Finite Group Databasebeta

← #7624390

Subgroups — #7624390

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62361C211yesmaximal, characteristic, complemented
2C2×#3380C211yesmaximal, characteristic, complemented
3#33956C214yesmaximal, complemented
4#33955C214yesmaximal, complemented
5#33943C214yesmaximal, complemented
6#33937C214yesmaximal, complemented
7#33805C212yesmaximal, complemented
8#33802C211yesmaximal, characteristic, complemented
9#33800C211yesmaximal, characteristic, complemented
10#33798C212yesmaximal, complemented
11#33674C212yesmaximal, complemented
12#33670C211yesmaximal, characteristic, complemented
13#33669C211yesmaximal, characteristic
14#33665C212yesmaximal, complemented
15#33497C211yesmaximal, characteristic, complemented
16#3507—21no—
17C2×#554—21no—
18#3396C2212yescomplemented
19#3393C2212yescomplemented
20#3383C2212yescomplemented
21#3380—22no—
22#3380—22no—
23#3380C2214yes—
24#3380C2214yes—
25#3380C2212yescomplemented
26#3365—22no—
27#3360—22no—
28#3250—21no—
29#3250—21no—
30#3246—21no—
31#3170C2211yescharacteristic
32#3169C2211yescharacteristic
33#3168C2212yes—
34#3168C2212yes—
35C2×#500C2211yescharacteristic
36C2×#499C2211yescharacteristic
37C2×#499C2211yescharacteristic
38C2×#478—21no—
39C2×#478—21no—
40C2×#476—21no—
41C2×C8×Q8C2211yescharacteristic
42C2×C4×D8C2211yescharacteristic
43C2×C4×D8C2211yescharacteristic
44C2×C4×D8C2211yescharacteristic
45#2255C2212yescomplemented
46#2255C2212yescomplemented
47#2255C2214yescomplemented
48#2253C2214yescomplemented
49#2245C2212yescomplemented
50#2244C2212yescomplemented
51#2243C2214yescomplemented
52#2242C2214yescomplemented
53#2236C2212yes—
54#2236C2212yes—
55#2235C2212yes—
56#2233C2212yes—
57#2233C2212yes—
58#2232C2212yes—
59#2229C2214yescomplemented
60#2229C2214yescomplemented
61#2228C2214yes—
62#2228C2214yes—
63#2228C2214yescomplemented
64#2228C2214yescomplemented
65#2226C2212yes—
66#2225C2214yescomplemented
67#2224C2212yes—
68#2224C2214yescomplemented
69#2210C2214yes—
70#2210C2214yes—
71#2208C2212yes—
72#2206C2212yes—
73#2187C2212yes—
74#2184C2212yes—
75#2183C2214yescomplemented
76#2183C2214yescomplemented
77#2177C2214yes—
78#2177C2214yes—
79#2175C2214yes—
80#2175C2214yes—
81C4×SD32C2214yes—
82C4×D16C2214yes—
83#2163C2211yescharacteristic
84#2163C2211yescharacteristic
85#2163C2212yes—
86#2162C2212yes—
87C22×D8C2311yescharacteristic
88#559—22no—
89#557—22no—
90#556—22no—
91#556—42no—
92#554—22no—
93#554—24no—
94#554—24no—
95#554—42no—
96#554—42no—
97#552—42no—
98#539—41no—
99#539—41no—
100#536—41no—
101#533—21no—
102#532—21no—
103C2×C4⋊Q8—21no—
104#529—22no—
105#529—22no—
106C2×#125—21no—
107C2×#125—21no—
108C2×C22⋊Q8—41no—
109C2×C22⋊Q8—41no—
110C2×C22⋊Q8—41no—
111#520—41no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C22×C4⋊C4—41no—
117C2×SD32C2314yes—
118C2×D16C2314yes—
119#500—22no—
120#500C2314yes—
121#500C2311yescharacteristic
122#500C2311yescharacteristic
123#499—21no—
124#499—22no—
125#499—21no—
126#499C2312yes—
127#499C2314yes—
128#499C2312yes—
129#499C2312yes—
130#499C2312yes—
131#495C2312yes—
132#495C2312yes—
133#494C2311yescharacteristic
134#493C2311yescharacteristic
135#488—22no—
136#488—22no—
137#487—21no—
138#486—21no—
139#479—22no—
140#479—22no—
141#478—24no—
142#478—24no—
143#478—24no—
144#478—24no—
145#478—22no—
146#478—22no—
147#476—22no—
148#476—24no—
149#476—24no—
150#475—22no—
151#467—21no—
152#467C2312yes—
153#466—21no—
154#466C2312yes—
155#465—22no—
156#465—22no—
157#464—22no—
158#464—22no—
159#463—22no—
160#463—22no—
161#461—22no—
162#460—22no—
163C8×Q8D412yes—
164C8×Q8D412yes—
165C8×Q8C2312yes—
166C8×Q8C2312yes—
