Finite Group Databasebeta

← #22574311

Subgroups — #22574311

Order 8576.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864128C213yesmaximal, complemented
2#16864122C211yesmaximal, characteristic, complemented
3#16864110C213yesmaximal, complemented
4#16864092C213yesmaximal, complemented
5#16864088C213yesmaximal, complemented
6#16864065C211yesmaximal, characteristic, complemented
7#16864057C211yesmaximal, characteristic, complemented
8#16864019C213yesmaximal, complemented
9#16863999C213yesmaximal, complemented
10#16863928C213yesmaximal, complemented
11#16863925C213yesmaximal, complemented
12#16863879C211yesmaximal, characteristic, complemented
13#16863874C211yesmaximal, characteristic, complemented
14C4×#11761629C211yesmaximal, characteristic, complemented
15#548×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17#14524042C2213yes—
18#14524039C2213yes—
19#14524038C2216yes—
20#14524036C2213yes—
21#14524036C2211yescharacteristic
22#14524035C2213yes—
23#14524034C2213yes—
24#14524034C2213yes—
25#14524032C2213yes—
26#14524030C2213yes—
27#14524029C2216yes—
28#14524027C2216yes—
29#14524025C2216yes—
30#14524025C2213yes—
31#14524025C2213yes—
32#14524023C2213yes—
33#14524019C2213yes—
34#14524017C2213yes—
35#14524017C2213yes—
36#14524015C2216yes—
37#14524014C2213yes—
38#14524014C2213yes—
39Q8×Dic67C2213yes—
40Q8×Dic67C2211yescharacteristic
41Q8×Dic67C2211yescharacteristic
42Q8×Dic67C2213yes—
43Q8×Dic67C2211yescharacteristic
44C4×Dic134C2213yes—
45C4×D268C2213yes—
46D4×Dic67C2213yes—
47D4×Dic67C2213yes—
48#14524005C2213yes—
49C4×C22⋊D67C2213yes—
50C4×C22⋊D67C2213yes—
51(C22⋊D67)⋊C4C2216yes—
52#14524002C2213yes—
53#14524002C2213yes—
54#14524002C2213yes—
55#14524001C2213yes—
56#14523999C2216yes—
57C2×C4⋊Dic67C2213yes—
58C42×D67C2213yes—
59C2×#41×C67C2211yescharacteristic
60C22×Q8×C67C2211yescharacteristic
61#124×C67C2213yes—
62C22⋊Q8×C67C2213yes—
63#121×C67C2213yes—
64C4×Q8×C67C2211yescharacteristic
65C4×D4×C67C2213yes—
66#11761629—24no—
67C2×Dic134C2313yes—
68C2×D268C2313yes—
69C2×C22⋊D67C2316yes—
70C2×C22⋊D67C2313yes—
71C2×C22⋊D67C2313yes—
72C22×Dic67C2316yes—
73C2×C4×D67C2313yes—
74C2×C4×D67C2313yes—
75C2×C4×D67C2313yes—
76C4⋊Dic67C2313yes—
77C4⋊Dic67C2313yes—
78C4⋊Dic67C2313yes—
79C4⋊Dic67C2311yescharacteristic
80C4⋊Dic67C2313yes—
81C4⋊Dic67C2313yes—
82C4⋊Dic67C2313yes—
83C4⋊Dic67C2313yes—
84Dic67⋊C4C2313yes—
85Dic67⋊C4C2316yes—
86Dic67⋊C4C2313yes—
87Dic67⋊C4C2311yescharacteristic
88Dic67⋊C4C2311yescharacteristic
89C22⋊Dic67C2313yes—
90C22⋊Dic67C2311yescharacteristic
91C22⋊Dic67C2313yes—
92C22⋊Dic67C2313yes—
93C22⋊Dic67C2313yes—
94C22⋊Dic67C2311yescharacteristic
95(C2×D67)⋊C4C2316yes—
96(C2×D67)⋊C4C2316yes—
97(C2×D67)⋊C4C2313yes—
