Finite Group Databasebeta

← #7624270

Subgroups — #7624270

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62249C211yesmaximal, characteristic
2C2×#3336C211yesmaximal, characteristic, complemented
3#33916C214yesmaximal, complemented
4#33915C214yesmaximal, complemented
5#33884C214yesmaximal, complemented
6#33882C214yesmaximal, complemented
7#33802C211yesmaximal, characteristic, complemented
8#33800C211yesmaximal, characteristic, complemented
9#33762C212yesmaximal, complemented
10#33756C212yesmaximal, complemented
11#33673C211yesmaximal, characteristic, complemented
12#33672C211yesmaximal, characteristic, complemented
13#33646C212yesmaximal, complemented
14#33645C212yesmaximal, complemented
15#33487C211yesmaximal, characteristic, complemented
16#3506—21no—
17C2×#552—21no—
18#3342C2212yes—
19#3341C2212yes—
20#3336—22no—
21#3336—22no—
22#3336C2214yes—
23#3336C2214yes—
24#3304—22no—
25#3303—22no—
26#3287C2212yes—
27#3283C2212yes—
28#3246—21no—
29#3191—21no—
30#3191—21no—
31#3170C2211yescharacteristic
32#3169C2211yescharacteristic
33C2×#499C2211yescharacteristic
34C2×#485—21no—
35C2×#483—21no—
36C2×#479—21no—
37#3050C2212yes—
38#3050C2212yes—
39C2×#467C2211yescharacteristic
40C2×#467C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C8×D4C2211yescharacteristic
43C2×#427C2211yescharacteristic
44C22×#107C2211yescharacteristic
45#2255C2212yescomplemented
46#2255C2212yescomplemented
47#2245C2212yescomplemented
48#2244C2212yescomplemented
49#2237C2214yescomplemented
50#2237C2214yescomplemented
51#2237C2214yescomplemented
52#2236C2212yes—
53#2236C2214yescomplemented
54#2235C2212yes—
55#2235C2212yes—
56#2234C2214yescomplemented
57#2233C2212yes—
58#2233C2214yescomplemented
59#2232C2212yes—
60#2232C2212yes—
61#2229C2214yescomplemented
62#2228C2214yescomplemented
63#2227C2212yescomplemented
64#2225C2212yescomplemented
65#2220C2214yescomplemented
66#2219C2214yescomplemented
67#2217C2214yescomplemented
68#2216C2214yescomplemented
69#2209C2212yescomplemented
70#2207C2212yescomplemented
71#2189C2214yescomplemented
72#2189C2214yescomplemented
73#2188C2212yes—
74#2185C2212yes—
75#2176C2214yescomplemented
76#2175C2214yescomplemented
77#2168C2214yes—
78#2168C2214yes—
79#2163C2211yescharacteristic
80#2163C2211yescharacteristic
81#2159C2214yescomplemented
82#2158C2214yescomplemented
83#2139C2214yescomplemented
84#2138C2214yescomplemented
85#2112C2212yes—
86C2×C22⋊C16C2212yes—
87C2×#135C2312yes—
88C2×#135C2312yes—
89C22×Q16C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92#559—42no—
93#556—22no—
94#555—22no—
95#553—42no—
96#552—24no—
97#552—24no—
98#552—42no—
99#552—42no—
100#550—22no—
101#547—22no—
102#538—41no—
103#538—41no—
104#535—41no—
105#533—21no—
106#532—21no—
107C2×#126—41no—
108C2×#125—21no—
109C2×#125—41no—
110#525—22no—
111#525—22no—
112C2×#123—41no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115#520—41no—
116C2×C4×Q8—21no—
117C2×C4×D4D411yescharacteristic
118C2×#117—21no—
119C2×#117—41no—
120C22×C4⋊C4—21no—
121C2×Q32C2314yes—
122C2×SD32C2314yes—
123C2×M32C2314yes—
124C22×C16C2314yes—
125#500C2311yescharacteristic
126#500C2311yescharacteristic
127#499—21no—
128#499—21no—
129#499C2312yes—
130#499C2312yes—
131#494C2311yescharacteristic
132#493C2311yescharacteristic
133#487—21no—
134#486—21no—
135#485—24no—
136#485—24no—
137#485—22no—
138#485—22no—
139#483—24no—
140#483—24no—
141#482—22no—
142#482—22no—
143#479—24no—
144#479—24no—
145#476—22no—
146#475—22no—
147#467—22no—
148#467—22no—
149#467—21no—
150#467C2314yes—
151#467C2314yes—
152#467C2312yes—
153#467C2312yes—
154#467C2312yes—
155#466—21no—
156#466C2312yes—
157#466C2312yes—
158#466C2312yes—
159#465—22no—
160#465—22no—
161#465—22no—
162#465—22no—
163#461—22no—
164#460—22no—
