Finite Group Databasebeta

← #27186217

Subgroups — #27186217

Order 12928.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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×#15692315C211yesmaximal, characteristic
2C2×#15692313C211yesmaximal, characteristic
3#20265054C211yesmaximal, characteristic
4C2×#13108083—21no—
5C2×C22⋊#87938—21no—
6#15692315—21no—
7#15692315C412yescomplemented
8#15692313—21no—
9#15692313C412yescomplemented
10C22×C22⋊D101C2211yescharacteristic
11C23×Dic101C411yescharacteristic, complemented
12C2×#13108051—21no—
13#15692162—21no—
14C2×C22⋊C4×C101C411yescharacteristic, complemented
15C22×#87938—41no—
16#13108083—22no—
17#13108083—41no—
18C2×#11717581—41no—
19C22⋊#87938—22no—
20C22⋊#87938—41no—
21C23×D101D411yescharacteristic
22C2×C22⋊D101—41no—
23C2×C22⋊D101—41no—
24C2×C22⋊D101C2×C411yescharacteristic
25C2×C22⋊D101C2×C411yescharacteristic
26C2×C22⋊D101D411yescharacteristic
27C2×C22⋊D101Q811yescharacteristic
28C22×Dic101—21no—
29C22×Dic101—42no—
30C22×Dic101—41no—
31C22×Dic101D411yescharacteristic
32#13108051—42no—
33C24×C101C2×C411yescharacteristic
34C22×C4×C101—21no—
35C22⋊C4×C101—41no—
36C23×C101—21no—
37C23×C101—21no—
38C23×C101—41no—
39C23×C101—41no—
40C23×C101C4211yescharacteristic
41C23×C101C22⋊C411yescharacteristic
42C23×C101C4⋊C411yescharacteristic
43C22×D101—21no—
44C22×D101—21no—
45C22×D101—41no—
46C22×D101—41no—
47C22×D101C22⋊C411yescharacteristic
48C22×D101C4⋊C411yescharacteristic
49C2×#87938—42no—
50C2×#87938—81no—
51C2×#87938—81no—
52C2×C4×C101—41no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C22⋊D101—41no—
56C22⋊D101—41no—
57C22⋊D101—41no—
58C22⋊D101—41no—
59C2×Dic101—21no—
60C2×Dic101—21no—
61C2×Dic101—22no—
62C2×Dic101—41no—
63C2×Dic101—42no—
64C2×Dic101—41no—
65C2×Dic101—42no—
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C4⋊C411yescharacteristic
68#11717581—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101—21no—
72C22×C101—21no—
73C22×C101—41no—
74C22×C101—41no—
75C22×C101—42no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101#9511yescharacteristic
79C22×C101#9911yescharacteristic
80C22×C101#9911yescharacteristic
81C2×D101—21no—
82C2×D101—21no—
83C2×D101—41no—
84C2×D101—41no—
85C2×D101—41no—
86C2×D101—41no—
87C2×D101—81no—
88#87938—82no—
89C4×C101—81no—
90Dic101—21no—
91Dic101—21no—
92Dic101—42no—
93C2×C101—21no—
94C2×C101—21no—
95C2×C101—41no—
96C2×C101—41no—
97C2×C101#34211yescharacteristic
98C2×C101AΣL(1,𝔽16)11yescharacteristic
99C2×C101#35211yescharacteristic
100D101—41no—
101D101—41no—
102#1390—1011nomaximal, Sylow
103C101#139011yescharacteristic, complemented, Sylow
104C2×#100—1011no—
105C2×#99—1011no—
106#386—1011no—
107C22×D4—1011no—
108C23×C4—1011no—
109C2×M16—2021no—
110C2×C22⋊C4—2021no—
111C2×C22⋊C4—2021no—
112C2×C22⋊C4#8793811yescharacteristic, complemented
113#100—1012no—
114#100—2021no—
115#99—1012no—
116#99—2021no—
117#95—2021no—
118C24—1011no—
119C24C2×#8793811yescharacteristic
120C2×D4—1011no—
121C2×D4—1011no—
122C2×D4—1011no—
123C2×D4—1011no—
124C2×D4—4041no—
125C2×D4—4041no—
126C22×C4—21no—
127C22×C4—1011no—
128C22×C4—4041no—
129C22×C4—4041no—
130C22×C4—4042no—
131C22×C4—2021no—
132M16—4041no—
133M16—2022no—
134C2×C8—4041no—
135C22⋊C4—41no—
136C22⋊C4—4042no—
137C22⋊C4—4041no—
138C22⋊C4—2022no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—1011no—
144C23—1011no—
145C23—4041no—
146C23—4041no—
147C23—2021no—
148C23—2021no—
149C23C4×#8793811yescharacteristic
150C23C22⋊#8793811yescharacteristic
151C23C4⋊#8793811yescharacteristic
152D4—4041no—
153D4—4041no—
154D4—4041no—
155D4—4041no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—8081no—
160C2×C4—8081no—
161C2×C4—1011no—
162C2×C4—1011no—
163C2×C4—4042no—
164C2×C4—4041no—
165C2×C4—4042no—
166C2×C4—4041no—
167C2×C4—4042no—
168C2×C4—2022no—
169C2×C4—2021no—
170C2×C4—2021no—
171C8—4042no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—8081no—
182C22—4041no—
183C22—4041no—
184C22—4041no—
185C22—4041no—
186C22—2021no—
187C22—2021no—
188C22(C2×#87938)⋊C411yescharacteristic
189C22#1569231311yescharacteristic
190C22#1569231411yescharacteristic
191C4—81no—
192C4—8082no—
193C4—4042no—
194C4—2021no—
195C4—2021no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—4041no—
201C2—4041no—
202C2#2026605511yescharacteristic
203C2#2026607811yescharacteristic
204C2#2026608011yescharacteristic
subgroupG/H
centerC22#15692314
commutatorC23×C101C42
FrattiniC24C2×#87938
FittingC2×C22⋊C4×C101C4
socleC22×C101#99
prime radical, cosocleC22×C22⋊D101C22
hypercenterC2×C22⋊C4#87938
hypocenterC101#1390
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #15692315C4semidirect
9. #15692313C4semidirect
11. C23×Dic101C4semidirect
14. C2×C22⋊C4×C101C4semidirect
103. C101#1390semidirect
112. C2×C22⋊C4#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923152
C2×#156923132
#202650542
#1390101