Finite Group Databasebeta

← #27172256

Subgroups — #27172256

Order 12928.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15692223C212yesmaximal, complemented
2C2×#15692170C212yesmaximal, complemented
3#20265032C211yesmaximal, characteristic, complemented
4#20264944C211yesmaximal, characteristic, complemented
5#534×C101C211yesmaximal, characteristic, complemented
6C24×D101C2211yescharacteristic
7C22×C22⋊D101—22no—
8#15692223—24no—
9#15692223—24no—
10C2×C22⋊Dic101C2211yescharacteristic
11C2×C22⋊Dic101C2212yes—
12C2×#13108051—22no—
13C2×#13108051—22no—
14#15692170—24no—
15C22×D4×C101C2212yes—
16#123×C101—22no—
17#121×C101—22no—
18#120×C101—22no—
19C2×C22⋊C4×C101C2211yescharacteristic
20C23×D101—22no—
21C23×D101—21no—
22C23×D101—22no—
23C23×D101—21no—
24C23×D101—44no—
25C23×D101D412yescomplemented
26C2×C22⋊D101—28no—
27C2×C22⋊D101—48no—
28C22×Dic101—22no—
29C22×Dic101—21no—
30C22⋊Dic101—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101D414yescomplemented
33#13108051—28no—
34#13108051—44no—
35C24×C101D412yescomplemented
36C24×C101C2311yescharacteristic
37C2×D4×C101—22no—
38C2×D4×C101—22no—
39C2×D4×C101—22no—
40C2×D4×C101—22no—
41C2×D4×C101—22no—
42C2×D4×C101—22no—
43C2×D4×C101—22no—
44C22×C4×C101D412yescomplemented
45C22×C4×C101D412yescomplemented
46C4⋊C4×C101—22no—
47C22⋊C4×C101—24no—
48C22⋊C4×C101—22no—
49C23×C101—22no—
50C23×C101—22no—
51C23×C101—22no—
52C23×C101—42no—
53C23×C101—44no—
54C23×C101C2×D411yescharacteristic
55C23×C101C2×D412yes—
56C23×C101C2×D412yes—
57C23×C101C2×D411yescharacteristic
58C23×C101#4111yescharacteristic
59C22×D101—24no—
60C22×D101—28no—
61C22×D101—24no—
62C22×D101—22no—
63C22×D101—44no—
64C22×D101—48no—
65C22×D101—44no—
66C22×D101—42no—
67C22×D101—44no—
68C22×D101—41no—
69C22×D101—42no—
70C22×D101—41no—
71D4×C101—42no—
72D4×C101—42no—
73D4×C101—42no—
74D4×C101—42no—
75D4×C101—82no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—42no—
81C22⋊D101—416no—
82C2×Dic101—24no—
83C2×Dic101—44no—
84C2×Dic101—42no—
85C2×Dic101—41no—
86C22×C101—22no—
87C22×C101—24no—
88C22×C101—41no—
89C22×C101—42no—
90C22×C101—41no—
91C22×C101—44no—
92C22×C101—42no—
93C22×C101—44no—
94C22×C101—82no—
95C22×C101#12012yes—
96C22×C101#12011yescharacteristic
97C22×C101#12111yescharacteristic
98C22×C101#12112yes—
99C22×C101#12711yescharacteristic
100C2×D101—28no—
101C2×D101—24no—
102C2×D101—44no—
103C2×D101—44no—
104C2×D101—42no—
105C2×D101—48no—
106C2×D101—42no—
107C2×D101—44no—
108C4×C101—42no—
109C4×C101—42no—
110Dic101—44no—
111Dic101—81no—
112C2×C101—22no—
113C2×C101—21no—
114C2×C101—21no—
115C2×C101—22no—
116C2×C101—82no—
117C2×C101#39211yescharacteristic
118C2×C101B(4,𝔽2)12yes—
119D101—44no—
120#2021—1011nomaximal, Sylow
121C101#202111yescharacteristic, complemented, Sylow
122#534D10111yescharacteristic, complemented
123C2×#120—1012no—
124C2×#99—1012no—
125#379—1011no—
126#342—1011no—
127C25—1011no—
128C22×D4—2022no—
129C22×D4C2×D10112yes—
130#123—22no—
131#121—22no—
132#120—22no—
133#120—2024no—
134#120—2024no—
135C2×C22⋊C4—1012no—
136C2×C22⋊C4—1011no—
137C2×C22⋊C4—2022no—
138C2×C22⋊C4—2022no—
139C2×C22⋊C4C2×D10111yescharacteristic
140#99—2024no—
141C24—1012no—
142C24—4044no—
143C24—2022no—
144C24—2021no—
145C24—2021no—
146C24—2022no—
147C24C22⋊D10112yescomplemented
148C24C22×D10111yescharacteristic
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—4048no—
157C2×D4—2028no—
158C22×C4—2021no—
159C22×C4—2022no—
160C22×C4C22⋊D10112yescomplemented
161C22×C4C22⋊D10112yescomplemented
162C4⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—1014no—
166C22⋊C4—4044no—
167C22⋊C4—2028no—
168C22⋊C4—2022no—
169C22⋊C4—2022no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—42no—
174C23—44no—
175C23—4042no—
176C23—4044no—
177C23—4048no—
178C23—4044no—
179C23—4044no—
180C23—4041no—
181C23—4042no—
182C23—4041no—
183C23—2024no—
184C23—2028no—
185C23—2024no—
186C23—2022no—
187C23D4×D10111yescharacteristic
188C23D4×D10112yes—
189C23C2×C22⋊D10112yes—
190C23C2×C22⋊D10111yescharacteristic
191C23#1310807311yescharacteristic
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4—82no—
197D4—40416no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—4042no—
204C2×C4—4041no—
205C2×C4—4044no—
206C2×C4—2024no—
207C22—22no—
208C22—24no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—44no—
213C22—42no—
214C22—44no—
215C22—82no—
216C22—4044no—
217C22—4044no—
218C22—4048no—
219C22—4042no—
220C22—4042no—
221C22—4044no—
222C22—2028no—
223C22—2024no—
224C22#1569222312yes—
225C22#1569222411yescharacteristic
226C22#1569222611yescharacteristic
227C22#1569222712yes—
228C22#1569225111yescharacteristic
229C4—42no—
230C4—42no—
231C4—8081no—
232C4—4044no—
233C2—21no—
234C2—22no—
235C2—22no—
236C2—21no—
237C2—82no—
238C2—4044no—
239C2#2026508511yescharacteristic
240C2#2026536412yes—
subgroupG/H
centerC22#15692224
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting#534×C101C2
socleC22×C101#120
hypercenter#534D101
hypocenterC101#2021
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#15692223C2semidirect
2. C2×#15692170C2semidirect
3. #20265032C2semidirect
4. #20264944C2semidirect
5. #534×C101C2semidirect
25. C23×D101D4semidirect
32. C22⋊Dic101D4semidirect
35. C24×C101D4semidirect
44. C22×C4×C101D4semidirect
45. C22×C4×C101D4semidirect
121. C101#2021semidirect
122. #534D101semidirect
147. C24C22⋊D101semidirect
160. C22×C4C22⋊D101semidirect
161. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922232
C2×#156921702
#202650322
#202649442
#534×C1012
#2021101