Finite Group Databasebeta

← #22570925

Subgroups — #22570925

Order 8576.

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

normal subgroups only

This group has 402 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, complemented
2#16863944C211yesmaximal, characteristic
3#16863801C211yesmaximal, characteristic, complemented
4#16863746C211yesmaximal, characteristic, complemented
5#16863742C211yesmaximal, characteristic, complemented
6#16863639C214yesmaximal, complemented
7#16863497C214yesmaximal, complemented
8C4×#11761611C211yesmaximal, characteristic, complemented
9#492×C67C211yesmaximal, characteristic, complemented
10C2×#11761627—21no—
11C2×#11761611C2212yes—
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524034—21no—
15#14524030—21no—
16#14524023—21no—
17#14524019C2212yes—
18#14524014—21no—
19#14524014C2212yes—
20Q8×Dic67C2212yes—
21#14524008—21no—
22D4×Dic67—21no—
23D4×Dic67C2212yes—
24#14524001—21no—
25C42×D67C2211yescharacteristic
26#14523977C2214yes—
27#14523967C2214yes—
28#14523964C2214yes—
29#14523953C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C2×SD16×C67C2214yes—
32C4⋊Q8×C67C2211yescharacteristic
33#127×C67C2211yescharacteristic
34C4×C8×C67C2211yescharacteristic
35#11761627—44no—
36C2×Dic134—21no—
37C2×Dic134—21no—
38C2×C22⋊D67—22no—
39C22×Dic67—22no—
40C2×C4×D67D411yescharacteristic
41C2×C4×D67D411yescharacteristic
42C2×C4×D67C2311yescharacteristic
43#11761611—24no—
44C2×C67⋊C8C2312yes—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—22no—
48C4⋊Dic67C2312yes—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51C22⋊Dic67—22no—
52C22⋊Dic67—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67C2312yes—
59C2×D4×C67—21no—
60C2×D4×C67—21no—
61C2×D4×C67C2312yes—
62SD16×C67—28no—
63C2×C8×C67C2312yes—
64C4⋊C4×C67—21no—
65C4⋊C4×C67—21no—
66C42×C67C2311yescharacteristic
67C23×C67—22no—
68C22×D67C2×D411yescharacteristic
69Q8×C67—24no—
70D4×C67—24no—
71D4×C67—42no—
72D4×C67—42no—
73C2×C4×C67—22no—
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2411yescharacteristic
77C22⋊D67—44no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—44no—
81C2×Dic67C2×D411yescharacteristic
82C2×Dic67C2×D411yescharacteristic
83C2×Dic67C2×D411yescharacteristic
84C4×D67—22no—
85C4×D67—22no—
86C4×D67—22no—
87Dic134—42no—
88Dic134—42no—
89C8×C67#4114yescomplemented
90C67⋊C8—22no—
91C22×C67—42no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67C22×D411yescharacteristic
95C2×D67—21no—
96C2×D67—21no—
97C2×D67—21no—
98C4×C67—42no—
99C4×C67AGL(1,ℤ/8)12yes—
100C4×C67#13712yes—
101C4×C67C2×#4112yes—
102Dic67—21no—
103Dic67—21no—
104Dic67—21no—
105Dic67—42no—
106Dic67—42no—
107C2×C67—82no—
108C2×C67#53211yescharacteristic
109C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C67C2×#13711yescharacteristic
111D67—41no—
112#3182—671nomaximal, Sylow
113C67#318211yescharacteristic, complemented, Sylow
114#533—671no—
115#532—671no—
116#500—671no—
117#492D6711yescharacteristic, complemented
118#487—671no—
119#486—671no—
120#468—674no—
121#440—674no—
122C4×M16—671no—
123C2×#41—1341no—
124C2×SD16C2×D6714yes—
125C2×M16—672no—
126C4⋊Q8—671no—
127C4⋊Q8—671no—
128C4⋊Q8C2×D6711yescharacteristic
129#127C2×D6711yescharacteristic
130#125—1341no—
131#124—1341no—
132C22⋊Q8—672no—
133C22⋊Q8—1341no—
134#121—672no—
135#121—1341no—
136C4×Q8—672no—
137C4×Q8—1341no—
138C4×D4—672no—
139C4×D4—1341no—
140#117—1341no—
141C2×C42—671no—
142#107—674no—
143#103—674no—
144#102—674no—
145#97—672no—
146C4×C8—671no—
147C4×C8C2×D6711yescharacteristic
148#41—2684no—
149C2×Q8—1341no—
150C2×Q8—1341no—
151C2×Q8C22×D6712yes—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—1342no—
155C2×D4C22×D6712yes—
156C22×C4—671no—
157C22×C4—671no—
158C22×C4—671no—
159C22×C4—1342no—
160SD16—28no—
161M16—1344no—
162C2×C8—672no—
163C2×C8C22×D6712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—672no—
167C4⋊C4—672no—
168C4⋊C4—1342no—
169C4⋊C4—1342no—
170C4⋊C4—1341no—
171C4⋊C4—1341no—
172C22⋊C4—1342no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C42—671no—
176C42—671no—
177C42—671no—
178C42—1342no—
179C42C22×D6711yescharacteristic
180C23—22no—
181C23—671no—
182Q8—24no—
183Q8—2682no—
184Q8—2682no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—2684no—
189C2×C4—22no—
190C2×C4—2684no—
191C2×C4—671no—
192C2×C4—671no—
193C2×C4—671no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4—1342no—
197C2×C4—1342no—
198C2×C4—1342no—
199C2×C4D4×D6711yescharacteristic
200C2×C4D4×D6711yescharacteristic
201C2×C4C23×D6711yescharacteristic
202C8—1342no—
203C8#1176162714yescomplemented
204C22—42no—
205C22—42no—
206C22—42no—
207C22—1341no—
208C22—1341no—
209C22—1341no—
210C22C2×D4×D6711yescharacteristic
211C4—42no—
212C4—2682no—
213C4—2682no—
214C4—1341no—
215C4—1341no—
216C4—1341no—
217C4#1452407012yes—
218C4#1452407212yes—
219C4C2×#1176162712yes—
220C2—82no—
221C2—2681no—
222C2#1686394411yescharacteristic
223C2C2×#1452407011yescharacteristic
224C2C2×#1452407211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#492×C67C2
socleC22×C67C22×D4
hypercenter#492D67
hypocenterC67#3182
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #16863951C2semidirect
3. #16863801C2semidirect
4. #16863746C2semidirect
5. #16863742C2semidirect
6. #16863639C2semidirect
7. #16863497C2semidirect
8. C4×#11761611C2semidirect
9. #492×C67C2semidirect
89. C8×C67#41semidirect
113. C67#3182semidirect
117. #492D67semidirect
203. C8#11761627semidirect