98(C2×D67)⋊C4C2313yes—
99C4×Dic67C2311yescharacteristic
100C4×Dic67C2313yes—
101C4×Dic67C2313yes—
102C4×Dic67C2316yes—
103C4×Dic67C2313yes—
104C4×Dic67C2313yes—
105#41×C67—24no—
106C2×Q8×C67—24no—
107C2×Q8×C67C2313yes—
108C2×Q8×C67C2311yescharacteristic
109C2×D4×C67C2313yes—
110C2×D4×C67C2313yes—
111C22×C4×C67C2313yes—
112C22×C4×C67C2313yes—
113C4⋊C4×C67C2313yes—
114C4⋊C4×C67C2313yes—
115C22⋊C4×C67C2311yescharacteristic
116C22⋊C4×C67C2313yes—
117C22⋊C4×C67C2316yes—
118C42×C67C2313yes—
119C23×C67C2413yes—
120C23×C67C2411yescharacteristic
121C22×D67C2413yes—
122Q8×C67—26no—
123Q8×C67#4114yes—
124D4×C67—26no—
125D4×C67—43no—
126C2×C4×C67—26no—
127C2×C4×C67—26no—
128C2×C4×C67C2411yescharacteristic
129C2×C4×C67C2413yes—
130C2×C4×C67C2413yes—
131C2×C4×C67C2411yescharacteristic
132C2×C4×C67C2413yes—
133C22⋊D67—46no—
134C22⋊D67—43no—
135C22⋊D67—43no—
136C2×Dic67—46no—
137C2×Dic67C2413yes—
138C2×Dic67C2411yescharacteristic
139C2×Dic67C2411yescharacteristic
140C2×Dic67C2411yescharacteristic
141C2×Dic67C2413yes—
142C2×Dic67C2411yescharacteristic
143C2×Dic67C2413yes—
144D268—26no—
145C4×D67—26no—
146C4×D67—43no—
147C4×D67—43no—
148Dic134—43no—
149C22×C67—21no—
150C22×C67—23no—
151C22×C67—23no—
152C22×C67—21no—
153C22×C67—23no—
154C22×C67—21no—
155C22×C67C2511yescharacteristic
156C2×D67—23no—
157C2×D67—23no—
158C2×D67—23no—
159C4×C67—21no—
160C4×C67—23no—
161C4×C67—23no—
162C4×C67—21no—
163C4×C67C2×#4116yes—
164Dic67—21no—
165Dic67—21no—
166Dic67—23no—
167Dic67—21no—
168Dic67—23no—
169Dic67—23no—
170Dic67—21no—
171C2×C67—41no—
172C2×C67—43no—
173C2×C67C22×#4111yescharacteristic
174C2×C67#58511yescharacteristic
175C2×C67#58511yescharacteristic
176D67—43no—
177#3524—671nomaximal, Sylow
178C67#352411yescharacteristic, complemented, Sylow
179#559—673no—
180#557—671no—
181#555—673no—
182#553—673no—
183#552—673no—
184#550—671no—
185#548—671no—
186#548D6711yescharacteristic, complemented
187#542—673no—
188#540—673no—
189#529—673no—
190#529—673no—
191#520—671no—
192#519—671no—
193C4×#41—671no—
194C2×#41—671no—
195C2×#41C2×D6711yescharacteristic
196C22×Q8C2×D6711yescharacteristic
197C4⋊Q8—673no—
198#127—673no—
199#126—676no—
200#126—671no—
201#126—673no—
202#125—673no—
203#125—673no—
204#125—673no—
205#124—673no—
206#124—673no—
207#124—676no—
208#124C2×D6713yes—
209#123—676no—
210#123—673no—
211#123—673no—
212#123—676no—
213C22⋊Q8—673no—
214C22⋊Q8—673no—
215C22⋊Q8C2×D6713yes—
216#121—673no—
217#121—673no—
218#121—676no—
219#121—673no—
220#121—673no—
221#121C2×D6713yes—
222C4×Q8—671no—
223C4×Q8—673no—
224C4×Q8—671no—
225C4×Q8—673no—
226C4×Q8—671no—
227C4×Q8—673no—
228C4×Q8C2×D6711yescharacteristic
229C4×D4—673no—
230C4×D4—673no—
231C4×D4—673no—
232C4×D4—673no—
233C4×D4—673no—
