Finite Group Databasebeta

← #22558745

Subgroups — #22558745

Order 8576.

222 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 69 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#16863930C211yesmaximal, characteristic
2#16863453C211yesmaximal, characteristic, complemented
3#16863132C212yesmaximal, complemented
4#16863129C212yesmaximal
5#382×C67C211yesmaximal, characteristic, complemented
6#14524037—21no—
7#14524035C411yescharacteristic, complemented
8#14524031C411yescharacteristic, complemented
9#14524026—21no—
10#14524022C2211yescharacteristic
11#14524017C2211yescharacteristic
12C4×Dic134C411yescharacteristic, complemented
13C4×D268C411yescharacteristic, complemented
14C4×C22⋊D67—21no—
15#14524000—21no—
16C4⋊(C67⋊C8)—21no—
17C22⋊(C67⋊C8)C2212yes—
18#14523954—21no—
19C2×C4⋊C4×C67—21no—
20C2×C42×C67C2211yescharacteristic
21#95×C67—21no—
22#95×C67C2212yes—
23C2×Dic134C2×C411yescharacteristic
24C2×D268C2×C411yescharacteristic
25C2×C22⋊D67—21no—
26C2×C4×D67—21no—
27C2×C67⋊C8—22no—
28C4⋊Dic67C2×C411yescharacteristic
29C4⋊Dic67C2×C411yescharacteristic
30Dic67⋊C4—21no—
31Dic67⋊C4—21no—
32Dic67⋊C4—21no—
33Dic67⋊C4—21no—
34C22⋊Dic67—21no—
35(C2×D67)⋊C4—21no—
36(C2×D67)⋊C4—21no—
37(C2×D67)⋊C4—21no—
38(C2×D67)⋊C4—21no—
39C4×Dic67—21no—
40C22×C4×C67—22no—
41C22×C4×C67D411yescharacteristic
42C22×C4×C67D411yescharacteristic
43C22×C4×C67C2311yescharacteristic
44C4⋊C4×C67—42no—
45C42×C67C2×C411yescharacteristic
46C42×C67C2×C411yescharacteristic
47C42×C67D411yescharacteristic
48C42×C67D411yescharacteristic
49C23×C67C2×D411yescharacteristic
50C22×D67—21no—
51C2×C4×C67—21no—
52C2×C4×C67—21no—
53C2×C4×C67—41no—
54C2×C4×C67—41no—
55C2×C4×C67—42no—
56C2×C4×C67—42no—
57C2×C4×C67—42no—
58C2×C4×C67C22⋊C411yescharacteristic
59C2×C4×C67C22⋊C411yescharacteristic
60C2×C4×C67C22⋊C411yescharacteristic
61C2×C4×C67C22⋊C411yescharacteristic
62C2×C4×C67C22×C411yescharacteristic
63C2×C4×C67C2×D411yescharacteristic
64C22⋊D67—41no—
65C22⋊D67—41no—
66C2×Dic67—21no—
67C2×Dic67—21no—
68C2×Dic67—21no—
69D268—41no—
70C4×D67—42no—
71Dic134—41no—
72C67⋊C8—42no—
73C22×C67—21no—
74C22×C67—21no—
75C22×C67C2×C22⋊C411yescharacteristic
76C22×C67AGL(1,ℤ/8)11yescharacteristic
77C22×C67#13711yescharacteristic
78C2×D67—41no—
79C2×D67—41no—
80C2×D67—41no—
81C4×C67—21no—
82C4×C67—21no—
83C4×C67—21no—
84C4×C67—21no—
85C4×C67—21no—
86C4×C67—21no—
87C4×C67—82no—
88Dic67—41no—
89Dic67—41no—
90Dic67—41no—
91C2×C67—21no—
92C2×C67—21no—
93C2×C67#41011yescharacteristic
94C2×C67#41711yescharacteristic
95C2×C67#42111yescharacteristic
96D67—81no—
97#1512—671nomaximal, Sylow
98C67#151211yescharacteristic, complemented, Sylow
99#529—671no—
100#432—671no—
101#382D6711yescharacteristic, complemented
102#328—672no—
103#327—672no—
104#126—1341no—
105#125—671no—
106#124—671no—
107#123—1341no—
108C22⋊Q8—671no—
109#121—671no—
110C4×Q8—671no—
111C4×D4—671no—
112C4×D4—1341no—
113#117—1341no—
114C2×C4⋊C4—21no—
115C2×C42C2×D6711yescharacteristic
116C4⋊C8—1341no—
117C22⋊C8—672no—
118#97—1341no—
119#95—21no—
120#95C2×D6712yes—
121C2×Q8—671no—
122C2×D4—671no—
123C2×D4—1341no—
124C22×C4—22no—
125C22×C4—1341no—
126C22×C4C22⋊D6711yescharacteristic
127C22×C4C22⋊D6711yescharacteristic
128C22×C4C22×D6711yescharacteristic
129C2×C8—1342no—
130C4⋊C4—42no—
131C4⋊C4—671no—
132C4⋊C4—671no—
133C4⋊C4—1341no—
134C4⋊C4—1341no—
135C4⋊C4—1341no—
136C4⋊C4—1341no—
137C22⋊C4—1341no—
138C22⋊C4—1341no—
139C22⋊C4—1341no—
140C22⋊C4—1341no—
141C22⋊C4—1341no—
142C42—1341no—
143C42C4×D6711yescharacteristic
144C42C4×D6711yescharacteristic
145C42D26811yescharacteristic
146C42D26811yescharacteristic
147C23—1341no—
148C23C2×C22⋊D6711yescharacteristic
149Q8—2681no—
150D4—2681no—
151D4—2681no—
152D4—2681no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—2682no—
161C2×C4—1341no—
162C2×C4—1341no—
163C2×C4—1341no—
164C2×C4(C2×D67)⋊C411yescharacteristic
165C2×C4(C2×D67)⋊C411yescharacteristic
166C2×C4(C2×D67)⋊C411yescharacteristic
167C2×C4(C2×D67)⋊C411yescharacteristic
168C2×C4C2×C4×D6711yescharacteristic
169C2×C4C2×D26811yescharacteristic
170C8—2682no—
171C22—21no—
172C22—21no—
173C22—2681no—
174C22—2681no—
175C22—2681no—
176C22C2×(C2×D67)⋊C411yescharacteristic
177C22#1452406911yescharacteristic
178C22#1452407411yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—82no—
186C4—2681no—
187C4—2681no—
188C4—2681no—
189C2—21no—
190C2—21no—
191C2—5361no—
192C2#1686336911yescharacteristic
193C2#1686339511yescharacteristic
194C2#1686341411yescharacteristic
subgroupG/H
centerC22C2×(C2×D67)⋊C4
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
Fitting#382×C67C2
socleC22×C67C2×C22⋊C4
hypercenter#382D67
hypocenterC67#1512
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #16863453C2semidirect
3. #16863132C2semidirect
5. #382×C67C2semidirect
7. #14524035C4semidirect
8. #14524031C4semidirect
12. C4×Dic134C4semidirect
13. C4×D268C4semidirect
98. C67#1512semidirect
101. #382D67semidirect