Finite Group Databasebeta

← #27140880

Subgroups — #27140880

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#20261114C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20260483C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#476×C97C211yesmaximal, characteristic, complemented
8C2×#13095501—22no—
9#15690810C2211yescharacteristic
10#15690796—22no—
11#15690784—21no—
12#15690781—21no—
13C4×D388C2211yescharacteristic
14#15690769—21no—
15#15690768—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194D412yes—
26C2×D388D411yescharacteristic
27C2×C22⋊D97—22no—
28C2×C4×D97—21no—
29#13095501—44no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D411yescharacteristic
35C22⋊Dic97—22no—
36#13095488—22no—
37#13095488—21no—
38C4×Dic97—22no—
39C2×Q8×C97—21no—
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C2×C8×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49C22×D97—21no—
50Q8×C97—42no—
51D4×C97—21no—
52D4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C22⋊D97—82no—
61C2×Dic97—21no—
62C2×Dic97—22no—
63D388—21no—
64D388—21no—
65C4×D97—41no—
66C4×D97—41no—
67Dic194—42no—
68C8×C97—41no—
69#11715895—22no—
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—81no—
81C4×C97#13511yescharacteristic
82C4×C97#13511yescharacteristic
83C4×C97#13711yescharacteristic
84C4×C97#13711yescharacteristic
85Dic97—41no—
86Dic97—42no—
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#476D9711yescharacteristic, complemented
98#333—971no—
99D4⋊C8—971no—
100D4⋊C8—971no—
101C2×Q16—1942no—
102C4⋊Q8—971no—
103C4⋊Q8C2×D9711yescharacteristic
104#124—1942no—
105#121—1941no—
106#121—1941no—
107C4×D4—971no—
108C4×D4C2×D9711yescharacteristic
109#117—1941no—
110#117—1941no—
111#106—21no—
112#106—1941no—
113C4⋊C8—971no—
114C4⋊C8C2×D9711yescharacteristic
115#102—21no—
116#102—1941no—
117C4×C8—971no—
118C2×Q8—21no—
119C2×Q8—972no—
120C2×D4—971no—
121C2×D4—1942no—
122C2×D4C22⋊D9711yescharacteristic
123C22×C4—21no—
124C22×C4—1941no—
125Q16—3884no—
126C2×C8—21no—
127C2×C8—1941no—
128C2×C8—1941no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—971no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4C22⋊D9711yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—1942no—
137C22⋊C4—1942no—
138C22⋊C4—1941no—
139C42—1942no—
140C42C22×D9711yescharacteristic
141C23—21no—
142C23—1941no—
143Q8—42no—
144Q8—3882no—
145D4—21no—
146D4—21no—
147D4—7762no—
148D4—1941no—
149D4—1941no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—3881no—
155C2×C4—3881no—
156C2×C4—1941no—
157C2×C4—1942no—
158C2×C4D4×D9711yescharacteristic
159C2×C4D4×D9711yescharacteristic
160C2×C4C2×C22⋊D9711yescharacteristic
161C8—41no—
162C8—3881no—
163C8—1942no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—3881no—
168C22—3881no—
169C22—3881no—
170C22#1569077811yescharacteristic
171C4—21no—
172C4—41no—
173C4—81no—
174C4—3881no—
175C4—3882no—
176C4#1569083011yescharacteristic
177C4#1569083111yescharacteristic
178C4#1569083911yescharacteristic
179C4#1569084111yescharacteristic
180C2—81no—
181C2—7761no—
182C2#2026089411yescharacteristic
183C2#2026089511yescharacteristic
184C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radical#20261450C2
hypercenter#476D97
hypocenterC97#1634
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20261114C2semidirect
3. #20261022C2semidirect
4. #20260483C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #476×C97C2semidirect
93. C97#1634semidirect
97. #476D97semidirect

Maximal subgroups

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

subgroupindex
#202614502
#202611142
#202610222
#202604832
D4⋊#117158952
#202604622
#476×C972
#163497