Finite Group Databasebeta

← #22571331

Subgroups — #22571331

Order 8576.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863947C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic, complemented
3#16863799C211yesmaximal, characteristic, complemented
4#16863744C211yesmaximal, characteristic, complemented
5#16863743C211yesmaximal, characteristic, complemented
6#16863711C214yesmaximal, complemented
7#16863675C214yesmaximal, complemented
8#16863447C211yesmaximal, characteristic, complemented
9#500×C67C211yesmaximal, characteristic, complemented
10C2×#11761629—21no—
11C2×Q8×D67—21no—
12#14524043C2211yescharacteristic
13#14524042—21no—
14#14524042C2212yes—
15#14524041—21no—
16#14524040C2211yescharacteristic
17#14524032—21no—
18#14524018—21no—
19#14524018C2212yes—
20#14524005—21no—
21#14524005C2212yes—
22C4⋊C4×D67—21no—
23C4⋊C4×D67C2212yes—
24C42×D67C2211yescharacteristic
25#14523976C2214yes—
26Dic67⋊C8C2212yes—
27#14523965C2214yes—
28#14523963C2214yes—
29(C2×D67)⋊C8C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#107×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35#11761629—44no—
36C2×Dic134—21no—
37C2×Dic134—22no—
38Q8×D67—44no—
39C2×D268—22no—
40C2×D268—21no—
41C2×D268—21no—
42C2×D268C2312yes—
43C2×C4×D67—22no—
44C2×C4×D67—22no—
45C2×C4×D67D411yescharacteristic
46C2×C4×D67D411yescharacteristic
47C2×C4×D67C2311yescharacteristic
48C2×C67⋊C8C2312yes—
49C4⋊Dic67—21no—
50C4⋊Dic67C2312yes—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67—21no—
59C2×Q8×C67—21no—
60C2×C8×C67C2312yes—
61C4⋊C4×C67—21no—
62C4⋊C4×C67—21no—
63C4⋊C4×C67C2312yes—
64C4⋊C4×C67C2312yes—
65C42×C67C2311yescharacteristic
66C22×D67—22no—
67C22×D67C2×D411yescharacteristic
68Q8×C67—42no—
69Q8×C67—42no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2411yescharacteristic
75C2×Dic67—22no—
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79D268—24no—
80D268—44no—
81D268—42no—
82D268—42no—
83C4×D67—22no—
84C4×D67—22no—
85C4×D67—44no—
86C4×D67—44no—
87C4×D67D814yes—
88Dic134—42no—
89Dic134—44no—
90C8×C67—22no—
91C67⋊C8—22no—
92C22×C67C22×D411yescharacteristic
93C2×D67—21no—
94C2×D67—21no—
95C2×D67—42no—
96C2×D67—42no—
97C2×D67—42no—
98C2×D67C2×D812yes—
99C4×C67—42no—
100C4×C67—42no—
101C4×C67C2×D812yes—
102C4×C67#13712yes—
103C4×C672−1+412yes—
104Dic67—21no—
105Dic67—21no—
106Dic67—42no—
107Dic67C2×D812yes—
108C2×C67#53611yescharacteristic
109C2×C67C22×D811yescharacteristic
110C2×C67C2×#13711yescharacteristic
111D67—22no—
112D67—82no—
113#3226—671nomaximal, Sylow
114C67#322611yescharacteristic, complemented, Sylow
115#532—671no—
116C2×C4⋊Q8—671no—
117#500—671no—
118#500D6711yescharacteristic, complemented
119#486—671no—
120#486—671no—
121#480—674no—
122#474—674no—
123#431—671no—
124C2×#41—1341no—
125C22×Q8—1341no—
126C4⋊Q8—672no—
127C4⋊Q8—671no—
128C4⋊Q8—1341no—
129C4⋊Q8—1341no—
130C4⋊Q8C2×D6711yescharacteristic
131C4⋊Q8C2×D6711yescharacteristic
132#127—671no—
133#124—1341no—
134#121—672no—
135#121—1341no—
136C4×D4—672no—
137C4×D4—1341no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—1341no—
140C2×C42—671no—
141#107—674no—
142#107C2×D6714yes—
143C4⋊C8—672no—
144#102—674no—
145#102—674no—
146C22⋊C8—672no—
147C4×C8—671no—
148C4×C8C2×D6711yescharacteristic
149#41—2684no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—2684no—
153C2×Q8—1342no—
154C2×Q8—1341no—
155C2×D4—672no—
156C2×D4—1342no—
157C2×D4—1341no—
158C2×D4—1341no—
159C22×C4—671no—
160C22×C4—671no—
161C22×C4—671no—
162C22×C4—1342no—
163C22×C4—1342no—
164C2×C8—672no—
165C2×C8C22×D6712yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—672no—
169C4⋊C4—1342no—
170C4⋊C4—1342no—
171C4⋊C4—1341no—
172C4⋊C4C22×D6712yes—
173C4⋊C4C22×D6712yes—
174C22⋊C4—1342no—
175C22⋊C4—1342no—
176C42—671no—
177C42—671no—
178C42—671no—
179C42C22×D6711yescharacteristic
180C23—671no—
181C23—1342no—
182Q8—42no—
183Q8—42no—
184Q8—2684no—
185Q8—2682no—
186D4—2684no—
187D4—2682no—
188D4—2682no—
189D4—1344no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—2684no—
193C2×C4—2684no—
194C2×C4—671no—
195C2×C4—674no—
196C2×C4—671no—
197C2×C4—671no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4D4×D6711yescharacteristic
202C2×C4D4×D6711yescharacteristic
203C2×C4C23×D6711yescharacteristic
204C8—22no—
205C8—1342no—
206C22—2682no—
207C22—2682no—
208C22—2682no—
209C22—672no—
210C22—1341no—
211C22—1341no—
212C22C2×D4×D6711yescharacteristic
213C4—42no—
214C4—42no—
215C4—2682no—
216C4—672no—
217C4—1341no—
218C4—1341no—
219C4D8×D6712yes—
220C4#1452407512yes—
221C4#1452409112yes—
222C2—5362no—
223C2—1342no—
224C2#1686397111yescharacteristic
225C2C2×D8×D6711yescharacteristic
226C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#500×C67C2
socleC22×C67C22×D4
hypercenter#500D67
hypocenterC67#3226
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D67C2semidirect
3. #16863799C2semidirect
4. #16863744C2semidirect
5. #16863743C2semidirect
6. #16863711C2semidirect
7. #16863675C2semidirect
8. #16863447C2semidirect
9. #500×C67C2semidirect
114. C67#3226semidirect
118. #500D67semidirect