Finite Group Databasebeta

← #7623838

Subgroups — #7623838

Order 512.

1301 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61287C211yesmaximal, characteristic, complemented
2#61284C211yesmaximal, characteristic, complemented
3#33924C212yesmaximal, complemented
4#33921C212yesmaximal, complemented
5#33917C214yesmaximal, complemented
6#33912C214yesmaximal, complemented
7#33908C212yesmaximal, complemented
8#33902C212yesmaximal, complemented
9#33731C211yesmaximal, characteristic, complemented
10#33653C212yesmaximal, complemented
11#33650C212yesmaximal, complemented
12#33626C211yesmaximal, characteristic, complemented
13C2×#2226C211yesmaximal, characteristic, complemented
14#33612C212yesmaximal, complemented
15C4×#509C211yesmaximal, characteristic, complemented
16#33438C211yesmaximal, characteristic, complemented
17#33411C212yesmaximal, complemented
18#3511—21no—
19#3506—21no—
20C2×#509C2211yescharacteristic
21C2×#504C2212yes—
22#3357—22no—
23#3355—22no—
24#3354—22no—
25#3353—22no—
26#3342C2212yescomplemented
27#3341C2212yes—
28#3341C2212yescomplemented
29#3339C2214yes—
30#3338C2212yescomplemented
31#3334C2212yes—
32#3334C2212yescomplemented
33#3160C2211yescharacteristic
34C2×#500C2211yescharacteristic
35#3098—22no—
36#3096—22no—
37#3079—21no—
38#3077—21no—
39#3061C2212yes—
40#3058C2212yes—
41C4×#135C2211yescharacteristic
42C2×C4×D8C2211yescharacteristic
43#2931C2211yescharacteristic
44#2913C2212yes—
45#2913C2212yes—
46#2255C2214yescomplemented
47#2254C2212yescomplemented
48#2252C2212yescomplemented
49#2237C2214yescomplemented
50#2235C2214yescomplemented
51#2234C2214yes—
52#2234C2214yes—
53#2234C2214yescomplemented
54#2233C2214yes—
55#2233C2214yescomplemented
56#2232C2214yes—
57#2229C2212yescomplemented
58#2228C2212yescomplemented
59#2227C2211yescharacteristic, complemented
60#2226C2212yes—
61#2226C2214yes—
62#2226C2212yes—
63#2226C2211yescharacteristic, complemented
64#2225C2211yescharacteristic, complemented
65#2224C2211yescharacteristic, complemented
66#2220C2214yes—
67#2218C2214yes—
68#2217C2214yes—
69#2215C2214yes—
70#2177C2212yes—
71#2175C2212yes—
72C4×SD32C2212yes—
73C4×D16C2212yes—
74#2169C2212yescomplemented
75#2168C2214yescomplemented
76D4×C16C2212yescomplemented
77#2159C2212yes—
78#2159C2212yes—
79#2159C2211yescharacteristic
80#2159C2214yescomplemented
81#2159C2214yescomplemented
82#2158C2212yes—
83#2158C2214yescomplemented
84C2×#367C2212yes—
85C2×#367C2214yescomplemented
86C2×C4⋊C16C2211yescharacteristic
87#2140C2211yescharacteristic
88#2139C2211yescharacteristic
89#2138C2212yes—
90C2×#357C2212yes—
91#2113C2212yes—
92C4×M32C2211yescharacteristic
93C2×#135C2311yescharacteristic
94C22×D8C2311yescharacteristic
95C2×#131C2312yes—
96#559—42no—
97#556—22no—
98#556—41no—
99#555—22no—
100#555—22no—
101#554—22no—
102#553—24no—
103#553—42no—
104#550—41no—
105#548—42no—
106#547—22no—
107#547—22no—
108#542—42no—
109#538—42no—
110#537—22no—
111#535—22no—
112#533—21no—
113#533—41no—
114#532—41no—
115#532D411yescharacteristic
116C2×C4⋊Q8—21no—
117#525—42no—
118#520—22no—
119#520—22no—
120C4×#41—21no—
121C2×C4×D4—21no—
122#509—24no—
123C2×SD32C2312yes—
124C2×D16C2312yes—
125#504—28no—
126C2×M32C2311yescharacteristic
127C2×M32C2314yes—
128C2×M32C2311yescharacteristic
129C22×C16C2312yes—
130C22×C16C2314yes—
131#500—21no—
132#500—21no—
133#500C2311yescharacteristic
134#500C2312yes—
135#500C2312yes—
136#500C2311yescharacteristic
137#499—21no—
138#499C2312yes—
139#498—21no—
140#485—24no—
141#484—24no—
142#482—24no—
143#482—24no—
144#482—24no—
145#481—24no—
146#480—24no—
147#480—24no—
148#478—24no—
149#477—22no—
150#476—22no—
151#476—22no—
152#475—22no—
153#474—22no—
154#474—22no—
155#469—24no—
156#468—24no—
157#467C2314yes—
158#466C2314yes—
159#466C2314yes—
160#466C2314yes—
161C4×Q16C2311yescharacteristic
162C4×Q16C2311yescharacteristic
163C4×SD16—21no—
164C4×SD16—21no—
165C4×D8—21no—
166C4×D8—21no—
167C4×D8C2312yes—
168C4×D8C2311yescharacteristic
