Finite Group Databasebeta

← #27169457

Subgroups — #27169457

Order 12928.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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×C22⋊C4×D101C211yesmaximal, characteristic
2#20265139C212yesmaximal, complemented
3#20264901C211yesmaximal, characteristic, complemented
4#20264900C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C24×D101C411yescharacteristic
7C22×C4×D101C411yescharacteristic
8C22×C4×D101C2211yescharacteristic
9C2×#13108057—22no—
10C2×C22⋊Dic101C411yescharacteristic
11C2×#13108051C411yescharacteristic
12C2×#13108051C2211yescharacteristic
13C22⋊C4×D101—22no—
14C22⋊C4×D101—22no—
15C22⋊#11717579C2212yescomplemented
16#15692167—22no—
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yescomplemented
19C23×D101—22no—
20C23×D101—21no—
21C23×D101—21no—
22C23×D101—22no—
23C23×D101—42no—
24C23×D101—42no—
25C23×D101C2×C411yescharacteristic
26C23×D101C2×C411yescharacteristic
27C22×Dic101C2×C411yescharacteristic
28C22×Dic101C2×C411yescharacteristic
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101—42no—
32C2×C4×D101—42no—
33C2×C4×D101D412yes—
34C2×C4×D101D412yes—
35#13108057—44no—
36C2×#11717579—22no—
37C22⋊Dic101—22no—
38#13108051—22no—
39#13108051—22no—
40C24×C101C2×C411yescharacteristic
41C22×C4×C101C2×C411yescharacteristic
42C22×C4×C101C2311yescharacteristic
43C2×C8×C101—22no—
44C22⋊C4×C101—22no—
45C23×C101—21no—
46C23×C101—21no—
47C23×C101—41no—
48C23×C101—41no—
49C23×C101M1612yescomplemented
50C23×C101C22×C411yescharacteristic
51C22×D101—21no—
52C22×D101—21no—
53C22×D101—22no—
54C22×D101—22no—
55C22×D101—24no—
56C22×D101—24no—
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—48no—
60C22×D101—48no—
61C22×D101—42no—
62C22×D101—42no—
63C22×D101—42no—
64C22×D101—42no—
65C22×D101—41no—
66C22×D101—41no—
67C22×D101C22⋊C411yescharacteristic
68C22×D101C22⋊C411yescharacteristic
69C22×D101M1612yescomplemented
70C2×C4×C101—21no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101C2×D412yes—
74C2×Dic101—21no—
75C2×Dic101—41no—
76C2×Dic101—41no—
77C2×Dic101C22⋊C412yes—
78C4×D101—24no—
79C4×D101—42no—
80C4×D101—42no—
81C4×D101—42no—
82C8×C101—42no—
83#11717579—42no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101—22no—
87C22×C101—42no—
88C22×C101—42no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101C2×C22⋊C411yescharacteristic
92C22×C101C2×M1611yescharacteristic
93C22×C101C2×M1611yescharacteristic
94C2×D101—24no—
95C2×D101—24no—
96C2×D101—21no—
97C2×D101—21no—
98C2×D101—21no—
99C2×D101—21no—
100C2×D101—22no—
101C2×D101—48no—
102C2×D101—48no—
103C2×D101—42no—
104C2×D101—41no—
105C2×D101—41no—
106C2×D101—42no—
107C2×D101#9912yes—
108C2×D101#10012yes—
109C4×C101—22no—
110C4×C101—41no—
111Dic101—22no—
112Dic101—41no—
113C2×C101—21no—
114C2×C101—21no—
115C2×C101—42no—
116C2×C101#40711yescharacteristic
117C2×C101C2×#9911yescharacteristic
118C2×C101C2×#10011yescharacteristic
119D101—22no—
120D101—22no—
121D101—42no—
122#1462—1011nomaximal, Sylow
123C101#146211yescharacteristic, complemented, Sylow
124C22×C22⋊C4—1011no—
125#407—1012no—
126#323—1011no—
127#323—1012no—
128#323D10111yescharacteristic, complemented
129C25—1011no—
130C23×C4—1011no—
131C23×C4—1011no—
132C2×M16—2022no—
133C2×C22⋊C4—1011no—
134C2×C22⋊C4—1011no—
135C2×C22⋊C4—1011no—
136C2×C22⋊C4—2022no—
137C2×C22⋊C4—2022no—
138C2×C22⋊C4C2×D10111yescharacteristic
139C22⋊C8—1012no—
140C22⋊C8—2022no—
141C22⋊C8C2×D10112yescomplemented
142C24—1011no—
143C24—1011no—
144C24—4042no—
145C24—4042no—
146C24—2021no—
147C24—2021no—
148C24—2022no—
149C24—2022no—
150C24C4×D10111yescharacteristic
151C22×C4—1011no—
152C22×C4—1012no—
153C22×C4—1012no—
154C22×C4—1011no—
155C22×C4—4042no—
156C22×C4—4042no—
157C22×C4—2021no—
158C22×C4—2021no—
159C22×C4C4×D10111yescharacteristic
160C22×C4C22×D10111yescharacteristic
161M16—4044no—
162C2×C8—22no—
163C2×C8—2022no—
164C22⋊C4—22no—
165C22⋊C4—2022no—
166C22⋊C4—2022no—
167C22⋊C4—2022no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—1011no—
173C23—1011no—
174C23—1012no—
175C23—4041no—
176C23—4041no—
177C23—4042no—
178C23—4042no—
179C23—4048no—
180C23—4048no—
181C23—4042no—
182C23—4042no—
183C23—2022no—
184C23—2022no—
185C23—2024no—
186C23—2024no—
187C23—2021no—
188C23—2021no—
189C23—2021no—
190C23—2021no—
191C23#1310805712yescomplemented
192C23C2×C4×D10111yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1012no—
197C2×C4—4042no—
198C2×C4—4042no—
199C2×C4—4042no—
200C2×C4—4041no—
201C2×C4—4041no—
202C2×C4—2024no—
203C2×C4—2021no—
204C2×C4D4×D10112yes—
205C8—42no—
206C8—4042no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—1012no—
215C22—1012no—
216C22—4042no—
217C22—4042no—
218C22—4048no—
219C22—4048no—
220C22—4041no—
221C22—4041no—
222C22—2024no—
223C22—2024no—
224C22—2022no—
225C22—2021no—
226C22—2021no—
227C22—2021no—
228C22—2021no—
229C22C22⋊C4×D10111yescharacteristic
230C22M16×D10111yescharacteristic
231C22C2×#1310805711yescharacteristic
232C4—22no—
233C4—41no—
234C4—4041no—
235C4—2022no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—4042no—
240C2—2022no—
241C2—2022no—
242C2#2026513911yescharacteristic
243C2#99×D10111yescharacteristic
244C2#100×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC24×D101C4
cosocleC22×C4×C101C23
hypercenter#323D101
hypocenterC101#1462
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265139C2semidirect
3. #20264901C2semidirect
4. #20264900C2semidirect
5. #323×C101C2semidirect
15. C22⋊#11717579C22semidirect
18. C22⋊C8×C101C22semidirect
49. C23×C101M16semidirect
69. C22×D101M16semidirect
123. C101#1462semidirect
128. #323D101semidirect
141. C22⋊C8C2×D101semidirect
191. C23#13108057semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D1012
#202651392
#202649012
#202649002
#323×C1012
#1462101