Finite Group Databasebeta

← #27157036

Subgroups — #27157036

Order 12416.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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#20261754C211yesmaximal, characteristic, complemented
2#20261753C211yesmaximal, characteristic, complemented
3#20261583C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5C4×D4×D97C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897C2212yescomplemented
8#15690867—21no—
9#15690867—21no—
10C4×#11715897—21no—
11C4×#11715897C2212yescomplemented
12C2×D4×D97C411yescharacteristic
13C22×C4×D97—21no—
14C4×D388C411yescharacteristic
15D4×Dic97C2211yescharacteristic
16#15690772C2211yescharacteristic
17C4×C22⋊D97—21no—
18#15690770—21no—
19C4⋊C4×D97C411yescharacteristic
20C22⋊C4×D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C23×D97—21no—
26C2×D388C2×C411yescharacteristic
27C2×C22⋊D97—21no—
28D4×D97—22no—
29D4×D97C814yescomplemented
30C22×Dic97—21no—
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—44no—
35C2×C4×D97C2×C411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40C22⋊Dic97—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97C2311yescharacteristic
46C2×D4×C97C2×C411yescharacteristic
47C22×C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C22⋊C4×C97—21no—
50C42×C97C2×C411yescharacteristic
51C23×C97—21no—
52C22×D97—21no—
53C22×D97—24no—
54C22×D97—42no—
55C22×D97—42no—
56C22×D97C22⋊C411yescharacteristic
57D4×C97—21no—
58D4×C97C2×C812yes—
59C2×C4×C97—21no—
60C2×C4×C97—42no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22×C411yescharacteristic
64C22⋊D97—24no—
65C2×Dic97—21no—
66C2×Dic97—42no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C2×D411yescharacteristic
69C2×Dic97C2×D411yescharacteristic
70D388—21no—
71D388C2×C812yes—
72C4×D97—22no—
73C4×D97—41no—
74C4×D97—42no—
75C4×D97—42no—
76C4×D97—42no—
77C4×D97C2×C812yes—
78C4×D97M1612yes—
79#11715897—42no—
80#11715897—42no—
81C22×C97—22no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—21no—
86C2×D97—21no—
87C2×D97—22no—
88C2×D97—48no—
89C2×D97—41no—
90C2×D97—41no—
91C2×D97C22⋊C812yes—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—41no—
95C4×C97C22×C811yescharacteristic
96C4×C97C2×M1611yescharacteristic
97Dic97—21no—
98Dic97—41no—
99Dic97C22⋊C812yes—
100Dic97AGL(1,ℤ/8)12yes—
101C2×C97—42no—
102C2×C97C2×C22⋊C811yescharacteristic
103C2×C97#41811yescharacteristic
104C2×C97#42111yescharacteristic
105D97—22no—
106D97—42no—
107#1478—971nomaximal, Sylow
108C97#147811yescharacteristic, complemented, Sylow
109C2×C4×D4—971no—
110#431—971no—
111#431—971no—
112D4⋊C8—972no—
113D4⋊C8—972no—
114C22×D4—971no—
115C23×C4—1941no—
116C4×D4—971no—
117C4×D4—971no—
118C4×D4—971no—
119C4×D4—1941no—
120C4×D4—1941no—
121C4×D4#8760411yescharacteristic
122C2×C4⋊C4—971no—
123C2×C22⋊C4—1941no—
124C2×C42—971no—
125C4⋊C8—972no—
126C4⋊C8—1941no—
127C22⋊C8—1941no—
128C22⋊C8—1941no—
129C4×C8—972no—
130C4×C8—1941no—
131C24—1941no—
132C2×D4—974no—
133C2×D4—971no—
134C2×D4—1942no—
135C2×D4—1941no—
136C2×D4C2×#8760411yescharacteristic
137C22×C4—21no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—3884no—
142C22×C4—1941no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4—1941no—
146C2×C8—1942no—
147C2×C8—1942no—
148C4⋊C4—971no—
149C4⋊C4—1941no—
150C4⋊C4C2×#8760411yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C42—971no—
156C42—971no—
157C42—971no—
158C42C2×#8760411yescharacteristic
159C23—21no—
160C23—971no—
161C23—3882no—
162C23—3882no—
163C23—1944no—
164C23—1941no—
165D4—21no—
166D4—972no—
167D4—1944no—
168D4—1941no—
169D4#1309551912yes—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—972no—
175C2×C4—972no—
176C2×C4—971no—
177C2×C4—3882no—
178C2×C4—3881no—
179C2×C4—3882no—
180C2×C4—3882no—
181C2×C4—3882no—
182C2×C4—1942no—
183C2×C4—1941no—
184C2×C4C22⋊#8760411yescharacteristic
185C2×C4C22⋊#8760411yescharacteristic
186C2×C4C22×#8760411yescharacteristic
187C8—3882no—
188C8—3882no—
189C22—22no—
190C22—41no—
191C22—41no—
192C22—972no—
193C22—3888no—
194C22—3881no—
195C22—3881no—
196C22—1942no—
197C22—1941no—
198C22—1941no—
199C22C2×C22⋊#8760411yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—972no—
204C4—972no—
205C4—3881no—
206C4—1941no—
207C4C2×#1309551911yescharacteristic
208C4D97⋊M1611yescharacteristic
209C2—42no—
210C2—3882no—
211C2—1942no—
212C2#2026170311yescharacteristic
213C2#2026172711yescharacteristic
214C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×D4×D97C4
cosocleC4×Dic97C23
hypercenterC4×D4#87604
hypocenterC97#1478
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261754C2semidirect
2. #20261753C2semidirect
3. #20261583C2semidirect
4. D4⋊#11715897C2semidirect
7. C4⋊#11715897C22semidirect
11. C4×#11715897C22semidirect
29. D4×D97C8semidirect
108. C97#1478semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202617532
#202615832
D4⋊#117158972
C4×D4×D972
#147897