Finite Group Databasebeta

← #27171585

Subgroups — #27171585

Order 12928.

598 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15692224C211yesmaximal, characteristic
2C2×#15692169C212yesmaximal, complemented
3#20265033C211yesmaximal, characteristic, complemented
4#20264943C212yesmaximal, complemented
5C22×C22⋊C4×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C412yescomplemented
7C22×C22⋊D101C2211yescharacteristic
8#15692224—22no—
9#15692224—21no—
10#15692224—21no—
11C2×C22⋊Dic101—22no—
12C2×C22⋊Dic101C412yescomplemented
13C2×C22⋊Dic101C2212yes—
14C2×C22⋊Dic101C2211yescharacteristic
15#15692169—24no—
16C25×C101C2211yescharacteristic
17C23×C4×C101—21no—
18C2×C22⋊C4×C101—21no—
19C2×C22⋊C4×C101—24no—
20C2×C22⋊C4×C101C2212yes—
21C23×D101C2×C411yescharacteristic, complemented
22C2×C22⋊D101—22no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—22no—
25C2×C22⋊D101—21no—
26C2×C22⋊D101—21no—
27C2×C22⋊D101—22no—
28C2×C22⋊D101—44no—
29C22×Dic101—22no—
30C22×Dic101C2×C411yescharacteristic, complemented
31C22×Dic101C2×C412yescomplemented
32C22⋊Dic101—22no—
33C22⋊Dic101—22no—
34C22⋊Dic101—21no—
35C22⋊Dic101—21no—
36C22⋊Dic101—22no—
37C22⋊Dic101—22no—
38C22⋊Dic101—44no—
39C24×C101—22no—
40C24×C101—22no—
41C24×C101—44no—
42C24×C101C2×C412yes—
43C24×C101D412yes—
44C24×C101D411yescharacteristic
45C24×C101D411yescharacteristic
46C24×C101C2311yescharacteristic
47C22×C4×C101—24no—
48C22×C4×C101—22no—
49C22×C4×C101—44no—
50C22⋊C4×C101—24no—
51C22⋊C4×C101—22no—
52C22⋊C4×C101D414yescomplemented
53C23×C101—24no—
54C23×C101—21no—
55C23×C101—21no—
56C23×C101—24no—
57C23×C101—24no—
58C23×C101—24no—
59C23×C101—21no—
60C23×C101—24no—
61C23×C101—21no—
62C23×C101—42no—
63C23×C101—42no—
64C23×C101—42no—
65C23×C101—42no—
66C23×C101—48no—
67C23×C101—48no—
68C23×C101C22⋊C412yes—
69C23×C101C22⋊C412yes—
70C23×C101C22×C411yescharacteristic
71C23×C101C2×D411yescharacteristic
72C23×C101C2×D412yes—
73C23×C101C2×D411yescharacteristic
74C23×C101#4112yes—
75C22×D101—21no—
76C22×D101—22no—
77C22×D101—42no—
78C2×C4×C101—24no—
79C2×C4×C101—48no—
80C2×C4×C101—44no—
81C22⋊D101—44no—
82C22⋊D101—44no—
83C22⋊D101—41no—
84C22⋊D101—42no—
85C22⋊D101—41no—
86C2×Dic101—21no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—42no—
90C2×Dic101—44no—
91C2×Dic101—42no—
92C22×C101—24no—
93C22×C101—24no—
94C22×C101—22no—
95C22×C101—22no—
96C22×C101—22no—
97C22×C101—22no—
98C22×C101—22no—
99C22×C101—22no—
100C22×C101—22no—
101C22×C101—22no—
102C22×C101—42no—
103C22×C101—42no—
104C22×C101—48no—
105C22×C101—48no—
106C22×C101—42no—
107C22×C101—42no—
108C22×C101#9914yescomplemented
109C22×C101C2×C22⋊C411yescharacteristic
110C22×C101C4×D412yes—
111C22×C101#12011yescharacteristic
112C22×C101#12112yes—
113C22×C101#12311yescharacteristic
114C2×D101—41no—
115C2×D101—42no—
116C2×D101—82no—
117C4×C101—44no—
118Dic101—41no—
119Dic101—42no—
120Dic101—82no—
121C2×C101—22no—
122C2×C101—21no—
123C2×C101—22no—
124C2×C101—24no—
125C2×C101—21no—
126C2×C101—42no—
127C2×C101#38611yescharacteristic
128C2×C101C2×#9912yes—
129D101—81no—
130#1898—1011nomaximal, Sylow
131C101#189811yescharacteristic, complemented, Sylow
132C2×#120—1011no—