167C4×Q16—21no—
168C4×Q16C2312yes—
169C4×Q16C2312yes—
170C4×Q16C2312yes—
171C4×SD16—22no—
172C4×SD16—22no—
173C4×D8—21no—
174C4×D8—21no—
175C4×D8—22no—
176C4×D8—22no—
177C4×D8—21no—
178C4×D8C2314yes—
179C4×D8C2312yes—
180C4×D8C2314yes—
181C4×D8C2312yes—
182C4×D8C2312yes—
183C4×D8C2312yes—
184C4×D8C2312yes—
185C2×#107—21no—
186C2×#107C2311yescharacteristic
187C2×#107C2311yescharacteristic
188C2×#107C2311yescharacteristic
189C2×#106—21no—
190C2×#106—21no—
191C2×C4⋊C8D411yescharacteristic
192C2×C4⋊C8D411yescharacteristic
193C2×C4⋊C8D411yescharacteristic
194C2×#102—21no—
195C2×#102—21no—
196C2×#102—21no—
197C2×C4×C8C2311yescharacteristic
198C2×C4×C8C2311yescharacteristic
199C2×C4×C8C2311yescharacteristic
200#367C2314yes—
201#367C2314yes—
202#367C2314yes—
203#366C2314yes—
204#366C2314yes—
205#366C2314yes—
206C4⋊C16C2314yes—
207C4⋊C16C2312yes—
208C4⋊C16C2312yes—
209C4⋊C16C2314yes—
210#358C2314yes—
211#358C2314yes—
212#358C2314yes—
213#357C2314yes—
214#357C2314yes—
215#357C2314yes—
216#357C2314yes—
217#357C2314yes—
218C22⋊C16C2312yes—
219C22⋊C16C2312yes—
220#346C2312yes—
221#346C2314yes—
222#346C2312yes—
223C4×C16C2314yes—
224C2×#41—41no—
225C22×Q8—41no—
226C22×Q8D811yescharacteristic
227C22×D4—21no—
228C23×C4—41no—
229C23×C4—41no—
230C23×C4—41no—
231C2×Q16—21no—
232C2×Q16C2412yes—
233C2×SD16—22no—
234C2×SD16—22no—
235C2×D8—24no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8C2412yes—
239C2×D8C2412yes—
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2411yescharacteristic
244C4⋊Q8—24no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250#127—21no—
251#126—42no—
252#126—42no—
253#126—42no—
254#126—42no—
255#125—22no—
256#125—24no—
257#125—22no—
258#125—22no—
259#125—22no—
260#125—42no—
261#125—42no—
262#125—42no—
263#125—41no—
264#125—41no—
265#125—41no—
266#124—22no—
267#124—22no—
268#124—42no—
269#124—41no—
270#123—42no—
271#123—42no—
272#123—42no—
273#123—42no—
274C22⋊Q8—22no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—42no—
278C22⋊Q8—42no—
279C22⋊Q8—41no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286#121—22no—
287#121—41no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—42no—
292C4×Q8—41no—
293C4×Q8D812yes—
294C4×Q8D812yes—
295C4×Q8C2×D412yes—
296C4×Q8C2×D412yes—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312#117—41no—
313#117—42no—
314#117—42no—
315#117—42no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—41no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4D811yescharacteristic
328C2×C4⋊C4D811yescharacteristic
329C2×C4⋊C4D811yescharacteristic
330C2×C22⋊C4—41no—
331C2×C22⋊C4—41no—
332C2×C22⋊C4—41no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336SD32—28no—
337D16—28no—
338C2×C16C2414yes—
339C2×C16C2414yes—
340#107—24no—
341#107—21no—
342#107—21no—
343#107—21no—
344#107—21no—
345#107—21no—
346#107C2412yes—
347#107C2412yes—
348#107C2412yes—
349#107C2412yes—
350#107C2412yes—
351#106—22no—
352#106—24no—
353#106—22no—
354#106—22no—
355#106—22no—
356C4⋊C8C2×D412yes—
357C4⋊C8C2×D412yes—
358C4⋊C8C2×D412yes—
359C4⋊C8C2×D412yes—
360C4⋊C8C2×D412yes—
361C4⋊C8C2×D412yes—
362#103—22no—
363#103—22no—
364#103—22no—
365#103—22no—
366#102—22no—
367#102—24no—
368#102—22no—
369#102—24no—
370#102—22no—
371#102—22no—
372#102—22no—
373#102—22no—
374C4×C8C2×D411yescharacteristic
375C4×C8C2×D412yes—
376C4×C8C2×D412yes—
377C4×C8C2×D411yescharacteristic
378C4×C8C2412yes—
379C4×C8C2411yescharacteristic
380C4×C8C2412yes—
381C4×C8C2411yescharacteristic
382C24—41no—
383#41—84no—
384C2×Q8—22no—
385C2×Q8—42no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—44no—
389C2×Q8—162no—
390C2×Q8C2×D812yes—
