Finite Group Databasebeta

← #27157081

Subgroups — #27157081

Order 12416.

276 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 72 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#20261738C211yesmaximal, characteristic, complemented
2C2×C4×#11715897C211yesmaximal, characteristic, complemented
3#20261583C212yesmaximal, complemented
4D4⋊#11715897C212yesmaximal, complemented
5#20261280C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8#15690877—21no—
9C4⋊#11715897C2212yes—
10C4×#11715897—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
23#13095520—42no—
24C2×#11715897—24no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27#13095510—42no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C22×Dic97—21no—
35C22×Dic97D411yescharacteristic
36C22×Dic97D411yescharacteristic
37C2×C4×D97—21no—
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40C22⋊Dic97—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C2×D4×C97—21no—
48C2×D4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C4⋊C4×C97C2×C411yescharacteristic
51C22⋊C4×C97—21no—
52C23×C97—21no—
53C23×C97C22⋊C411yescharacteristic
54C22×D97—21no—
55D4×C97—22no—
56D4×C97—42no—
57C2×C4×C97—22no—
58C2×C4×C97—21no—
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C22⋊D97—42no—
62C22⋊D97—42no—
63C22⋊D97—42no—
64C2×Dic97—22no—
65C2×Dic97—21no—
66C2×Dic97—41no—
67C2×Dic97—41no—
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97D812yes—
71C2×Dic97SD1612yes—
72C2×Dic97C2×D411yescharacteristic
73C2×Dic97C2×D411yescharacteristic
74D388—22no—
75C4×D97—41no—
76C4×D97—41no—
77Dic194—42no—
78#11715897—24no—
79#11715897—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97#10212yes—
86C22×C97C2×C22⋊C411yescharacteristic
87C2×D97—41no—
88C2×D97—41no—
89C2×D97—41no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97#13112yes—
93Dic97—21no—
94Dic97—41no—
95Dic97#10212yes—
96Dic97#10412yes—
97Dic97C2×D811yescharacteristic
98Dic97C2×SD1611yescharacteristic
99C2×C97—22no—
100C2×C97—81no—
101C2×C97#40811yescharacteristic
102C2×C97C2×#10211yescharacteristic
103C2×C97C2×#10411yescharacteristic
104D97—81no—
105#1492—971nomaximal, Sylow
106C97#149211yescharacteristic, complemented, Sylow
107#532—971no—
108#423—971no—
109C2×C4×C8—971no—
110D4⋊C8—972no—
111D4⋊C8—972no—
112C2×#41—1941no—
113C2×M16—1941no—
114C22×C8—1941no—
115C4⋊Q8—971no—
116#127—971no—
117#124—1941no—
118#121—971no—
119#121—1941no—
120#121#8760411yescharacteristic
121C4×D4—971no—
122C4×D4—971no—
123C4×D4—1941no—
124C2×C42—971no—
125C4⋊C8—972no—
126C4⋊C8—1941no—
127C4×C8—972no—
128C4×C8—1941no—
129#41—3882no—
130#41—3882no—
131C2×Q8—1941no—
132C2×D4—21no—
133C2×D4—971no—
134C2×D4—1941no—
135C2×D4—1941no—
136C2×D4—1941no—
137C2×D4C2×#8760411yescharacteristic
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—1941no—
141C22×C4—1941no—
142C22×C4C2×#8760411yescharacteristic
143M16—3882no—
144C2×C8—1942no—
145C2×C8—1942no—
146C2×C8—1944no—
147C4⋊C4—971no—
148C4⋊C4—1941no—
149C4⋊C4C2×#8760411yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—1941no—
152C22⋊C4—1941no—
153C22⋊C4—1941no—
154C42—971no—
155C42—971no—
156C42—971no—
157C42—971no—
158C23—21no—
159C23—1941no—
160C23C22⋊#8760411yescharacteristic
161Q8—3882no—
162D4—22no—
163D4—42no—
164D4—3882no—
165D4—3882no—
166D4—3882no—
167D4—1942no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—972no—
171C2×C4—972no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3881no—
179C2×C4—3881no—
180C2×C4—1942no—
181C2×C4—1941no—
182C2×C4C22⋊#8760411yescharacteristic
183C2×C4C22×#8760411yescharacteristic
184C8—3882no—
185C8—1944no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—3881no—
192C22—3881no—
193C22—3881no—
194C22#1569087212yes—
195C22C2×C22⋊#8760411yescharacteristic
196C4—21no—
197C4—41no—
198C4—971no—
199C4—971no—
200C4—972no—
201C4—972no—
202C4—3881no—
203C4—1941no—
204C4#1569089712yes—
205C2—22no—
206C2—81no—
207C2—7761no—
208C2#2026170811yescharacteristic
209C2C2×#1569087211yescharacteristic
210C2D97⋊#10411yescharacteristic
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#1492
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. #20261738C2semidirect
2. C2×C4×#11715897C2semidirect
3. #20261583C2semidirect
4. D4⋊#11715897C2semidirect
106. C97#1492semidirect

Maximal subgroups

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

subgroupindex
#202617382
C2×C4×#117158972
#202615832
D4⋊#117158972
#202612802
#149297