Finite Group Databasebeta

← #27172258

Subgroups — #27172258

Order 12928.

638 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 73 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#20265739C211yesmaximal, characteristic, complemented
2C2×#15692224C212yesmaximal, complemented
3C2×#15692169C212yesmaximal, complemented
4#20264944C211yesmaximal, characteristic, complemented
5#379×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101—22no—
7C22×C22⋊D101C2212yes—
8#15692239—22no—
9#15692229—22no—
10#15692225—22no—
11#15692224—24no—
12#15692224—24no—
13C2×C22⋊Dic101—22no—
14C2×C22⋊Dic101C2212yes—
15#15692169—24no—
16C25×C101C2211yescharacteristic
17C2×C22⋊C4×C101—22no—
18C2×C22⋊C4×C101C2211yescharacteristic
19C2×C22⋊C4×C101C2211yescharacteristic
20C23×D101D412yescomplemented
21C2×D404—22no—
22C2×C22⋊D101—28no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—22no—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—22no—
27C2×C22⋊D101—22no—
28C2×C22⋊D101—22no—
29C2×C22⋊D101—48no—
30C22×Dic101—22no—
31C22×Dic101D412yescomplemented
32C4⋊Dic101—22no—
33C22⋊Dic101—28no—
34C22⋊Dic101—22no—
35C22⋊Dic101D414yescomplemented
36#13108051—24no—
37C24×C101—22no—
38C24×C101—21no—
39C24×C101—22no—
40C24×C101—21no—
41C24×C101—44no—
42C24×C101D412yescomplemented
43C24×C101C2311yescharacteristic
44C22×C4×C101—21no—
45C22×C4×C101D412yescomplemented
46C22⋊C4×C101—22no—
47C22⋊C4×C101—22no—
48C22⋊C4×C101—44no—
49C23×C101—24no—
50C23×C101—28no—
51C23×C101—24no—
52C23×C101—22no—
53C23×C101—44no—
54C23×C101—42no—
55C23×C101—44no—
56C23×C101—42no—
57C23×C101—48no—
58C23×C101—44no—
59C23×C101—41no—
60C23×C101—42no—
61C23×C101—41no—
62C23×C101C2×D412yes—
63C23×C101C2×D412yes—
64C23×C101C2×D411yescharacteristic
65C23×C101C2×D411yescharacteristic
66C23×C101#4111yescharacteristic
67C22×D101—22no—
68C22×D101—22no—
69C22×D101—22no—
70C22×D101—44no—
71C2×C4×C101—22no—
72C2×C4×C101—42no—
73C2×C4×C101—42no—
74C2×C4×C101—41no—
75C22⋊D101—416no—
76C22⋊D101—42no—
77C22⋊D101—42no—
78C22⋊D101—42no—
79C22⋊D101—42no—
80C2×Dic101—22no—
81C2×Dic101—22no—
82C2×Dic101—24no—
83C2×Dic101—22no—
84C2×Dic101—44no—
85D404—82no—
86C22×C101—24no—
87C22×C101—22no—
88C22×C101—28no—
89C22×C101—24no—
90C22×C101—42no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—44no—
94C22×C101—44no—
95C22×C101—48no—
96C22×C101—42no—
97C22×C101—42no—
98C22×C101—44no—
99C22×C101#12011yescharacteristic
100C22×C101#12012yes—
101C22×C101#12112yes—
102C22×C101#12111yescharacteristic
103C22×C101#12711yescharacteristic
104C2×D101—44no—
105C2×D101—42no—
106C2×D101—44no—
107C2×D101—82no—
108C4×C101—42no—
109C4×C101—81no—
110Dic101—44no—
111Dic101—42no—
112C2×C101—22no—
113C2×C101—21no—
114C2×C101—22no—
115C2×C101—21no—
116C2×C101—44no—
117C2×C101#39211yescharacteristic
118C2×C101B(4,𝔽2)12yes—
119D101—82no—
120#2021—1011nomaximal, Sylow
121C101#202111yescharacteristic, complemented, Sylow
122#534—1011no—
123C2×#120—1012no—
124C2×#99—1012no—
125#379D10111yescharacteristic, complemented
126#342—1011no—
127C25C2×D10111yescharacteristic
128C22×D4—1012no—
129C22×D4—2022no—
130#123—2022no—
131#121—2022no—
132#120—2022no—
133#120—2024no—
134#120—2024no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—1012no—
137C2×C22⋊C4—2022no—
138C2×C22⋊C4C2×D10111yescharacteristic
139C2×C22⋊C4C2×D10111yescharacteristic
140#99—2024no—
141C24—22no—
142C24—21no—
143C24—22no—
144C24—21no—
145C24—44no—
146C24—1012no—
147C24D40412yescomplemented
148C24C22×D10111yescharacteristic
149C2×D4—4048no—
150C2×D4—2022no—
151C2×D4—2028no—
152C2×D4—2022no—
153C2×D4—2022no—
154C2×D4—2022no—
155C2×D4—2022no—
156C2×D4—2022no—
157C2×D4—2022no—
158C22×C4—21no—
159C22×C4—1012no—
160C22×C4—2022no—
161C22×C4C22⋊D10112yescomplemented
162C4⋊C4—2022no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—44no—
166C22⋊C4—1014no—
167C22⋊C4—2024no—
168C22⋊C4—2028no—
169C22⋊C4—2022no—
170C23—24no—
171C23—28no—
172C23—24no—
173C23—22no—
174C23—44no—
175C23—42no—
176C23—42no—
177C23—44no—
178C23—48no—
179C23—44no—
180C23—41no—
181C23—42no—
182C23—41no—
183C23—4044no—
184C23—2022no—
185C23—2022no—
186C23—2022no—
187C23D4×D10112yes—
188C23D4×D10112yes—
189C23C2×C22⋊D10111yescharacteristic
190C23C2×D40411yescharacteristic
191C23#1310807411yescharacteristic
192D4—8082no—
193D4—40416no—
194D4—4042no—
195D4—4042no—
196D4—4042no—
197D4—4042no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—4044no—
203C2×C4—2022no—
204C2×C4—2024no—
205C2×C4—2022no—
206C2×C4—2022no—
207C22—22no—
208C22—24no—
209C22—28no—
210C22—24no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—44no—
215C22—44no—
216C22—42no—
217C22—48no—
218C22—42no—
219C22—44no—
220C22—8082no—
221C22—4044no—
222C22—4042no—
223C22—4044no—
224C22#1569222311yescharacteristic
225C22#1569222512yes—
226C22#1569222812yes—
227C22#1569222911yescharacteristic
228C22#1569225111yescharacteristic
229C4—42no—
230C4—81no—
231C4—4044no—
232C4—4042no—
233C2—21no—
234C2—22no—
235C2—21no—
236C2—22no—
237C2—44no—
238C2—8082no—
239C2#2026508611yescharacteristic
240C2#2026536312yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting#379×C101C2
socleC22×C101#120
hypercenter#379D101
hypocenterC101#2021
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
1. #20265739C2semidirect
2. C2×#15692224C2semidirect
3. C2×#15692169C2semidirect
4. #20264944C2semidirect
5. #379×C101C2semidirect
20. C23×D101D4semidirect
31. C22×Dic101D4semidirect
35. C22⋊Dic101D4semidirect
42. C24×C101D4semidirect
45. C22×C4×C101D4semidirect
121. C101#2021semidirect
125. #379D101semidirect
147. C24D404semidirect
161. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657392
C2×#156922242
C2×#156921692
#202649442
#379×C1012
#2021101