Finite Group Databasebeta

← #6495696

Subgroups — #6495696

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#33121C212yesmaximal, complemented
4#32918C212yesmaximal, complemented
5#29983C212yesmaximal, complemented
6#29801C212yesmaximal, complemented
7#29571C214yesmaximal, complemented
8#29371C212yesmaximal, complemented
9#29366C212yesmaximal, complemented
10#29337C211yesmaximal, characteristic
11#29332C211yesmaximal, characteristic, complemented
12#28313C211yesmaximal, characteristic, complemented
13#28305C211yesmaximal, characteristic
14#24074C212yesmaximal, complemented
15#23946C212yesmaximal, complemented
16#22159C214yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#519C2211yescharacteristic
20C2×#481C2212yes—
21C2×#483C2212yes—
22C2×#464C2214yes—
23C2×#451C2212yes—
24C2×#448C2212yes—
25#2868C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2211yescharacteristic
28#2856C2211yescharacteristic
29#2640C2212yes—
30#2620C2212yes—
31#2512—22no—
32#2491—22no—
33#2452—22no—
34#2422—22no—
35#2356—22no—
36#2355—22no—
37#2343—22no—
38#2343—22no—
39#2326C2212yes—
40#2320C2212yes—
41#2320C2212yes—
42#2317C2211yescharacteristic
43#2310C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2296C2211yescharacteristic
47#2295C2212yes—
48#2295C2212yes—
49C2×#391C2211yescharacteristic
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2099C2212yes—
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2058C2214yes—
57#2043C2212yes—
58#2042C2214yes—
59#2041C2212yes—
60#2024C2214yes—
61#2016C2212yes—
62#2015C2214yes—
63#2014C2212yes—
64#1983C2214yes—
65#1971C2214yes—
66#1968C2214yes—
67#1948C2214yes—
68#1936C2214yes—
69#1933C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1880C2214yes—
73#1878C2214yes—
74#1878C2214yes—
75#1877C2214yes—
76#1875C2212yes—
77#1875C2212yes—
78#1873C2212yes—
79#1873C2212yes—
80#1865C2214yes—
81#1864C2214yes—
82#1839C2212yes—
83#1831C2212yes—
84#1818C2212yes—
85#1803C2214yes—
86#1798C2212yes—
87#1768C2212yes—
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yes—
91C23×Q8C2311yescharacteristic
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94#542—44no—
95#542—44no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#126—22no—
101C2×#124—22no—
102C2×#124C2312yes—
103C2×#124C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121—22no—
108C2×#121C2312yes—
109#519—28no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#483—44no—
124#481—44no—
125#464—216no—
126#451—44no—
127#448—44no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143#397—24no—
144#396—24no—
145#394—24no—
146#393—24no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#388—24no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—21no—
179#385—21no—
180#385—24no—
181#384—21no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#383—22no—
187#383—22no—
188#382—21no—
189#382—22no—
190#382—22no—
191#381—21no—
192#381—21no—
193#380—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4D412yes—
206#376—22no—
207#376—22no—
208#376C2311yescharacteristic
209#376C2311yescharacteristic
210C2×#95—22no—
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—24no—
226C22×Q8—42no—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4C2×D412yes—
236C23×C4C2411yescharacteristic
237C2×Q16—28no—
238C2×Q16—44no—
239C2×Q16—44no—
240C2×SD16—28no—
241C2×SD16—44no—
242C2×SD16—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—84no—
249#127—42no—
250#126—48no—
251#124—28no—
252#124—42no—
253#124—42no—
254#124—48no—
255#123—48no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—48no—
259#121—24no—
260#121—24no—
261#121—84no—
262C4×Q8—216no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8#4118yes—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—84no—
272#117—216no—
273#117—24no—
274#117—24no—
275#117—24no—
276#117—42no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D411yescharacteristic
312C2×C22⋊C4C2×D411yescharacteristic
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#106—44no—
333#106—44no—
334C4⋊C8—28no—
335#103—28no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#102—28no—
341#102—44no—
342#102—44no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345#97—42no—
346C4×C8—42no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24—22no—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—416no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—84no—
382C2×Q8C2×#4114yes—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—84no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—84no—
410C22×C4—82no—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4C2511yescharacteristic
422Q16—88no—
423SD16—88no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4C2×#4114yes—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—24no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C42—24no—
469C42—28no—
470C42—28no—
471C42—24no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—44no—
483C42—82no—
484C42—82no—
485C42C2×#4114yes—
486C23—22no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23#53512yes—
496C23C23×D411yescharacteristic
497Q8—28no—
498Q8—48no—
499Q8—44no—
500Q8—44no—
501Q8—44no—
502Q8—164no—
503D4—44no—
504D4—44no—
505D4—88no—
506D4—164no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
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—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—164no—
550C2×C4—162no—
551C2×C4#52514yes—
552C2×C4#53512yes—
553C2×C4#57714yes—
554C2×C4C22×#4112yes—
555C2×C4C2×H(5,𝔽2)12yes—
556C2×C4#58512yes—
557C8—44no—
558C8—44no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—84no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22C2×#52511yescharacteristic
570C22C2×#53511yescharacteristic
571C22#344911yescharacteristic
572C22C2×#57712yes—
573C22#358612yes—
574C4—24no—
575C4—24no—
576C4—22no—
577C4—42no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—42no—
583C4—44no—
584C4—44no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4#349414yes—
591C2—42no—
592C2—162no—
593C2#3530111yescharacteristic
594C2#6059912yes—
595C2#6081811yescharacteristic
596C2#6082811yescharacteristic
597C2#6097712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #34254C2semidirect
3. #33121C2semidirect
4. #32918C2semidirect
5. #29983C2semidirect
6. #29801C2semidirect
7. #29571C2semidirect
8. #29371C2semidirect
9. #29366C2semidirect
11. #29332C2semidirect
12. #28313C2semidirect
14. #24074C2semidirect
15. #23946C2semidirect
16. #22159C2semidirect
17. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342542
#331212
#329182
#299832
#298012
#295712
#293712
#293662
#293372
#293322
#283132
#283052
#240742
#239462
#221592
#207842