Finite Group Databasebeta

← #6651607

Subgroups — #6651607

Order 512.

1555 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35111C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33099C211yesmaximal, characteristic, complemented
4#33095C211yesmaximal, characteristic
5#32689C214yesmaximal, complemented
6#32638C214yesmaximal, complemented
7#32582C214yesmaximal, complemented
8#32580C214yesmaximal, complemented
9#29642C211yesmaximal, characteristic, complemented
10#29639C211yesmaximal, characteristic
11#27617C212yesmaximal, complemented
12#24607C212yesmaximal
13#24606C212yesmaximal, complemented
14#24067C212yesmaximal
15#21822C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#432C2212yes—
22#2855C2211yescharacteristic
23#2855C2211yescharacteristic
24#2854C2211yescharacteristic
25#2854C2211yescharacteristic
26#2623—21no—
27#2614—21no—
28#2604—22no—
29#2599—22no—
30#2592C2212yes—
31#2584C2212yes—
32#2581C2212yes—
33#2580C2212yes—
34#2567—22no—
35#2567—22no—
36#2567C2214yes—
37#2567C2214yes—
38C2×#395—21no—
39C2×#395—21no—
40#2322—21no—
41#2322—21no—
42C2×#391—21no—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#384—21no—
46#2283C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2101C2214yes—
49#2088C2214yes—
50#2086C2214yes—
51#2085C2214yes—
52#2067C2214yes—
53#2066C2214yes—
54#2059C2214yes—
55#2058C2214yes—
56#2057C2214yes—
57#2042C2214yes—
58#2024C2214yes—
59#2023C2214yes—
60#1985C2214yes—
61#1985C2214yes—
62#1944C2214yes—
63#1943C2214yes—
64#1936C2214yes—
65#1935C2214yes—
66#1928C2214yes—
67#1926C2214yes—
68#1921C2214yes—
69#1919C2214yes—
70#1919C2214yes—
71#1918C2214yes—
72#1874C2212yes—
73#1872C2212yes—
74#1865C2214yes—
75#1862C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1817C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8D411yescharacteristic
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—44no—
87C2×C4⋊Q8C2314yes—
88#529—44no—
89C2×#126—21no—
90C2×#125C2311yescharacteristic
91C2×#124C2311yescharacteristic
92C2×#123—21no—
93C2×#123—21no—
94C2×#123—21no—
95C2×C22⋊Q8C2311yescharacteristic
96C2×#121C2311yescharacteristic
97C2×C4×Q8—44no—
98C2×C4×Q8D412yes—
99C2×C4×Q8D412yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—21no—
104C2×C4×D4C2311yescharacteristic
105C2×#117—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#484—44no—
119#481—44no—
120#432—44no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2314yes—
133#400—22no—
134#399—22no—
135#398—22no—
136#397—22no—
137#395—22no—
138#395—22no—
139#395—24no—
140#395—24no—
141#395—24no—
142#395—24no—
143#393—22no—
144#393—22no—
145#391—24no—
146#391—24no—
147#391C2312yes—
148#391C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#386—21no—
174#386—21no—
175#385—21no—
176#385—21no—
177#385—22no—
178#385—22no—
179#384—24no—
180#384—24no—
181#383—21no—
182#382C2312yes—
183#381—21no—
184#381—21no—
185#381—21no—
186#380C2311yescharacteristic
187#380C2311yescharacteristic
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—21no—
192#376—21no—
193C2×#95—21no—
194C2×#95—22no—
195C2×#95—22no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207#336C2314yes—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—44no—
213C22×Q8—84no—
214C22×Q8C2×D412yes—
215C22×Q8C2×D412yes—
216C22×Q8C2411yescharacteristic
217C22×D4—21no—
218C22×D4C2411yescharacteristic
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—44no—
232C4⋊Q8—44no—
233C4⋊Q8—48no—
234#126—44no—
235#125—42no—
236#124—42no—
237#123—44no—
238#123—44no—
239#123—44no—
240C22⋊Q8—24no—
241#121—24no—
242C4×Q8—44no—
243C4×Q8—44no—
244C4×Q8—42no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×D4—42no—
250C4×D4—44no—
251#117—44no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—84no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2411yescharacteristic
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—21no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—44no—
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2412yes—
314#107—28no—
315#107—44no—
316#106—28no—
317#106—44no—
318C4⋊C8—44no—
319#103—28no—
320#103—44no—
321#102—28no—
322#102—44no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325#97—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—21no—
332#95—21no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344C24—21no—
345C24C22×D411yescharacteristic
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—416no—
355C2×Q8—48no—
356C2×Q8—84no—
357C2×Q8—84no—
358C2×Q8—816no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—82no—
394C22×C4#13714yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C4C2511yescharacteristic
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C42−1+418yes—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C42—22no—
453C42—24no—
454C42—22no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—48no—
463C42—42no—
464C42—82no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23#53412yes—
473C23C2×#13712yes—
474C23#57512yes—
475C23C23×D411yescharacteristic
476Q8—48no—
477Q8—48no—
478Q8—44no—
479Q8—816no—
480D4—44no—
481D4—84no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—416no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—88no—
533C2×C4#53412yes—
534C2×C4C2×#13714yes—
535C2×C4C2×H(5,𝔽2)12yes—
536C2×C4C2×2−1+414yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—81no—
544C22—81no—
545C22—82no—
546C22—82no—
547C22—81no—
548C22#319614yes—
549C22C2×#53411yescharacteristic
550C22#348612yes—
551C22C22×#13711yescharacteristic
552C22C2×#57511yescharacteristic
553C22#358512yes—
554C4—24no—
555C4—22no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—42no—
560C4—41no—
561C4—42no—
562C4—41no—
563C4—48no—
564C4—81no—
565C4—81no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4#354414yes—
571C2—24no—
572C2—161no—
573C2#3725811yescharacteristic
574C2C2×#319611yescharacteristic
575C2#6075311yescharacteristic
576C2#6183012yes—
577C2#6183112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #33099C2semidirect
5. #32689C2semidirect
6. #32638C2semidirect
7. #32582C2semidirect
8. #32580C2semidirect
9. #29642C2semidirect
11. #27617C2semidirect
13. #24606C2semidirect
15. #21822C2semidirect

Maximal subgroups

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

subgroupindex
#351112
C2×#25672
#330992
#330952
#326892
#326382
#325822
#325802
#296422
#296392
#276172
#246072
#246062
#240672
#218222