Finite Group Databasebeta

← #22574202

Subgroups — #22574202

Order 8576.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864122C211yesmaximal, characteristic, complemented
2#16864118C213yesmaximal, complemented
3#16864101C213yesmaximal, complemented
4#16864094C213yesmaximal, complemented
5#16864087C213yesmaximal, complemented
6Q8×C22⋊D67C211yesmaximal, characteristic, complemented
7#16864058C211yesmaximal, characteristic, complemented
8#16864057C211yesmaximal, characteristic, complemented
9#16864018C213yesmaximal, complemented
10#16864004C213yesmaximal, complemented
11#16863930C213yesmaximal, complemented
12#16863930C213yesmaximal, complemented
13#16863878C211yesmaximal, characteristic, complemented
14#16863872C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C2×Q8×D67C2211yescharacteristic
18#14524042C2213yes—
19#14524041C2213yes—
20#14524038C2216yes—
21#14524037C2216yes—
22#14524037C2213yes—
23#14524037C2213yes—
24#14524035C2213yes—
25#14524035C2213yes—
26#14524034C2213yes—
27#14524033C2216yes—
28#14524032C2213yes—
29#14524031C2213yes—
30#14524031C2213yes—
31#14524028C2216yes—
32#14524026C2211yescharacteristic
33#14524026C2213yes—
34#14524023C2213yes—
35#14524022C2213yes—
36#14524022C2213yes—
37#14524021C2216yes—
38#14524019C2213yes—
39#14524019C2213yes—
40#14524018C2213yes—
41#14524017C2213yes—
42#14524017C2213yes—
43Q8×Dic67C2211yescharacteristic
44Q8×Dic67C2211yescharacteristic
45C4×Dic134C2213yes—
46C4×Dic134C2213yes—
47#14524008C2213yes—
48C4×D268C2213yes—
49C4×D268C2213yes—
50#14524005C2213yes—
51C4×C22⋊D67C2213yes—
52C4×C22⋊D67C2213yes—
53C4×C22⋊D67C2216yes—
54#14524001C2213yes—
55#14524000C2213yes—
56#14524000C2213yes—
57#14524000C2216yes—
58C4⋊C4×D67C2213yes—
59C22×Q8×C67C2211yescharacteristic
60C4×Q8×C67C2213yes—
61C4×Q8×C67C2211yescharacteristic
62C4×Q8×C67C2213yes—
63C4×Q8×C67C2211yescharacteristic
64C2×C4⋊C4×C67C2213yes—
65C2×C42×C67C2213yes—
66#11761629—24no—
67C2×Dic134C2313yes—
68C2×Dic134C2313yes—
69Q8×D67—24no—
70C2×D268C2313yes—
71C2×D268C2313yes—
72C2×C22⋊D67C2313yes—
73C2×C22⋊D67C2311yescharacteristic
74C2×C4×D67C2313yes—
75C2×C4×D67C2313yes—
76C2×C4×D67C2316yes—
77C4⋊Dic67C2313yes—
78C4⋊Dic67C2313yes—
79C4⋊Dic67C2313yes—
80C4⋊Dic67C2313yes—
81Dic67⋊C4C2313yes—
82Dic67⋊C4C2311yescharacteristic
83Dic67⋊C4C2313yes—
84Dic67⋊C4C2313yes—
85Dic67⋊C4C2313yes—
86Dic67⋊C4C2316yes—
87Dic67⋊C4C2313yes—
88Dic67⋊C4C2316yes—
89Dic67⋊C4C2313yes—
90Dic67⋊C4C2311yescharacteristic
91C22⋊Dic67C2311yescharacteristic
92C22⋊Dic67C2313yes—
93(C2×D67)⋊C4C2311yescharacteristic
94(C2×D67)⋊C4C2313yes—
95(C2×D67)⋊C4C2313yes—
96(C2×D67)⋊C4C2313yes—
97(C2×D67)⋊C4C2313yes—
98(C2×D67)⋊C4C2316yes—
99(C2×D67)⋊C4C2316yes—
100(C2×D67)⋊C4C2313yes—
