Finite Group Databasebeta

← #27141213

Subgroups — #27141213

Order 12416.

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

normal subgroups only

This group has 198 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#20261111C211yesmaximal, characteristic, complemented
2#20260930C211yesmaximal, characteristic, complemented
3#20260813C211yesmaximal, characteristic, complemented
4#20260485C211yesmaximal, characteristic, complemented
5#20260479C211yesmaximal, characteristic, complemented
6#20260478C211yesmaximal, characteristic, complemented
7#459×C97C211yesmaximal, characteristic, complemented
8C2×#13095496—21no—
9C2×#13095496—21no—
10C2×#13095496—21no—
11C2×#13095495—21no—
12#15690807C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14C4⋊#11715895C2211yescharacteristic
15#15690730—21no—
16C22⋊#11715895—21no—
17C4×#11715895C2211yescharacteristic
18C2×D8×C97—21no—
19#127×C97C2211yescharacteristic
20C4×D4×C97C2211yescharacteristic
21C4⋊C8×C97C2211yescharacteristic
22#102×C97—21no—
23C2×D388—21no—
24C2×D388—21no—
25C2×D388—21no—
26#13095496—42no—
27#13095496—42no—
28#13095496—42no—
29#13095495—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895D411yescharacteristic
33C2×#11715895D411yescharacteristic
34C2×D4×C97—21no—
35C2×D4×C97—21no—
36C2×D4×C97—21no—
37C2×D4×C97D411yescharacteristic
38C22×C4×C97—21no—
39D8×C97—42no—
40C2×C8×C97—21no—
41C4⋊C4×C97D411yescharacteristic
42C22⋊C4×C97—21no—
43C42×C97C2311yescharacteristic
44C23×C97—21no—
45C23×C97—41no—
46C22×D97—41no—
47D4×C97—21no—
48D4×C97—21no—
49D4×C97—41no—
50D4×C97—41no—
51D4×C97—41no—
52D4×C97—41no—
53D4×C97—81no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97#4111yescharacteristic
60D388—41no—
61D388—41no—
62D388—41no—
63D388—41no—
64D388—81no—
65C8×C97—41no—
66#11715895—21no—
67#11715895—41no—
68#11715895—41no—
69#11715895D812yescomplemented
70C22×C97—41no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97—81no—
74C22×C97—81no—
75C22×C97—81no—
76C22×C97#12111yescharacteristic
77C2×D97—81no—
78C2×D97—81no—
79C2×D97—81no—
80C4×C97—21no—
81C4×C97—41no—
82C4×C97C2×D811yescharacteristic
83C4×C97AGL(1,ℤ/8)11yescharacteristic
84C4×C97AGL(1,ℤ/8)11yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C2×C97—81no—
87C2×C97—161no—
88C2×C97#45911yescharacteristic
89C2×C97#46911yescharacteristic
90C2×C97#47211yescharacteristic
91D97—161no—
92#1687—971nomaximal, Sylow
93C97#168711yescharacteristic, complemented, Sylow
94#493—971no—
95#459—971no—
96#459D9711yescharacteristic, complemented
97#436—971no—
98#334—971no—
99#331—971no—
100#331—971no—
101C2×D8—21no—
102C2×D8—1941no—
103C2×D8—1941no—
104C2×D8—1941no—
105C2×M16—1941no—
106#127—971no—
107#127C2×D9711yescharacteristic
108C4×D4C2×D9711yescharacteristic
109C4⋊C8—971no—
110C4⋊C8—971no—
111C4⋊C8C2×D9711yescharacteristic
112#102—21no—
113#102—1941no—
114C22⋊C8—1941no—
115C4×C8—971no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—1941no—
120C2×D4—1941no—
121C2×D4—1941no—
122C2×D4C22⋊D9711yescharacteristic
123C22×C4—21no—
124D8—42no—
125D8—3882no—
126D8—3882no—
127D8—3882no—
128M16—3882no—
129C2×C8—21no—
130C2×C8—971no—
131C2×C8—971no—
132C2×C8—1941no—
133C2×C8—1941no—
134C4⋊C4C22⋊D9711yescharacteristic
135C22⋊C4—21no—
136C42C22×D9711yescharacteristic
137C23—21no—
138C23—41no—
139C23—3881no—
140D4—21no—
141D4—21no—
142D4—41no—
143D4—41no—
144D4—41no—
145D4—41no—
146D4—81no—
147D4—7761no—
148D4—3881no—
149D4—3881no—
150D4—3881no—
151D4—3881no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4D4×D9711yescharacteristic
156C2×C4C2×C22⋊D9711yescharacteristic
157C2×C4#1309551011yescharacteristic
158C8—41no—
159C8—972no—
160C8—3881no—
161C8—3881no—
162C8—1941no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—7761no—
170C22—7761no—
171C22—7761no—
172C22#1569078211yescharacteristic
173C4—21no—
174C4—41no—
175C4D8×D9711yescharacteristic
176C4#1569083411yescharacteristic
177C4#1569083511yescharacteristic
178C4#1569083611yescharacteristic
179C2—81no—
180C2—161no—
181C2—15521no—
182C2#2026092811yescharacteristic
183C2#2026098611yescharacteristic
184C2#2026100311yescharacteristic
subgroupG/H
centerC22#15690782
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#459×C97C2
socleC22×C97#121
hypercenter#459D97
hypocenterC97#1687
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20261111C2semidirect
2. #20260930C2semidirect
3. #20260813C2semidirect
4. #20260485C2semidirect
5. #20260479C2semidirect
6. #20260478C2semidirect
7. #459×C97C2semidirect
69. #11715895D8semidirect
93. C97#1687semidirect
96. #459D97semidirect

Maximal subgroups

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

subgroupindex
#202611112
#202609302
#202608132
#202604852
#202604792
#202604782
#459×C972
#168797