Finite Group Databasebeta

← #6798809

Subgroups — #6798809

Order 512.

1406 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#36132C211yesmaximal, characteristic
3#33295C211yesmaximal, characteristic, complemented
4#31749C214yesmaximal, complemented
5#31747C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#29214C211yesmaximal, characteristic, complemented
8#28725C211yesmaximal, characteristic, complemented
9#27881C212yesmaximal, complemented
10#26038C214yesmaximal, complemented
11#25910C212yesmaximal, complemented
12#24980C214yesmaximal
13#21839C212yesmaximal, complemented
14#21657C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic
16C2×#533C2212yes—
17C2×#491C2211yescharacteristic
18C2×#491C2212yes—
19C2×#432C2212yes—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2551C2212yes—
25#2536C2212yes—
26#2508—22no—
27#2495—22no—
28#2489C2214yes—
29#2485—22no—
30#2465—22no—
31#2465—22no—
32#2452—22no—
33#2439C2212yes—
34#2439C2212yes—
35#2365—22no—
36#2356C2212yes—
37#2354C2212yes—
38#2346—22no—
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125C2211yescharacteristic
41#2285C2212yes—
42C42×D4C2211yescharacteristic
43#2077C2214yes—
44#2076C2214yes—
45#2074C2212yes—
46#2074C2212yes—
47#2068C2214yes—
48#2066C2214yes—
49#2059C2214yes—
50#2036C2214yes—
51#2035C2214yes—
52#2024C2214yes—
53#2024C2214yes—
54#1992C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1992C2211yescharacteristic
59#1991C2212yes—
60#1991C2212yes—
61#1988C2214yes—
62#1987C2214yes—
63#1960C2211yescharacteristic
64#1955C2212yes—
65#1949C2214yes—
66#1940C2212yes—
67#1933C2214yes—
68#1923C2214yes—
69#1920C2214yes—
70#1918C2214yes—
71#1873C2214yes—
72#1873C2214yes—
73#1847C2212yes—
74#1836C2214yes—
75#1836C2211yescharacteristic
76#1825C2214yes—
77#1825C2214yes—
78#1803C2214yes—
79#1767C2211yescharacteristic
80#1763C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83#533—28no—
84#533—44no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125C2312yes—
90C2×#125C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8C2314yes—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×D4D412yes—
99C2×C4×D4C2311yescharacteristic
100C2×#117—22no—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C22×C42C2312yes—
104#491—24no—
105#491—44no—
106#432—44no—
107C2×#107C2314yes—
108C2×#106C2314yes—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×#97C2312yes—
118C2×#97C2311yescharacteristic
119C2×#97C2312yes—
120C2×#97C2311yescharacteristic
121C2×C4×C8C2312yes—
122#400—24no—
123#398—22no—
124#397—24no—
125#396—22no—
126#396—22no—
127#395—22no—
128#395—22no—
129#394—22no—
130#393—22no—
131#393—22no—
132#391—22no—
133#391—22no—
134#391C2312yes—
135#391C2314yes—
136#390—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#388—24no—
149#388—24no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#385—22no—
159#385—22no—
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2312yes—
163#383—22no—
164#382—22no—
165#382—24no—
166#382C2311yescharacteristic
167#382C2311yescharacteristic
168#382C2311yescharacteristic
169#382C2312yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174#380C2314yes—
175#380C2312yes—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2311yescharacteristic
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4C2312yes—
187#376—22no—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C2412yes—
204C22×D4C2×D411yescharacteristic
205C23×C4C2×D412yes—
206C23×C4C2×D412yes—
207C23×C4C2412yes—
208C22×C8C2412yes—
209C22×C8C2414yes—
210C22×C8C2412yes—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—48no—
218#125—24no—
219#125—22no—
220#125—24no—
221#125—42no—
222#125—42no—
223#125#4114yes—
224C22⋊Q8—48no—
225C22⋊Q8—48no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C4×Q8—28no—
229C4×Q8—48no—
230C4×Q8—48no—
231C4×D4—24no—
232C4×D4—44no—
233#117—24no—
234#117—24no—
235#117—42no—
236#117—42no—
237#117—48no—
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—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
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—22no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287#107—44no—
288#106—44no—
289C4⋊C8—24no—
290C4⋊C8—44no—
291#103—28no—
292#103—44no—
293#103—44no—
294#103—44no—
295C22⋊C8—44no—
296C22⋊C8—44no—
297#97—22no—
298#97—22no—
299#97—24no—
300#97—24no—
301C4×C8—42no—
302#95—24no—
303#95—22no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95C2×D411yescharacteristic
319#95C2×D411yescharacteristic
320#95C2×D412yes—
321#95C2×D412yes—
322C24C22×D412yes—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—24no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×D4—22no—
334C2×D4—42no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C2511yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C4⋊C4—22no—
370C4⋊C4—28no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4C2×#4114yes—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—48no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—84no—
407C42—24no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—42no—
413C42—42no—
414C42—44no—
415C42—44no—
416C42—44no—
417C42—44no—
418C42—48no—
419C42C2×#4112yes—
420C42H(5,𝔽2)12yes—
421C422−1+414yes—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23C23×D411yescharacteristic
430Q8—44no—
431Q8—44no—
432Q8—88no—
433Q8—88no—
434D4—44no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4#53612yes—
475C2×C4#53612yes—
476C2×C4#54614yes—
477C2×C4#57514yes—
478C2×C4C22×#4111yescharacteristic
479C2×C4C2×H(5,𝔽2)11yescharacteristic
480C2×C4C2×2−1+412yes—
481C2×C4#58512yes—
482C8—42no—
483C8—42no—
484C8—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22C2×#53611yescharacteristic
493C22C2×#54611yescharacteristic
494C22#350011yescharacteristic
495C22C2×#57512yes—
496C22#358612yes—
497C4—22no—
498C4—22no—
499C4—24no—
500C4—24no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—44no—
505C4—82no—
506C4—82no—
507C4—84no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4#351812yes—
512C4#354712yes—
513C2—82no—
514C2#3895811yescharacteristic
515C2#6086311yescharacteristic
516C2#6088411yescharacteristic
517C2#6116312yes—
518C2#6258112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33295C2semidirect
4. #31749C2semidirect
5. #31747C2semidirect
6. #29852C2semidirect
7. #29214C2semidirect
8. #28725C2semidirect
9. #27881C2semidirect
10. #26038C2semidirect
11. #25910C2semidirect
13. #21839C2semidirect
14. #21657C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#361322
#332952
#317492
#317472
#298522
#292142
#287252
#278812
#260382
#259102
#249802
#218392
#216572
#212712