Finite Group Databasebeta

← #27140406

Subgroups — #27140406

Order 12416.

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

normal subgroups only

This group has 240 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
1C4⋊Q8×D97C211yesmaximal, characteristic
2#20260787C211yesmaximal, characteristic
3#20260757C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5#20260481C211yesmaximal, characteristic, complemented
6#20260480C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9C2×#13095494—21no—
10#15690810C2211yescharacteristic
11#15690809—21no—
12#15690809C412yes—
13#15690808—21no—
14C4⋊C4×D97—21no—
15C4⋊C4×D97C412yes—
16C42×D97C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21C8×Dic97—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
33#13095494—42no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97C2×C412yes—
38#13095490—22no—
39#13095490—22no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C2×Q8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C412yes—
48C42×C97C2311yescharacteristic
49C22×D97C22⋊C411yescharacteristic
50Q8×C97—41no—
51Q8×C97—41no—
52C2×C4×C97—22no—
53C2×C4×C97C22×C411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—22no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C22⋊C411yescharacteristic
60C4×D97—21no—
61C4×D97—21no—
62C4×D97—22no—
63C4×D97—44no—
64C4×D97D812yes—
65C4×D97SD1612yes—
66Dic194—41no—
67Dic194—41no—
68Dic194—44no—
69C8×C97—41no—
70C8×C97—41no—
71#11715895—41no—
72#11715895—41no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97#10212yes—
77C4×C97—21no—
78C4×C97—42no—
79C4×C97C2×D811yescharacteristic
80C4×C97C2×SD1611yescharacteristic
81C4×C97#13711yescharacteristic
82C4×C97#13711yescharacteristic
83Dic97—21no—
84Dic97—42no—
85Dic97#10112yes—
86Dic97#10212yes—
87C2×C97C2×#10111yescharacteristic
88C2×C97C2×#10211yescharacteristic
89C2×C97#41911yescharacteristic
90D97—22no—
91#1546—971nomaximal, Sylow
92C97#154611yescharacteristic, complemented, Sylow
93C2×C4⋊Q8—971no—
94#431—971no—
95#423—971no—
96#332—971no—
97#332—971no—
98#332—971no—
99#332D9711yescharacteristic, complemented
100C22×Q8—1941no—
101C2×M16—1941no—
102C4⋊Q8—972no—
103C4⋊Q8—971no—
104C4⋊Q8—1941no—
105C4⋊Q8—1941no—
106C4⋊Q8C2×D9711yescharacteristic
107C2×C4⋊C4—972no—
108C2×C4⋊C4—1941no—
109C2×C42—971no—
110C4⋊C8—971no—
111C4⋊C8—971no—
112C4⋊C8—1941no—
113C4⋊C8C2×D9711yescharacteristic
114C4⋊C8C2×D9711yescharacteristic
115C22⋊C8—1941no—
116C4×C8—1941no—
117C2×Q8—21no—
118C2×Q8—3882no—
119C2×Q8—3882no—
120C2×Q8—1942no—
121C2×Q8—1941no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—1942no—
126M16—3882no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—1941no—
130C2×C8—1941no—
131C4⋊C4—21no—
132C4⋊C4—972no—
133C4⋊C4—1942no—
134C4⋊C4—1942no—
135C4⋊C4—1941no—
136C4⋊C4C4×D9712yes—
137C42—971no—
138C42—971no—
139C42—971no—
140C42C22×D9711yescharacteristic
141C23—971no—
142Q8—41no—
143Q8—41no—
144Q8—3884no—
145Q8—3881no—
146Q8—3881no—
147C2×C4—22no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—972no—
151C2×C4—972no—
152C2×C4—971no—
153C2×C4—3884no—
154C2×C4—1942no—
155C2×C4—1942no—
156C2×C4—1941no—
157C2×C4—1941no—
158C2×C4C2×C4×D9711yescharacteristic
159C2×C4D4×D9711yescharacteristic
160C2×C4D4×D9711yescharacteristic
161C8—41no—
162C8—41no—
163C8—3881no—
164C8—3881no—
165C22—972no—
166C22—1941no—
167C22—1941no—
168C22C22⋊C4×D9711yescharacteristic
169C4—21no—
170C4—42no—
171C4—972no—
172C4—972no—
173C4—3882no—
174C4—1941no—
175C4D8×D9711yescharacteristic
176C4SD16×D9711yescharacteristic
177C4#1569083911yescharacteristic
178C4#1569084211yescharacteristic
179C2—1942no—
180C2#101×D9711yescharacteristic
181C2#102×D9711yescharacteristic
182C2#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#1546
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
92. C97#1546semidirect
99. #332D97semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D972
#202607872
#202607572
#202604822
#202604812
#202604802
#332×C972
#154697