Finite Group Databasebeta

← #22570611

Subgroups — #22570611

Order 8576.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#16863947C211yesmaximal, characteristic
2#125×D67C211yesmaximal, characteristic, complemented
3#16863764C211yesmaximal, characteristic, complemented
4#16863762C211yesmaximal, characteristic, complemented
5#16863761C211yesmaximal, characteristic, complemented
6#16863752C214yesmaximal, complemented
7#16863396C214yesmaximal, complemented
8#16863348C211yesmaximal, characteristic, complemented
9#491×C67C211yesmaximal, characteristic, complemented
10C2×#11761629C2212yes—
11C2×#11761611C2212yes—
12#14524040C2211yescharacteristic
13#14524035C2211yescharacteristic
14#14524034C2212yes—
15#14524033—22no—
16#14524032C2212yes—
17#14524018—22no—
18#14524005—22no—
19C4⋊C4×D67—22no—
20C4⋊C4×D67C2212yes—
21C42×D67C2211yescharacteristic
22#14523967C2214yes—
23#14523965C2214yes—
24#14523963C2214yes—
25#14523954C2211yescharacteristic
26C8×Dic67C2212yes—
27C4⋊Q8×C67C2211yescharacteristic
28#125×C67C2211yescharacteristic
29#103×C67C2214yes—
30#97×C67C2211yescharacteristic
31#11761629—28no—
32C2×D268—22no—
33C2×D268—22no—
34C2×D268C2312yes—
35C2×C4×D67—22no—
36C2×C4×D67—22no—
37C2×C4×D67D412yes—
38C2×C4×D67C2311yescharacteristic
39#11761611—24no—
40C2×C67⋊C8C2312yes—
41C4⋊Dic67—22no—
42C4⋊Dic67C2312yes—
43Dic67⋊C4—22no—
44Dic67⋊C4—22no—
45Dic67⋊C4—22no—
46(C2×D67)⋊C4—24no—
47C4×Dic67D412yes—
48C4×Dic67C2311yescharacteristic
49C2×Q8×C67C2312yes—
50C2×C8×C67C2312yes—
51C4⋊C4×C67—22no—
52C4⋊C4×C67—22no—
53C4⋊C4×C67C2312yes—
54C42×C67C2311yescharacteristic
55C22×D67—22no—
56C22×D67C2×D411yescharacteristic
57Q8×C67—24no—
58C2×C4×C67—22no—
59C2×C4×C67—22no—
60C2×C4×C67C2×D412yes—
61C2×C4×C67C2411yescharacteristic
62C2×Dic67—22no—
63C2×Dic67C2×D412yes—
64C2×Dic67C2×D411yescharacteristic
65D268—24no—
66D268—44no—
67D268—44no—
68C4×D67—42no—
69C4×D67—44no—
70C4×D67—44no—
71C4×D67#4114yes—
72C8×C67—22no—
73C67⋊C8—22no—
74C22×C67C22×D411yescharacteristic
75C2×D67—22no—
76C2×D67—42no—
77C2×D67—44no—
78C2×D67#12412yes—
79C4×C67—22no—
80C4×C67—42no—
81C4×C67—42no—
82C4×C67C2×#4112yes—
83Dic67—22no—
84Dic67—42no—
85Dic67#12412yes—
86C2×C67C2×#12411yescharacteristic
87C2×C67#57512yes—
88D67—22no—
89D67—82no—
90#3142—671nomaximal, Sylow
91C67#314211yescharacteristic, complemented, Sylow
92#532—671no—
93C2×#125—671no—
94#491—671no—
95#491D6711yescharacteristic, complemented
96#490—671no—
97#490—671no—
98#488—674no—
99#417—674no—
100C4×M16—671no—
101C2×#41—672no—
102C2×M16—672no—
103C4⋊Q8C2×D6711yescharacteristic
104#127—671no—
105#125—672no—
106#125—671no—
107#125—1342no—
108#125C2×D6711yescharacteristic
109#124—672no—
110#121—1342no—
111C4×D4—1342no—
112C2×C4⋊C4—672no—
113C2×C4⋊C4—1342no—
114C2×C42—671no—
115#103—674no—
116#103C2×D6714yes—
117#102—674no—
118#102—674no—
119#97—671no—
120#97C2×D6711yescharacteristic
121C4×C8—672no—
122#41—1348no—
123C2×Q8C22×D6712yes—
124C2×D4—672no—
125C2×D4—1342no—
126C2×D4—1342no—
127C22×C4—672no—
128C22×C4—671no—
129C22×C4—1342no—
130C22×C4—1342no—
131M16—1344no—
132C2×C8—672no—
133C2×C8C22×D6712yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—672no—
137C4⋊C4—1342no—
138C4⋊C4—1342no—
139C4⋊C4—1342no—
140C4⋊C4—1342no—
141C4⋊C4C22×D6712yes—
142C22⋊C4—1344no—
143C42—672no—
144C42—671no—
145C42C22×D6711yescharacteristic
146C23—671no—
147C23—1342no—
148Q8—24no—
149D4—2684no—
150D4—2684no—
151D4—1344no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—2682no—
155C2×C4—2684no—
156C2×C4—2684no—
157C2×C4—671no—
158C2×C4—674no—
159C2×C4—672no—
160C2×C4—1342no—
161C2×C4D4×D6712yes—
162C2×C4C23×D6711yescharacteristic
163C8—22no—
164C8—1342no—
165C22—2682no—
166C22—2684no—
167C22—672no—
168C22—1342no—
169C22C2×D4×D6711yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—2682no—
174C4—672no—
175C4—1342no—
176C4#41×D6712yes—
177C2—5362no—
178C2—1342no—
179C2#124×D6711yescharacteristic
180C2#1686419512yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#491×C67C2
socleC22×C67C22×D4
hypercenter#491D67
hypocenterC67#3142
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #125×D67C2semidirect
3. #16863764C2semidirect
4. #16863762C2semidirect
5. #16863761C2semidirect
6. #16863752C2semidirect
7. #16863396C2semidirect
8. #16863348C2semidirect
9. #491×C67C2semidirect
91. C67#3142semidirect
95. #491D67semidirect