Finite Group Databasebeta

← #27157095

Subgroups — #27157095

Order 12416.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261756C211yesmaximal, characteristic, complemented
2#20261751C211yesmaximal, characteristic, complemented
3#20261583C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5#20261280C211yesmaximal, characteristic
6C4⋊#11715897—21no—
7C4⋊#11715897C2212yes—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10#15690865—21no—
11C4×#11715897C2212yes—
12C2×#13095510—21no—
13#15690809C411yescharacteristic
14#15690806C411yescharacteristic
15#15690796—21no—
16#15690784C411yescharacteristic
17#15690782—21no—
18D4×Dic97C2211yescharacteristic
19#15690772C2211yescharacteristic
20#15690770—21no—
21C2×C4×Dic97C2211yescharacteristic
22#121×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25#13095510—44no—
26C2×Dic194—21no—
27C2×D388C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C2×C4×D97—21no—
35C4⋊Dic97C2×C411yescharacteristic
36#13095490—21no—
37C22⋊Dic97—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C2×D4×C97—21no—
45C2×D4×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C23×C97—21no—
50C23×C97C22⋊C411yescharacteristic
51C22×D97—21no—
52D4×C97—22no—
53D4×C97—42no—
54C2×C4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—42no—
59C22⋊D97—42no—
60C22⋊D97—42no—
61C2×Dic97—22no—
62C2×Dic97—21no—
63C2×Dic97—41no—
64C2×Dic97—42no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C2×Dic97C2×D411yescharacteristic
69D388—22no—
70C4×D97—42no—
71Dic194—42no—
72#11715897—42no—
73#11715897—42no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C2×D97—41no—
82C2×D97—41no—
83C2×D97—41no—
84C4×C97—21no—
85C4×C97—41no—
86C4×C97#13112yes—
87Dic97—21no—
88Dic97—21no—
89Dic97—21no—
90Dic97—41no—
91Dic97AGL(1,ℤ/8)12yes—
92C2×C97—41no—
93C2×C97—81no—
94C2×C97#40811yescharacteristic
95C2×C97#41811yescharacteristic
96C2×C97#42111yescharacteristic
97D97—81no—
98#1496—971nomaximal, Sylow
99C97#149611yescharacteristic, complemented, Sylow
100#532—971no—
101#432—971no—
102#431—971no—
103D4⋊C8—972no—
104D4⋊C8—972no—
105C2×#41—1941no—
106C4⋊Q8—971no—
107#127—971no—
108#124—1941no—
109#121—971no—
110#121—1941no—
111#121#8760411yescharacteristic
112C4×D4—971no—
113C4×D4—971no—
114C4×D4—1941no—
115C2×C42—971no—
116C4⋊C8—972no—
117C4⋊C8—1941no—
118C22⋊C8—1941no—
119C22⋊C8—1941no—
120#97—1941no—
121C4×C8—972no—
122#41—3884no—
123C2×Q8—1941no—
124C2×D4—21no—
125C2×D4—971no—
126C2×D4—1941no—
127C2×D4—1941no—
128C2×D4—1941no—
129C2×D4C2×#8760411yescharacteristic
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—1941no—
133C22×C4—1941no—
134C22×C4C2×#8760411yescharacteristic
135C2×C8—1942no—
136C2×C8—1942no—
137C4⋊C4—971no—
138C4⋊C4—1941no—
139C4⋊C4C2×#8760411yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—1941no—
142C22⋊C4—1941no—
143C22⋊C4—1941no—
144C42—971no—
145C42—971no—
146C42—971no—
147C42—971no—
148C23—21no—
149C23—1941no—
150C23C22⋊#8760411yescharacteristic
151Q8—3882no—
152D4—22no—
153D4—42no—
154D4—3882no—
155D4—3882no—
156D4—3882no—
157D4—1942no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—3881no—
167C2×C4—1942no—
168C2×C4—1941no—
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22×#8760411yescharacteristic
171C8—3882no—
172C8—3882no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—3881no—
180C22—3881no—
181C22—3881no—
182C22C2×C22⋊#8760411yescharacteristic
183C4—21no—
184C4—41no—
185C4—972no—
186C4—3881no—
187C4—1941no—
188C4—1941no—
189C4—1941no—
190C4#1569089712yes—
191C2—41no—
192C2—81no—
193C2—7761no—
194C2#2026170811yescharacteristic
195C2#2026172711yescharacteristic
196C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261280C2
cosocleC4×Dic97C23
hypercenter#121#87604
hypocenterC97#1496
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. #20261751C2semidirect
3. #20261583C2semidirect
4. D4⋊#11715897C2semidirect
99. C97#1496semidirect

Maximal subgroups

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

subgroupindex
#202617562
#202617512
#202615832
D4⋊#117158972
#202612802
#149697