Finite Group Databasebeta

← #22574321

Subgroups — #22574321

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#16864131C213yesmaximal, complemented
2#16864112C213yesmaximal, complemented
3#16864099C213yesmaximal, complemented
4#16864084C213yesmaximal, complemented
5#16864068C211yesmaximal, characteristic, complemented
6#16864058C211yesmaximal, characteristic, complemented
7#16864058C211yesmaximal, characteristic, complemented
8#16864020C213yesmaximal, complemented
9#16864002C213yesmaximal, complemented
10#16863933C213yesmaximal, complemented
11#16863931C213yesmaximal, complemented
12#16863878C211yesmaximal, characteristic, complemented
13#16863872C211yesmaximal, characteristic, complemented
14C4×#11761629C211yesmaximal, characteristic, complemented
15#557×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×#11761629C2211yescharacteristic
18C2×Q8×D67C2211yescharacteristic
19#14524040C2213yes—
20#14524038C2216yes—
21#14524038C2213yes—
22#14524038C2211yescharacteristic
23#14524034C2213yes—
24#14524032C2216yes—
25#14524032C2213yes—
26#14524031C2213yes—
27#14524031C2213yes—
28#14524028C2216yes—
29#14524028C2216yes—
30#14524028C2213yes—
31#14524028C2213yes—
32#14524023C2213yes—
33#14524023C2213yes—
34#14524019C2213yes—
35#14524018C2213yes—
36#14524018C2213yes—
37#14524018C2213yes—
38#14524018C2216yes—
39#14524018C2213yes—
40#14524018C2213yes—
41Q8×Dic67C2211yescharacteristic
42C4×Dic134C2213yes—
43#14524008C2213yes—
44C4×D268C2213yes—
45C4×D268C2213yes—
46C4×D268C2213yes—
47C4×D268C2213yes—
48C4×D268C2213yes—
49#14524005C2213yes—
50#14524005C2213yes—
51#14524005C2216yes—
52#14524001C2213yes—
53#14524001C2216yes—
54#14524001C2213yes—
55#14524000C2213yes—
56#14524000C2213yes—
57C4⋊C4×D67C2213yes—
58C42×D67C2213yes—
59C4⋊Q8×C67C2213yes—
60#125×C67C2213yes—
61#125×C67C2213yes—
62C4×Q8×C67C2211yescharacteristic
63C4×Q8×C67C2211yescharacteristic
64C4×Q8×C67C2211yescharacteristic
65C4×Q8×C67C2213yes—
66#11761629—24no—
67#11761629—24no—
68C2×Dic134C2313yes—
69Q8×D67—24no—
70C2×D268C2313yes—
71C2×D268C2313yes—
72C2×D268C2316yes—
73C2×D268C2313yes—
74C2×D268C2313yes—
75C2×D268C2313yes—
76C2×C4×D67C2313yes—
77C2×C4×D67C2313yes—
78C2×C4×D67C2313yes—
79C2×C4×D67C2313yes—
80C2×C4×D67C2313yes—
81C2×C4×D67C2316yes—
82C4⋊Dic67C2313yes—
83C4⋊Dic67C2313yes—
84C4⋊Dic67C2311yescharacteristic
85C4⋊Dic67C2313yes—
86Dic67⋊C4C2313yes—
87Dic67⋊C4C2311yescharacteristic
88Dic67⋊C4C2313yes—
89Dic67⋊C4C2313yes—
90(C2×D67)⋊C4C2311yescharacteristic
91(C2×D67)⋊C4C2313yes—
92(C2×D67)⋊C4C2313yes—
93(C2×D67)⋊C4C2313yes—
94(C2×D67)⋊C4C2313yes—
95(C2×D67)⋊C4C2316yes—
96(C2×D67)⋊C4C2316yes—
97(C2×D67)⋊C4C2313yes—
98(C2×D67)⋊C4C2316yes—
