Finite Group Databasebeta

← #22574299

Subgroups — #22574299

Order 8576.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864124C213yesmaximal, complemented
2#16864122C211yesmaximal, characteristic, complemented
3#16864114C213yesmaximal, complemented
4#16864087C213yesmaximal, complemented
5#16864085C213yesmaximal, complemented
6Q8×D268C211yesmaximal, characteristic, complemented
7Q8×C22⋊D67C211yesmaximal, characteristic, complemented
8Q8×C22⋊D67C211yesmaximal, characteristic, complemented
9#16864008C213yesmaximal, complemented
10#16864004C213yesmaximal, complemented
11#16863950C213yesmaximal, complemented
12#16863924C213yesmaximal
13#16863872C211yesmaximal, characteristic, complemented
14C4×Q8×D67C211yesmaximal, characteristic, complemented
15#519×C67C211yesmaximal, characteristic, complemented
16C2×Q8×D67C2211yescharacteristic
17C2×Q8×D67C2211yescharacteristic
18#14524043C2213yes—
19#14524042C2213yes—
20#14524041C2213yes—
21#14524041C2213yes—
22#14524038C2213yes—
23#14524037C2216yes—
24#14524036C2213yes—
25#14524036C2216yes—
26#14524035C2216yes—
27#14524035C2213yes—
28#14524034C2213yes—
29#14524033C2216yes—
30#14524033C2216yes—
31#14524033C2213yes—
32#14524029C2213yes—
33#14524028C2213yes—
34#14524025C2213yes—
35#14524023C2213yes—
36#14524023C2213yes—
37#14524022C2213yes—
38#14524022C2213yes—
39#14524021C2213yes—
40#14524019C2213yes—
41#14524019C2213yes—
42#14524016C2211yescharacteristic
43Q8×Dic67C2211yescharacteristic
44Q8×Dic67C2211yescharacteristic
45C4×Dic134C2213yes—
46C4×Dic134C2213yes—
47C4×Dic134C2216yes—
48C4×Dic134C2213yes—
49#14524008C2213yes—
50#14524008C2213yes—
51C4×D268C2213yes—
52C4×C22⋊D67C2213yes—
53C4×C22⋊D67C2213yes—
54#14524001C2213yes—
55#14524000C2213yes—
56#14524000C2216yes—
57C4⋊C4×D67C2213yes—
58C42×D67C2213yes—
59C22×Q8×C67C2211yescharacteristic
60C4×Q8×C67C2213yes—
61C4×Q8×C67C2211yescharacteristic
62C4×Q8×C67C2213yes—
63C4×Q8×C67C2211yescharacteristic
64#117×C67C2213yes—
65#117×C67C2213yes—
66C2×Dic134C2313yes—
67C2×Dic134C2313yes—
68C2×Dic134C2313yes—
69Q8×D67—24no—
70Q8×D67—24no—
71C2×D268C2311yescharacteristic
72C2×C22⋊D67C2311yescharacteristic
73C2×C22⋊D67C2311yescharacteristic
74C2×C4×D67C2311yescharacteristic
75C2×C4×D67C2313yes—
76C2×C4×D67C2313yes—
77C2×C4×D67C2313yes—
78C4⋊Dic67C2313yes—
79C4⋊Dic67C2313yes—
80C4⋊Dic67C2313yes—
81C4⋊Dic67C2316yes—
82C4⋊Dic67C2313yes—
83Dic67⋊C4C2316yes—
84Dic67⋊C4C2313yes—
85Dic67⋊C4C2313yes—
86Dic67⋊C4C2316yes—
87Dic67⋊C4C2316yes—
88Dic67⋊C4C2313yes—
89Dic67⋊C4C2313yes—
90Dic67⋊C4C2311yescharacteristic
91Dic67⋊C4C2316yes—
92Dic67⋊C4C2313yes—
93C22⋊Dic67C2313yes—
94C22⋊Dic67C2313yes—
95(C2×D67)⋊C4C2311yescharacteristic
96(C2×D67)⋊C4C2313yes—
