Finite Group Databasebeta

← #27186995

Subgroups — #27186995

Order 12928.

380 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 78 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×#15692314C212yesmaximal
2#20266119C212yesmaximal, complemented
3#20266055C212yesmaximal, complemented
4C2×#15692217C211yesmaximal, characteristic
5C2×C4⋊#87938—22no—
6C2×C22⋊#87938C2214yes—
7#15692314—24no—
8(C2×#87938)⋊C4—22no—
9C22×D404C2211yescharacteristic
10C22×C4×D101C411yescharacteristic, complemented
11C22×C4×D101C2211yescharacteristic
12#15692217—22no—
13#15692217—22no—
14C2×#13108051C411yescharacteristic, complemented
15C2×#13108051C2211yescharacteristic
16C2×C4×Dic101C411yescharacteristic, complemented
17C2×C4⋊C4×C101C411yescharacteristic, complemented
18C22×#87938—24no—
19C4⋊#87938—44no—
20C22⋊#87938—24no—
21C22⋊#87938—24no—
22C23×D101C2×C411yescharacteristic
23C23×D101C2311yescharacteristic
24C2×D404—22no—
25C2×D404—22no—
26C2×D404D412yes—
27C2×D404Q812yes—
28C22×Dic101C2×C411yescharacteristic
29C22×Dic101C2×C411yescharacteristic
30C2×C4×D101—21no—
31C2×C4×D101—21no—
32C2×C4×D101—42no—
33C2×C4×D101—42no—
34C2×C4×D101D412yes—
35C2×C4×D101D412yes—
36#13108051—22no—
37#13108051—22no—
38C4×Dic101—21no—
39C4×Dic101—21no—
40C22×C4×C101C2×C411yescharacteristic
41C22×C4×C101C2×C411yescharacteristic
42C22×C4×C101C2×C411yescharacteristic
43C4⋊C4×C101—22no—
44C23×C101C22×C411yescharacteristic
45C22×D101—21no—
46C22×D101—21no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101—41no—
50C22×D101—41no—
51C22×D101C4⋊C412yescomplemented
52C22×D101C2×D412yes—
53C22×D101C2×D411yescharacteristic
54C22×D101C2×Q811yescharacteristic
55C22×D101#4112yes—
56C2×#87938—44no—
57C2×#87938—44no—
58C2×#87938—44no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101C22⋊C412yes—
65C2×C4×C101C4⋊C412yescomplemented
66C2×Dic101—22no—
67C2×Dic101—21no—
68C2×Dic101—41no—
69C2×Dic101C22⋊C411yescharacteristic
70C2×Dic101C22⋊C411yescharacteristic
71D404—24no—
72D404—24no—
73C4×D101—24no—
74C4×D101—42no—
75C4×D101—42no—
76C4×D101—42no—
77C22×C101—21no—
78C22×C101—21no—
79C22×C101C2×C22⋊C411yescharacteristic
80C22×C101C2×C4⋊C411yescharacteristic
81C22×C101#11711yescharacteristic
82C2×D101—22no—
83C2×D101—22no—
84C2×D101—22no—
85C2×D101—41no—
86C2×D101—41no—
87C2×D101—42no—
88C2×D101—42no—
89C2×D101—41no—
90C2×D101—41no—
91C2×D101—42no—
92C2×D101#12112yes—
93C2×D101C22⋊Q812yes—
94#87938—84no—
95C4×C101—22no—
96C4×C101—22no—
97C4×C101—41no—
98Dic101—21no—
99Dic101—41no—
100Dic101#9912yescomplemented
101C2×C101—21no—
102C2×C101—21no—
103C2×C101#38011yescharacteristic
104C2×C101C2×#9911yescharacteristic
105C2×C101#41011yescharacteristic
106D101—22no—
107D101—22no—
108D101—42no—
109#1799—1011nomaximal, Sylow
110C101#179911yescharacteristic, complemented, Sylow
111C2×C4×D4—1011no—
112C2×#99—1012no—
113#380—1012no—
114#342—1012no—
115C22×D4—1011no—
116C23×C4—1011no—
117C23×C4—1011no—
118C4×D4—2022no—
119C4×D4—2022no—
120C2×C4⋊C4—2022no—
121C2×C4⋊C4#8793811yescharacteristic, complemented
122C2×C22⋊C4—1011no—
123C2×C22⋊C4—1011no—
124C2×C22⋊C4—1014no—
125C2×C42—1011no—
126#99—2024no—
127#95—2022no—
128C24—1011no—
129C24—1011no—
130C2×D4—1012no—
131C2×D4—1012no—
132C2×D4—2022no—
133C2×D4—2022no—
134C22×C4—1011no—
135C22×C4—1012no—
136C22×C4—1012no—
137C22×C4—1011no—
138C22×C4—4042no—
139C22×C4—4042no—
140C22×C4—2021no—
141C22×C4—2021no—
142C22×C4—2024no—
143C22×C4C2×#8793811yescharacteristic
144C22×C4C2×#8793811yescharacteristic
145C22×C4C2×#8793811yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—4044no—
148C22⋊C4—2022no—
149C22⋊C4—2024no—
150C22⋊C4—2024no—
151C22⋊C4—2022no—
152C42—2021no—
153C42—2021no—
154C23—1011no—
155C23—1011no—
156C23—1012no—
157C23—1012no—
158C23—1012no—
159C23—4041no—
160C23—4041no—
161C23—4041no—
162C23—4041no—
163C23—2021no—
164C23—2021no—
165C23C22×#8793811yescharacteristic
166D4—2024no—
167D4—2024no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—4042no—
176C2×C4—4042no—
177C2×C4—4042no—
178C2×C4—4044no—
179C2×C4—4044no—
180C2×C4—4044no—
181C2×C4—4041no—
182C2×C4—2024no—
183C2×C4—2022no—
184C2×C4—2021no—
185C2×C4C22⋊#8793812yes—
186C2×C4C4⋊#8793812yescomplemented
187C22—21no—
188C22—21no—
189C22—1012no—
190C22—1012no—
191C22—4042no—
192C22—4042no—
193C22—4041no—
194C22—4041no—
195C22—4042no—
196C22—4041no—
197C22—4041no—
198C22—2022no—
199C22—2022no—
200C22—2022no—
201C22C2×C22⋊#8793811yescharacteristic
202C22C2×C4⋊#8793811yescharacteristic
203C22#1569233411yescharacteristic
204C4—22no—
205C4—22no—
206C4—41no—
207C4—8084no—
208C4—1012no—
209C4—4041no—
210C4—2021no—
211C2—21no—
212C2—21no—
213C2—4042no—
214C2—2022no—
215C2—2022no—
216C2#2026611911yescharacteristic
217C2D101⋊#9911yescharacteristic
218C2#2026615811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C4⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D404C22
cosocleC23×D101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1799
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20266119C2semidirect
3. #20266055C2semidirect
10. C22×C4×D101C4semidirect
14. C2×#13108051C4semidirect
16. C2×C4×Dic101C4semidirect
17. C2×C4⋊C4×C101C4semidirect
51. C22×D101C4⋊C4semidirect
65. C2×C4×C101C4⋊C4semidirect
100. Dic101#99semidirect
110. C101#1799semidirect
121. C2×C4⋊C4#87938semidirect
186. C2×C4C4⋊#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923142
#202661192
#202660552
C2×#156922172
#1799101