Finite Group Databasebeta

← #6457538

Subgroups — #6457538

Order 512.

1496 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35627C211yesmaximal, characteristic, complemented
2#35558C211yesmaximal, characteristic
3#28056C214yesmaximal, complemented
4#28049C214yesmaximal, complemented
5#28044C214yesmaximal, complemented
6#28037C214yesmaximal, complemented
7C2×#2092C211yesmaximal, characteristic
8#23667C211yesmaximal, characteristic, complemented
9#23656C211yesmaximal, characteristic, complemented
10#23612C211yesmaximal, characteristic
11#21155C211yesmaximal, characteristic, complemented
12#21095C211yesmaximal, characteristic
13#21091C211yesmaximal, characteristic
14#21089C211yesmaximal, characteristic
15#21065C211yesmaximal, characteristic
16#21038C211yesmaximal, characteristic, complemented
17#20765C211yesmaximal, characteristic, complemented
18#20455C211yesmaximal, characteristic, complemented
19#20439C211yesmaximal, characteristic, complemented
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2571—22no—
33#2570—22no—
34#2513—22no—
35#2513C2214yes—
36#2512—22no—
37#2512C2214yes—
38#2509C2214yescomplemented
39#2508C2214yescomplemented
40#2505—24no—
41#2480—24no—
42#2317C2211yescharacteristic
43#2317C2211yescharacteristic
44#2317C2211yescharacteristic
45#2316C2211yescharacteristic
46#2309C2211yescharacteristic
47#2309C2211yescharacteristic
48#2296C2211yescharacteristic
49#2296C2211yescharacteristic
50C2×#390C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#382C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2092C2212yes—
56#2092C2212yes—
57#2092C2212yes—
58#2092C2212yes—
59#2087C2214yescomplemented
60#2086C2214yescomplemented
61#2085C2212yescomplemented
62#2085C2212yescomplemented
63#2084C2212yescomplemented
64#2083C2212yescomplemented
65#2067C2214yes—
66#2065C2214yes—
67#1943C2214yescomplemented
68#1941C2214yescomplemented
69#1936C2214yes—
70#1936C2214yes—
71#1936C2214yes—
72#1934C2214yes—
73#1934C2214yes—
74#1934C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1920C2214yescomplemented
78#1919C2214yescomplemented
79#1836C2214yescomplemented
80#1835C2214yescomplemented
81#1776C2214yescomplemented
82C4×#107C2214yescomplemented
83#1771C2212yescomplemented
84#1771C2212yescomplemented
85#1770C2212yescomplemented
86C8×C4⋊C4C2212yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×D4C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C2×#126—24no—
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—22no—
113C22×C42C2311yescharacteristic
114#427—24no—
115#427—24no—
116C2×#107—24no—
117C2×#107C2312yes—
118C2×#107C2312yes—
119C2×#107C2312yes—
120C2×#107C2312yes—
121C2×#106—24no—
122C2×#106C2312yes—
123C2×#106C2312yes—
124C2×#106C2312yes—
125C2×#106C2312yes—
126#424—24no—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#418—24no—
130#418—24no—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139#405—24no—
140#405—24no—
141C4×M16—24no—
142C2×#97C2312yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#400—24no—
150#400—24no—
151#398—24no—
152#397—24no—
153#397—24no—
154#397—24no—
155#396—24no—
156#390—21no—
157#390—22no—
158#390—21no—
159#390—22no—
160#390C2314yes—
161#390C2314yes—
162#390C2312yes—
163#390C2312yes—
164#389—22no—
165#389—24no—
166#389D412yes—
167#389D412yes—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#387—24no—
178#387—24no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#385D414yes—
184#384—21no—
185#384—21no—
186#384—21no—
187#384C2312yes—
188#382—21no—
189#382—21no—
190#382—21no—
191#382—22no—
192#382—24no—
193#382C2312yes—
194#382C2312yes—
195#382C2312yes—
196#380—22no—
197#380—22no—
198#379—22no—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C2×#95—22no—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2311yescharacteristic
222#336C2311yescharacteristic
223#336C2311yescharacteristic
224#336C2311yescharacteristic
225#336C2312yes—
226#336C2312yes—
227C25—21no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246C2×M16#4114yes—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8#4114yes—
