Finite Group Databasebeta

← #7622790

Subgroups — #7622790

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60331C211yesmaximal, characteristic
2#33508C213yesmaximal, complemented
3#33507C213yesmaximal, complemented
4#33504C213yesmaximal, complemented
5#33502C211yesmaximal, characteristic, complemented
6#33499C213yesmaximal, complemented
7#33498C211yesmaximal, characteristic, complemented
8#33496C211yesmaximal, characteristic, complemented
9#33485C211yesmaximal, characteristic, complemented
10#33483C213yesmaximal, complemented
11#33480C213yesmaximal, complemented
12#33479C213yesmaximal, complemented
13Q8×M32C211yesmaximal, characteristic, complemented
14#33476C213yesmaximal, complemented
15C2×Q8×C16C211yesmaximal, characteristic, complemented
16#3482C411yescharacteristic
17#2994C413yes—
18#2994C413yes—
19#2991C411yescharacteristic
20#2990C413yes—
21#2986C413yes—
22#2984C411yescharacteristic
23#2984C411yescharacteristic
24#2966C2213yes—
25#2966C2213yes—
26C8×#41C2213yes—
27C8×#41C2211yescharacteristic
28#2960C2213yes—
29C2×C8×Q8C2211yescharacteristic
30C2×C8×Q8C2211yescharacteristic
31#2183C2213yes—
32#2183C2213yes—
33#2183C2213yes—
34#2183C2216yes—
35#2183C2213yes—
36#2183C2213yes—
37#2183C2213yes—
38Q8×C16C2213yes—
39Q8×C16C2211yescharacteristic
40Q8×C16C2211yescharacteristic
41Q8×C16C2213yes—
42Q8×C16C2213yes—
43Q8×C16C2211yescharacteristic
44Q8×C16C2213yes—
45Q8×C16C2211yescharacteristic
46#2169C2213yes—
47#2169C2213yes—
48#2169C2213yes—
49#2169C2213yes—
50#2168C2211yescharacteristic
51#2168C2213yes—
52#2168C2213yes—
53#2168C2213yes—
54#2168C2216yes—
55D4×C16C2213yes—
56D4×C16C2216yes—
57D4×C16C2213yes—
58#2164C2216yes—
59#2164C2216yes—
60#2163C2213yes—
61#2163C2213yes—
62#2162C2213yes—
63#2162C2213yes—
64#2162C2213yes—
65#2162C2213yes—
66#2151C2216yes—
67#2150C2213yes—
68C2×C4⋊C16C2213yes—
69C2×C4⋊C16C2213yes—
70#2108C2216yes—
71C4×M32C2213yes—
72C4×M32C2213yes—
73C2×C4×C16C2213yes—
74#557C811yescharacteristic
75#556C813yes—
76#552C813yes—
77#552C813yes—
78D4×Q8C811yescharacteristic
79#548C811yescharacteristic
80#548C811yescharacteristic
81#547C813yes—
82#533C2×C413yes—
83#529C2×C413yes—
84#529C2×C413yes—
85C4×#41C2×C413yes—
86C4×#41C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×M32C2311yescharacteristic
90C2×M32C2313yes—
91C2×M32C2313yes—
92C2×M32C2313yes—
93C22×C16C2311yescharacteristic
94C22×C16C2313yes—
95C22×C16C2316yes—
96#446C2×C413yes—
97#446C2×C413yes—
98#446C2×C416yes—
99#446C2×C413yes—
100#446C2×C413yes—
101C8×Q8C2×C411yescharacteristic
102C8×Q8C2×C411yescharacteristic
103C8×Q8C2×C413yes—
104C8×Q8C2×C413yes—
105C8×Q8C2×C411yescharacteristic
106C8×Q8C2×C413yes—
107C8×Q8C2311yescharacteristic
108C8×Q8C2313yes—
109C8×Q8C2313yes—
110C8×Q8C2313yes—
111#436C2×C413yes—
112#436C2×C413yes—
113#435C2×C416yes—
114#435C2×C416yes—
115C8×D4C2×C413yes—
116C8×D4C2×C413yes—
117C8×D4C2×C413yes—
118C8×D4C2×C413yes—
119C8×D4C2313yes—
120C8×D4C2313yes—
121C8×D4C2311yescharacteristic
122C8×D4C2313yes—
123#433C2×C416yes—
124#433C2×C416yes—
125#432C2×C413yes—
126#432C2×C413yes—
127#431C2×C413yes—
128#431C2×C413yes—
129#431C2×C413yes—
130#431C2×C413yes—
131#424C2×C416yes—
132#423C2×C413yes—
133C2×C4⋊C8C2×C413yes—
134C2×C4⋊C8C2×C413yes—
135#405C2316yes—
136C2×C4×C8C2313yes—
137C2×C4×C8C2313yes—
138C2×C4×C8C2313yes—