97(C2×D67)⋊C4C2313yes—
98(C2×D67)⋊C4C2313yes—
99(C2×D67)⋊C4C2313yes—
100(C2×D67)⋊C4C2313yes—
101C4×Dic67C2313yes—
102C4×Dic67C2313yes—
103C4×Dic67C2313yes—
104C4×Dic67C2316yes—
105C4×Dic67C2313yes—
106C2×Q8×C67—24no—
107C2×Q8×C67C2313yes—
108C2×Q8×C67C2311yescharacteristic
109C22×C4×C67C2313yes—
110C4⋊C4×C67C2313yes—
111C4⋊C4×C67C2313yes—
112C4⋊C4×C67C2313yes—
113C4⋊C4×C67C2313yes—
114C22⋊C4×C67C2311yescharacteristic
115C22⋊C4×C67C2313yes—
116C42×C67C2313yes—
117C42×C67C2316yes—
118C42×C67C2313yes—
119C23×C67C2411yescharacteristic
120C22×D67C2411yescharacteristic
121C22×D67C2411yescharacteristic
122Q8×C67—26no—
123Q8×C67#4114yes—
124C2×C4×C67—26no—
125C2×C4×C67C2411yescharacteristic
126C2×C4×C67C2413yes—
127C2×C4×C67C2413yes—
128C2×C4×C67C2411yescharacteristic
129C2×C4×C67C2413yes—
130C2×C4×C67C2413yes—
131C22⋊D67—22no—
132C22⋊D67—22no—
133C2×Dic67C2413yes—
134C2×Dic67C2411yescharacteristic
135C2×Dic67C2413yes—
136C2×Dic67C2413yes—
137C2×Dic67C2413yes—
138C2×Dic67C2411yescharacteristic
139D268—22no—
140C4×D67—26no—
141C4×D67—26no—
142C4×D67—26no—
143C4×D67—22no—
144Dic134—26no—
145Dic134—26no—
146Dic134—26no—
147C22×C67—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67C2511yescharacteristic
151C2×D67—21no—
152C2×D67—21no—
153C2×D67—21no—
154C2×D67—21no—
155C2×D67—21no—
156C2×D672−1+412yes—
157C4×C67—23no—
158C4×C67—23no—
159C4×C67—23no—
160C4×C67—21no—
161C4×C67—21no—
162C4×C67C2×#4116yes—
163Dic67—23no—
164Dic67—23no—
165Dic67—23no—
166Dic67—23no—
167Dic67—21no—
168Dic672−1+412yes—
169C2×C67—41no—
170C2×C67C22×#4111yescharacteristic
171C2×C67C2×2−1+411yescharacteristic
172C2×C67C2×2−1+411yescharacteristic
173D67—22no—
174D67—41no—
175#3523—671nomaximal, Sylow
176C67#352311yescharacteristic, complemented, Sylow
177#557—671no—
178#557—673no—
179#556—673no—
180#552—673no—
181#552—673no—
182D4×Q8—671no—
183D4×Q8—671no—
184D4×Q8—671no—
185#541—673no—
186#541—673no—
187#533—673no—
188#529—673no—
189#519—671no—
190#519D6711yescharacteristic, complemented
191C2×C4×Q8—671no—
192C22×Q8—671no—
193C22×Q8—671no—
194C22×Q8C2×D6711yescharacteristic
195C4⋊Q8—673no—
196C4⋊Q8—673no—
197C4⋊Q8—673no—
198C4⋊Q8—673no—
199#126—673no—
200#126—673no—
201#126—676no—
202#126—676no—
203#125—673no—
204#125—676no—
205#125—676no—
206#125—673no—
207#125—676no—
208#125—673no—
209#124—673no—
210#123—673no—
211#123—673no—
212C22⋊Q8—673no—
213C22⋊Q8—673no—
214C22⋊Q8—673no—
215C22⋊Q8—673no—
216C22⋊Q8—673no—
217C22⋊Q8—673no—
218C22⋊Q8—673no—
219#121—671no—
220C4×Q8—671no—
221C4×Q8—671no—
222C4×Q8—673no—
223C4×Q8—673no—
224C4×Q8—673no—
225C4×Q8—673no—
226C4×Q8—673no—
227C4×Q8—676no—
228C4×Q8C2×D6711yescharacteristic
229C4×Q8C2×D6713yes—
