Finite Group Databasebeta

← #27140813

Subgroups — #27140813

Order 12416.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic194C211yesmaximal, characteristic, complemented
2#20261107C211yesmaximal, characteristic
3#20260932C211yesmaximal, characteristic
4#20260483C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260465C211yesmaximal, characteristic, complemented
7#460×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9C2×#13095499—22no—
10#15690810—21no—
11#15690810C2211yescharacteristic
12#15690809—22no—
13#15690809—21no—
14Q8×Dic97—21no—
15C4×Dic194—21no—
16C4×Dic194C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18#15690731—21no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21#127×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—22no—
26C2×Dic194D411yescharacteristic
27C2×Dic194D412yes—
28#13095499—44no—
29#13095499—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C4⋊Dic97—22no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36#13095490—22no—
37#13095490—21no—
38C4×Dic97—22no—
39C4×Dic97—21no—
40C2×Q8×C97D411yescharacteristic
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44SD16×C97—42no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—41no—
51Q8×C97—21no—
52Q8×C97SD1612yescomplemented
53D4×C97—42no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×Dic97—21no—
64C2×Dic97—22no—
65C2×Dic97—21no—
66Dic194—21no—
67Dic194—42no—
68Dic194—82no—
69Dic194SD1612yescomplemented
70C8×C97—41no—
71#11715895—22no—
72#11715895—41no—
73C22×C97—81no—
74C22×C97—81no—
75C22×C97—81no—
76C22×C97#12011yescharacteristic
77C4×C97—22no—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97C2×SD1611yescharacteristic
81C4×C97C2×SD1611yescharacteristic
82C4×C97#13711yescharacteristic
83C4×C97#13711yescharacteristic
84Dic97—22no—
85Dic97—42no—
86Dic97—41no—
87C2×C97—161no—
88C2×C97#44811yescharacteristic
89C2×C97#44811yescharacteristic
90C2×C97#45311yescharacteristic
91#1623—971nomaximal, Sylow
92C97#162311yescharacteristic, complemented, Sylow
93Q82—971no—
94#492—971no—
95#460—971no—
96#460D9711yescharacteristic
97#333—971no—
98Q8⋊C8—971no—
99Q8⋊C8—971no—
100C2×SD16—21no—
101C2×SD16—1941no—
102C2×SD16—1942no—
103C4⋊Q8—971no—
104C4⋊Q8—1941no—
105C4⋊Q8—1942no—
106C4⋊Q8—1941no—
107#127C2×D9711yescharacteristic
108C4×Q8—971no—
109C4×Q8—1941no—
110C4×Q8—1941no—
111C4×Q8C2×D9711yescharacteristic
112C4⋊C8—971no—
113C4⋊C8C2×D9711yescharacteristic
114#102—21no—
115#102—1941no—
116C4×C8—971no—
117C2×Q8—971no—
118C2×Q8—972no—
119C2×Q8—1942no—
120C2×Q8C22⋊D9711yescharacteristic
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124SD16—42no—
125SD16—3882no—
126SD16—3884no—
127C2×C8—21no—
128C2×C8—1941no—
129C2×C8—1941no—
130C4⋊C4—21no—
131C4⋊C4—971no—
132C4⋊C4—1942no—
133C4⋊C4—1942no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C4⋊C4C22⋊D9711yescharacteristic
138C42—21no—
139C42—1942no—
140C42—1941no—
141C42C22×D9711yescharacteristic
142C23—41no—
143Q8—21no—
144Q8—7762no—
145Q8—972no—
146Q8—3882no—
147Q8—1941no—
148Q8#1309549912yescomplemented
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—1941no—
157C2×C4—1942no—
158C2×C4—1941no—
159C2×C4D4×D9711yescharacteristic
160C2×C4D4×D9711yescharacteristic
161C2×C4C2×C22⋊D9711yescharacteristic
162C8—41no—
163C8—3881no—
164C8—1942no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#1569077811yescharacteristic
169C4—22no—
170C4—21no—
171C4—41no—
172C4—3881no—
173C4—3882no—
174C4—1942no—
175C4SD16×D9711yescharacteristic
176C4C2×#1309549911yescharacteristic
177C4#1569083911yescharacteristic
178C4#1569084011yescharacteristic
179C2—161no—
180C2#2026087211yescharacteristic
181C2#2026087311yescharacteristic
182C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#460×C97C2
socleC22×C97#120
prime radical#127×C97C22
hypercenter#460D97
hypocenterC97#1623
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×Dic194C2semidirect
4. #20260483C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260465C2semidirect
52. Q8×C97SD16semidirect
69. Dic194SD16semidirect
92. C97#1623semidirect
148. Q8#13095499semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1942
#202611072
#202609322
#202604832
Q8⋊#117158952
#202604652
#460×C972
#162397