Finite Group Databasebeta

← #22574209

Subgroups — #22574209

Order 8576.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864122C211yesmaximal, characteristic, complemented
2#16864096C213yesmaximal, complemented
3#16864094C213yesmaximal, complemented
4#16864068C211yesmaximal, characteristic, complemented
5#16864067C213yesmaximal, complemented
6#16864065C211yesmaximal, characteristic, complemented
7#16864065C211yesmaximal, characteristic, complemented
8#16864047C213yesmaximal, complemented
9#16864001C213yesmaximal, complemented
10#16863997C213yesmaximal, complemented
11#16863943C213yesmaximal, complemented
12#16863924C213yesmaximal, complemented
13#16863878C211yesmaximal, characteristic, complemented
14C4×#11761629C211yesmaximal, characteristic, complemented
15#520×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×#11761629C2211yescharacteristic
18C2×#11761628C2216yes—
19#14524042C2213yes—
20#14524040C2213yes—
21#14524039C2213yes—
22#14524039C2213yes—
23#14524038C2213yes—
24#14524036C2213yes—
25#14524035C2213yes—
26#14524034C2213yes—
27#14524033C2211yescharacteristic
28#14524029C2213yes—
29#14524028C2213yes—
30#14524028C2216yes—
31#14524026C2216yes—
32#14524025C2213yes—
33#14524021C2213yes—
34#14524018C2213yes—
35#14524018C2213yes—
36#14524017C2213yes—
37#14524017C2213yes—
38#14524017C2216yes—
39#14524016C2216yes—
40#14524016C2216yes—
41#14524016C2213yes—
42#14524014C2213yes—
43#14524014C2213yes—
44Q8×Dic67C2211yescharacteristic
45Q8×Dic67C2211yescharacteristic
46C4×Dic134C2213yes—
47C4×D268C2213yes—
48C4×D268C2213yes—
49C4×D268C2213yes—
50#14524005C2213yes—
51#14524005C2213yes—
52C4×C22⋊D67C2213yes—
53C4×C22⋊D67C2213yes—
54C4×C22⋊D67C2216yes—
55#14524001C2213yes—
56#14524000C2213yes—
57C4⋊C4×D67C2213yes—
58C42×D67C2213yes—
59C2×#41×C67C2211yescharacteristic
60C4×Q8×C67C2211yescharacteristic
61C4×Q8×C67C2211yescharacteristic
62C4×D4×C67C2213yes—
63C4×D4×C67C2213yes—
64#117×C67C2213yes—
65C2×C4⋊C4×C67C2213yes—
66#11761629—24no—
67#11761629—24no—
68#11761628—412no—
69C2×Dic134C2313yes—
70C2×D268C2313yes—
71C2×D268C2313yes—
72C2×D268C2313yes—
73C2×D268C2316yes—
74C2×C22⋊D67C2313yes—
75C2×C22⋊D67C2316yes—
76C2×C22⋊D67C2313yes—
77C2×C22⋊D67C2316yes—
78C2×C4×D67C2313yes—
79C2×C4×D67C2313yes—
80C2×C4×D67C2313yes—
81C2×C4×D67C2313yes—
82C2×C4×D67C2316yes—
83C4⋊Dic67C2313yes—
84C4⋊Dic67C2313yes—
85C4⋊Dic67C2311yescharacteristic
86C4⋊Dic67C2313yes—
87Dic67⋊C4C2313yes—
88Dic67⋊C4C2311yescharacteristic
89Dic67⋊C4C2311yescharacteristic
90Dic67⋊C4C2311yescharacteristic
91Dic67⋊C4C2316yes—
92Dic67⋊C4C2311yescharacteristic
93Dic67⋊C4C2313yes—
94C22⋊Dic67C2313yes—
95C22⋊Dic67C2313yes—
96(C2×D67)⋊C4C2313yes—
97(C2×D67)⋊C4C2313yes—
98(C2×D67)⋊C4C2316yes—
99(C2×D67)⋊C4C2313yes—
100(C2×D67)⋊C4C2313yes—
101(C2×D67)⋊C4C2313yes—