252C22×C8#4114yes—
253C22×C8C2412yes—
254C22×C8C2411yescharacteristic
255C22×C8C2411yescharacteristic
256C22×C8C2411yescharacteristic
257C22×C8C2411yescharacteristic
258C22×C8C2412yes—
259#126—88no—
260#117—22no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—22no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117#4114yes—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—84no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—24no—
328#107—24no—
329#107—24no—
330#107—24no—
331#106—24no—
332#106—24no—
333#106—24no—
334#106—24no—
335C4⋊C8—24no—
336C4⋊C8—24no—
337#102—24no—
338#102—24no—
339#102—24no—
340#102—24no—
341#102—28no—
342#102—28no—
343#97—24no—
344#97—24no—
345#97—24no—
346C4×C8—24no—
347C4×C8—24no—
348C4×C8—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95C2×D414yes—
362#95C2×D414yes—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—82no—
368C24—82no—
369C24C22×D411yescharacteristic
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D414yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C22×C4C2511yescharacteristic
412M16—28no—
413M16—44no—
414C2×C8—28no—
415C2×C8—28no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8C2×#4114yes—
430C2×C8C2×#4112yes—
431C2×C8C2×#4112yes—
432C2×C8C2×#4112yes—
433C2×C8C2×#4112yes—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4C2×#4114yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C42—24no—
477C42—24no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—22no—
486C42—42no—
487C42—42no—
488C42—42no—
489C42—84no—
490C42—84no—
491C23—22no—
492C23—22no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—88no—
499C23—84no—
500C23—162no—
501C23—162no—
502C23—164no—
503C23#53212yes—
504C23#54712yes—
505C23#54812yes—
506C23C23×D411yescharacteristic
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—84no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—84no—
544C2×C4—162no—
545C2×C4—162no—
546C2×C4—162no—
547C2×C4#52914yes—
548C2×C4#53212yes—
549C2×C4#54114yes—
550C2×C4#54712yes—
551C2×C4#54812yes—
552C2×C4#55214yes—
553C2×C4#55214yes—
554C2×C4C22×#4111yescharacteristic
555C2×C4C22×#4111yescharacteristic
556C2×C4C22×#4111yescharacteristic
557C2×C4C22×#4111yescharacteristic
558C2×C4C2×2−1+411yescharacteristic
559C2×C4C2×2−1+411yescharacteristic
560C8—22no—
561C8—22no—
562C8—22no—
563C8—22no—
564C8—24no—
565C8—42no—
566C22—22no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22—82no—
575C22—168no—
576C22#251914yes—
577C22C2×#53211yescharacteristic
578C22C2×#54711yescharacteristic
579C22C2×#54811yescharacteristic
580C22#358412yes—
581C22#358512yes—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—42no—
586C4—42no—
587C4—44no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4#251914yes—
595C4C2×#52911yescharacteristic
596C4C2×#54111yescharacteristic
597C4C2×#55211yescharacteristic
598C4C2×#55211yescharacteristic
599C2—24no—
600C2—162no—
601C2C2×#251911yescharacteristic
602C2#6071311yescharacteristic
603C2#6071911yescharacteristic
604C2#6127612yes—
605C2#6134912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35627C2semidirect
3. #28056C2semidirect
4. #28049C2semidirect
5. #28044C2semidirect
6. #28037C2semidirect
8. #23667C2semidirect
9. #23656C2semidirect
11. #21155C2semidirect
16. #21038C2semidirect
17. #20765C2semidirect
18. #20455C2semidirect
19. #20439C2semidirect
38. #2509C22semidirect
39. #2508C22semidirect
59. #2087C22semidirect
60. #2086C22semidirect
61. #2085C22semidirect
62. #2085C22semidirect
63. #2084C22semidirect
64. #2083C22semidirect
67. #1943C22semidirect
68. #1941C22semidirect
77. #1920C22semidirect
78. #1919C22semidirect
79. #1836C22semidirect
80. #1835C22semidirect
81. #1776C22semidirect
82. C4×#107C22semidirect
83. #1771C22semidirect
84. #1771C22semidirect
85. #1770C22semidirect
86. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#356272
#355582
#280562
#280492
#280442
#280372
C2×#20922
#236672
#236562
#236122
#211552
#210952
#210912
#210892
#210652
#210382
#207652
#204552
#204392