteristic
266C22×C8C2411yescharacteristic
267C4⋊Q8—21no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—21no—
272C4⋊Q8—22no—
273C4⋊Q8—41no—
274C4⋊Q8—42no—
275C4⋊Q8—41no—
276C4⋊Q8—41no—
277#126—44no—
278#126—44no—
279#126—44no—
280#126—44no—
281#126—42no—
282#126—42no—
283#126—42no—
284#126—41no—
285#126—42no—
286#125—21no—
287#125—22no—
288#125—42no—
289#125—42no—
290#125—42no—
291#125—41no—
292#125—42no—
293#125—41no—
294#125—42no—
295#124—24no—
296#124—24no—
297#124—21no—
298#124—22no—
299#124—42no—
300#124—42no—
301#123—42no—
302#123—41no—
303C22⋊Q8—21no—
304C22⋊Q8—22no—
305C22⋊Q8—42no—
306C22⋊Q8—42no—
307C22⋊Q8—44no—
308C22⋊Q8—41no—
309C22⋊Q8—42no—
310C22⋊Q8—41no—
311C22⋊Q8—42no—
312C22⋊Q8—41no—
313C22⋊Q8—42no—
314#121—21no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—21no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—41no—
323C4×Q8—41no—
324C4×Q8—42no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×Q8—41no—
328C4×Q8—42no—
329C4×Q8D811yescharacteristic
330C4×Q8D811yescharacteristic
331C4×Q8D812yes—
332C4×Q8C2×D411yescharacteristic
333C4×Q8C2×D411yescharacteristic
334C4×Q8C2×D412yes—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—21no—
338C4×D4—22no—
339C4×D4—41no—
340C4×D4—42no—
341C4×D4—41no—
342C4×D4—41no—
343#117—42no—
344#117—41no—
345#117—42no—
346#117—42no—
347#117—44no—
348#117—41no—
349#117—42no—
350#117—41no—
351#117—42no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4D811yescharacteristic
357C2×C4⋊C4D812yes—
358C2×C22⋊C4—42no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—42no—
361C2×C42—42no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364Q32—44no—
365Q32—44no—
366Q32—44no—
367SD32—44no—
368SD32—44no—
369SD32—42no—
370D16—42no—
371C2×C16C2412yes—
372C2×C16C2412yes—
373C2×C16C2414yes—
374#107—21no—
375#107—21no—
376#107—22no—
377#107C2411yescharacteristic
378#107C2412yes—
379#107C2411yescharacteristic
380#107C2412yes—
381#106—21no—
382#106—22no—
383C4⋊C8C2×D411yescharacteristic
384C4⋊C8C2×D411yescharacteristic
385C4⋊C8C2×D412yes—
386C4⋊C8C2×D412yes—
387C4⋊C8C2×D412yes—
388C4⋊C8C2×D412yes—
389C4⋊C8C2×D412yes—
390#103—24no—
391#103—24no—
392#103—21no—
393#103—22no—
394#103—22no—
395#103—22no—
396#103—22no—
397#103—21no—
398#103—22no—
399#102—22no—
400#102—22no—
401#102—21no—
402#102—22no—
403#102—21no—
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D412yes—
407C4×C8C2×D412yes—
408C4×C8C2411yescharacteristic
409C4×C8C2412yes—
410C4×C8C2411yescharacteristic
411C4×C8C2412yes—
412C24—41no—
413C2×Q8—22no—
414C2×Q8—21no—
415C2×Q8—24no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—41no—
421C2×Q8—42no—
422C2×Q8—84no—
423C2×Q8C2×D811yescharacteristic
424C2×Q8C2×D811yescharacteristic
425C2×Q8C2×D812yes—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—41no—
431C2×D4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—84no—
439C22×C4C2×D812yes—
440C22×C4C2×D811yescharacteristic
441C22×C4C2×D811yescharacteristic
442C22×C4C2×D812yes—
443C22×C4C22×D411yescharacteristic
444Q16—24no—
445Q16—22no—
446Q16—24no—
447Q16—24no—
448Q16—44no—
449Q16—82no—
450Q16—81no—
451Q16—82no—
452Q16—82no—
453SD16—82no—
454SD16—81no—
455SD16—84no—
456SD16—84no—
457D8—22no—
458D8—22no—
459D8—22no—
460D8—42no—
461C2×C8—22no—
462C2×C8—42no—