165#452—22no—
166#452—22no—
167#449—22no—
168#449—22no—
169C4×Q16—21no—
170C4×Q16—21no—
171C4×Q16—21no—
172C4×Q16C2312yes—
173C4×Q16C2312yes—
174C4×Q16C2312yes—
175C4×D8—21no—
176C4×D8C2312yes—
177C8×D4D412yes—
178C8×D4D412yes—
179C8×D4C2312yes—
180C8×D4C2312yes—
181#427—22no—
182#427C2314yes—
183#427C2312yes—
184#427C2312yes—
185C2×#107—21no—
186C2×#107—22no—
187C2×#107C2314yes—
188C2×#107C2311yescharacteristic
189C2×#107C2311yescharacteristic
190C2×#107C2311yescharacteristic
191C2×#106—21no—
192C2×#106—21no—
193C2×C4⋊C8D411yescharacteristic
194#416—22no—
195#416—22no—
196C2×#103—21no—
197C2×#103—21no—
198C2×#103—21no—
199C2×C22⋊C8D411yescharacteristic
200C2×C22⋊C8D411yescharacteristic
201C2×C4×C8C2311yescharacteristic
202#367C2314yes—
203#367C2314yes—
204#367C2314yes—
205#366C2314yes—
206#366C2314yes—
207#366C2314yes—
208C4⋊C16C2312yes—
209C4⋊C16C2312yes—
210#358C2314yes—
211#358C2314yes—
212#358C2314yes—
213#358C2314yes—
214#358C2314yes—
215#357C2314yes—
216#357C2314yes—
217#357C2314yes—
218C22⋊C16C2312yes—
219C22⋊C16C2312yes—
220C22⋊C16C2314yes—
221C22⋊C16C2314yes—
222#346C2312yes—
223#346C2312yes—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—41no—
227C22×Q8—21no—
228C22×D4D811yescharacteristic
229C23×C4—41no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232#135—44no—
233#135—44no—
234C2×Q16—21no—
235C2×Q16—21no—
236C2×Q16—42no—
237C2×Q16C2412yes—
238C2×Q16C2412yes—
239C2×SD16—22no—
240C2×SD16—22no—
241C2×D8—21no—
242C2×D8C2412yes—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C22×C8C2×D411yescharacteristic
250C22×C8C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256#127—21no—
257#127—42no—
258#126—44no—
259#126—44no—
260#126—42no—
261#126—42no—
262#125—22no—
263#125—22no—
264#125—44no—
265#125—44no—
266#125—41no—
267#125—41no—
268#125—41no—
269#124—42no—
270#124—42no—
271#124—42no—
272#124—41no—
273#123—44no—
274#123—44no—
275#123—42no—
276#123—42no—
277#123—42no—
278#123—42no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286C22⋊Q8—41no—
287#121—22no—
288#121—22no—
289#121—22no—
290#121—42no—
291#121—42no—
292#121—41no—
293#121—42no—
294#121—42no—
295#120—42no—
296#120—42no—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—42no—
301C4×Q8—41no—
302C4×Q8—41no—
303C4×Q8—41no—
304C4×D4—22no—
305C4×D4—42no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—41no—
311C4×D4D812yes—
312C4×D4D812yes—
313C4×D4C2×D412yes—
314C4×D4C2×D412yes—
315#117—24no—
316#117—22no—
317#117—22no—
318#117—42no—
319#117—41no—
320#117—42no—
321#117—44no—
322#117—44no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—41no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4D811yescharacteristic
336C2×C22⋊C4—41no—
337C2×C22⋊C4—41no—
338C2×C22⋊C4—41no—
339C2×C22⋊C4D811yescharacteristic
340C2×C22⋊C4D811yescharacteristic
341C2×C42—41no—
342C2×C42—41no—
343C2×C42C2×D411yescharacteristic
344Q32—44no—
345SD32—44no—
346M32—44no—
347C2×C16—44no—
348C2×C16C2414yes—
349C2×C16C2414yes—
350#107—21no—
351#107—21no—
352#107—24no—
353#107—21no—
354#107—21no—
355#107—21no—
356#107C2412yes—
357#107C2412yes—
358#107C2412yes—
359#107C2412yes—
360#107C2412yes—
361#106—24no—
362#106—22no—
363#106—22no—
364#106—22no—
365#106—22no—
366C4⋊C8C2×D412yes—
367C4⋊C8C2×D412yes—
368#103—22no—
369#103—22no—
370#103—24no—
371#103—24no—
372#103—22no—
373#103—22no—
374#103—22no—
375#103—22no—
376#102—22no—
377#102—22no—
378#102—22no—
379#102—22no—
380C22⋊C8C2×D412yes—
381C22⋊C8C2×D412yes—
382C22⋊C8C2×D412yes—
383C22⋊C8C2×D412yes—
384C4×C8C2×D411yescharacteristic
385C4×C8C2×D411yescharacteristic
386C4×C8C2411yescharacteristic
387C4×C8C2411yescharacteristic
388C24C2×D811yescharacteristic
389C24C2×D811yescharacteristic
390#41—84no—
391#41—82no—
392#41—82no—
