Finite Group Databasebeta

← #27140231

Subgroups — #27140231

Order 12416.

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

normal subgroups only

This group has 284 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#20260788C211yesmaximal, characteristic, complemented
2C4×C4⋊Dic97C211yesmaximal, characteristic, complemented
3#20260469C214yesmaximal, complemented
4#532×C97C211yesmaximal, characteristic
5C2×C4⋊Dic97—21no—
6C2×C4×Dic97—21no—
7C4⋊#11715895—21no—
8C22⋊#11715895C2212yes—
9C4×#11715895—21no—
10#15690717C2212yes—
11C2×#41×C97—21no—
12C4⋊Q8×C97C411yescharacteristic, complemented
13#127×C97C411yescharacteristic, complemented
14#124×C97—21no—
15#121×C97—21no—
16#121×C97C2212yes—
17C4×D4×C97—21no—
18C4×D4×C97C412yescomplemented
19C2×C42×C97C2211yescharacteristic
20C22×Dic97—22no—
21C2×#11715895—22no—
22C4⋊Dic97—24no—
23C4×Dic97—42no—
24#41×C97—42no—
25#41×C97—42no—
26C2×Q8×C97—21no—
27C2×D4×C97—22no—
28C2×D4×C97—21no—
29C2×D4×C97—21no—
30C2×D4×C97C2×C412yes—
31C22×C4×C97—22no—
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97C2311yescharacteristic
35C4⋊C4×C97—21no—
36C4⋊C4×C97C2×C412yes—
37C22⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C42×C97C2×C411yescharacteristic
40C42×C97C2×C411yescharacteristic
41C42×C97D411yescharacteristic
42C42×C97D411yescharacteristic
43C23×C97—22no—
44C23×C97C2×D411yescharacteristic
45Q8×C97—42no—
46D4×C97—42no—
47D4×C97—42no—
48D4×C97—42no—
49D4×C97—42no—
50D4×C97—42no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—41no—
54C2×C4×C97—44no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97D812yes—
60C2×C4×C97SD1612yes—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×Dic97—24no—
64C2×Dic97—42no—
65C2×Dic97—42no—
66#11715895—42no—
67C22×C97—21no—
68C22×C97—21no—
69C22×C97—42no—
70C22×C97—42no—
71C22×C97—42no—
72C22×C97C2×C22⋊C411yescharacteristic
73C22×C97C2×D811yescharacteristic
74C22×C97C2×SD1611yescharacteristic
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—42no—
79C4×C97#10212yes—
80C4×C97#10212yes—
81C4×C97#10412yes—
82Dic97—44no—
83C2×C97—21no—
84C2×C97—21no—
85C2×C97—82no—
86C2×C97#41011yescharacteristic
87C2×C97C2×#10211yescharacteristic
88C2×C97C2×#10411yescharacteristic
89#1502—971nomaximal, Sylow
90C97#150211yescharacteristic, complemented, Sylow
91#532D9711yescharacteristic
92#431—971no—
93C4×C4⋊C4—971no—
94#327—974no—
95C2×#41—21no—
96C4⋊Q8Dic9711yescharacteristic, complemented
97#127Dic9711yescharacteristic, complemented
98#124—21no—
99#121—21no—
100#121C2×D9712yes—
101C4×D4—21no—
102C4×D4Dic9712yescomplemented
103C2×C4⋊C4—1941no—
104C2×C42—1941no—
105C2×C42C2×D9711yescharacteristic
106C4⋊C8—1941no—
107C22⋊C8—972no—
108C4×C8—1941no—
109#95—972no—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×D4—22no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C2×Dic9712yes—
117C22×C4—22no—
118C22×C4—1942no—
119C22×C4C22⋊D9711yescharacteristic
120C22×C4C22⋊D9711yescharacteristic
121C22×C4C22×D9711yescharacteristic
122C2×C8—1942no—
123C4⋊C4—21no—
124C4⋊C4—1944no—
125C4⋊C4C2×Dic9712yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C42—3882no—
129C42C2×Dic9711yescharacteristic
130C42C2×Dic9711yescharacteristic
131C42C22⋊D9711yescharacteristic
132C42C22⋊D9711yescharacteristic
133C23—22no—
134C23C2×C22⋊D9711yescharacteristic
135Q8—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—42no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—44no—
145C2×C4—3882no—
146C2×C4—3882no—
147C2×C4—1944no—
148C2×C4C22⋊Dic9711yescharacteristic
149C2×C4C22⋊Dic9711yescharacteristic
150C2×C4C22⋊Dic9711yescharacteristic
151C2×C4C22⋊Dic9711yescharacteristic
152C2×C4#1309549612yes—
153C2×C4#1309549912yes—
154C2×C4C22×Dic9711yescharacteristic
155C2×C4C2×C22⋊D9711yescharacteristic
156C8—3882no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22C2×C22⋊Dic9711yescharacteristic
163C22C2×#1309549611yescharacteristic
164C22C2×#1309549911yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—3884no—
170C4#1569073112yes—
171C4#1569073112yes—
172C4#1569073712yes—
173C2—21no—
174C2—21no—
175C2—82no—
176C2#2026070911yescharacteristic
177C2C2×#1569073111yescharacteristic
178C2C2×#1569073711yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#532×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#532D97
hypocenterC97#1502
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20260788C2semidirect
2. C4×C4⋊Dic97C2semidirect
3. #20260469C2semidirect
12. C4⋊Q8×C97C4semidirect
13. #127×C97C4semidirect
18. C4×D4×C97C4semidirect
90. C97#1502semidirect
96. C4⋊Q8Dic97semidirect
97. #127Dic97semidirect
102. C4×D4Dic97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202607882
C4×C4⋊Dic972
#202604692
#532×C972
#150297