Finite Group Databasebeta

← #27140653

Subgroups — #27140653

Order 12416.

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

normal subgroups only

This group has 368 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#20261297C211yesmaximal, characteristic, complemented
2#20260869C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#392×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97—21no—
7C2×#13095496—22no—
8#15690794—22no—
9#15690784—21no—
10#15690784C2212yes—
11#15690780—22no—
12#15690731—22no—
13C22⋊#11715895C2212yescomplemented
14C22×D4×C97—21no—
15C22×D4×C97C2211yescharacteristic, complemented
16C2×C22⋊C4×C97—21no—
17C2×C22⋊C4×C97C2211yescharacteristic
18#95×C97C2211yescharacteristic, complemented
19C23×D97—21no—
20C2×D388—21no—
21C2×D388D412yescomplemented
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97—42no—
25C2×C22⋊D97—42no—
26C22×Dic97—21no—
27#13095496—44no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97D412yescomplemented
31#13095488—22no—
32#13095488—22no—
33C24×C97—21no—
34C24×C97D411yescharacteristic
35C2×D4×C97—22no—
36C2×D4×C97—42no—
37C2×D4×C97—42no—
38C2×D4×C97—42no—
39C2×D4×C97—42no—
40C22×C4×C97—21no—
41C22×C4×C97D411yescharacteristic
42C22×C4×C97C2311yescharacteristic
43C22⋊C4×C97—22no—
44C22⋊C4×C97—42no—
45C23×C97—21no—
46C23×C97—21no—
47C23×C97—22no—
48C23×C97—41no—
49C23×C97—41no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97—41no—
53C23×C97—41no—
54C23×C97—41no—
55C23×C97—41no—
56C23×C97D812yescomplemented
57C23×C97C2×D411yescharacteristic
58C22×D97—22no—
59C22×D97—42no—
60C22×D97—81no—
61C22×D97—81no—
62D4×C97—24no—
63D4×C97—42no—
64D4×C97—48no—
65C2×C4×C97—22no—
66C2×C4×C97—21no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97—41no—
71C2×C4×C97C2×D412yes—
72C22⋊D97—44no—
73C22⋊D97—42no—
74C22⋊D97—42no—
75C2×Dic97—22no—
76C2×Dic97—42no—
77D388—42no—
78D388—81no—
79#11715895—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—22no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97—42no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—42no—
89C22×C97—81no—
90C22×C97—81no—
91C22×C97—81no—
92C22×C97—81no—
93C22×C97—81no—
94C22×C97#12011yescharacteristic
95C22×C97C2×D811yescharacteristic
96C22×C97AGL(1,ℤ/8)11yescharacteristic
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—42no—
100C2×D97—84no—
101C4×C97—22no—
102C4×C97—42no—
103C4×C97—41no—
104Dic97—42no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—42no—
108C2×C97—82no—
109C2×C97#44711yescharacteristic
110C2×C97#45311yescharacteristic
111C2×C97B(4,𝔽2)11yescharacteristic
112D97—82no—
113#1595—971nomaximal, Sylow
114C97#159511yescharacteristic, complemented, Sylow
115#534—971no—
116#447—972no—
117#392D9711yescharacteristic, complemented
118#327—972no—
119#323—971no—
120C22×D4—21no—
121C22×D4—1941no—
122C22×D4C2×D9711yescharacteristic, complemented
123C2×D8—1942no—
124#123—1942no—
125#121—972no—
126#121—1941no—
127#120—1942no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C2×D9711yescharacteristic
130#102—1942no—
131C22⋊C8—972no—
132#95C2×D9711yescharacteristic, complemented
133C24—21no—
134C24—1941no—
135C24C22⋊D9711yescharacteristic
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—972no—
142C2×D4—3882no—
143C2×D4—3882no—
144C2×D4—1941no—
145C2×D4—1942no—
146C2×D4—1942no—
147C22×C4—21no—
148C22×C4—1941no—
149C22×C4C22⋊D9711yescharacteristic
150C22×C4C22×D9711yescharacteristic
151D8—3884no—
152C2×C8—1942no—
153C4⋊C4—972no—
154C4⋊C4—1941no—
155C22⋊C4—22no—
156C22⋊C4—42no—
157C22⋊C4—1942no—
158C22⋊C4—1942no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—7761no—
171C23—7761no—
172C23—3882no—
173C23—1942no—
174C23#1309549612yescomplemented
175C23C2×C22⋊D9711yescharacteristic
176D4—24no—
177D4—42no—
178D4—48no—
179D4—7761no—
180D4—3882no—
181D4—3884no—
182D4—3882no—
183D4—3882no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—3882no—
191C2×C4—1942no—
192C2×C4D4×D9712yes—
193C8—3882no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—7764no—
209C22—3882no—
210C22—3882no—
211C22—3882no—
212C22#1569077811yescharacteristic
213C22C2×#1309549611yescharacteristic
214C22#1569083511yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—3882no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—7762no—
224C2#2026086811yescharacteristic
225C2#2026090211yescharacteristic
226C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#392×C97C2
socleC22×C97#120
hypercenter#392D97
hypocenterC97#1595
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #20261297C2semidirect
2. #20260869C2semidirect
3. #20260468C2semidirect
4. #20260459C2semidirect
5. #392×C97C2semidirect
13. C22⋊#11715895C22semidirect
15. C22×D4×C97C22semidirect
18. #95×C97C22semidirect
21. C2×D388D4semidirect
30. C4⋊Dic97D4semidirect
56. C23×C97D8semidirect
114. C97#1595semidirect
117. #392D97semidirect
122. C22×D4C2×D97semidirect
132. #95C2×D97semidirect
174. C23#13095496semidirect

Maximal subgroups

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

subgroupindex
#202612972
#202608692
#202604682
#202604592
#392×C972
#159597