Finite Group Databasebeta

← #22571390

Subgroups — #22571390

Order 8576.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863953C211yesmaximal, characteristic, complemented
2#16863930C211yesmaximal, characteristic
3#16863798C212yesmaximal, complemented
4#16863751C212yesmaximal, complemented
5#16863738C212yesmaximal, complemented
6#16863721C212yesmaximal
7#16863692C211yesmaximal, characteristic, complemented
8#16863687C211yesmaximal, characteristic, complemented
9#16863682C211yesmaximal, characteristic, complemented
10#16863677C211yesmaximal, characteristic, complemented
11#431×C67C211yesmaximal, characteristic, complemented
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524037—21no—
15#14524035C2212yes—
16#14524035C2211yescharacteristic
17#14524031C2211yescharacteristic
18#14524026—21no—
19#14524022—21no—
20#14524022—21no—
21#14524022C2211yescharacteristic
22#14524017C2211yescharacteristic
23C4×Dic134—21no—
24C4×Dic134—21no—
25C4×Dic134C2211yescharacteristic
26C4×D268C2211yescharacteristic
27C4×C22⋊D67—21no—
28#14524002C2212yes—
29#14524000—21no—
30#14523977C2212yes—
31#14523977C2212yes—
32#14523975C2212yes—
33#14523975C2212yes—
34#14523968C2212yes—
35#14523968C2212yes—
36#14523965C2212yes—
37#14523965C2212yes—
38C2×C42×C67C2211yescharacteristic
39C4⋊C8×C67C2211yescharacteristic
40C4⋊C8×C67C2211yescharacteristic
41C22⋊C8×C67C2212yes—
42C4×C8×C67C2212yes—
43C2×Dic134—21no—
44C2×Dic134—21no—
45C2×Dic134C2311yescharacteristic
46C2×D268C2311yescharacteristic
47C2×C22⋊D67—21no—
48C2×C4×D67—21no—
49C4⋊Dic67—21no—
50C4⋊Dic67—21no—
51C4⋊Dic67C2312yes—
52C4⋊Dic67C2312yes—
53C4⋊Dic67C2311yescharacteristic
54C4⋊Dic67C2311yescharacteristic
55Dic67⋊C4—21no—
56Dic67⋊C4—21no—
57Dic67⋊C4—21no—
58Dic67⋊C4—22no—
59Dic67⋊C4—22no—
60Dic67⋊C4—21no—
61C22⋊Dic67—22no—
62C22⋊Dic67—21no—
63(C2×D67)⋊C4—21no—
64(C2×D67)⋊C4—21no—
65(C2×D67)⋊C4—21no—
66(C2×D67)⋊C4—21no—
67C4×Dic67—21no—
68C4×Dic67—22no—
69C22×C4×C67D411yescharacteristic
70C22×C4×C67D411yescharacteristic
71C22×C4×C67C2311yescharacteristic
72C2×C8×C67C2312yes—
73C2×C8×C67C2312yes—
74C42×C67D411yescharacteristic
75C42×C67D411yescharacteristic
76C42×C67C2311yescharacteristic
77C42×C67C2311yescharacteristic
78C23×C67C2×D411yescharacteristic
79C22×D67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—41no—
84C2×C4×C67C2×D411yescharacteristic
85C2×C4×C67C2×D411yescharacteristic
86C2×C4×C67C2×D411yescharacteristic
87C2×C4×C67C2×D411yescharacteristic
88C2×C4×C67C2×D411yescharacteristic
89C2×C4×C67C2411yescharacteristic
90C22⋊D67—42no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—21no—
96D268—21no—
97D268—21no—
98C4×D67—42no—
99Dic134—21no—
100Dic134—21no—
101Dic134—42no—
102Dic134—42no—
103C8×C67—22no—
104C8×C67—22no—
105C22×C67—21no—
106C22×C67—21no—
107C22×C67—21no—
108C22×C67C22×D411yescharacteristic
109C2×D67—41no—
110C2×D67—41no—
111C2×D67—41no—
112C4×C67—21no—
113C4×C67—21no—
114C4×C67—21no—
115C4×C67—21no—
116C4×C67#13512yes—
117C4×C672−1+411yescharacteristic
118C4×C672−1+411yescharacteristic
119Dic67—41no—
120Dic67—42no—
121Dic67—41no—
122Dic67—42no—