101(C2×D67)⋊C4C2313yes—
102(C2×D67)⋊C4C2311yescharacteristic
103C4×Dic67C2316yes—
104C4×Dic67C2313yes—
105C4×Dic67C2313yes—
106C2×Q8×C67—24no—
107C2×Q8×C67C2313yes—
108C2×Q8×C67C2311yescharacteristic
109C22×C4×C67C2311yescharacteristic
110C22×C4×C67C2313yes—
111C22×C4×C67C2313yes—
112C4⋊C4×C67C2313yes—
113C4⋊C4×C67C2313yes—
114C4⋊C4×C67C2316yes—
115C42×C67C2313yes—
116C42×C67C2313yes—
117C42×C67C2313yes—
118C42×C67C2313yes—
119C23×C67C2411yescharacteristic
120C22×D67C2411yescharacteristic
121C22×D67C2413yes—
122Q8×C67—26no—
123Q8×C67#4114yes—
124C2×C4×C67—22no—
125C2×C4×C67—26no—
126C2×C4×C67—43no—
127C2×C4×C67C2411yescharacteristic
128C2×C4×C67C2411yescharacteristic
129C2×C4×C67C2413yes—
130C2×C4×C67C2413yes—
131C2×C4×C67C2413yes—
132C2×C4×C67C2413yes—
133C22⋊D67—26no—
134C22⋊D67—22no—
135C2×Dic67C2411yescharacteristic
136C2×Dic67C2413yes—
137C2×Dic67C2411yescharacteristic
138C2×Dic67C2413yes—
139C2×Dic67C2413yes—
140C2×Dic67C2411yescharacteristic
141D268—26no—
142D268—43no—
143C4×D67—26no—
144C4×D67—26no—
145C4×D67—46no—
146Dic134—26no—
147Dic134—43no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C672−1+412yes—
151C22×C67C2511yescharacteristic
152C2×D67—21no—
153C2×D67—23no—
154C2×D67—21no—
155C2×D67—23no—
156C2×D67—23no—
157C2×D67—21no—
158C4×C67—23no—
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—23no—
162C4×C67—23no—
163C4×C67C2×#4116yes—
164Dic67—23no—
165Dic67—21no—
166Dic67—23no—
167Dic67—21no—
168Dic67—21no—
169Dic67—23no—
170C2×C67—22no—
171C2×C67C22×#4111yescharacteristic
172C2×C67C2×2−1+411yescharacteristic
173C2×C67#58511yescharacteristic
174D67—41no—
175D67—43no—
176#3516—671nomaximal, Sylow
177C67#351611yescharacteristic, complemented, Sylow
178#557—671no—
179#556—673no—
180#554—673no—
181#553—673no—
182#552—673no—
183D4×Q8—671no—
184#548—671no—
185#548—671no—
186#542—673no—
187#541—673no—
188#529—673no—
189#529—673no—
190#520—671no—
191#519—671no—
192C2×C4×Q8D6711yescharacteristic, complemented
193C2×#41—671no—
194C22×Q8—671no—
195C22×Q8C2×D6711yescharacteristic
196C4⋊Q8—673no—
197C4⋊Q8—673no—
198#126—676no—
199#126—676no—
200#126—673no—
201#126—673no—
202#125—676no—
203#125—673no—
204#125—673no—
205#125—673no—
206#124—673no—
207#124—673no—
208#124—673no—
209#123—671no—
210#123—673no—
211#123—676no—
212C22⋊Q8—673no—
213C22⋊Q8—673no—
214C22⋊Q8—673no—
215C22⋊Q8—673no—
216C22⋊Q8—673no—
217C22⋊Q8—676no—
218#121—673no—
219#121—673no—
220#121—673no—
221C4×Q8—671no—
222C4×Q8—671no—
223C4×Q8—673no—
224C4×Q8—673no—
225C4×Q8—673no—
226C4×Q8C2×D6711yescharacteristic
227C4×Q8C2×D6713yes—
228C4×Q8C2×D6711yescharacteristic
229C4×Q8C2×D6713yes—
230C4×D4—673no—
231C4×D4—673no—
232C4×D4—676no—
233C4×D4—673no—
234C4×D4—673no—
235C4×D4—673no—
