Finite Group Databasebeta

← #6434540

Subgroups — #6434540

Order 512.

1796 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35287C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic, complemented
3#30129C212yesmaximal, complemented
4#30126C212yesmaximal, complemented
5#30090C212yesmaximal, complemented
6#30053C212yesmaximal, complemented
7#29965C212yesmaximal, complemented
8#29907C212yesmaximal, complemented
9#29725C212yesmaximal, complemented
10#29628C212yesmaximal, complemented
11#23825C211yesmaximal, characteristic, complemented
12#23819C211yesmaximal, characteristic, complemented
13#23802C211yesmaximal, characteristic, complemented
14#23799C211yesmaximal, characteristic, complemented
15#23693C214yesmaximal, complemented
16#21464C214yesmaximal, complemented
17#20780C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520C2212yes—
20C2×#449C2214yes—
21C2×#442C2214yes—
22C2×C4×SD16C2214yes—
23#2860C2212yes—
24#2854C2212yes—
25#2647C2211yescharacteristic
26#2641C2211yescharacteristic
27#2629C2211yescharacteristic
28#2619C2211yescharacteristic
29#2593—22no—
30#2554—22no—
31#2468—22no—
32#2457—22no—
33#2361—22no—
34#2361—22no—
35#2354—22no—
36D4×C4⋊C4—22no—
37#2326C2212yes—
38#2326C2212yes—
39#2324C2212yes—
40#2319C2212yes—
41#2311C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2096C2212yes—
53#2094C2212yes—
54#2094C2212yes—
55#2094C2212yes—
56#2084C2214yes—
57#2074C2212yes—
58#2073C2212yes—
59#2071C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2055C2214yes—
63#2054C2214yes—
64#2043C2212yes—
65#2042C2214yes—
66#2041C2212yes—
67#2026C2212yes—
68#2025C2212yes—
69#2014C2212yes—
70#2013C2212yes—
71#2012C2214yes—
72#2011C2212yes—
73#2011C2212yes—
74#1880C2214yes—
75#1879C2214yes—
76#1874C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1853C2214yes—
80#1840C2212yes—
81#1831C2212yes—
82#1817C2212yes—
83#1798C2212yes—
84#1788C2214yes—
85#1768C2212yes—
86#1763C2214yes—
87C4×#102C2214yes—
88#1750C2212yes—
89#1745C2211yescharacteristic
90#1744C2211yescharacteristic
91C22×#41C2312yes—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94#532—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#126—22no—
99C2×#125C2311yescharacteristic
100C2×#124C2312yes—
101C2×#123—22no—
102C2×#121—22no—
103C2×#121C2312yes—
104#520—48no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123#449—216no—
124#442—48no—
125C4×SD16—48no—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#394—22no—
145#394—22no—
146#394—22no—
147#392—22no—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—24no—
176#387—22no—
177#386—22no—
178#386—24no—
179#386—22no—
180#385—22no—
181#385—24no—
182#385—24no—
183#384—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#382C2311yescharacteristic
189#381—22no—
190#380—24no—
191#380—22no—
192#380C2312yes—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C2×#41—24no—
216C2×#41—44no—
217C2×#41—44no—
218C2×#41#4118yes—
219C22×Q8C2412yes—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2412yes—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4C2×D412yes—
232C23×C4C2411yescharacteristic
233C2×SD16—28no—
234C2×SD16—44no—
235C2×SD16—44no—
236C2×D8—28no—
237C2×D8—44no—
238C2×D8—44no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—24no—
242#127—24no—
243#127—42no—
244#126—84no—
245#125—42no—
246#124—24no—
247#124—24no—
248#123—84no—
249#121—216no—
250#121—28no—
251C4×Q8—28no—
252C4×Q8—44no—
253C4×D4—216no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—44no—
257C4×D4—48no—
258C4×D4—48no—
259C4×D4—84no—
260C4×D4—84no—
261C4×D4—84no—
262#117—24no—
263#117—42no—
264#117—48no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—84no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C22⋊C4C2×D411yescharacteristic
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#106—44no—
324#106—44no—
325C4⋊C8—44no—
326#103—28no—
327#103—44no—
328#103—44no—
329#102—28no—
330#102—44no—
331#102—44no—
332#102—44no—
333#102—44no—
334C22⋊C8—28no—
335#97—24no—
336C4×C8—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24—22no—
352C24C22×D411yescharacteristic
353#41—416no—
354#41—416no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8C2×#4114yes—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4C2×#4114yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C2×#4114yes—
412C22×C4C2511yescharacteristic
413SD16—416no—
414D8—416no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—28no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—84no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C22⋊C4—216no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—82no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—42no—
471C42—42no—
472C42—44no—
473C42—44no—
474C42—82no—
475C42H(5,𝔽2)14yes—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—24no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23#52512yes—
493C23C23×D411yescharacteristic
494Q8—28no—
495Q8—44no—
496D4—28no—
497D4—416no—
498D4—416no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—88no—
503D4—164no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—164no—
543C2×C4—164no—
544C2×C4—164no—
545C2×C4—164no—
546C2×C4#52512yes—
547C2×C4#53512yes—
548C2×C4#53512yes—
549C2×C4#57714yes—
550C2×C4C22×#4112yes—
551C2×C4C2×H(5,𝔽2)12yes—
552C2×C4#58512yes—
553C8—44no—
554C8—44no—
555C22—22no—
556C22—24no—
557C22—42no—
558C22—41no—
559C22—41no—
560C22—44no—
561C22—48no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22C2×#52511yescharacteristic
573C22C2×#53511yescharacteristic
574C22#344911yescharacteristic
575C22C2×#57712yes—
576C22#358512yes—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4#345914yes—
593C2—42no—
594C2—84no—
595C2—162no—
596C2#3459311yescharacteristic
597C2#6058812yes—
598C2#6082011yescharacteristic
599C2#6083011yescharacteristic
600C2#6098912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35287C2semidirect
2. #34207C2semidirect
3. #30129C2semidirect
4. #30126C2semidirect
5. #30090C2semidirect
6. #30053C2semidirect
7. #29965C2semidirect
8. #29907C2semidirect
9. #29725C2semidirect
10. #29628C2semidirect
11. #23825C2semidirect
12. #23819C2semidirect
13. #23802C2semidirect
14. #23799C2semidirect
15. #23693C2semidirect
16. #21464C2semidirect
17. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#352872
#342072
#301292
#301262
#300902
#300532
#299652
#299072
#297252
#296282
#238252
#238192
#238022
#237992
#236932
#214642
#207802