139C4⋊C16C2313yes—
140C4⋊C16C2313yes—
141C4⋊C16C2313yes—
142C4⋊C16C2313yes—
143C4⋊C16C2316yes—
144C4⋊C16C2316yes—
145C4⋊C16C2313yes—
146C4⋊C16C2313yes—
147C4⋊C16C2311yescharacteristic
148C4⋊C16C2313yes—
149C4⋊C16C2313yes—
150C4⋊C16C2313yes—
151C22⋊C16C2311yescharacteristic
152C22⋊C16C2313yes—
153C22⋊C16C2311yescharacteristic
154C22⋊C16C2313yes—
155C22⋊C16C2313yes—
156C22⋊C16C2313yes—
157C22⋊C16C2316yes—
158#346C2311yescharacteristic
159#346C2313yes—
160#346C2313yes—
161#346C2313yes—
162#346C2316yes—
163C4×C16C2313yes—
164C4×C16C2313yes—
165C4×C16C2313yes—
166C4×C16C2313yes—
167C4×C16C2313yes—
168C4×C16C2313yes—
169C4×C16C2316yes—
170C2×#41C2×C811yescharacteristic
171C2×#41C2×C813yes—
172C22×Q8C2×C811yescharacteristic
173C22×Q8C2×C811yescharacteristic
174C2×M16C22×C416yes—
175C22×C8C22×C413yes—
176C22×C8C22×C413yes—
177C22×C8C22×C413yes—
178C22×C8C2411yescharacteristic
179C22×C8C2411yescharacteristic
180C22×C8C2413yes—
181C4⋊Q8C2×C813yes—
182C4⋊Q8C2×C813yes—
183#126C2×C816yes—
184#126C2×C816yes—
185#125C2×C813yes—
186#125C2×C813yes—
187#125C2×C816yes—
188#124C2×C813yes—
189#124C2×C813yes—
190#123C2×C816yes—
191#123C2×C816yes—
192C22⋊Q8C2×C813yes—
193C22⋊Q8C2×C813yes—
194C22⋊Q8C2×C816yes—
195C22⋊Q8C2×C813yes—
196C22⋊Q8C2×C813yes—
197#121C2×C813yes—
198#121C2×C813yes—
199C4×Q8C2×C811yescharacteristic
200C4×Q8C2×C811yescharacteristic
201C4×Q8C2×C813yes—
202C4×Q8C2×C813yes—
203C4×Q8C2×C811yescharacteristic
204C4×Q8C2×C813yes—
205C4×Q8C22×C411yescharacteristic
206C4×Q8C22×C413yes—
207C4×Q8C22×C413yes—
208C4×Q8C22×C413yes—
209C4×D4C2×C813yes—
210C4×D4C2×C813yes—
211C4×D4C2×C813yes—
212C4×D4C2×C813yes—
213C4×D4C22×C411yescharacteristic
214C4×D4C22×C413yes—
215C4×D4C22×C413yes—
216C4×D4C22×C413yes—
217#117C2×C813yes—
218#117C2×C813yes—
219#117C2×C813yes—
220#117C2×C813yes—
221#117C22×C416yes—
222C2×C4⋊C4C2×C813yes—
223C2×C4⋊C4C2×C813yes—
224C2×C42C22×C413yes—
225C2×C42C22×C413yes—
226C2×C42C22×C413yes—
227M32—22no—
228M32—26no—
229M32—43no—
230M32—43no—
231C2×C16—22no—
232C2×C16—46no—
233C2×C16—43no—
234C2×C16C2411yescharacteristic
235C2×C16C2413yes—
236C2×C16C2411yescharacteristic
237C2×C16C2413yes—
238C2×C16C2411yescharacteristic
239C2×C16C2413yes—
240C2×C16C2413yes—
241C2×C16C2411yescharacteristic
242C4⋊C8C22×C413yes—
243C4⋊C8C22×C413yes—
244C4⋊C8C22×C413yes—
245C4⋊C8C22×C416yes—
246C4⋊C8C22×C416yes—
247C4⋊C8C22×C413yes—
248C4⋊C8C22×C413yes—
249C4⋊C8C22×C413yes—
250C4⋊C8C22×C411yescharacteristic
251C4⋊C8C22×C413yes—
252C4⋊C8C22×C413yes—
253C4⋊C8C22×C413yes—
254C22⋊C8C22×C411yescharacteristic
255C22⋊C8C22×C411yescharacteristic
256C22⋊C8C22×C413yes—
257C22⋊C8C22×C413yes—
258C22⋊C8C22×C416yes—
259C22⋊C8C22×C413yes—
260C22⋊C8C22×C413yes—
261#97C22×C416yes—
262#97C22×C416yes—
263C4×C8C22×C413yes—
264C4×C8C22×C413yes—
265C4×C8C22×C413yes—
266C4×C8C22×C413yes—
267C4×C8C22×C413yes—
268C4×C8C22×C413yes—
269C4×C8C2413yes—
270C4×C8C2413yes—
271C4×C8C2413yes—
272C4×C8C2411yescharacteristic
273#41—24no—
274#41—46no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8C22×C813yes—
278C2×Q8C22×C813yes—
279C2×Q8C22×C813yes—
280C2×Q8C22×C811yescharacteristic
281C2×D4C22×C811yescharacteristic
282C2×D4C22×C813yes—
283C2×D4C22×C813yes—
284C2×D4C22×C813yes—
285C22×C4C22×C813yes—
286C22×C4C22×C813yes—
287C22×C4C22×C813yes—
288C22×C4C22×C813yes—