234C4×D4—676no—
235C4×D4—673no—
236C4×D4C2×D6713yes—
237#117—673no—
238#117—673no—
239#117—673no—
240#117—673no—
241#117—676no—
242C2×C4⋊C4—673no—
243C2×C42—673no—
244#41—24no—
245#41—1344no—
246C2×Q8—24no—
247C2×Q8—673no—
248C2×Q8C22×D6713yes—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—673no—
251C2×D4—676no—
252C2×D4—673no—
253C2×D4—673no—
254C2×D4C22×D6713yes—
255C2×D4C22×D6713yes—
256C22×C4—676no—
257C22×C4—673no—
258C22×C4—673no—
259C22×C4—673no—
260C22×C4C22×D6713yes—
261C22×C4C22×D6713yes—
262C4⋊C4—673no—
263C4⋊C4—673no—
264C4⋊C4—673no—
265C4⋊C4—671no—
266C4⋊C4—673no—
267C4⋊C4—673no—
268C4⋊C4—673no—
269C4⋊C4—673no—
270C4⋊C4—676no—
271C4⋊C4—671no—
272C4⋊C4—673no—
273C4⋊C4—671no—
274C4⋊C4—673no—
275C4⋊C4C22×D6713yes—
276C4⋊C4C22×D6713yes—
277C22⋊C4—671no—
278C22⋊C4—673no—
279C22⋊C4—673no—
280C22⋊C4—673no—
281C22⋊C4—673no—
282C22⋊C4—671no—
283C22⋊C4—676no—
284C22⋊C4—673no—
285C22⋊C4—676no—
286C22⋊C4—673no—
287C22⋊C4C22×D6716yes—
288C22⋊C4C22×D6713yes—
289C22⋊C4C22×D6711yescharacteristic
290C42—671no—
291C42—673no—
292C42—673no—
293C42—676no—
294C42—673no—
295C42—673no—
296C42C22×D6713yes—
297C23—673no—
298C23C23×D6711yescharacteristic
299C23C23×D6713yes—
300Q8—26no—
301Q8—2683no—
302Q8#1176162714yes—
303D4—26no—
304D4—43no—
305D4—2686no—
306D4—2683no—
307D4—2683no—
308D4—1346no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—2683no—
312C2×C4—2683no—
313C2×C4—2686no—
314C2×C4—671no—
315C2×C4—671no—
316C2×C4—673no—
317C2×C4—671no—
318C2×C4—673no—
319C2×C4—671no—
320C2×C4—673no—
321C2×C4—1346no—
322C2×C4C23×D6713yes—
323C2×C4C23×D6711yescharacteristic
324C2×C4C23×D6713yes—
325C2×C4C23×D6711yescharacteristic
326C2×C4C23×D6713yes—
327C22—21no—
328C22—23no—
329C22—21no—
330C22—23no—
331C22—23no—
332C22—21no—
333C22—1343no—
334C22—1343no—
335C22—1343no—
336C22C24×D6711yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—1341no—
342C4—1341no—
343C4—1343no—
344C4—1341no—
345C4—1343no—
346C4—1343no—
347C4—1341no—
348C4C2×#1176162716yes—
349C2—41no—
350C2—43no—
351C2—2683no—
352C2C22×#1176162711yescharacteristic
353C2#1686423211yescharacteristic
354C2#1686423311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#548×C67C2
hypercenter#548D67
hypocenterC67#3524
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16864128C2semidirect
2. #16864122C2semidirect
3. #16864110C2semidirect
4. #16864092C2semidirect
5. #16864088C2semidirect
6. #16864065C2semidirect
7. #16864057C2semidirect
8. #16864019C2semidirect
9. #16863999C2semidirect
10. #16863928C2semidirect
11. #16863925C2semidirect
12. #16863879C2semidirect
13. #16863874C2semidirect
14. C4×#11761629C2semidirect
15. #548×C67C2semidirect
178. C67#3524semidirect
186. #548D67semidirect