Finite Group Databasebeta

← #6660672

Subgroups — #6660672

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35309C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic
3#32787C212yesmaximal, complemented
4#32774C212yesmaximal, complemented
5#30367C212yesmaximal, complemented
6#30358C212yesmaximal, complemented
7#29948C212yesmaximal, complemented
8#29901C214yesmaximal
9#29701C212yesmaximal, complemented
10#29341C212yesmaximal, complemented
11#29306C212yesmaximal, complemented
12#27681C212yesmaximal, complemented
13#24715C214yesmaximal, complemented
14#24306C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#479C2214yes—
20C2×#433C2212yes—
21#2857C2212yes—
22#2857C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2851C2211yescharacteristic
26#2629C2212yes—
27#2629C2211yescharacteristic
28#2629C2212yes—
29#2621C2211yescharacteristic
30#2616C2212yes—
31#2582—22no—
32#2538—22no—
33#2468—22no—
34#2466—22no—
35#2462—22no—
36#2460—22no—
37#2417—22no—
38#2417—22no—
39#2323C2212yes—
40#2316C2212yes—
41#2314C2211yescharacteristic
42#2311C2212yes—
43C4×C22⋊Q8C2212yes—
44C2×#384C2211yescharacteristic
45#2101C2214yes—
46#2100C2212yes—
47#2100C2212yes—
48#2095C2212yes—
49#2095C2212yes—
50#2083C2214yes—
51#2077C2214yes—
52#2076C2214yes—
53#2075C2214yes—
54#2075C2214yes—
55#2059C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2043C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2024C2212yes—
64#2024C2212yes—
65#1989C2214yes—
66#1947C2214yes—
67#1943C2214yes—
68#1942C2214yes—
69#1935C2214yes—
70#1933C2214yes—
71#1873C2214yes—
72#1872C2214yes—
73#1854C2214yes—
74#1852C2214yes—
75#1817C2212yes—
76#1803C2214yes—
77#1798C2212yes—
78#1797C2212yes—
79#1780C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#541—44no—
85#533—44no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#126—22no—
89C2×#125—22no—
90C2×#125C2312yes—
91C2×#125C2311yescharacteristic
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C4×#41—44no—
99C2×C4×Q8—22no—
100C2×C4×Q8D412yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#479—48no—
115#433—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131#398—24no—
132#398—24no—
133#398—22no—
134#398—22no—
135#397—22no—
136#396—24no—
137#396—24no—
138#396—22no—
139#395—22no—
140#394—22no—
141#394—22no—
142#394—22no—
143#391—22no—
144#391—22no—
145#391—22no—
146#391C2312yes—
147#391C2312yes—
148#390—22no—
149#389—24no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2314yes—
156#389C2314yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#386—22no—
168#386—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2312yes—
175#384C2312yes—
176#382C2311yescharacteristic
177#381—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380—24no—
182#380C2314yes—
183#380C2314yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2312yes—
199#336C2312yes—
200#336C2314yes—
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—42no—
204C22×Q8—22no—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2412yes—
207C22×D4C2×D411yescharacteristic
208C22×D4C2×D411yescharacteristic
209C22×D4C2×D411yescharacteristic
210C23×C4—22no—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8C2414yes—
218C22×C8C2414yes—
219C4⋊Q8—44no—
220C4⋊Q8—42no—
221#126—48no—
222#125—28no—
223#125—42no—
224#125—48no—
225C22⋊Q8—28no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C22⋊Q8—48no—
229C22⋊Q8—84no—
230C22⋊Q8—84no—
231C4×Q8—28no—
232C4×Q8—44no—
233C4×Q8—48no—
234C4×D4—44no—
235C4×D4—44no—
236C4×D4—44no—
237#117—24no—
238#117—24no—
239#117—42no—
240#117—42no—
241#117—42no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—84no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#107—44no—
296#107—44no—
297#106—44no—
298C4⋊C8—28no—
299#103—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303C22⋊C8—28no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—44no—
307#97—42no—
308C4×C8—42no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324C24C22×D411yescharacteristic
325C24C22×D411yescharacteristic
326C24C22×D411yescharacteristic
327#41—416no—
328C2×Q8—22no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—84no—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D412yes—
378C22×C42−1+414yes—
379C22×C4C2511yescharacteristic
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C42—24no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—42no—
430C42—42no—
431C42—44no—
432C42—44no—
433C42—82no—
434C42H(5,𝔽2)14yes—
435C23—22no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—22no—
448C23—22no—
449C23#53612yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—48no—
453Q8—88no—
454D4—44no—
455D4—44no—
456D4—44no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—48no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—84no—
498C2×C4—84no—
499C2×C4—164no—
500C2×C4#53414yes—
501C2×C4#53612yes—
502C2×C4#57814yes—
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4C2×2−1+412yes—
505C2×C4#58512yes—
506C8—44no—
507C8—44no—
508C22—22no—
509C22—24no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22C2×#53411yescharacteristic
524C22C2×#53611yescharacteristic
525C22#344911yescharacteristic
526C22C2×#57812yes—
527C22#358512yes—
528C4—22no—
529C4—24no—
530C4—24no—
531C4—41no—
532C4—41no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4—82no—
539C4—84no—
540C4—82no—
541C4#354114yes—
542C2—42no—
543C2—81no—
544C2—81no—
545C2#3743811yescharacteristic
546C2#6076412yes—
547C2#6089312yes—
548C2#6101311yescharacteristic
549C2#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. #32787C2semidirect
4. #32774C2semidirect
5. #30367C2semidirect
6. #30358C2semidirect
7. #29948C2semidirect
9. #29701C2semidirect
10. #29341C2semidirect
11. #29306C2semidirect
12. #27681C2semidirect
13. #24715C2semidirect
14. #24306C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#352802
#327872
#327742
#303672
#303582
#299482
#299012
#297012
#293412
#293062
#276812
#247152
#243062
#207202