236#117—673no—
237#117—673no—
238#117—673no—
239#117—676no—
240C2×C4⋊C4—673no—
241C2×C4⋊C4C2×D6713yes—
242C2×C42C2×D6713yes—
243#41—1344no—
244C2×Q8—24no—
245C2×Q8—673no—
246C2×Q8—673no—
247C2×Q8—1344no—
248C2×Q8C22×D6713yes—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—673no—
251C2×D4—671no—
252C2×D4—673no—
253C2×D4—673no—
254C22×C4—676no—
255C22×C4—673no—
256C22×C4—673no—
257C22×C4C22×D6711yescharacteristic
258C22×C4C22×D6713yes—
259C22×C4C22×D6713yes—
260C4⋊C4—673no—
261C4⋊C4—673no—
262C4⋊C4—671no—
263C4⋊C4—673no—
264C4⋊C4—673no—
265C4⋊C4—673no—
266C4⋊C4—673no—
267C4⋊C4—676no—
268C4⋊C4—673no—
269C4⋊C4—673no—
270C4⋊C4—676no—
271C4⋊C4—673no—
272C4⋊C4—671no—
273C4⋊C4—673no—
274C4⋊C4C22×D6713yes—
275C4⋊C4C22×D6713yes—
276C4⋊C4C22×D6716yes—
277C22⋊C4—671no—
278C22⋊C4—673no—
279C22⋊C4—673no—
280C22⋊C4—671no—
281C22⋊C4—673no—
282C22⋊C4—673no—
283C22⋊C4—673no—
284C22⋊C4—676no—
285C22⋊C4—671no—
286C22⋊C4—676no—
287C22⋊C4—673no—
288C22⋊C4—673no—
289C42—676no—
290C42—673no—
291C42—673no—
292C42C22×D6713yes—
293C42C22×D6713yes—
294C42C22×D6713yes—
295C42C22×D6713yes—
296C23—671no—
297C23—673no—
298C23C23×D6711yescharacteristic
299Q8—26no—
300Q8—2683no—
301Q8—1346no—
302Q8#1176162814yes—
303D4—2683no—
304D4—1346no—
305D4—1342no—
306D4—1346no—
307C2×C4—22no—
308C2×C4—26no—
309C2×C4—43no—
310C2×C4—2686no—
311C2×C4—671no—
312C2×C4—673no—
313C2×C4—671no—
314C2×C4—673no—
315C2×C4—673no—
316C2×C4—671no—
317C2×C4—1346no—
318C2×C4—1346no—
319C2×C4C23×D6713yes—
320C2×C4C23×D6711yescharacteristic
321C2×C4C23×D6711yescharacteristic
322C2×C4C23×D6713yes—
323C2×C4C23×D6713yes—
324C2×C4C23×D6713yes—
325C22—21no—
326C22—21no—
327C22—1341no—
328C22—1343no—
329C22—1341no—
330C22—1343no—
331C22—1343no—
332C22—1341no—
333C22#1452409112yes—
334C22C24×D6711yescharacteristic
335C4—23no—
336C4—21no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—1341no—
341C4—1343no—
342C4—1343no—
343C4—1341no—
344C4—1343no—
345C4—1341no—
346C4C2×#1176162816yes—
347C2—22no—
348C2—2681no—
349C2—2683no—
350C2C22×#1176162811yescharacteristic
351C2C2×#1452409111yescharacteristic
352C2#1686423211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
FittingC2×C4×Q8×C67C2
hypercenterC2×C4×Q8D67
hypocenterC67#3516
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #16864122C2semidirect
2. #16864118C2semidirect
3. #16864101C2semidirect
4. #16864094C2semidirect
5. #16864087C2semidirect
6. Q8×C22⋊D67C2semidirect
7. #16864058C2semidirect
8. #16864057C2semidirect
9. #16864018C2semidirect
10. #16864004C2semidirect
11. #16863930C2semidirect
12. #16863930C2semidirect
13. #16863878C2semidirect
14. #16863872C2semidirect
15. C2×C4×Q8×C67C2semidirect
177. C67#3516semidirect
192. C2×C4×Q8D67semidirect