169C4×D8C2312yes—
170C4×D8C2311yescharacteristic
171#436D412yes—
172#436D412yes—
173C8×D4C2312yes—
174C8×D4C2312yes—
175#428—22no—
176#428—22no—
177#428—24no—
178#428—24no—
179#427—21no—
180#427C2314yes—
181#427C2314yes—
182#427C2312yes—
183#427C2311yescharacteristic
184C2×#107—21no—
185C2×#107C2314yes—
186C2×#107C2314yes—
187C2×#107C2312yes—
188C2×#107C2311yescharacteristic
189#423D411yescharacteristic
190#418—22no—
191#417—22no—
192#416—21no—
193C2×#102—21no—
194#408D412yes—
195C2×C4×C8C2311yescharacteristic
196#367C2314yes—
197#367C2314yes—
198#367C2312yes—
199#367C2312yes—
200#367C2314yes—
201#367C2312yes—
202#367C2314yes—
203#366C2312yes—
204#366C2312yes—
205#366C2312yes—
206#366C2314yes—
207#366C2314yes—
208C4⋊C16C2311yescharacteristic
209C4⋊C16C2312yes—
210C4⋊C16C2311yescharacteristic
211#358C2312yes—
212#358C2312yes—
213#358C2312yes—
214#357C2314yes—
215#357C2314yes—
216#357C2312yes—
217#357C2312yes—
218#357C2312yes—
219#357C2312yes—
220#357C2312yes—
221C22⋊C16C2314yes—
222C22⋊C16C2314yes—
223#346C2312yes—
224C4×C16C2312yes—
225C2×#41—21no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41C2×D412yes—
229C22×Q8—41no—
230C22×D4—21no—
231C23×C4—41no—
232#135—22no—
233#135—22no—
234C2×Q16C2411yescharacteristic
235C2×SD16—21no—
236C2×D8—21no—
237C2×D8—22no—
238C2×D8—22no—
239C2×D8C2412yes—
240C2×D8C2411yescharacteristic
241#131—24no—
242#131—24no—
243C2×M16C2×D412yes—
244C2×M16C2×D414yes—
245C22×C8C2×D411yescharacteristic
246C22×C8C2414yes—
247C22×C8C2411yescharacteristic
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—41no—
253C4⋊Q8—41no—
254C4⋊Q8D811yescharacteristic
255#127—42no—
256#127D811yescharacteristic
257#126—44no—
258#126—44no—
259#125—22no—
260#125—42no—
261#124—42no—
262#124—42no—
263#124—44no—
264#124—44no—
265#124D812yes—
266#123—44no—
267#123—44no—
268#123—44no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—41no—
272C22⋊Q8—41no—
273C22⋊Q8—42no—
274C22⋊Q8—42no—
275C22⋊Q8—44no—
276#121—24no—
277#121—24no—
278#121—24no—
279#121—42no—
280#121—42no—
281#121—42no—
282#121—44no—
283#121—44no—
284#121D812yes—
285#121D812yes—
286#120—44no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—42no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—41no—
293C4×Q8—41no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—44no—
299C4×D4—42no—
300C4×D4—44no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—41no—
304C4×D4—41no—
305C4×D4—44no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4C2×D412yes—
309C4×D4C2×D412yes—
310#117—21no—
311#117—22no—
312#117—24no—
313#117—24no—
314#117—42no—
315#117—42no—
316#117—41no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—41no—
322C2×C22⋊C4—41no—
323C2×C42—41no—
324C2×C42C2×D411yescharacteristic
325SD32—24no—
326D16—24no—
327M32—22no—
328M32—28no—
329M32#4114yescomplemented
330C2×C16—24no—
331C2×C16—28no—
332C2×C16C2412yes—
333C2×C16C2412yes—
334C2×C16C2412yes—
335C2×C16C2412yes—
336#107—22no—
337#107—22no—
338#107—21no—
339#107—22no—
340#107—24no—
341#107—24no—
342#107—21no—
343#107—21no—
344#107C2411yescharacteristic
345#107C2411yescharacteristic
346#107C2414yes—
347#107C2412yes—
348#107C2412yes—
349#107C2412yes—
350#106—22no—
351#106—21no—
352#106—22no—
353#106—24no—
354#106—24no—
355C4⋊C8C2×D412yes—
356C4⋊C8C2×D412yes—
357#103—21no—
358#103—21no—
359#103—24no—
360#102—22no—
361#102—22no—
362#102—21no—
363#102—21no—
364#102—24no—
365#102—24no—
366#102—24no—
367C22⋊C8C2×D414yes—
368C22⋊C8C2×D414yes—
369C4×C8C2×D411yescharacteristic
370C4×C8C2×D411yescharacteristic
371C4×C8C2411yescharacteristic
372C4×C8C2411yescharacteristic
373C24—41no—
374#41—24no—
375#41—24no—
376#41—42no—
377#41—42no—
378#41—164no—
379#41—164no—
380C2×Q8—21no—
381C2×Q8—41no—
382C2×Q8—42no—
383C2×Q8—41no—
384C2×Q8—42no—
385C2×Q8—162no—
386C2×Q8C2×D812yes—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—41no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—41no—
395C2×D4—42no—
