Finite Group Databasebeta

← #22572254

Subgroups — #22572254

Order 8576.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic134C211yesmaximal, characteristic
2#16863766C211yesmaximal, characteristic, complemented
3#16863737C211yesmaximal, characteristic, complemented
4#16863726C211yesmaximal, characteristic, complemented
5#16863700C211yesmaximal, characteristic, complemented
6#16863652C211yesmaximal, characteristic, complemented
7#16863635C211yesmaximal, characteristic, complemented
8#16863606C211yesmaximal, characteristic, complemented
9#16863551C211yesmaximal, characteristic, complemented
10#16863551C211yesmaximal, characteristic, complemented
11#16863501C211yesmaximal, characteristic, complemented
12#16863468C211yesmaximal, characteristic
13#16863404C211yesmaximal, characteristic, complemented
14C2×#14523966C211yesmaximal, characteristic, complemented
15#552×C67C211yesmaximal, characteristic
16C22×Dic134C2211yescharacteristic
17C22×Dic134C2211yescharacteristic
18C2×#11761618C2211yescharacteristic
19C2×#11761618C2211yescharacteristic
20C2×#11761618C2211yescharacteristic
21C2×#11761612C2211yescharacteristic
22C22×C67⋊C8C2211yescharacteristic
23#14524043C2211yescharacteristic
24#14524042—21no—
25#14524022C2211yescharacteristic
26#14524022C2211yescharacteristic
27#14524020—21no—
28#14524020—21no—
29C4×Dic134C2211yescharacteristic
30D4×Dic67—21no—
31#14523976C2211yescharacteristic
32#14523974C2211yescharacteristic
33#14523974C2211yescharacteristic
34C4⋊(C67⋊C8)C2211yescharacteristic
35#14523967C2211yescharacteristic
36#14523966C2211yescharacteristic
37#14523966C2211yescharacteristic
38#14523966C2211yescharacteristic
39#14523966C2211yescharacteristic
40#14523966C2211yescharacteristic
41#14523966C2211yescharacteristic
42#14523966C2211yescharacteristic
43#14523966C2211yescharacteristic
44#14523966C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic
47#14523954C2211yescharacteristic
48#126×C67—21no—
49#125×C67—21no—
50#125×C67C2211yescharacteristic
51#123×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53C22⋊Q8×C67C2211yescharacteristic
54C4×Q8×C67C2211yescharacteristic
55C4×D4×C67C2211yescharacteristic
56#117×C67—21no—
57#117×C67C2211yescharacteristic
58C2×C4⋊C4×C67C2211yescharacteristic
59C2×Dic134—22no—
60C2×Dic134—22no—
61C2×Dic134—21no—
62C2×Dic134—21no—
63C2×Dic134—41no—
64C2×Dic134—41no—
65C2×Dic134C2311yescharacteristic
66C2×Dic134C2311yescharacteristic
67C2×Dic134C2311yescharacteristic
68C22×Dic67—21no—
69C22×Dic67—21no—
70#11761618—22no—
71#11761618—41no—
72#11761618—41no—
73#11761612—41no—
74C2×C67⋊C8—41no—
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C4⋊Dic67—21no—
80C4⋊Dic67—21no—
81C4⋊Dic67C2311yescharacteristic
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85C22⋊Dic67—21no—
86C22⋊Dic67—21no—
87C4×Dic67—21no—
88C2×Q8×C67C2311yescharacteristic
89C2×D4×C67D411yescharacteristic
90C22×C4×C67—21no—
91C22×C4×C67C2311yescharacteristic
92C22×C4×C67C2311yescharacteristic
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67—21no—
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67D411yescharacteristic
100C4⋊C4×C67C2311yescharacteristic
101C4⋊C4×C67C2311yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67D411yescharacteristic
107C22⋊C4×C67D411yescharacteristic
108C42×C67—21no—
109C42×C67—21no—
110C42×C67C2311yescharacteristic
111C23×C67C2×D411yescharacteristic
112C23×C67C2×D411yescharacteristic
113Q8×C67—21no—
114Q8×C67—21no—
115D4×C67—22no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—42no—
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2411yescharacteristic
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—42no—
135C2×Dic67—42no—
136Dic134—21no—
137Dic134—21no—
138Dic134—21no—
139Dic134—21no—
140Dic134—21no—
141Dic134—42no—
142Dic134—42no—
143Dic134#4112yes—
144C67⋊C8—21no—
145C67⋊C8—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67#13712yes—
154C22×C67C22×D411yescharacteristic
