Finite Group Databasebeta

← #6436970

Subgroups — #6436970

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35309C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29948C212yesmaximal, complemented
4#29944C212yesmaximal, complemented
5#29930C212yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#29911C212yesmaximal, complemented
8#29909C212yesmaximal, complemented
9#29901C214yesmaximal
10#23773C214yesmaximal, complemented
11#23762C212yesmaximal, complemented
12#23757C212yesmaximal, complemented
13#21398C212yesmaximal, complemented
14#21368C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#433C2212yes—
19#2857C2211yescharacteristic
20#2855C2212yes—
21#2851C2211yescharacteristic
22#2629C2211yescharacteristic
23#2621C2211yescharacteristic
24#2616C2212yes—
25#2468—22no—
26#2466—22no—
27#2462—22no—
28#2460—22no—
29#2355—22no—
30#2354—22no—
31#2352—22no—
32#2352—22no—
33#2323C2212yes—
34#2314C2211yescharacteristic
35#2314C2212yes—
36#2314C2212yes—
37#2311C2212yes—
38C4×C22⋊Q8C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2097C2214yes—
48#2095C2212yes—
49#2095C2212yes—
50#2084C2212yes—
51#2083C2212yes—
52#2083C2214yes—
53#2077C2214yes—
54#2076C2214yes—
55#2075C2214yes—
56#2075C2214yes—
57#2072C2212yes—
58#2072C2212yes—
59#2072C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#1989C2214yes—
65#1981C2214yes—
66#1949C2214yes—
67#1947C2214yes—
68#1873C2214yes—
69#1872C2214yes—
70#1854C2214yes—
71#1852C2214yes—
72#1817C2212yes—
73#1798C2212yes—
74#1797C2212yes—
75#1780C2212yes—
76#1776C2214yes—
77C4×#107C2214yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#541—28no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#126—22no—
87C2×#125—22no—
88C2×#125C2311yescharacteristic
89C2×C22⋊Q8C2312yes—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C4×#41—28no—
93C2×C4×Q8D412yes—
94C2×C4×Q8C2312yes—
95C2×C4×D4D412yes—
96C2×C4×D4D412yes—
97C2×C4×D4D412yes—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4—22no—
105C22×C4⋊C4C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109#433—28no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×C22⋊C8C2314yes—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#398—22no—
126#398—22no—
127#397—22no—
128#396—22no—
129#395—22no—
130#394—22no—
131#394—22no—
132#394—22no—
133#391—22no—
134#391C2312yes—
135#391C2312yes—
136#390—22no—
137#389—22no—
138#389—22no—
139#389C2312yes—
140#389C2312yes—
141#388—24no—
142#388—24no—
143#388—22no—
144#388—22no—
145#388—22no—
146#387—24no—
147#387—24no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#386—22no—
155#386—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384—22no—
160#384C2312yes—
161#384C2311yescharacteristic
162#384C2312yes—
163#384C2312yes—
164#382—22no—
165#382—22no—
166#382C2312yes—
167#382C2311yescharacteristic
168#381—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178C4×C4⋊C4—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—24no—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199C2×#41—24no—
200C2×#41—42no—
201C2×#41—42no—
202C22×Q8C2×D411yescharacteristic
203C22×Q8C2412yes—
204C22×D4C2×D411yescharacteristic
205C22×D4C2×D411yescharacteristic
206C22×D4C2×D411yescharacteristic
207C23×C4—22no—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2×D412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—24no—
217#126—48no—
218#125—24no—
219#125—48no—
220C22⋊Q8—28no—
221C22⋊Q8—42no—
222C22⋊Q8—44no—
223C22⋊Q8—42no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—28no—
229#117—24no—
230#117—24no—
231#117—44no—
232#117—84no—
233#117—84no—
234#117#4118yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—84no—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D412yes—
270C2×C22⋊C4C2×D412yes—
271C2×C22⋊C4C2×D412yes—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287#107—44no—
288#107—44no—
289#107—44no—
290#106—44no—
291C4⋊C8—28no—
292#103—44no—
293#103—44no—
294#103—44no—
295#103—44no—
296C22⋊C8—28no—
297C22⋊C8—44no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300#97—24no—
301C4×C8—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318C24C22×D411yescharacteristic
319C24C22×D411yescharacteristic
320C24C22×D411yescharacteristic
321#41—416no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—84no—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C2×#4114yes—
371C22×C4C2511yescharacteristic
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4#12318yes—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4#12318yes—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—44no—
422C42—44no—
423C42—82no—
424C42—84no—
425C42—84no—
426C42C2×#4114yes—
427C42C2×#4114yes—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23C2×#12312yes—
443C23C23×D411yescharacteristic
444Q8—44no—
445Q8—44no—
446Q8—44no—
447D4—44no—
448D4—44no—
449D4—44no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—164no—
492C2×C4C2×#12312yes—
493C2×C4C2×#12314yes—
494C2×C4C2×#12314yes—
495C2×C4#57814yes—
496C2×C4C22×#4112yes—
497C2×C4#58512yes—
498C2×C4#58512yes—
499C8—44no—
500C8—44no—
501C22—22no—
502C22—24no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22C22×#12311yescharacteristic
517C22#344911yescharacteristic
518C22#344911yescharacteristic
519C22C2×#57812yes—
520C22#358512yes—
521C4—22no—
522C4—24no—
523C4—24no—
524C4—24no—
525C4—41no—
526C4—41no—
527C4—42no—
528C4—84no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—84no—
533C4—82no—
534C4#345514yes—
535C2—42no—
536C2—81no—
537C2—81no—
538C2#3463511yescharacteristic
539C2#6052612yes—
540C2#6099312yes—
541C2#6101311yescharacteristic
542C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29948C2semidirect
4. #29944C2semidirect
5. #29930C2semidirect
6. #29923C2semidirect
7. #29911C2semidirect
8. #29909C2semidirect
10. #23773C2semidirect
11. #23762C2semidirect
12. #23757C2semidirect
13. #21398C2semidirect
14. #21368C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#341882
#299482
#299442
#299302
#299232
#299112
#299092
#299012
#237732
#237622
#237572
#213982
#213682
#207202