396C2×D4C2×D814yes—
397C2×D4C2×D812yes—
398C2×D4C2×D812yes—
399C2×D4C2×D812yes—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—162no—
410C22×C4C2×D811yescharacteristic
411C22×C4C2×D811yescharacteristic
412C22×C4C22×D414yes—
413C22×C4C22×D411yescharacteristic
414Q16—21no—
415Q16—21no—
416SD16—42no—
417D8—21no—
418D8—22no—
419D8—22no—
420D8—21no—
421D8—42no—
422M16—24no—
423M16—28no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8C22×D411yescharacteristic
428C2×C8C22×D411yescharacteristic
429C2×C8C22×D414yes—
430C2×C8C22×D411yescharacteristic
431C2×C8C2×#4112yes—
432C2×C82−1+412yes—
433C2×C8C2511yescharacteristic
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—24no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—41no—
453C4⋊C4C2×D812yes—
454C4⋊C4C2×D812yes—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4C2×D814yes—
468C22⋊C4C2×D814yes—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—44no—
473C42—44no—
474C42—42no—
475C42C2×D811yescharacteristic
476C42C2×D811yescharacteristic
477C42C22×D411yescharacteristic
478C42C22×D411yescharacteristic
479C16—22no—
480C16—22no—
481C16—22no—
482C16C2×#4114yes—
483C23—41no—
484C23—42no—
485C23—82no—
486C23—161no—
487C23—161no—
488C23C22×D814yes—
489C23C22×D811yescharacteristic
490Q8—24no—
491Q8—41no—
492Q8—41no—
493Q8—84no—
494Q8—82no—
495Q8—161no—
496Q8—162no—
497D4—28no—
498D4—24no—
499D4—24no—
500D4—42no—
501D4—42no—
502D4—41no—
503D4—42no—
504D4—41no—
505D4—82no—
506D4—82no—
507D4—164no—
508D4—164no—
509D4—164no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—84no—
525C2×C4—162no—
526C2×C4—161no—
527C2×C4—162no—
528C2×C4—164no—
529C2×C4—164no—
530C2×C4#54712yes—
531C2×C4C22×D814yes—
532C2×C4C22×D811yescharacteristic
533C2×C4C22×D811yescharacteristic
534C2×C4C22×D811yescharacteristic
535C2×C4C22×D811yescharacteristic
536C2×C4C22×D811yescharacteristic
537C2×C4#57812yes—
538C2×C4C23×D411yescharacteristic
539C8—24no—
540C8—21no—
541C8—21no—
542C8#54712yes—
543C8C22×#4111yescharacteristic
544C8C2×2−1+411yescharacteristic
545C22—21no—
546C22—21no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—81no—
551C22—82no—
552C22—82no—
553C22—81no—
554C22—81no—
555C22—82no—
556C22—164no—
557C22#333412yes—
558C22C23×D811yescharacteristic
559C4—24no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—81no—
564C4—81no—
565C4—82no—
566C4—82no—
567C4—81no—
568C4—84no—
569C4—82no—
570C4#333412yes—
571C4#341612yes—
572C4C2×#54711yescharacteristic
573C4C2×#57811yescharacteristic
574C2—22no—
575C2—44no—
576C2—161no—
577C2—162no—
578C2C2×#333411yescharacteristic
579C2C2×#341611yescharacteristic
580C2#6263611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ #2931 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61287C2semidirect
2. #61284C2semidirect
3. #33924C2semidirect
4. #33921C2semidirect
5. #33917C2semidirect
6. #33912C2semidirect
7. #33908C2semidirect
8. #33902C2semidirect
9. #33731C2semidirect
10. #33653C2semidirect
11. #33650C2semidirect
12. #33626C2semidirect
13. C2×#2226C2semidirect
14. #33612C2semidirect
15. C4×#509C2semidirect
16. #33438C2semidirect
17. #33411C2semidirect
26. #3342C22semidirect
28. #3341C22semidirect
30. #3338C22semidirect
32. #3334C22semidirect
46. #2255C22semidirect
47. #2254C22semidirect
48. #2252C22semidirect
49. #2237C22semidirect
50. #2235C22semidirect
53. #2234C22semidirect
55. #2233C22semidirect
57. #2229C22semidirect
58. #2228C22semidirect
59. #2227C22semidirect
63. #2226C22semidirect
64. #2225C22semidirect
65. #2224C22semidirect
74. #2169C22semidirect
75. #2168C22semidirect
76. D4×C16C22semidirect
80. #2159C22semidirect
81. #2159C22semidirect
83. #2158C22semidirect
85. C2×#367C22semidirect
329. M32#41semidirect

Maximal subgroups

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

subgroupindex
#612872
#612842
#339242
#339212
#339172
#339122
#339082
#339022
#337312
#336532
#336502
#336262
C2×#22262
#336122
C4×#5092
#334382
#334112