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—21no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67C2×#4111yescharacteristic
164C4×C67H(5,𝔽2)11yescharacteristic
165Dic67—22no—
166Dic67—41no—
167Dic67—41no—
168Dic67—41no—
169C2×C67—22no—
170C2×C67—41no—
171C2×C67#54611yescharacteristic
172C2×C67C2×#13711yescharacteristic
173C2×C67#57811yescharacteristic
174#3312—671nomaximal, Sylow
175C67#331211yescharacteristic, complemented, Sylow
176#552D6711yescharacteristic
177D4×Q8—671no—
178#491—671no—
179#485—671no—
180#483—671no—
181#479—671no—
182#470—671no—
183#467—671no—
184#462—671no—
185#451—671no—
186#451—671no—
187#441—671no—
188#435—671no—
189#419—671no—
190C2×#103—671no—
191C22×Q8—671no—
192C22×Q8—671no—
193C2×Q16—671no—
194C2×Q16—671no—
195C2×Q16—671no—
196C2×M16—671no—
197C22×C8—671no—
198C4⋊Q8—671no—
199C4⋊Q8—1341no—
200#126—21no—
201#125—21no—
202#125C2×D6711yescharacteristic
203#123—21no—
204C22⋊Q8—671no—
205C22⋊Q8—671no—
206C22⋊Q8—1341no—
207C22⋊Q8—1341no—
208C22⋊Q8C2×D6711yescharacteristic
209C22⋊Q8C2×D6711yescharacteristic
210C4×Q8—671no—
211C4×Q8C2×D6711yescharacteristic
212C4×D4—1341no—
213C4×D4C2×D6711yescharacteristic
214#117—21no—
215#117C2×D6711yescharacteristic
216C2×C4⋊C4C2×D6711yescharacteristic
217#107—671no—
218#106—671no—
219#106—671no—
220C4⋊C8—671no—
221#103—671no—
222#103—671no—
223#103—671no—
224#103—671no—
225#103—671no—
226#103—671no—
227#103—671no—
228#103—671no—
229#103—671no—
230#103—671no—
231C22⋊C8—671no—
232C22⋊C8—671no—
233#97—671no—
234C2×Q8—2681no—
235C2×Q8—2681no—
236C2×Q8—671no—
237C2×Q8—671no—
238C2×Q8—671no—
239C2×Q8—1342no—
240C2×Q8—1342no—
241C2×Q8—1341no—
242C2×Q8—1341no—
243C2×Q8C22×D6711yescharacteristic
244C2×D4C22⋊D6711yescharacteristic
245C22×C4—21no—
246C22×C4—1341no—
247C22×C4—1341no—
248C22×C4C22×D6711yescharacteristic
249C22×C4C22×D6711yescharacteristic
250Q16—2681no—
251Q16—2681no—
252Q16—1342no—
253M16—2681no—
254C2×C8—2681no—
255C2×C8—671no—
256C2×C8—671no—
257C2×C8—671no—
258C2×C8—671no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—671no—
266C4⋊C4—1341no—
267C4⋊C4—1341no—
268C4⋊C4—1341no—
269C4⋊C4—1341no—
270C4⋊C4—1341no—
271C4⋊C4C22⋊D6711yescharacteristic
272C4⋊C4C22×D6711yescharacteristic
273C4⋊C4C22×D6711yescharacteristic
274C4⋊C4C22×D6711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1341no—
279C22⋊C4—1341no—
280C22⋊C4C22⋊D6711yescharacteristic
281C22⋊C4C22⋊D6711yescharacteristic
282C42—21no—
283C42—21no—
284C42—1341no—
285C42C22×D6711yescharacteristic
286C23C2×C22⋊D6711yescharacteristic
287C23C2×C22⋊D6711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2682no—
291Q8—2682no—
292Q8—672no—
293Q8—1341no—
294Q8—1341no—
295Q8—1341no—
296Q8—1341no—
297Q8—1341no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—2682no—
309C2×C4—2682no—
310C2×C4—1341no—
311C2×C4—1341no—
312C2×C4—1341no—
313C2×C4—1341no—
314C2×C4C2×C22⋊D6711yescharacteristic
315C2×C4C2×C22⋊D6711yescharacteristic
316C2×C4C2×C22⋊D6711yescharacteristic
317C2×C4C2×C22⋊D6711yescharacteristic
318C2×C4C23×D6711yescharacteristic
319C8—1341no—
320C8—1341no—
321C8—1341no—
322C8—1341no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1452407312yes—
329C22C22×C22⋊D6711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—2681no—
339C4—2681no—
340C4—2681no—
341C4—1342no—
342C4#41×D6711yescharacteristic
343C4#1452408911yescharacteristic
344C2—22no—
345C2—41no—
346C2#1686403411yescharacteristic
347C2C2×#1452407311yescharacteristic
348C2#1686421111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#552×C67C2
socleC22×C67C22×D4
hypercenter#552D67
hypocenterC67#3312
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #16863766C2semidirect
3. #16863737C2semidirect
4. #16863726C2semidirect
5. #16863700C2semidirect
6. #16863652C2semidirect
7. #16863635C2semidirect
8. #16863606C2semidirect
9. #16863551C2semidirect
10. #16863551C2semidirect
11. #16863501C2semidirect
13. #16863404C2semidirect
14. C2×#14523966C2semidirect
175. C67#3312semidirect