Finite Group Databasebeta

← #27140883

Subgroups — #27140883

Order 12416.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261450C211yesmaximal, characteristic
2#20261115C211yesmaximal, characteristic, complemented
3#20261026C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic, complemented
5#20260483C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×Dic388—22no—
9#15690810C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690796—22no—
12#15690784—21no—
13#15690781—21no—
14C4×D388C2211yescharacteristic
15#15690769—21no—
16#15690768—21no—
17#15690742—21no—
18#15690741—21no—
19C4⋊#11715895C2211yescharacteristic
20#15690732—21no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194D412yes—
27C2×D388D411yescharacteristic
28C2×C22⋊D97—22no—
29C2×C4×D97—21no—
30Dic388—44no—
31C2×#11715895—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37C22⋊Dic97—22no—
38#13095488—22no—
39#13095488—21no—
40C4×Dic97—22no—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97—21no—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49C22×D97—21no—
50D4×C97—21no—
51D4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C22⋊D97—82no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61C2×Dic97—41no—
62D388—21no—
63D388—21no—
64C4×D97—41no—
65C4×D97—41no—
66Dic194—42no—
67Dic194—42no—
68C8×C97—22no—
69C8×C97—41no—
70#11715895—41no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97#12011yescharacteristic
75C2×D97—41no—
76C2×D97—41no—
77C2×D97—41no—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97#13511yescharacteristic
81C4×C97#13511yescharacteristic
82C4×C97#13711yescharacteristic
83C4×C97#13711yescharacteristic
84Dic97—41no—
85Dic97—42no—
86Dic97—81no—
87C2×C97—81no—
88C2×C97#45211yescharacteristic
89C2×C97#45211yescharacteristic
90C2×C97#45311yescharacteristic
91D97—81no—
92#1634—971nomaximal, Sylow
93C97#163411yescharacteristic, complemented, Sylow
94#555—971no—
95#494—971no—
96#476—971no—
97#476—971no—
98#333—971no—
99D4⋊C8—971no—
100D4⋊C8D9711yescharacteristic, complemented
101C2×Q16—1942no—
102C4⋊Q8—971no—
103C4⋊Q8—971no—
104#124—1942no—
105#121—1941no—
106#121—1941no—
107C4×D4—971no—
108C4×D4C2×D9711yescharacteristic
109#117—1941no—
110#117—1941no—
111#106—1941no—
112#106—1941no—
113C4⋊C8—971no—
114C4⋊C8C2×D9711yescharacteristic
115#102—1941no—
116#102—1941no—
117C4×C8C2×D9711yescharacteristic
118C2×Q8—972no—
119C2×Q8—1941no—
120C2×D4—971no—
121C2×D4—1942no—
122C2×D4D38811yescharacteristic
123C22×C4—21no—
124C22×C4—1941no—
125Q16—3884no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—1941no—
129C4⋊C4—971no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4D38811yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1942no—
137C22⋊C4—1942no—
138C22⋊C4—1941no—
139C42—1942no—
140C42C22×D9711yescharacteristic
141C23—21no—
142C23—1941no—
143Q8—3882no—
144Q8—3882no—
145D4—21no—
146D4—21no—
147D4—7762no—
148D4—1941no—
149D4—1941no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—3881no—
156C2×C4—1941no—
157C2×C4—1942no—
158C2×C4D4×D9711yescharacteristic
159C2×C4D4×D9711yescharacteristic
160C2×C4C2×D38811yescharacteristic
161C8—22no—
162C8—41no—
163C8—3881no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—3881no—
168C22—3881no—
169C22—3881no—
170C22#1569078011yescharacteristic
171C4—21no—
172C4—41no—
173C4—7761no—
174C4—3881no—
175C4—3882no—
176C4#1569082911yescharacteristic
177C4#1569083211yescharacteristic
178C4#1569083911yescharacteristic
179C4#1569084311yescharacteristic
180C2—81no—
181C2—7761no—
182C2#2026089811yescharacteristic
183C2#2026089911yescharacteristic
184C2#2026090411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radical#20261450C2
hypercenterD4⋊C8D97
hypocenterC97#1634
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261115C2semidirect
3. #20261026C2semidirect
4. #20261025C2semidirect
5. #20260483C2semidirect
6. #20260464C2semidirect
7. D4⋊C8×C97C2semidirect
93. C97#1634semidirect
100. D4⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202614502
#202611152
#202610262
#202610252
#202604832
#202604642
D4⋊C8×C972
#163497