99(C2×D67)⋊C4C2311yescharacteristic
100(C2×D67)⋊C4C2311yescharacteristic
101(C2×D67)⋊C4C2313yes—
102(C2×D67)⋊C4C2313yes—
103C4×Dic67C2311yescharacteristic
104C4×Dic67C2313yes—
105C4×Dic67C2313yes—
106C4×Dic67C2316yes—
107C2×Q8×C67C2313yes—
108C2×Q8×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C4⋊C4×C67C2313yes—
111C4⋊C4×C67C2313yes—
112C4⋊C4×C67C2313yes—
113C4⋊C4×C67C2313yes—
114C4⋊C4×C67C2313yes—
115C4⋊C4×C67C2316yes—
116C42×C67C2313yes—
117C42×C67C2313yes—
118C42×C67C2313yes—
119C22×D67C2411yescharacteristic
120C22×D67C2413yes—
121C22×D67C2413yes—
122Q8×C67—43no—
123Q8×C67#4114yes—
124C2×C4×C67C2411yescharacteristic
125C2×C4×C67C2411yescharacteristic
126C2×C4×C67C2411yescharacteristic
127C2×C4×C67C2413yes—
128C2×C4×C67C2413yes—
129C2×C4×C67C2413yes—
130C2×C4×C67C2413yes—
131C2×Dic67C2411yescharacteristic
132C2×Dic67C2413yes—
133C2×Dic67C2411yescharacteristic
134C2×Dic67C2413yes—
135C2×Dic67C2411yescharacteristic
136D268—26no—
137D268—26no—
138D268—43no—
139D268—43no—
140D268—43no—
141D268—46no—
142C4×D67—26no—
143C4×D67—26no—
144C4×D67—26no—
145C4×D67—43no—
146C4×D67—46no—
147C4×D67—43no—
148Dic134—26no—
149C22×C67C2511yescharacteristic
150C2×D67—21no—
151C2×D67—23no—
152C2×D67—23no—
153C2×D67—23no—
154C2×D67—23no—
155C2×D67—23no—
156C2×D67—21no—
157C2×D67—23no—
158C2×D67—21no—
159C4×C67—21no—
160C4×C67—23no—
161C4×C67—23no—
162C4×C67—21no—
163C4×C67—23no—
164C4×C67—21no—
165C4×C67C2×#4116yes—
166Dic67—23no—
167Dic67—21no—
168Dic67—21no—
169Dic67—21no—
170Dic67—23no—
171C2×C67C22×#4111yescharacteristic
172C2×C67#58511yescharacteristic
173C2×C67#58511yescharacteristic
174D67—41no—
175D67—43no—
176D67—43no—
177#3524—671nomaximal, Sylow
178C67#352411yescharacteristic, complemented, Sylow
179#559—673no—
180#557D6711yescharacteristic, complemented
181#555—673no—
182#553—673no—
183#552—673no—
184#550—671no—
185#548—671no—
186#548—671no—
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×#41—671no—
196C22×Q8—671no—
197C4⋊Q8C2×D6713yes—
198#127—673no—
199#126—671no—
200#126—673no—
201#126—676no—
202#125—673no—
203#125C2×D6713yes—
204#125C2×D6713yes—
205#124—673no—
206#124—676no—
207#124—673no—
208#124—673no—
209#123—676no—
210#123—673no—
211#123—673no—
212#123—676no—
213C22⋊Q8—673no—
214C22⋊Q8—673no—
215C22⋊Q8—673no—
216#121—673no—
217#121—673no—
218#121—673no—
219#121—673no—
220#121—676no—
221#121—673no—
222C4×Q8—671no—
223C4×Q8—673no—
224C4×Q8—673no—
225C4×Q8C2×D6711yescharacteristic
226C4×Q8C2×D6711yescharacteristic
227C4×Q8C2×D6711yescharacteristic
228C4×Q8C2×D6713yes—
229C4×D4—673no—
230C4×D4—673no—
231C4×D4—676no—
232C4×D4—673no—
233C4×D4—673no—
234C4×D4—673no—
235C4×D4—673no—
