Finite Group Databasebeta

← #27140303

Subgroups — #27140303

Order 12416.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#15690784C211yesmaximal, characteristic
2C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
3#20260468C214yesmaximal, complemented
4#380×C97C211yesmaximal, characteristic, complemented
5C22×D388C411yescharacteristic, complemented
6C22×C22⋊D97—21no—
7C22×#11715895—21no—
8#15690784—21no—
9#15690784—21no—
10#15690784C412yescomplemented
11#15690784C2212yes—
12C2×C4⋊Dic97C411yescharacteristic, complemented
13C2×#13095488—21no—
14C22⋊#11715895—21no—
15C22⋊#11715895C2212yescomplemented
16C23×C4×C97C2211yescharacteristic
17C2×C4⋊C4×C97—21no—
18C2×C22⋊C4×C97—21no—
19#95×C97C2212yescomplemented
20C23×D97—21no—
21C2×D388—21no—
22C2×D388—42no—
23C2×D388C2×C412yes—
24C2×C22⋊D97—22no—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—42no—
27C2×C22⋊D97—42no—
28C22×Dic97—21no—
29C2×#11715895—22no—
30C2×#11715895—42no—
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C412yes—
33#13095488—22no—
34#13095488—22no—
35C24×C97D411yescharacteristic
36C22×C4×C97—22no—
37C22×C4×C97—42no—
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97D411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—42no—
45C22⋊C4×C97—42no—
46C23×C97—41no—
47C23×C97—41no—
48C23×C97C22⋊C411yescharacteristic
49C23×C97C22⋊C411yescharacteristic
50C23×C97D812yes—
51C23×C97SD1612yes—
52C23×C97C2×D411yescharacteristic
53C22×D97—22no—
54C22×D97—42no—
55C22×D97—42no—
56C22×D97—42no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—42no—
64C2×C4×C97—42no—
65C2×C4×C97—42no—
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22⋊C411yescharacteristic
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C22⋊D97—44no—
71C22⋊D97—42no—
72C22⋊D97—42no—
73C2×Dic97—22no—
74C2×Dic97—42no—
75D388—42no—
76D388—42no—
77#11715895—42no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97#10212yes—
87C22×C97#10212yes—
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97C2×D811yescharacteristic
90C22×C97C2×SD1611yescharacteristic
91C2×D97—42no—
92C2×D97—42no—
93C2×D97—42no—
94C2×D97—84no—
95C4×C97—21no—
96C4×C97—21no—
97C4×C97—21no—
98C4×C97—82no—
99C4×C97#9912yes—
100Dic97—42no—
101C2×C97—21no—
102C2×C97—22no—
103C2×C97—22no—
104C2×C97—21no—
105C2×C97C2×#9911yescharacteristic
106C2×C97C2×#10211yescharacteristic
107C2×C97#42111yescharacteristic
108D97—82no—
109#1519—971nomaximal, Sylow
110C97#151911yescharacteristic, complemented, Sylow
111C2×#121—971no—
112C2×C22⋊C8—971no—
113#380D9711yescharacteristic, complemented
114#327—974no—
115C22×D4—971no—
116C22×D4—1941no—
117C23×C4C2×D9711yescharacteristic
118C22×C8—1941no—
119#121—972no—
120#121—972no—
121#121—1941no—
122#121—1941no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—971no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—1941no—
127C22⋊C8—972no—
128C22⋊C8—1941no—
129#95C2×D9712yescomplemented
130C24—1941no—
131C24C22⋊D9711yescharacteristic
132C2×D4—972no—
133C2×D4—3882no—
134C2×D4—3882no—
135C2×D4—3882no—
136C2×D4—1942no—
137C2×D4—1942no—
138C2×D4—1941no—
139C22×C4—22no—
140C22×C4—42no—
141C22×C4—1941no—
142C22×C4C4×D9711yescharacteristic
143C22×C4C4×D9711yescharacteristic
144C22×C4D38811yescharacteristic
145C22×C4D38811yescharacteristic
146C22×C4C22⋊D9711yescharacteristic
147C22×C4C22×D9711yescharacteristic
148C2×C8—3882no—
149C2×C8—1942no—
150C4⋊C4—42no—
151C4⋊C4—972no—
152C4⋊C4—1941no—
153C22⋊C4—42no—
154C22⋊C4—1942no—
155C22⋊C4—1942no—
156C23—41no—
157C23—41no—
158C23—3882no—
159C23—3882no—
160C23—3882no—
161C23—1942no—
162C23#1309548811yescharacteristic
163C23#1309548811yescharacteristic
164C23#1309549612yes—
165C23#1309549812yes—
166C23C2×C22⋊D9711yescharacteristic
167D4—3882no—
168D4—3882no—
169D4—3882no—
170D4—3884no—
171D4—3882no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—3882no—
182C2×C4—1942no—
183C2×C4#1309548811yescharacteristic
184C2×C4#1309548811yescharacteristic
185C2×C4C2×C4×D9711yescharacteristic
186C2×C4C2×D38811yescharacteristic
187C8—3882no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—7764no—
197C22—3882no—
198C22—3882no—
199C22—3882no—
200C22#1569073012yes—
201C22#1569073012yes—
202C22C2×#1309548811yescharacteristic
203C22C2×#1309549611yescharacteristic
204C22C2×#1309549811yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—82no—
209C4—3882no—
210C4#1569072412yes—
211C2—21no—
212C2—22no—
213C2—21no—
214C2—22no—
215C2—7762no—
216C2C2×#1569072411yescharacteristic
217C2C2×#1569073011yescharacteristic
218C2#2026075311yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#380×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690784C2
cosocleC22×C4×C97C23
hypercenter#380D97
hypocenterC97#1519
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11715895C2semidirect
3. #20260468C2semidirect
4. #380×C97C2semidirect
5. C22×D388C4semidirect
10. #15690784C4semidirect
12. C2×C4⋊Dic97C4semidirect
15. C22⋊#11715895C22semidirect
19. #95×C97C22semidirect
110. C97#1519semidirect
113. #380D97semidirect
129. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907842
C2×C22⋊#117158952
#202604682
#380×C972
#151997