Finite Group Databasebeta

← #22560437

Subgroups — #22560437

Order 8576.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#16863879C211yesmaximal, characteristic
2#16863438C211yesmaximal, characteristic, complemented
3#16863397C211yesmaximal, characteristic
4#16863266C211yesmaximal, characteristic, complemented
5#16863164C211yesmaximal, characteristic, complemented
6#16863156C211yesmaximal, characteristic, complemented
7C2×#104×C67C211yesmaximal, characteristic
8C2×#11761612C2211yescharacteristic
9Q8×Dic67—21no—
10D4×Dic67—22no—
11D4×Dic67—21no—
12#14524002—21no—
13#14524002C2211yescharacteristic
14C2×C4⋊Dic67—21no—
15C2×C4⋊Dic67—22no—
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2211yescharacteristic
18#14523977—21no—
19#14523975—21no—
20#14523967—21no—
21#14523964—21no—
22C2×#41×C67C2211yescharacteristic
23C2×M16×C67C2211yescharacteristic
24C2×C42×C67C2211yescharacteristic
25#104×C67C414yescomplemented
26C22×Dic67—22no—
27C22×Dic67—21no—
28C22×Dic67D412yes—
29#11761612—21no—
30#11761612—21no—
31C2×C67⋊C8—21no—
32C4⋊Dic67—22no—
33C4⋊Dic67—21no—
34C4⋊Dic67—21no—
35C4⋊Dic67—21no—
36C4⋊Dic67—21no—
37C4⋊Dic67—44no—
38C4⋊Dic67D411yescharacteristic
39C4⋊Dic67D411yescharacteristic
40C22⋊Dic67—22no—
41C22⋊Dic67—21no—
42C4×Dic67—22no—
43C4×Dic67—21no—
44#41×C67—21no—
45#41×C67—21no—
46#41×C67C2×C412yes—
47#41×C67Q812yes—
48C2×Q8×C67D411yescharacteristic
49C2×D4×C67—21no—
50C2×D4×C67D411yescharacteristic
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67C2311yescharacteristic
54M16×C67—21no—
55M16×C67C2×C412yescomplemented
56C2×C8×C67—21no—
57C42×C67—21no—
58C42×C67C2×C412yescomplemented
59C23×C67—21no—
60C23×C67C2×D411yescharacteristic
61Q8×C67—21no—
62Q8×C67C4⋊C412yescomplemented
63D4×C67—22no—
64D4×C67—22no—
65D4×C67—21no—
66D4×C67C4⋊C412yescomplemented
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67C22×C411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×Q811yescharacteristic
77C2×C4×C67#4111yescharacteristic
78C2×C4×C67#4111yescharacteristic
79C2×Dic67—21no—
80C2×Dic67—22no—
81C2×Dic67—21no—
82C2×Dic67—42no—
83C2×Dic67—42no—
84C2×Dic67—41no—
85C2×Dic67—82no—
86C8×C67—22no—
87C67⋊C8—41no—
88C22×C67—21no—
89C22×C67—21no—
90C22×C67—22no—
91C22×C67—41no—
92C22×C67—41no—
93C22×C67C4×D411yescharacteristic
94C22×C67#12011yescharacteristic
95C22×C67C22⋊Q811yescharacteristic
96C4×C67—24no—
97C4×C67—22no—
98C4×C67C2×C4⋊C411yescharacteristic
99C4×C67C4×D411yescharacteristic
100C4×C67C22⋊Q811yescharacteristic
101C4×C67#12311yescharacteristic
102Dic67—42no—
103Dic67—41no—
104Dic67—41no—
105Dic67—81no—
106C2×C67—21no—
107C2×C67—21no—
108C2×C67—42no—
109C2×C67#38511yescharacteristic
110C2×C67#45311yescharacteristic
111C2×C67#45611yescharacteristic
112#1859—671nomaximal, Sylow
113C67#185911yescharacteristic, complemented, Sylow
114#520—671no—
115#428—671no—
116C2×#104D6711yescharacteristic
117#417—671no—
118#384—671no—
119#343—671no—
120#339—671no—
121C2×#41C2×D6711yescharacteristic
122C2×M16—671no—
123C2×M16C2×D6711yescharacteristic
124C4×Q8—1341no—
125C4×D4—1342no—
126C4×D4—1341no—
127#117—671no—
128#117—1341no—
129C2×C4⋊C4—671no—
130C2×C4⋊C4—671no—
131C2×C4⋊C4—1342no—
132C2×C4⋊C4—1341no—
133C2×C42C2×D6711yescharacteristic
134#107—1341no—
135#106—1341no—
136#104Dic6714yescomplemented
137#103—1341no—
138#102—1341no—
139#41—21no—
140#41—21no—
141#41Dic13412yes—
142#41C2×Dic6712yes—
143C2×Q8D26811yescharacteristic
144C2×D4—21no—
145C2×D4D26811yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—672no—
149C22×C4—1341no—
150C22×C4—1342no—
151C22×C4C22×D6711yescharacteristic
152M16—21no—
153M16—1341no—
154M16—1341no—
155M16C2×Dic6712yescomplemented
156C2×C8—21no—
157C2×C8—1341no—
158C4⋊C4—2684no—
159C4⋊C4—671no—
160C4⋊C4—671no—
161C4⋊C4—1342no—
162C4⋊C4—1341no—
163C4⋊C4—1341no—
164C4⋊C4—1341no—
165C4⋊C4—1341no—
166C22⋊C4—1342no—
167C22⋊C4—1341no—
168C42—21no—
169C42—1342no—
170C42—1341no—
171C42C2×Dic6712yescomplemented
172C23—21no—
173C23D4×D6711yescharacteristic
174Q8—21no—
175Q8C4⋊Dic6712yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊Dic6712yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—2682no—
187C2×C4—2682no—
188C2×C4—2681no—
189C2×C4—5362no—
190C2×C4—1341no—
191C2×C4—1341no—
192C2×C4—1342no—
193C2×C4C22×Dic6711yescharacteristic
194C2×C4D4×D6711yescharacteristic
195C2×C4C2×D26811yescharacteristic
196C2×C4C2×Dic13411yescharacteristic
197C2×C4#1176162711yescharacteristic
198C2×C4#1176162711yescharacteristic
199C8—22no—
200C8—2681no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22D4×Dic6711yescharacteristic
207C22#1452401311yescharacteristic
208C22#1452402011yescharacteristic
209C4—24no—
210C4—22no—
211C4—2682no—
212C4—2681no—
213C4—2681no—
214C4—5361no—
215C4C2×C4⋊Dic6711yescharacteristic
216C4D4×Dic6711yescharacteristic
217C4#1452402011yescharacteristic
218C4#1452402711yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#1686327111yescharacteristic
223C2#1686356511yescharacteristic
224C2#1686358011yescharacteristic
subgroupG/H
centerC22#14524013
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
FittingC2×#104×C67C2
socleC22×C67#120
hypercenterC2×#104D67
hypocenterC67#1859
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #16863438C2semidirect
4. #16863266C2semidirect
5. #16863164C2semidirect
6. #16863156C2semidirect
25. #104×C67C4semidirect
55. M16×C67C2×C4semidirect
58. C42×C67C2×C4semidirect
62. Q8×C67C4⋊C4semidirect
66. D4×C67C4⋊C4semidirect
113. C67#1859semidirect
136. #104Dic67semidirect
155. M16C2×Dic67semidirect
171. C42C2×Dic67semidirect
175. Q8C4⋊Dic67semidirect
179. D4C4⋊Dic67semidirect