Finite Group Databasebeta

← #22560809

Subgroups — #22560809

Order 8576.

186 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 75 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#16863434C211yesmaximal, characteristic
2#16863422C211yesmaximal, characteristic
3#16863351C211yesmaximal, characteristic
4#16863161C211yesmaximal, characteristic
5#16863156C211yesmaximal, characteristic
6#16863156C211yesmaximal, characteristic
7#336×C67C211yesmaximal, characteristic
8C2×#11761612C2211yescharacteristic
9C2×#11761612C2211yescharacteristic
10#14524002C2211yescharacteristic
11#14524002C2211yescharacteristic
12#14523977C411yescharacteristic, complemented
13#14523977C411yescharacteristic, complemented
14#14523975C411yescharacteristic, complemented
15#14523975C411yescharacteristic, complemented
16Dic67⋊C8—21no—
17Dic67⋊C8—21no—
18#14523953—21no—
19C8×Dic67—21no—
20C22×C8×C67C2211yescharacteristic
21C22×C8×C67C2211yescharacteristic
22C2×C42×C67C2211yescharacteristic
23#11761612—21no—
24#11761612—21no—
25#11761612—21no—
26#11761612Q812yes—
27C2×C67⋊C8—21no—
28C2×C67⋊C8—21no—
29C4⋊Dic67C2×C411yescharacteristic
30C4⋊Dic67C2×C411yescharacteristic
31C4⋊Dic67C2×C411yescharacteristic
32C4⋊Dic67C2×C411yescharacteristic
33C22⋊Dic67—21no—
34C22⋊Dic67—21no—
35C4×Dic67—21no—
36C4×Dic67—21no—
37C22×C4×C67—21no—
38C22×C4×C67C2311yescharacteristic
39C2×C8×C67—21no—
40C2×C8×C67—21no—
41C2×C8×C67—21no—
42C2×C8×C67—21no—
43C2×C8×C67C2×C411yescharacteristic
44C2×C8×C67C2×C411yescharacteristic
45C2×C8×C67D411yescharacteristic
46C2×C8×C67D411yescharacteristic
47C42×C67—21no—
48C42×C67—21no—
49C23×C67#4111yescharacteristic
50C2×C4×C67—22no—
51C2×C4×C67—41no—
52C2×C4×C67—41no—
53C2×C4×C67C22×C411yescharacteristic
54C2×C4×C67C2×D411yescharacteristic
55C2×C4×C67C2×Q811yescharacteristic
56C2×C4×C67#4111yescharacteristic
57C2×C4×C67#4111yescharacteristic
58C2×C4×C67#4111yescharacteristic
59C2×Dic67—21no—
60C2×Dic67—21no—
61C2×Dic67—21no—
62C2×Dic67—21no—
63C8×C67—21no—
64C8×C67—21no—
65C8×C67—21no—
66C8×C67—21no—
67C67⋊C8—22no—
68C67⋊C8—41no—
69C22×C67—21no—
70C22×C67—21no—
71C22×C67C4×D411yescharacteristic
72C22×C67#12511yescharacteristic
73C22×C67#12611yescharacteristic
74C4×C67—42no—
75C4×C67#11711yescharacteristic
76C4×C67C4×Q811yescharacteristic
77C4×C67C22⋊Q811yescharacteristic
78C4×C67#12311yescharacteristic
79Dic67—41no—
80Dic67—41no—
81Dic67—41no—
82Dic67—41no—
83C2×C67—21no—
84C2×C67—21no—
85C2×C67#38711yescharacteristic
86C2×C67#44311yescharacteristic
87C2×C67#44411yescharacteristic
88#1922—671nomaximal, Sylow
89C67#192211yescharacteristic, complemented, Sylow
90#427—671no—
91#424—671no—
92#405—671no—
93#341—671no—
94#339—671no—
95#339—671no—
96#336D6711yescharacteristic
97C2×M16—671no—
98C2×M16—671no—
99C22×C8C2×D6711yescharacteristic
100C22×C8C2×D6711yescharacteristic
101#117—671no—
102#117—671no—
103C2×C42C2×D6711yescharacteristic
104#107—671no—
105#107—671no—
106#106—671no—
107#106—671no—
108C4⋊C8—1341no—
109C4⋊C8—1341no—
110#97—1341no—
111C4×C8—1341no—
112C22×C4—21no—
113C22×C4C22×D6711yescharacteristic
114M16—672no—
115M16—1341no—
116M16—1341no—
117M16—1341no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—1341no—
123C2×C8—1341no—
124C2×C8C4×D6711yescharacteristic
125C2×C8C4×D6711yescharacteristic
126C2×C8D26811yescharacteristic
127C2×C8D26811yescharacteristic
128C4⋊C4—671no—
129C4⋊C4—671no—
130C4⋊C4—671no—
131C4⋊C4—671no—
132C22⋊C4—1341no—
133C22⋊C4—1341no—
134C42—21no—
135C42—21no—
136C42—1341no—
137C42—1341no—
138C23#1176162811yescharacteristic
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—1341no—
143C2×C4—1341no—
144C2×C4—1341no—
145C2×C4—1341no—
146C2×C4C2×C4×D6711yescharacteristic
147C2×C4C2×D26811yescharacteristic
148C2×C4Q8×D6711yescharacteristic
149C2×C4#1176162711yescharacteristic
150C2×C4#1176162711yescharacteristic
151C2×C4#1176162711yescharacteristic
152C8—21no—
153C8—21no—
154C8—21no—
155C8—21no—
156C8—2681no—
157C8—1342no—
158C22—21no—
159C22—21no—
160C22C4×D26811yescharacteristic
161C22#1452403311yescharacteristic
162C22#1452403611yescharacteristic
163C4—42no—
164C4—2681no—
165C4—2681no—
166C4—2681no—
167C4—2681no—
168C4#1452399911yescharacteristic
169C4#1452400811yescharacteristic
170C4#1452402311yescharacteristic
171C4#1452402711yescharacteristic
172C2—21no—
173C2—21no—
174C2#1686328111yescharacteristic
175C2#1686351411yescharacteristic
176C2#1686351911yescharacteristic
subgroupG/H
centerC2×C4C2×D268
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
Fitting#336×C67C2
socleC22×C67C4×D4
hypercenter#336D67
hypocenterC67#1922
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #336

Complements

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

NH ≅ G/Ntype
12. #14523977C4semidirect
13. #14523977C4semidirect
14. #14523975C4semidirect
15. #14523975C4semidirect
89. C67#1922semidirect