123Dic67—41no—
124C2×C67—41no—
125C2×C67#53611yescharacteristic
126C2×C67C2×#13511yescharacteristic
127C2×C67#57511yescharacteristic
128D67—81no—
129#3233—671nomaximal, Sylow
130C67#323311yescharacteristic, complemented, Sylow
131#533—671no—
132#529—671no—
133#499—672no—
134#488—672no—
135#485—672no—
136#482—672no—
137#477—671no—
138#476—671no—
139#475—671no—
140#474—671no—
141#431D6711yescharacteristic, complemented
142C4⋊Q8—671no—
143C4⋊Q8—671no—
144#126—1341no—
145#125—671no—
146#125—672no—
147#124—671no—
148#123—1341no—
149C22⋊Q8—671no—
150C22⋊Q8—1341no—
151C22⋊Q8—1341no—
152#121—671no—
153C4×Q8—671no—
154C4×Q8—1341no—
155C4×Q8—1341no—
156C4×D4—671no—
157C4×D4—1341no—
158#117—672no—
159#117—1341no—
160C2×C42C2×D6711yescharacteristic
161#107—672no—
162#107—672no—
163#106—672no—
164#106—672no—
165C4⋊C8C2×D6711yescharacteristic
166C4⋊C8C2×D6711yescharacteristic
167#103—672no—
168#103—672no—
169#102—672no—
170#102—672no—
171C22⋊C8C2×D6712yes—
172C4×C8C2×D6712yes—
173C2×Q8—671no—
174C2×Q8—1341no—
175C2×Q8—1341no—
176C2×D4—671no—
177C2×D4—1341no—
178C22×C4—1341no—
179C22×C4D26811yescharacteristic
180C22×C4D26811yescharacteristic
181C22×C4C22×D6711yescharacteristic
182C2×C8C22×D6712yes—
183C2×C8C22×D6712yes—
184C4⋊C4—672no—
185C4⋊C4—672no—
186C4⋊C4—671no—
187C4⋊C4—671no—
188C4⋊C4—1341no—
189C4⋊C4—1341no—
190C4⋊C4—1341no—
191C4⋊C4—1341no—
192C4⋊C4—1342no—
193C4⋊C4—1342no—
194C4⋊C4—1341no—
195C4⋊C4—1341no—
196C22⋊C4—1342no—
197C22⋊C4—1341no—
198C22⋊C4—1341no—
199C22⋊C4—1341no—
200C22⋊C4—1341no—
201C22⋊C4—1341no—
202C42—1342no—
203C42—1341no—
204C42D26811yescharacteristic
205C42D26811yescharacteristic
206C42C22×D6711yescharacteristic
207C42C22×D6711yescharacteristic
208C23—1341no—
209C23C2×D26811yescharacteristic
210Q8—2682no—
211Q8—2682no—
212Q8—1341no—
213Q8—1341no—
214D4—2682no—
215D4—1341no—
216D4—1341no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—2682no—
222C2×C4—1341no—
223C2×C4—1341no—
224C2×C4—1342no—
225C2×C4—1342no—
226C2×C4—1341no—
227C2×C4C2×D26811yescharacteristic
228C2×C4C2×D26811yescharacteristic
229C2×C4C2×D26811yescharacteristic
230C2×C4C2×D26811yescharacteristic
231C2×C4C2×D26811yescharacteristic
232C2×C4C23×D6711yescharacteristic
233C8—22no—
234C8—22no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—2681no—
239C22—2681no—
240C22—2681no—
241C22C22×D26811yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—2681no—
247C4—2682no—
248C4—2681no—
249C4—2682no—
250C4—2681no—
251C4#1452406512yes—
252C4#1452409011yescharacteristic
253C4#1452409011yescharacteristic
254C2—41no—
255C2—5361no—
256C2#1686397211yescharacteristic
257C2C2×#1452406511yescharacteristic
258C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#431×C67C2
socleC22×C67C22×D4
hypercenter#431D67
hypocenterC67#3233
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #16863953C2semidirect
3. #16863798C2semidirect
4. #16863751C2semidirect
5. #16863738C2semidirect
7. #16863692C2semidirect
8. #16863687C2semidirect
9. #16863682C2semidirect
10. #16863677C2semidirect
11. #431×C67C2semidirect
130. C67#3233semidirect
141. #431D67semidirect