289C22×C4C22×C813yes—
290C22×C4C22×C813yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C23×C413yes—
294M16—46no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—23no—
298C2×C8—21no—
299C2×C8—23no—
300C2×C8—21no—
301C2×C8—43no—
302C2×C8—43no—
303C2×C8—43no—
304C2×C8C23×C411yescharacteristic
305C2×C8C23×C411yescharacteristic
306C2×C8C23×C411yescharacteristic
307C2×C8C23×C413yes—
308C2×C8C23×C413yes—
309C2×C8C23×C413yes—
310C2×C8C23×C413yes—
311C2×C8C2511yescharacteristic
312C4⋊C4C22×C813yes—
313C4⋊C4C22×C813yes—
314C4⋊C4C22×C813yes—
315C4⋊C4C22×C816yes—
316C4⋊C4C22×C816yes—
317C4⋊C4C22×C811yescharacteristic
318C4⋊C4C22×C813yes—
319C4⋊C4C22×C813yes—
320C4⋊C4C22×C813yes—
321C4⋊C4C22×C813yes—
322C4⋊C4C22×C813yes—
323C4⋊C4C22×C813yes—
324C4⋊C4C22×C816yes—
325C4⋊C4C22×C816yes—
326C22⋊C4C22×C811yescharacteristic
327C22⋊C4C22×C811yescharacteristic
328C22⋊C4C22×C813yes—
329C22⋊C4C22×C813yes—
330C22⋊C4C22×C816yes—
331C22⋊C4C22×C813yes—
332C22⋊C4C22×C813yes—
333C22⋊C4C22×C816yes—
334C22⋊C4C22×C816yes—
335C42C22×C813yes—
336C42C22×C813yes—
337C42C22×C813yes—
338C42C22×C813yes—
339C42C22×C813yes—
340C42C22×C813yes—
341C42C23×C411yescharacteristic
342C42C23×C413yes—
343C42C23×C413yes—
344C42C23×C413yes—
345C16—21no—
346C16—21no—
347C16—23no—
348C16—23no—
349C16—23no—
350C16—23no—
351C16—21no—
352C162−1+412yes—
353C23C23×C813yes—
354C23C23×C811yescharacteristic
355C23C23×C811yescharacteristic
356Q8—26no—
357Q8—26no—
358Q8—43no—
359Q8#50414yes—
360D4—26no—
361D4—22no—
362D4—26no—
363D4—43no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—21no—
369C2×C4—23no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—23no—
374C2×C4—43no—
375C2×C4—43no—
376C2×C4C23×C811yescharacteristic
377C2×C4C23×C811yescharacteristic
378C2×C4C23×C811yescharacteristic
379C2×C4C23×C813yes—
380C2×C4C23×C811yescharacteristic
381C2×C4C23×C813yes—
382C2×C4C23×C813yes—
383C2×C4C23×C813yes—
384C2×C4C23×C813yes—
385C2×C4C23×C813yes—
386C2×C4C23×C813yes—
387C2×C4C24×C411yescharacteristic
388C8—21no—
389C8—23no—
390C8—21no—
391C8—21no—
392C8—23no—
393C8—23no—
394C8—23no—
395C8C2×2−1+411yescharacteristic
396C8#58511yescharacteristic
397C22—21no—
398C22—21no—
399C22—21no—
400C22—23no—
401C22—21no—
402C22—23no—
403C22—21no—
404C22—23no—
405C22#340712yes—
406C22C24×C811yescharacteristic
407C4—21no—
408C4—23no—
409C4—23no—
410C4—23no—
411C4—23no—
412C4—21no—
413C4—23no—
414C4—21no—
415C4—21no—
416C4—23no—
417C4C2×#50416yes—
418C4C4×2−1+411yescharacteristic
419C4#357311yescharacteristic
420C2—22no—
421C2—41no—
422C2—43no—
423C2C8×2−1+411yescharacteristic
424C2C22×#50411yescharacteristic
425C2C2×#340711yescharacteristic
subgroupG/H
center, Frattini, cosocleC2×C8C25
commutator, socleC22C24×C8
prime radical#3482C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33508C2semidirect
3. #33507C2semidirect
4. #33504C2semidirect
5. #33502C2semidirect
6. #33499C2semidirect
7. #33498C2semidirect
8. #33496C2semidirect
9. #33485C2semidirect
10. #33483C2semidirect
11. #33480C2semidirect
12. #33479C2semidirect
13. Q8×M32C2semidirect
14. #33476C2semidirect
15. C2×Q8×C16C2semidirect

Maximal subgroups

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

subgroupindex
#603312
#335082
#335072
#335042
#335022
#334992
#334982
#334962
#334852
#334832
#334802
#334792
Q8×M322
#334762
C2×Q8×C162