Finite Group Databasebeta

← #27172264

Subgroups — #27172264

Order 12928.

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

normal subgroups only

This group has 430 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
1#20265739C211yesmaximal, characteristic
2C2×#15692239C212yesmaximal, complemented
3C2×#15692169C212yesmaximal, complemented
4#20265038C211yesmaximal, characteristic, complemented
5#342×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C2212yes—
7C23×Dic101C2211yescharacteristic
8#15692239—22no—
9#15692239—24no—
10#15692239—24no—
11#15692229—22no—
12#15692225—22no—
13C2×C4⋊Dic101—22no—
14C2×C22⋊Dic101C2211yescharacteristic
15C2×#13108051—22no—
16#15692169—24no—
17#15692162—22no—
18C2×C22⋊C4×C101C2211yescharacteristic
19C2×C22⋊C4×C101C2212yes—
20C23×D101D412yes—
21C2×D404—22no—
22C2×C22⋊D101—22no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—22no—
25C2×C22⋊D101—22no—
26C2×C22⋊D101—22no—
27C2×C22⋊D101—22no—
28C22×Dic101—21no—
29C22×Dic101—22no—
30C22×Dic101—22no—
31C22×Dic101—21no—
32C22×Dic101—21no—
33C22×Dic101D412yes—
34C22×Dic101D412yescomplemented
35C4⋊Dic101—28no—
36C4⋊Dic101—22no—
37C22⋊Dic101—22no—
38#13108051—28no—
39#13108051—24no—
40C24×C101C2311yescharacteristic
41C22×C4×C101—22no—
42C22×C4×C101D412yes—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C22⋊C4×C101D414yescomplemented
46C23×C101—42no—
47C23×C101C2×D412yes—
48C23×C101C2×D411yescharacteristic
49C23×C101C2×D411yescharacteristic
50C23×C101C2×D412yes—
51C23×C101#4111yescharacteristic
52C22×D101—22no—
53C22×D101—22no—
54C22×D101—22no—
55C22×D101—44no—
56C2×C4×C101—22no—
57C2×C4×C101—24no—
58C2×C4×C101—44no—
59C2×C4×C101—42no—
60C22⋊D101—42no—
61C22⋊D101—42no—
62C22⋊D101—42no—
63C22⋊D101—42no—
64C2×Dic101—24no—
65C2×Dic101—22no—
66C2×Dic101—22no—
67C2×Dic101—22no—
68C2×Dic101—22no—
69C2×Dic101—41no—
70C2×Dic101—42no—
71C2×Dic101—41no—
72C2×Dic101—42no—
73C2×Dic101—41no—
74D404—82no—
75C22×C101—22no—
76C22×C101—24no—
77C22×C101—41no—
78C22×C101—42no—
79C22×C101—41no—
80C22×C101#12012yes—
81C22×C101#12011yescharacteristic
82C22×C101#12111yescharacteristic
83C22×C101#12112yes—
84C22×C101#12711yescharacteristic
85C2×D101—44no—
86C2×D101—42no—
87C2×D101—44no—
88C2×D101—82no—
89C4×C101—44no—
90C4×C101—42no—
91Dic101—42no—
92Dic101—42no—
93Dic101—81no—
94C2×C101—21no—
95C2×C101—22no—
96C2×C101—22no—
97C2×C101—21no—
98C2×C101#39211yescharacteristic
99C2×C101#45812yes—
100D101—82no—
101#2022—1011nomaximal, Sylow
102C101#202211yescharacteristic, complemented, Sylow
103#534—1011no—
104C2×#123—1012no—
105C2×#99—1012no—
106#381—1011no—
107#342D10111yescharacteristic, complemented
108C22×D4—1012no—
109C23×C4—1011no—
110#123—2024no—
111#123—2022no—
112#123—2024no—
113#121—2022no—
114#120—2022no—
115C2×C4⋊C4—2022no—
116C2×C22⋊C4—1011no—
117C2×C22⋊C4—2022no—
118C2×C22⋊C4C2×D10111yescharacteristic
119C2×C22⋊C4C2×D10112yes—
120#99—2024no—
121#95—2022no—
122C24—1012no—
123C24C22×D10111yescharacteristic
124C2×D4—2022no—
125C2×D4—2022no—
126C2×D4—2022no—
127C2×D4—2022no—
128C2×D4—2022no—
129C2×D4—2022no—
130C2×D4—2022no—
131C22×C4—22no—
132C22×C4—1012no—
133C22×C4—1012no—
134C22×C4—2021no—
135C22×C4—2022no—
136C22×C4—2021no—
137C22×C4—2022no—
138C22×C4—2021no—
139C22×C4D40412yes—
140C4⋊C4—2028no—
141C4⋊C4—2022no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—2028no—
145C22⋊C4—2024no—
146C22⋊C4—2022no—
147C22⋊C4D40414yescomplemented
148C23—42no—
149C23—4044no—
150C23—2022no—
151C23—2022no—
152C23—2022no—
153C23D4×D10112yes—
154C23D4×D10111yescharacteristic
155C23C2×D40411yescharacteristic
156C23C2×D40412yes—
157C23#1310807511yescharacteristic
158D4—8082no—
159D4—4042no—
160D4—4042no—
161D4—4042no—
162D4—4042no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—4042no—
168C2×C4—4041no—
169C2×C4—4041no—
170C2×C4—4042no—
171C2×C4—4041no—
172C2×C4—2024no—
173C2×C4—2022no—
174C2×C4—2022no—
175C2×C4—2022no—
176C2×C4—2022no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—8082no—
183C22—4044no—
184C22—4042no—
185C22—4044no—
186C22#1569222512yes—
187C22#1569222511yescharacteristic
188C22#1569223012yes—
189C22#1569223011yescharacteristic
190C22#1569225211yescharacteristic
191C4—44no—
192C4—42no—
193C4—8081no—
194C4—4042no—
195C4—4042no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—8082no—
201C2#2026508711yescharacteristic
202C2#2026536812yes—
subgroupG/H
centerC22#15692225
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101#120
prime radical#20265739C2
hypercenter#342D101
hypocenterC101#2022
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#15692239C2semidirect
3. C2×#15692169C2semidirect
4. #20265038C2semidirect
5. #342×C101C2semidirect
34. C22×Dic101D4semidirect
45. C22⋊C4×C101D4semidirect
102. C101#2022semidirect
107. #342D101semidirect
147. C22⋊C4D404semidirect

Maximal subgroups

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

subgroupindex
#202657392
C2×#156922392
C2×#156921692
#202650382
#342×C1012
#2022101