102(C2×D67)⋊C4C2313yes—
103C4×Dic67C2311yescharacteristic
104C4×Dic67C2313yes—
105C4×Dic67C2313yes—
106C4×Dic67C2313yes—
107#41×C67—24no—
108C2×Q8×C67C2311yescharacteristic
109C2×D4×C67C2313yes—
110C22×C4×C67C2313yes—
111C22×C4×C67C2316yes—
112C4⋊C4×C67C2313yes—
113C4⋊C4×C67C2313yes—
114C4⋊C4×C67C2313yes—
115C4⋊C4×C67C2311yescharacteristic
116C22⋊C4×C67C2313yes—
117C22⋊C4×C67C2313yes—
118C42×C67C2313yes—
119C42×C67C2313yes—
120C23×C67C2413yes—
121C22×D67C2413yes—
122C22×D67C2413yes—
123Q8×C67#4114yescomplemented
124D4×C67—26no—
125C2×C4×C67—26no—
126C2×C4×C67—46no—
127C2×C4×C67C2411yescharacteristic
128C2×C4×C67C2411yescharacteristic
129C2×C4×C67C2413yes—
130C2×C4×C67C2413yes—
131C2×C4×C67C2411yescharacteristic
132C2×C4×C67C2413yes—
133C22⋊D67—26no—
134C22⋊D67—46no—
135C22⋊D67—43no—
136C22⋊D67—46no—
137C2×Dic67C2411yescharacteristic
138C2×Dic67C2411yescharacteristic
139C2×Dic67C2413yes—
140C2×Dic67C2413yes—
141C2×Dic67C2411yescharacteristic
142C2×Dic67C2411yescharacteristic
143D268—26no—
144D268—26no—
145D268—43no—
146D268—46no—
147C4×D67—26no—
148C4×D67—26no—
149C4×D67—26no—
150C4×D67—43no—
151C4×D67—46no—
152Dic134—26no—
153C22×C67—23no—
154C22×C67—23no—
155C22×C67—23no—
156C22×C67C2511yescharacteristic
157C2×D67—23no—
158C2×D67—23no—
159C2×D67—23no—
160C2×D67—23no—
161C2×D67—23no—
162C2×D67—23no—
163C4×C67—21no—
164C4×C67—23no—
165C4×C67—23no—
166C4×C67—21no—
167C4×C67—21no—
168C4×C67C2×#4116yes—
169Dic67—21no—
170Dic67—21no—
171Dic67—23no—
172Dic67—23no—
173Dic67—21no—
174Dic67H(5,𝔽2)12yes—
175C2×C67—43no—
176C2×C67C22×#4111yescharacteristic
177C2×C67C2×H(5,𝔽2)11yescharacteristic
178C2×C67#58511yescharacteristic
179D67—43no—
180D67—43no—
181#3517—671nomaximal, Sylow
182C67#351711yescharacteristic, complemented, Sylow
183#557—671no—
184#553—673no—
185#553—673no—
186#550—671no—
187#550—671no—
188#550—671no—
189#550—673no—
190#547—673no—
191#540—673no—
192#540—673no—
193#532—673no—
194#529—673no—
195#520—671no—
196#520D6711yescharacteristic, complemented
197C4×#41—671no—
198C2×#41—671no—
199C2×#41—671no—
200C2×#41—676no—
201C2×#41C2×D6711yescharacteristic
202C4⋊Q8—673no—
203#127—673no—
204#127—673no—
205#127—673no—
206#126—673no—
207#126—673no—
208#125—671no—
209#125—673no—
210#125—673no—
211#124—673no—
212#123—676no—
213#123—673no—
214#123—676no—
215#123—673no—
216C22⋊Q8—673no—
217#121—673no—
218#121—673no—
219#121—673no—
220#121—676no—
221#121—673no—
222#121—673no—
223#121—673no—
224#121—673no—
225#121—676no—
226#121—676no—
227C4×Q8—671no—
228C4×Q8—671no—
229C4×Q8—673no—
230C4×Q8C2×D6711yescharacteristic
231C4×Q8C2×D6711yescharacteristic
232C4×D4—673no—
233C4×D4—673no—
234C4×D4—676no—
235C4×D4—673no—
236C4×D4—673no—
237C4×D4—673no—
238C4×D4—673no—
239C4×D4—673no—
240C4×D4C2×D6713yes—
241C4×D4C2×D6713yes—