393#41—82no—
394#41—82no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—41no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C2×D812yes—
412C2×D4C2×D812yes—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—84no—
426C22×C4C2×D811yescharacteristic
427C22×C4C2×D811yescharacteristic
428C22×C4C2×D811yescharacteristic
429C22×C4C2×D811yescharacteristic
430C22×C4C2×D812yes—
431C22×C4C2×D812yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435Q16—22no—
436Q16—22no—
437Q16—24no—
438Q16—42no—
439Q16—81no—
440SD16—82no—
441SD16—82no—
442D8—24no—
443D8—81no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8C22×D411yescharacteristic
453C2×C8C22×D411yescharacteristic
454C2×C8C22×D411yescharacteristic
455C2×C8C22×D412yes—
456C2×C8C22×D412yes—
457C2×C8C2511yescharacteristic
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—44no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4C2×D812yes—
484C4⋊C4C2×D812yes—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4C2×D812yes—
498C22⋊C4C2×D812yes—
499C22⋊C4C2×D812yes—
500C22⋊C4C2×D812yes—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42C2×D811yescharacteristic
508C42C2×D811yescharacteristic
509C42C22×D411yescharacteristic
510C42C22×D411yescharacteristic
511C16—24no—
512C16—24no—
513C23—22no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23C22×D812yes—
521C23C22×D812yes—
522C23C22×D811yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—82no—
528Q8—82no—
529Q8—161no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—82no—
535D4—84no—
536D4—161no—
537D4—162no—
538D4—162no—
539D4—162no—
540D4—162no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—82no—
562C2×C4—88no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—162no—
567C2×C4—162no—
568C2×C4C22×D812yes—
569C2×C4C22×D811yescharacteristic
570C2×C4C22×D811yescharacteristic
571C2×C4C22×D811yescharacteristic
572C2×C4C22×D812yes—
573C2×C4C22×D811yescharacteristic
574C2×C4C22×D811yescharacteristic
575C2×C4C23×D411yescharacteristic
576C8—21no—
577C8—21no—
578C8—21no—
579C8—22no—
580C8—22no—
581C8#58512yes—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22#341412yes—
596C22C23×D811yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—22no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4#348511yescharacteristic
612C4#358611yescharacteristic
613C2—22no—
614C2—42no—
615C2—42no—
616C2—162no—
617C2#6169411yescharacteristic
618C2C2×#341411yescharacteristic
619C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62249C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3336C2semidirect
3. #33916C2semidirect
4. #33915C2semidirect
5. #33884C2semidirect
6. #33882C2semidirect
7. #33802C2semidirect
8. #33800C2semidirect
9. #33762C2semidirect
10. #33756C2semidirect
11. #33673C2semidirect
12. #33672C2semidirect
13. #33646C2semidirect
14. #33645C2semidirect
15. #33487C2semidirect
45. #2255C22semidirect
46. #2255C22semidirect
47. #2245C22semidirect
48. #2244C22semidirect
49. #2237C22semidirect
50. #2237C22semidirect
51. #2237C22semidirect
53. #2236C22semidirect
56. #2234C22semidirect
58. #2233C22semidirect
61. #2229C22semidirect
62. #2228C22semidirect
63. #2227C22semidirect
64. #2225C22semidirect
65. #2220C22semidirect
66. #2219C22semidirect
67. #2217C22semidirect
68. #2216C22semidirect
69. #2209C22semidirect
70. #2207C22semidirect
71. #2189C22semidirect
72. #2189C22semidirect
75. #2176C22semidirect
76. #2175C22semidirect
81. #2159C22semidirect
82. #2158C22semidirect
83. #2139C22semidirect
84. #2138C22semidirect

Maximal subgroups

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

subgroupindex
#622492
C2×#33362
#339162
#339152
#338842
#338822
#338022
#338002
#337622
#337562
#336732
#336722
#336462
#336452
#334872