230C4×Q8C2×D6711yescharacteristic
231C4×Q8C2×D6713yes—
232C4×D4—673no—
233C4×D4—673no—
234C4×D4—673no—
235#117—673no—
236#117—673no—
237#117—676no—
238#117C2×D6713yes—
239#117C2×D6713yes—
240C2×C4⋊C4—673no—
241C2×C42—673no—
242C2×Q8—24no—
243C2×Q8—673no—
244C2×Q8—673no—
245C2×Q8—673no—
246C2×Q8—1344no—
247C2×Q8—1344no—
248C2×Q8C22×D6713yes—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—671no—
251C2×D4—671no—
252C2×D4—671no—
253C22×C4—671no—
254C22×C4—673no—
255C22×C4—673no—
256C22×C4—673no—
257C22×C4C22×D6713yes—
258C4⋊C4—673no—
259C4⋊C4—676no—
260C4⋊C4—673no—
261C4⋊C4—673no—
262C4⋊C4—673no—
263C4⋊C4—676no—
264C4⋊C4—676no—
265C4⋊C4—673no—
266C4⋊C4—673no—
267C4⋊C4—673no—
268C4⋊C4—671no—
269C4⋊C4—676no—
270C4⋊C4—676no—
271C4⋊C4—673no—
272C4⋊C4—673no—
273C4⋊C4C22×D6713yes—
274C4⋊C4C22×D6713yes—
275C4⋊C4C22×D6713yes—
276C4⋊C4C22×D6713yes—
277C22⋊C4—673no—
278C22⋊C4—673no—
279C22⋊C4—671no—
280C22⋊C4—673no—
281C22⋊C4—673no—
282C22⋊C4—673no—
283C22⋊C4—673no—
284C22⋊C4—673no—
285C22⋊C4C22×D6711yescharacteristic
286C22⋊C4C22×D6713yes—
287C42—673no—
288C42—673no—
289C42—673no—
290C42—676no—
291C42—673no—
292C42C22×D6713yes—
293C42C22×D6716yes—
294C42C22×D6713yes—
295C23—671no—
296C23—671no—
297C23C23×D6711yescharacteristic
298Q8—26no—
299Q8—1346no—
300Q8—1346no—
301Q8—1346no—
302Q8#1176162814yes—
303D4—1342no—
304D4—1342no—
305D4—1342no—
306C2×C4—26no—
307C2×C4—671no—
308C2×C4—673no—
309C2×C4—671no—
310C2×C4—673no—
311C2×C4—673no—
312C2×C4—673no—
313C2×C4—1342no—
314C2×C4—1346no—
315C2×C4—1346no—
316C2×C4—1346no—
317C2×C4C23×D6713yes—
318C2×C4C23×D6711yescharacteristic
319C2×C4C23×D6713yes—
320C2×C4C23×D6711yescharacteristic
321C2×C4C23×D6713yes—
322C2×C4C23×D6713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—672no—
327C22—1341no—
328C22—1341no—
329C22—1341no—
330C22—1341no—
331C22—1341no—
332C22C24×D6711yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—672no—
339C4—1343no—
340C4—1343no—
341C4—1343no—
342C4—1341no—
343C4—1343no—
344C4C2×#1176162816yes—
345C2—41no—
346C2—2681no—
347C2—1342no—
348C2C22×#1176162811yescharacteristic
349C22−1+4×D6711yescharacteristic
350C22−1+4×D6711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#519×C67C2
hypercenter#519D67
hypocenterC67#3523
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #16864124C2semidirect
2. #16864122C2semidirect
3. #16864114C2semidirect
4. #16864087C2semidirect
5. #16864085C2semidirect
6. Q8×D268C2semidirect
7. Q8×C22⋊D67C2semidirect
8. Q8×C22⋊D67C2semidirect
9. #16864008C2semidirect
10. #16864004C2semidirect
11. #16863950C2semidirect
13. #16863872C2semidirect
14. C4×Q8×D67C2semidirect
15. #519×C67C2semidirect
176. C67#3523semidirect
190. #519D67semidirect