133C22×C22⋊C4D10111yescharacteristic, complemented
134C2×#99—1012no—
135#379—1011no—
136#342—1012no—
137C25C2×D10111yescharacteristic
138C22×D4—1012no—
139C22×D4—1011no—
140C23×C4—21no—
141#120—2022no—
142#120—2021no—
143#120—2021no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—1012no—
147C2×C22⋊C4—1011no—
148C2×C22⋊C4—1012no—
149C2×C22⋊C4—2022no—
150C2×C22⋊C4C2×D10112yes—
151#99—2024no—
152C24—22no—
153C24—22no—
154C24—44no—
155C24—1011no—
156C24C4×D10112yes—
157C24D40412yes—
158C24C22⋊D10111yescharacteristic
159C24C22⋊D10111yescharacteristic
160C24C22×D10111yescharacteristic
161C2×D4—4044no—
162C2×D4—2022no—
163C2×D4—2022no—
164C2×D4—2022no—
165C2×D4—2021no—
166C2×D4—2021no—
167C2×D4—2022no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—44no—
171C22×C4—1012no—
172C22×C4—1011no—
173C22×C4—2022no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—4044no—
177C22⋊C4—2022no—
178C22⋊C4—2022no—
179C22⋊C4—2022no—
180C22⋊C4—2022no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C22⋊C4C22⋊D10114yescomplemented
184C23—24no—
185C23—21no—
186C23—21no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—21no—
191C23—24no—
192C23—21no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—48no—
198C23—48no—
199C23—4042no—
200C23—2022no—
201C23—2021no—
202C23#1310805112yes—
203C23#1310805112yes—
204C23C2×C4×D10111yescharacteristic
205C23C2×C22⋊D10111yescharacteristic
206C23C2×C22⋊D10112yes—
207C23C2×D40411yescharacteristic
208C23#1310807412yes—
209D4—4044no—
210D4—4044no—
211D4—4041no—
212D4—4042no—
213D4—4041no—
214C2×C4—24no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—4044no—
218C2×C4—4042no—
219C2×C4—4042no—
220C2×C4—2021no—
221C2×C4—2022no—
222C2×C4—2022no—
223C22—24no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—48no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—8082no—
240C22—4041no—
241C22—4042no—
242C22#1569216914yescomplemented
243C22C2×#1310805111yescharacteristic
244C22C4×C22⋊D10112yes—
245C22#1569222411yescharacteristic
246C22#1569222912yes—
247C22#1569223811yescharacteristic
248C4—44no—
249C4—8082no—
250C4—4041no—
251C4—4042no—
252C2—21no—
253C2—22no—
254C2—24no—
255C2—21no—
256C2—22no—
257C2—42no—
258C2—8081no—
259C2#2026505611yescharacteristic
260C2C2×#1569216912yes—
subgroupG/H
centerC22C2×#13108051
commutatorC23×C101C22×C4
FrattiniC24C22×D101
FittingC22×C22⋊C4×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×#15692224C2
cosocleC24×C101C23
hypercenterC22×C22⋊C4D101
hypocenterC101#1898
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15692169C2semidirect
3. #20265033C2semidirect
4. #20264943C2semidirect
5. C22×C22⋊C4×C101C2semidirect
6. C22×C22⋊D101C4semidirect
12. C2×C22⋊Dic101C4semidirect
21. C23×D101C2×C4semidirect
30. C22×Dic101C2×C4semidirect
31. C22×Dic101C2×C4semidirect
52. C22⋊C4×C101D4semidirect
108. C22×C101#99semidirect
131. C101#1898semidirect
133. C22×C22⋊C4D101semidirect
183. C22⋊C4C22⋊D101semidirect
242. C22#15692169semidirect

Maximal subgroups

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

subgroupindex
C2×#156922242
C2×#156921692
#202650332
#202649432
C22×C22⋊C4×C1012
#1898101