Finite Group Databasebeta

← #27140451

Subgroups — #27140451

Order 12416.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4⋊Q8×D97C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic
3#20260791C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5#20260481C211yesmaximal, characteristic, complemented
6#20260480C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9#15690810C2211yescharacteristic
10#15690809—21no—
11#15690809C412yes—
12#15690808—21no—
13C4⋊C4×D97—21no—
14C4⋊C4×D97C412yes—
15C42×D97C2211yescharacteristic
16C4⋊#11715895C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18#15690722—21no—
19#15690722—21no—
20#15690720—21no—
21#15690720—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194—22no—
27Q8×D97—42no—
28Q8×D97—42no—
29C2×C4×D97—22no—
30C2×C4×D97C2×C411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C412yes—
37#13095490—22no—
38#13095490—22no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C2×Q8×C97—21no—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C412yes—
47C42×C97C2311yescharacteristic
48C22×D97C22⋊C411yescharacteristic
49Q8×C97—41no—
50Q8×C97—41no—
51C2×C4×C97—22no—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×Dic97—22no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C4×D97—22no—
60C4×D97—21no—
61C4×D97—21no—
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—44no—
65Dic194—44no—
66Dic194—41no—
67Dic194—41no—
68C8×C97—41no—
69C8×C97—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—21no—
74C2×D97—21no—
75C2×D97#10112yes—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97AGL(1,ℤ/8)11yescharacteristic
79C4×C97AGL(1,ℤ/8)11yescharacteristic
80C4×C97#13711yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—21no—
83Dic97—21no—
84Dic97—21no—
85Dic97—42no—
86C2×C97C2×#10111yescharacteristic
87C2×C97#41811yescharacteristic
88C2×C97#41911yescharacteristic
89D97—22no—
90#1556—971nomaximal, Sylow
91C97#155611yescharacteristic, complemented, Sylow
92C2×C4⋊Q8—971no—
93#432—971no—
94#432—971no—
95#332—971no—
96#332—971no—
97#332—971no—
98#332D9711yescharacteristic, complemented
99C22×Q8—1941no—
100C4⋊Q8—972no—
101C4⋊Q8—971no—
102C4⋊Q8—1941no—
103C4⋊Q8—1941no—
104C4⋊Q8C2×D9711yescharacteristic
105C2×C4⋊C4—972no—
106C2×C4⋊C4—1941no—
107C2×C42—971no—
108C4⋊C8—971no—
109C4⋊C8—971no—
110C4⋊C8C2×D9711yescharacteristic
111C4⋊C8C2×D9711yescharacteristic
112C22⋊C8—1941no—
113C22⋊C8—1941no—
114#97—1941no—
115#97—1941no—
116C2×Q8—21no—
117C2×Q8—3882no—
118C2×Q8—3882no—
119C2×Q8—1942no—
120C2×Q8—1941no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—1942no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—1941no—
128C2×C8—1941no—
129C4⋊C4—21no—
130C4⋊C4—972no—
131C4⋊C4—1942no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C4⋊C4C4×D9712yes—
135C42—971no—
136C42—971no—
137C42—971no—
138C42C22×D9711yescharacteristic
139C23—971no—
140Q8—41no—
141Q8—41no—
142Q8—3884no—
143Q8—3881no—
144Q8—3881no—
145C2×C4—22no—
146C2×C4—971no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—3884no—
150C2×C4—1942no—
151C2×C4—1942no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4C2×C4×D9711yescharacteristic
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C8—41no—
160C8—41no—
161C8—3881no—
162C8—3881no—
163C22—972no—
164C22—1941no—
165C22—1941no—
166C22C22⋊C4×D9711yescharacteristic
167C4—21no—
168C4—42no—
169C4—3882no—
170C4—1941no—
171C4—1941no—
172C4—1941no—
173C4#1569083411yescharacteristic
174C4#1569083711yescharacteristic
175C4#1569083911yescharacteristic
176C4#1569084211yescharacteristic
177C2—1942no—
178C2#101×D9711yescharacteristic
179C2#2026073811yescharacteristic
180C2#2026074211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#332D97
hypocenterC97#1556
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #20260482C2semidirect
5. #20260481C2semidirect
6. #20260480C2semidirect
7. #332×C97C2semidirect
91. C97#1556semidirect
98. #332D97semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D972
#202607912
#202604822
#202604812
#202604802
#332×C972
#155697