236C4×D4—673no—
237#117—673no—
238#117—673no—
239#117—676no—
240#117—673no—
241#117—673no—
242C2×C4⋊C4—673no—
243C2×C42—673no—
244#41—1344no—
245#41—1344no—
246C2×Q8—673no—
247C2×Q8—1344no—
248C2×Q8C22×D6713yes—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—673no—
251C2×D4—673no—
252C2×D4—676no—
253C2×D4—673no—
254C2×D4—673no—
255C2×D4—673no—
256C22×C4—673no—
257C22×C4—673no—
258C22×C4—676no—
259C22×C4—673no—
260C22×C4—673no—
261C22×C4—673no—
262C4⋊C4—673no—
263C4⋊C4—673no—
264C4⋊C4—671no—
265C4⋊C4—673no—
266C4⋊C4—673no—
267C4⋊C4—673no—
268C4⋊C4—671no—
269C4⋊C4—673no—
270C4⋊C4C22×D6713yes—
271C4⋊C4C22×D6713yes—
272C4⋊C4C22×D6713yes—
273C4⋊C4C22×D6716yes—
274C4⋊C4C22×D6713yes—
275C4⋊C4C22×D6711yescharacteristic
276C4⋊C4C22×D6713yes—
277C22⋊C4—671no—
278C22⋊C4—673no—
279C22⋊C4—673no—
280C22⋊C4—673no—
281C22⋊C4—676no—
282C22⋊C4—673no—
283C22⋊C4—676no—
284C22⋊C4—671no—
285C22⋊C4—673no—
286C22⋊C4—671no—
287C22⋊C4—673no—
288C22⋊C4—676no—
289C22⋊C4—673no—
290C42—671no—
291C42—673no—
292C42—673no—
293C42—676no—
294C42C22×D6713yes—
295C42C22×D6713yes—
296C42C22×D6713yes—
297C23—671no—
298C23—673no—
299C23—673no—
300Q8—43no—
301Q8—1346no—
302Q8#1176162914yes—
303D4—2683no—
304D4—2686no—
305D4—2683no—
306D4—2683no—
307D4—1346no—
308D4—1346no—
309C2×C4—2683no—
310C2×C4—2683no—
311C2×C4—2686no—
312C2×C4—671no—
313C2×C4—671no—
314C2×C4—673no—
315C2×C4—673no—
316C2×C4—671no—
317C2×C4—1346no—
318C2×C4—1346no—
319C2×C4—1346no—
320C2×C4C23×D6713yes—
321C2×C4C23×D6711yescharacteristic
322C2×C4C23×D6713yes—
323C2×C4C23×D6711yescharacteristic
324C2×C4C23×D6711yescharacteristic
325C2×C4C23×D6713yes—
326C2×C4C23×D6713yes—
327C22—1341no—
328C22—1343no—
329C22—1343no—
330C22—1343no—
331C22—1341no—
332C22—1343no—
333C22—1343no—
334C22—1343no—
335C22—1341no—
336C22C24×D6711yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—1341no—
344C4—1341no—
345C4—1343no—
346C4—1341no—
347C4—1343no—
348C4C2×#1176162916yes—
349C2—2681no—
350C2—2683no—
351C2—2683no—
352C2C22×#1176162911yescharacteristic
353C2#1686423211yescharacteristic
354C2#1686423311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#557×C67C2
hypercenter#557D67
hypocenterC67#3524
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #16864131C2semidirect
2. #16864112C2semidirect
3. #16864099C2semidirect
4. #16864084C2semidirect
5. #16864068C2semidirect
6. #16864058C2semidirect
7. #16864058C2semidirect
8. #16864020C2semidirect
9. #16864002C2semidirect
10. #16863933C2semidirect
11. #16863931C2semidirect
12. #16863878C2semidirect
13. #16863872C2semidirect
14. C4×#11761629C2semidirect
15. #557×C67C2semidirect
178. C67#3524semidirect
180. #557D67semidirect