Finite Group Databasebeta

← #27140807

Subgroups — #27140807

Order 12416.

236 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261445C211yesmaximal, characteristic, complemented
2#20261107C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic
4#20260483C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#460×C97C211yesmaximal, characteristic, complemented
8C2×#13095499—22no—
9C2×#13095496—21no—
10#15690810C2211yescharacteristic
11#15690802—21no—
12#15690788—22no—
13#15690786—21no—
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16C4⋊C4×D97—21no—
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×Dic194D412yes—
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28C2×C4×D97—22no—
29#13095499—44no—
30#13095496—42no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36#13095490—22no—
37#13095490—22no—
38#13095488—22no—
39#13095488—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—
51C22×D97—21no—
52Q8×C97—21no—
53Q8×C97—21no—
54D4×C97—42no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×Dic97—21no—
65C2×Dic97—22no—
66D388—21no—
67D388SD1612yescomplemented
68C4×D97—41no—
69C4×D97—41no—
70C4×D97—82no—
71Dic194—42no—
72C8×C97—41no—
73#11715895—22no—
74#11715895—41no—
75C22×C97—81no—
76C22×C97—81no—
77C22×C97—81no—
78C22×C97#12011yescharacteristic
79C2×D97—22no—
80C2×D97—41no—
81C2×D97—41no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97—41no—
85C4×C97C2×SD1611yescharacteristic
86C4×C97#13511yescharacteristic
87C4×C97AGL(1,ℤ/8)11yescharacteristic
88C4×C97#13711yescharacteristic
89Dic97—41no—
90Dic97—42no—
91C2×C97—161no—
92C2×C97#44811yescharacteristic
93C2×C97#44911yescharacteristic
94C2×C97#45311yescharacteristic
95D97—42no—
96#1622—971nomaximal, Sylow
97C97#162211yescharacteristic, complemented, Sylow
98#554—971no—
99#492—971no—
100#460D9711yescharacteristic, complemented
101#459—971no—
102#333—971no—
103Q8⋊C8—971no—
104D4⋊C8—971no—
105C2×SD16—21no—
106C2×SD16—1942no—
107C2×D8—1941no—
108C4⋊Q8—971no—
109#127C2×D9711yescharacteristic
110#125—1941no—
111C22⋊Q8—1942no—
112C22⋊Q8—1941no—
113C4×Q8C2×D9711yescharacteristic
114C4×D4—971no—
115C4×D4—1941no—
116C2×C4⋊C4—1941no—
117C4⋊C8—971no—
118C4⋊C8C2×D9711yescharacteristic
119#102—21no—
120#102—1941no—
121C4×C8—971no—
122C2×Q8—972no—
123C2×Q8C22⋊D9711yescharacteristic
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—971no—
128C22×C4—1942no—
129C22×C4—1941no—
130SD16—42no—
131SD16—3884no—
132D8—3882no—
133C2×C8—21no—
134C2×C8—1941no—
135C2×C8—1941no—
136C4⋊C4—21no—
137C4⋊C4—971no—
138C4⋊C4—1942no—
139C4⋊C4—1942no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C4⋊C4C22⋊D9711yescharacteristic
143C22⋊C4—1942no—
144C22⋊C4—1941no—
145C42—21no—
146C42C22×D9711yescharacteristic
147C23—41no—
148C23—1941no—
149Q8—21no—
150Q8—21no—
151Q8—3882no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—972no—
158D4—1941no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—7762no—
162C2×C4—3881no—
163C2×C4—3881no—
164C2×C4—1941no—
165C2×C4—1942no—
166C2×C4D4×D9711yescharacteristic
167C2×C4D4×D9711yescharacteristic
168C2×C4C2×C22⋊D9711yescharacteristic
169C8—41no—
170C8—3881no—
171C8—1942no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—3881no—
176C22—3881no—
177C22—1942no—
178C22#1569077811yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—3881no—
183C4—3882no—
184C4SD16×D9711yescharacteristic
185C4#1569083011yescharacteristic
186C4#1569083611yescharacteristic
187C4#1569083911yescharacteristic
188C2—161no—
189C2—3882no—
190C2#2026087211yescharacteristic
191C2#2026087811yescharacteristic
192C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#460×C97C2
socleC22×C97#120
prime radical#20260927C2
hypercenter#460D97
hypocenterC97#1622
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. #20261445C2semidirect
2. #20261107C2semidirect
4. #20260483C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #460×C97C2semidirect
67. D388SD16semidirect
97. C97#1622semidirect
100. #460D97semidirect

Maximal subgroups

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

subgroupindex
#202614452
#202611072
#202609272
#202604832
Q8⋊#117158952
#202604622
#460×C972
#162297