391C2×Q8C2×D812yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—84no—
411C22×C4—162no—
412C22×C4—162no—
413C22×C4C2×D811yescharacteristic
414C22×C4C2×D811yescharacteristic
415C22×C4C2×D811yescharacteristic
416C22×C4C2×D811yescharacteristic
417C22×C4C2×D811yescharacteristic
418C22×C4C2×D811yescharacteristic
419C22×C4C22×D411yescharacteristic
420Q16—24no—
421Q16—81no—
422SD16—82no—
423SD16—82no—
424D8—24no—
425D8—22no—
426D8—24no—
427D8—81no—
428D82−1+414yes—
429C2×C8—22no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8C22×D411yescharacteristic
434C2×C8C22×D411yescharacteristic
435C2×C8C22×D412yes—
436C2×C8C22×D412yes—
437C2×C8C22×D411yescharacteristic
438C2×C8C2511yescharacteristic
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4C2×D812yes—
467C4⋊C4C2×D812yes—
468C4⋊C4C2×D812yes—
469C4⋊C4C2×D812yes—
470C4⋊C4C2×D812yes—
471C4⋊C4C2×D812yes—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C42—42no—
486C42—42no—
487C42—42no—
488C42C2×D812yes—
489C42C2×D812yes—
490C42C2×D811yescharacteristic
491C42C2×D811yescharacteristic
492C42C22×D411yescharacteristic
493C42C22×D412yes—
494C42C22×D412yes—
495C42C22×D411yescharacteristic
496C16—24no—
497C16—24no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—82no—
502C23—82no—
503C23C22×D811yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—82no—
509Q8—161no—
510Q8—164no—
511D4—24no—
512D4—44no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—82no—
517D4—84no—
518D4—161no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—88no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—162no—
539C2×C4—162no—
540C2×C4—164no—
541C2×C4—162no—
542C2×C4—162no—
543C2×C4C22×D811yescharacteristic
544C2×C4C22×D812yes—
545C2×C4C22×D812yes—
546C2×C4C22×D811yescharacteristic
547C2×C4C22×D812yes—
548C2×C4C22×D812yes—
549C2×C4C22×D811yescharacteristic
550C2×C4C22×D811yescharacteristic
551C2×C4C22×D811yescharacteristic
552C2×C4C23×D411yescharacteristic
553C8—21no—
554C8—22no—
555C8—22no—
556C8—21no—
557C8—21no—
558C8C2×2−1+412yes—
559C22—21no—
560C22—21no—
561C22—44no—
562C22—88no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22#341412yes—
569C22C23×D811yescharacteristic
570C4—21no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—22no—
575C4—22no—
576C4—21no—
577C4—21no—
578C4—22no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4#348611yescharacteristic
586C4#358611yescharacteristic
587C2—22no—
588C2—84no—
589C2—162no—
590C2#6181711yescharacteristic
591C2C2×#341411yescharacteristic
592C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#33669C2
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. #62361C2semidirect
2. C2×#3380C2semidirect
3. #33956C2semidirect
4. #33955C2semidirect
5. #33943C2semidirect
6. #33937C2semidirect
7. #33805C2semidirect
8. #33802C2semidirect
9. #33800C2semidirect
10. #33798C2semidirect
11. #33674C2semidirect
12. #33670C2semidirect
14. #33665C2semidirect
15. #33497C2semidirect
18. #3396C22semidirect
19. #3393C22semidirect
20. #3383C22semidirect
25. #3380C22semidirect
45. #2255C22semidirect
46. #2255C22semidirect
47. #2255C22semidirect
48. #2253C22semidirect
49. #2245C22semidirect
50. #2244C22semidirect
51. #2243C22semidirect
52. #2242C22semidirect
59. #2229C22semidirect
60. #2229C22semidirect
63. #2228C22semidirect
64. #2228C22semidirect
66. #2225C22semidirect
68. #2224C22semidirect
75. #2183C22semidirect
76. #2183C22semidirect

Maximal subgroups

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

subgroupindex
#623612
C2×#33802
#339562
#339552
#339432
#339372
#338052
#338022
#338002
#337982
#336742
#336702
#336692
#336652
#334972