Finite Group Databasebeta

← #27157053

Subgroups — #27157053

Order 12416.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×C4⋊#11715897C211yesmaximal, characteristic, complemented
2C4×#13095520C211yesmaximal, characteristic, complemented
3Q8⋊#11715897C212yesmaximal, complemented
4#20261583C212yesmaximal, complemented
5#20261271C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8C4⋊#11715897—21no—
9C4⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897C2212yes—
12#15690805—21no—
13#15690801C411yescharacteristic
14#15690799C411yescharacteristic
15#15690794—21no—
16#15690784C411yescharacteristic
17Q8×Dic97C2211yescharacteristic
18#15690772C2211yescharacteristic
19#15690770—21no—
20#15690768—21no—
21C2×C4×Dic97C2211yescharacteristic
22C22⋊Q8×C97C411yescharacteristic
23#13095520—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—42no—
27C2×D388C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C22×Dic97D411yescharacteristic
30C22×Dic97D411yescharacteristic
31C2×C4×D97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C2×Q8×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C22⋊C4×C97—21no—
50C23×C97C22⋊C411yescharacteristic
51C22×D97—21no—
52Q8×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C22⋊D97—42no—
59C2×Dic97—22no—
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—21no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66C2×Dic97C2×D411yescharacteristic
67D388—22no—
68C4×D97—41no—
69C4×D97—41no—
70#11715897—24no—
71#11715897—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97#10412yes—
75C22×C97C2×C22⋊C411yescharacteristic
76C2×D97—41no—
77C2×D97—41no—
78C2×D97—41no—
79C4×C97—21no—
80C4×C97—41no—
81C4×C97—41no—
82C4×C97#13112yes—
83Dic97—21no—
84Dic97—21no—
85Dic97—21no—
86Dic97—41no—
87Dic97AGL(1,ℤ/8)11yescharacteristic
88Dic97#13711yescharacteristic
89C2×C97—22no—
90C2×C97#40811yescharacteristic
91C2×C97#41711yescharacteristic
92C2×C97C2×#10411yescharacteristic
93D97—81no—
94#1483—971nomaximal, Sylow
95C97#148311yescharacteristic, complemented, Sylow
96#529—971no—
97C2×C4⋊C8—971no—
98C4×M16—971no—
99Q8⋊C8—972no—
100D4⋊C8—972no—
101C2×M16—1941no—
102C22×C8—1941no—
103#126—1941no—
104#125—971no—
105#124—971no—
106#123—1941no—
107C22⋊Q8#8760411yescharacteristic
108#121—971no—
109C4×Q8—971no—
110C4×D4—971no—
111C4×D4—1941no—
112#117—1941no—
113C2×C42—971no—
114C4⋊C8—972no—
115C4⋊C8—1941no—
116#97—1941no—
117C4×C8—972no—
118C2×Q8C2×#8760411yescharacteristic
119C2×D4—971no—
120C2×D4—1941no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—1941no—
124C22×C4C2×#8760411yescharacteristic
125M16—1944no—
126C2×C8—3882no—
127C2×C8—1942no—
128C2×C8—1942no—
129C4⋊C4—21no—
130C4⋊C4—971no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4C2×#8760411yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1941no—
137C22⋊C4—1941no—
138C22⋊C4—1941no—
139C22⋊C4—1941no—
140C42—971no—
141C42—971no—
142C42—971no—
143C42—971no—
144C42—1941no—
145C23—1941no—
146C23C22⋊#8760411yescharacteristic
147Q8—22no—
148D4—3882no—
149D4—1942no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—971no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4—1941no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4—1942no—
163C2×C4C22⋊#8760411yescharacteristic
164C2×C4C22×#8760411yescharacteristic
165C8—3882no—
166C8—1944no—
167C22—21no—
168C22—21no—
169C22—3881no—
170C22—3881no—
171C22—3881no—
172C22#1569087512yes—
173C22C2×C22⋊#8760411yescharacteristic
174C4—21no—
175C4—41no—
176C4—41no—
177C4—971no—
178C4—971no—
179C4—3881no—
180C4—1941no—
181C4—1941no—
182C4—1941no—
183C4#1569089712yes—
184C2—22no—
185C2—7761no—
186C2#2026170811yescharacteristic
187C2#2026172611yescharacteristic
188C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690784C4
cosocleC4×Dic97C23
hypercenterC22⋊Q8#87604
hypocenterC97#1483
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#11715897C2semidirect
2. C4×#13095520C2semidirect
3. Q8⋊#11715897C2semidirect
4. #20261583C2semidirect
95. C97#1483semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#117158972
C4×#130955202
Q8⋊#117158972
#202615832
#202612712
#148397