Finite Group Databasebeta

← #22571348

Subgroups — #22571348

Order 8576.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#16863951C211yesmaximal, characteristic, complemented
2#16863947C211yesmaximal, characteristic
3#16863792C211yesmaximal, characteristic, complemented
4#16863744C211yesmaximal, characteristic, complemented
5#16863743C211yesmaximal, characteristic, complemented
6#16863723C214yesmaximal, complemented
7#16863699C214yesmaximal, complemented
8#16863447C211yesmaximal, characteristic, complemented
9#500×C67C211yesmaximal, characteristic, complemented
10C2×#11761629—21no—
11#14524043C2211yescharacteristic
12#14524040C2211yescharacteristic
13#14524034C2212yes—
14#14524032—21no—
15#14524023—21no—
16#14524019—21no—
17#14524018—21no—
18#14524018C2212yes—
19Q8×Dic67—21no—
20#14524008—21no—
21#14524005—21no—
22#14524005C2212yes—
23#14524001C2212yes—
24C42×D67C2211yescharacteristic
25#14523974C2214yes—
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×D268—22no—
38C2×D268—21no—
39C2×D268—21no—
40C2×D268C2312yes—
41C2×C4×D67—22no—
42C2×C4×D67D411yescharacteristic
43C2×C4×D67D411yescharacteristic
44C2×C4×D67C2311yescharacteristic
45C2×C67⋊C8C2312yes—
46C4⋊Dic67—21no—
47C4⋊Dic67—22no—
48C4⋊Dic67C2312yes—
49Dic67⋊C4—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—22no—
52(C2×D67)⋊C4—22no—
53C4×Dic67—22no—
54C4×Dic67D411yescharacteristic
55C4×Dic67D411yescharacteristic
56C4×Dic67C2311yescharacteristic
57C2×Q8×C67—21no—
58C2×Q8×C67—21no—
59C2×C8×C67C2312yes—
60C4⋊C4×C67—21no—
61C4⋊C4×C67—21no—
62C4⋊C4×C67C2312yes—
63C4⋊C4×C67C2312yes—
64C42×C67C2311yescharacteristic
65C22×D67—22no—
66C22×D67C2×D411yescharacteristic
67Q8×C67—42no—
68Q8×C67—42no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2411yescharacteristic
74C2×Dic67—22no—
75C2×Dic67C2×D411yescharacteristic
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78D268—24no—
79D268—44no—
80D268—42no—
81D268—42no—
82C4×D67—22no—
83C4×D67—22no—
84C4×D67—22no—
85C4×D67—44no—
86Dic134—42no—
87C8×C67—22no—
88C67⋊C8—22no—
89C22×C67C22×D411yescharacteristic
90C2×D67—21no—
91C2×D67—21no—
92C2×D67—21no—
93C2×D67—42no—
94C2×D67—42no—
95C2×D67—42no—
96C4×C67—42no—
97C4×C67—42no—
98C4×C67#13512yes—
99C4×C67AGL(1,ℤ/8)12yes—
100C4×C672−1+412yes—
101Dic67—21no—
102Dic67—21no—
103Dic67—21no—
104Dic67—42no—
105C2×C67#53611yescharacteristic
106C2×C67C2×#13511yescharacteristic
107C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
108D67—41no—
109D67—82no—
110#3228—671nomaximal, Sylow
111C67#322811yescharacteristic, complemented, Sylow
112#533—671no—
113#532—671no—
114#500D6711yescharacteristic, complemented
115#498—671no—
116#486—671no—
117#486—671no—
118#482—674no—
119#478—674no—
120#431—671no—
121C2×#41—1341no—
122C4⋊Q8—671no—
123C4⋊Q8C2×D6711yescharacteristic
124C4⋊Q8C2×D6711yescharacteristic
125#127—671no—
126#125—672no—
127#124—1341no—
128C22⋊Q8—1341no—
129C22⋊Q8—1341no—
130#121—672no—
131#121—1341no—
132C4×Q8—1341no—
133C4×Q8—1341no—
134C4×D4—672no—
135C4×D4—1341no—
136#117—672no—
137C2×C42—671no—
138#107C2×D6714yes—
139#106—674no—
140C4⋊C8—672no—
141#102—674no—
142#102—674no—
143C22⋊C8—672no—
144C4×C8—671no—
145C4×C8C2×D6711yescharacteristic
146#41—2684no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1341no—
150C2×D4—672no—
151C2×D4—1342no—
152C2×D4—1341no—
153C2×D4—1341no—
154C22×C4—671no—
155C22×C4—671no—
156C22×C4—671no—
157C22×C4—1342no—
158C2×C8—672no—
159C2×C8C22×D6712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—672no—
163C4⋊C4—1342no—
164C4⋊C4—1342no—
165C4⋊C4—1341no—
166C4⋊C4C22×D6712yes—
167C4⋊C4C22×D6712yes—
168C22⋊C4—1342no—
169C22⋊C4—1342no—
170C22⋊C4—1342no—
171C42—671no—
172C42—671no—
173C42—671no—
174C42—1342no—
175C42C22×D6711yescharacteristic
176C23—671no—
177C23—1342no—
178Q8—42no—
179Q8—42no—
180Q8—2682no—
181D4—2684no—
182D4—2682no—
183D4—2682no—
184D4—1344no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—2684no—
188C2×C4—671no—
189C2×C4—671no—
190C2×C4—671no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4D4×D6711yescharacteristic
196C2×C4D4×D6711yescharacteristic
197C2×C4C23×D6711yescharacteristic
198C8—22no—
199C8—1342no—
200C22—2682no—
201C22—2682no—
202C22—2682no—
203C22—1341no—
204C22—1341no—
205C22—1341no—
206C22C2×D4×D6711yescharacteristic
207C4—42no—
208C4—42no—
209C4—2682no—
210C4—1341no—
211C4—1341no—
212C4—1341no—
213C4#1452406612yes—
214C4#1452406712yes—
215C4#1452409112yes—
216C2—2681no—
217C2—5362no—
218C2#1686397111yescharacteristic
219C2C2×#1452406611yescharacteristic
220C2C2×#1452406711yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#500×C67C2
socleC22×C67C22×D4
hypercenter#500D67
hypocenterC67#3228
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
1. #16863951C2semidirect
3. #16863792C2semidirect
4. #16863744C2semidirect
5. #16863743C2semidirect
6. #16863723C2semidirect
7. #16863699C2semidirect
8. #16863447C2semidirect
9. #500×C67C2semidirect
111. C67#3228semidirect
114. #500D67semidirect