463C2×C8—41no—
464C2×C8C22×D411yescharacteristic
465C2×C8C22×D411yescharacteristic
466C2×C8C22×D411yescharacteristic
467C2×C8C22×D412yes—
468C2×C8C22×D412yes—
469C2×C8C2511yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—41no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4C2×D811yescharacteristic
499C4⋊C4C2×D811yescharacteristic
500C4⋊C4C2×D812yes—
501C4⋊C4C2×D812yes—
502C4⋊C4C2×D812yes—
503C4⋊C4C2×D812yes—
504C4⋊C4C2×D812yes—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—41no—
520C42—44no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42C2×D811yescharacteristic
528C42C2×D812yes—
529C42C2×D811yescharacteristic
530C42C2×D812yes—
531C42C22×D411yescharacteristic
532C42C22×D411yescharacteristic
533C42C22×D412yes—
534C42C22×D412yes—
535C16—22no—
536C16—24no—
537C16—22no—
538C23—41no—
539C23—42no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23C22×D811yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—42no—
547Q8—44no—
548Q8—42no—
549Q8—41no—
550Q8—44no—
551Q8—42no—
552Q8—41no—
553Q8—84no—
554Q8—84no—
555Q8—82no—
556D4—42no—
557D4—41no—
558D4—42no—
559D4—41no—
560D4—82no—
561D4—82no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—82no—
581C2×C4C22×D812yes—
582C2×C4C22×D812yes—
583C2×C4C22×D811yescharacteristic
584C2×C4C22×D812yes—
585C2×C4C22×D811yescharacteristic
586C2×C4C22×D811yescharacteristic
587C2×C4C22×D812yes—
588C2×C4C22×D811yescharacteristic
589C2×C4C22×D811yescharacteristic
590C2×C4C23×D411yescharacteristic
591C8—22no—
592C8—22no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8#58512yes—
597C22—21no—
598C22—21no—
599C22—81no—
600C22—82no—
601C22—81no—
602C22—82no—
603C22—82no—
604C22—81no—
605C22—164no—
606C22#341712yes—
607C22C23×D811yescharacteristic
608C4—21no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—81no—
620C4—84no—
621C4—81no—
622C4—82no—
623C4#341412yes—
624C4#348711yescharacteristic
625C4#358511yescharacteristic
626C2—22no—
627C2—161no—
628C2—162no—
629C2#6188011yescharacteristic
630C2C2×#341411yescharacteristic
631C2C2×#341711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#2184C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62364C2semidirect
2. C2×#3393C2semidirect
3. #33969C2semidirect
4. #33965C2semidirect
5. #33960C2semidirect
6. #33944C2semidirect
7. #33934C2semidirect
8. #33931C2semidirect
9. #33784C2semidirect
10. #33781C2semidirect
12. #33775C2semidirect
13. #33677C2semidirect
14. #33670C2semidirect
16. #33664C2semidirect
17. #33498C2semidirect
20. #3395C22semidirect
27. #3393C22semidirect
28. #3393C22semidirect
29. #3383C22semidirect
30. #3383C22semidirect
31. Q8×D8C22semidirect
49. #2251C22semidirect
51. #2247C22semidirect
52. #2245C22semidirect
55. #2241C22semidirect
59. #2229C22semidirect
61. #2227C22semidirect
66. #2211C22semidirect
69. #2209C22semidirect
70. #2209C22semidirect
71. #2209C22semidirect
73. #2208C22semidirect
81. #2183C22semidirect
82. #2183C22semidirect
83. Q8×C16C22semidirect
84. #2176C22semidirect
86. #2175C22semidirect
87. C4×Q32C22semidirect
88. C4×Q32C22semidirect

Maximal subgroups

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

subgroupindex
#623642
C2×#33932
#339692
#339652
#339602
#339442
#339342
#339312
#337842
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