242#117—673no—
243#117—673no—
244#117C2×D6713yes—
245C2×C4⋊C4—673no—
246C2×C4⋊C4C2×D6713yes—
247C2×C42—673no—
248#41—24no—
249#41—26812no—
250#41—1344no—
251#41—1344no—
252C2×Q8—673no—
253C2×Q8C22×D6711yescharacteristic
254C2×D4—676no—
255C2×D4—673no—
256C2×D4—676no—
257C2×D4—673no—
258C2×D4—673no—
259C2×D4—676no—
260C2×D4—673no—
261C2×D4—673no—
262C2×D4C22×D6713yes—
263C22×C4—673no—
264C22×C4—676no—
265C22×C4—673no—
266C22×C4—673no—
267C22×C4—673no—
268C22×C4C22×D6713yes—
269C22×C4C22×D6716yes—
270C4⋊C4—673no—
271C4⋊C4—673no—
272C4⋊C4—671no—
273C4⋊C4—673no—
274C4⋊C4—671no—
275C4⋊C4—671no—
276C4⋊C4—671no—
277C4⋊C4—676no—
278C4⋊C4—671no—
279C4⋊C4—673no—
280C4⋊C4—673no—
281C4⋊C4C22×D6713yes—
282C4⋊C4C22×D6713yes—
283C4⋊C4C22×D6713yes—
284C4⋊C4C22×D6711yescharacteristic
285C22⋊C4—673no—
286C22⋊C4—673no—
287C22⋊C4—673no—
288C22⋊C4—676no—
289C22⋊C4—673no—
290C22⋊C4—673no—
291C22⋊C4—673no—
292C22⋊C4—673no—
293C22⋊C4—673no—
294C22⋊C4C22×D6713yes—
295C22⋊C4C22×D6713yes—
296C42—671no—
297C42—673no—
298C42—673no—
299C42—673no—
300C42C22×D6713yes—
301C42C22×D6713yes—
302C23—673no—
303C23—673no—
304C23C23×D6713yes—
305Q8—1346no—
306Q8#1176162814yescomplemented
307D4—26no—
308D4—2686no—
309D4—2683no—
310D4—2686no—
311D4—2686no—
312D4—2683no—
313D4—1346no—
314D4—1346no—
315D4—1346no—
316C2×C4—26no—
317C2×C4—46no—
318C2×C4—2686no—
319C2×C4—2683no—
320C2×C4—671no—
321C2×C4—671no—
322C2×C4—671no—
323C2×C4—673no—
324C2×C4—673no—
325C2×C4—671no—
326C2×C4—1346no—
327C2×C4—1346no—
328C2×C4—1346no—
329C2×C4C23×D6711yescharacteristic
330C2×C4C23×D6713yes—
331C2×C4C23×D6711yescharacteristic
332C2×C4C23×D6713yes—
333C2×C4C23×D6711yescharacteristic
334C2×C4C23×D6713yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1343no—
339C22—1343no—
340C22—1343no—
341C22—1343no—
342C22—1343no—
343C22—1343no—
344C22C24×D6711yescharacteristic
345C4—21no—
346C4—23no—
347C4—23no—
348C4—21no—
349C4—21no—
350C4—672no—
351C4—1341no—
352C4—1341no—
353C4—1343no—
354C4—1343no—
355C4—1341no—
356C4C2×#1176162816yes—
357C2—43no—
358C2—2683no—
359C2—2683no—
360C2C22×#1176162811yescharacteristic
361C2H(5,𝔽2)×D6711yescharacteristic
362C2#1686423311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#520×C67C2
hypercenter#520D67
hypocenterC67#3517
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16864122C2semidirect
2. #16864096C2semidirect
3. #16864094C2semidirect
4. #16864068C2semidirect
5. #16864067C2semidirect
6. #16864065C2semidirect
7. #16864065C2semidirect
8. #16864047C2semidirect
9. #16864001C2semidirect
10. #16863997C2semidirect
11. #16863943C2semidirect
12. #16863924C2semidirect
13. #16863878C2semidirect
14. C4×#11761629C2semidirect
15. #520×C67C2semidirect
123. Q8×C67#41semidirect
182. C67#3517semidirect
196. #520D67semidirect
306. Q8#11761628semidirect