Finite Group Databasebeta

← #22571153

Subgroups — #22571153

Order 8576.

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

normal subgroups only

This group has 384 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#16863951C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic
3#16863775C211yesmaximal, characteristic, complemented
4#16863747C211yesmaximal, characteristic, complemented
5#16863746C211yesmaximal, characteristic, complemented
6#16863604C214yesmaximal, complemented
7#16863550C214yesmaximal
8#16863447C211yesmaximal, characteristic
9#494×C67C211yesmaximal, characteristic, complemented
10C2×Q8×D67C2212yes—
11C2×#11761618C2214yes—
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524042—21no—
15#14524042—21no—
16#14524041C2212yes—
17#14524034—21no—
18#14524023—21no—
19#14524019C2212yes—
20Q8×Dic67C2212yes—
21#14524008—21no—
22#14524001—21no—
23C4⋊C4×D67—21no—
24C4⋊C4×D67—21no—
25C42×D67C2211yescharacteristic
26Dic67⋊C8C2212yes—
27#14523968C2214yes—
28#14523967C2214yes—
29(C2×D67)⋊C8C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C2×Q16×C67C2214yes—
32C4⋊Q8×C67C2211yescharacteristic
33C4⋊Q8×C67C2211yescharacteristic
34C4×C8×C67C2211yescharacteristic
35C2×Dic134—21no—
36C2×Dic134—22no—
37C2×Dic134C2312yes—
38Q8×D67—44no—
39C2×C4×D67—22no—
40C2×C4×D67D411yescharacteristic
41C2×C4×D67D411yescharacteristic
42C2×C4×D67C2311yescharacteristic
43#11761618—44no—
44C2×C67⋊C8C2312yes—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67C2312yes—
59C2×Q8×C67C2312yes—
60Q16×C67—44no—
61C2×C8×C67C2312yes—
62C4⋊C4×C67—21no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67—21no—
65C4⋊C4×C67—21no—
66C42×C67C2311yescharacteristic
67C22×D67C2×D411yescharacteristic
68Q8×C67—24no—
69Q8×C67—24no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2411yescharacteristic
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79C2×Dic67C2×D411yescharacteristic
80C4×D67—22no—
81C4×D67—22no—
82C4×D67—44no—
83C4×D67Q1614yescomplemented
84Dic134—24no—
85Dic134—44no—
86Dic134—42no—
87C8×C67—22no—
88C67⋊C8—22no—
89C22×C67C22×D411yescharacteristic
90C2×D67—21no—
91C2×D67—21no—
92C2×D67C2×Q1612yes—
93C4×C67—42no—
94C4×C67—42no—
95C4×C67C2×Q1612yes—
96C4×C67#13712yes—
97C4×C67H(5,𝔽2)12yes—
98Dic67—21no—
99Dic67—21no—
100Dic67—42no—
101Dic67—42no—
102Dic67C2×Q1612yes—
103C2×C67#53511yescharacteristic
104C2×C67C22×Q1611yescharacteristic
105C2×C67C2×#13711yescharacteristic
106D67—22no—
107#3209—671nomaximal, Sylow
108C67#320911yescharacteristic, complemented, Sylow
109#533—671no—
110C2×C4⋊Q8—671no—
111#494—671no—
112#494D6711yescharacteristic, complemented
113#487—671no—
114#487—671no—
115#462—674no—
116#451—674no—
117#431—671no—
118C22×Q8—672no—
119C2×Q16—674no—
120C2×Q16C2×D6714yes—
121C4⋊Q8—672no—
122C4⋊Q8—671no—
123C4⋊Q8—671no—
124C4⋊Q8—1341no—
125C4⋊Q8—1341no—
126C4⋊Q8C2×D6711yescharacteristic
127C4⋊Q8C2×D6711yescharacteristic
128#125—1341no—
129C22⋊Q8—672no—
130C22⋊Q8—1341no—
131C4×Q8—672no—
132C4×Q8—1341no—
133#117—1341no—
134C2×C4⋊C4—1341no—
135C2×C4⋊C4—1341no—
136C2×C42—671no—
137C4⋊C8—672no—
138#103—674no—
139#103—674no—
140C22⋊C8—672no—
141C4×C8—671no—
142C4×C8C2×D6711yescharacteristic
143C2×Q8—2684no—
144C2×Q8—672no—
145C2×Q8—1342no—
146C2×Q8—1341no—
147C2×Q8C22×D6712yes—
148C2×Q8C22×D6712yes—
149C22×C4—671no—
150C22×C4—671no—
151C22×C4—671no—
152C22×C4—1342no—
153Q16—44no—
154Q16—2684no—
155C2×C8—672no—
156C2×C8C22×D6712yes—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—672no—
162C4⋊C4—1342no—
163C4⋊C4—1342no—
164C4⋊C4—1342no—
165C4⋊C4—1342no—
166C4⋊C4—1341no—
167C4⋊C4—1341no—
168C4⋊C4—1341no—
169C22⋊C4—1342no—
170C42—671no—
171C42—671no—
172C42—671no—
173C42—1342no—
174C42C22×D6711yescharacteristic
175C23—671no—
176Q8—24no—
177Q8—24no—
178Q8—2684no—
179Q8—2682no—
180Q8—1344no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—2684no—
184C2×C4—671no—
185C2×C4—674no—
186C2×C4—671no—
187C2×C4—671no—
188C2×C4—1342no—
189C2×C4—1342no—
190C2×C4—1342no—
191C2×C4—1342no—
192C2×C4D4×D6711yescharacteristic
193C2×C4D4×D6711yescharacteristic
194C2×C4C23×D6711yescharacteristic
195C8—22no—
196C8—1342no—
197C22—672no—
198C22—1341no—
199C22—1341no—
200C22C2×D4×D6711yescharacteristic
201C4—42no—
202C4—42no—
203C4—2682no—
204C4—2682no—
205C4—672no—
206C4—1341no—
207C4—1341no—
208C4Q16×D6712yes—
209C4#1452407512yes—
210C4#1452408812yes—
211C2—1342no—
212C2#1686395911yescharacteristic
213C2C2×Q16×D6711yescharacteristic
214C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#494×C67C2
socleC22×C67C22×D4
hypercenter#494D67
hypocenterC67#3209
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
3. #16863775C2semidirect
4. #16863747C2semidirect
5. #16863746C2semidirect
6. #16863604C2semidirect
9. #494×C67C2semidirect
83. C4×D67Q16semidirect
108. C67#3209semidirect
112. #494D67semidirect