Finite Group Databasebeta

← #27171664

Subgroups — #27171664

Order 12928.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265744C211yesmaximal, characteristic
2#20265195C212yesmaximal, complemented
3C2×#15692172C211yesmaximal, characteristic, complemented
4#20264933C212yesmaximal, complemented
5#518×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7C2×#13108058C2212yes—
8#15692251C412yescomplemented
9#15692242C412yescomplemented
10#15692226—22no—
11#15692224—22no—
12#15692214C2211yescharacteristic
13#15692183—24no—
14#15692172—22no—
15C22×D4×C101C2211yescharacteristic
16C4×D4×C101—24no—
17#117×C101C2212yes—
18C2×C22⋊C4×C101—22no—
19#13108074—22no—
20#13108074—22no—
21C2×Dic202C2×C411yescharacteristic, complemented
22C2×D404C2×C411yescharacteristic, complemented
23C2×C22⋊D101—21no—
24C2×C22⋊D101—24no—
25C2×C4×D101—21no—
26#13108058—22no—
27#13108058—22no—
28C2×#11717579—22no—
29C4⋊Dic101—21no—
30C22⋊Dic101—24no—
31C22⋊Dic101—21no—
32C4×Dic101C2×C412yescomplemented
33C24×C101D412yes—
34C2×D4×C101—22no—
35C2×D4×C101—22no—
36C2×D4×C101C2×C412yes—
37C2×D4×C101D412yes—
38C22×C4×C101—24no—
39C22×C4×C101C2311yescharacteristic
40C4⋊C4×C101—22no—
41C22⋊C4×C101—24no—
42C22⋊C4×C101—22no—
43C42×C101D414yescomplemented
44C23×C101—22no—
45C23×C101—22no—
46C23×C101—82no—
47C23×C101C22⋊C414yes—
48C23×C101C2×D411yescharacteristic
49C22×D101—21no—
50D4×C101—24no—
51D4×C101—24no—
52C2×C4×C101—24no—
53C2×C4×C101—84no—
54C2×C4×C101C22×C411yescharacteristic
55C2×C4×C101C2×D412yes—
56C2×C4×C101C2×D411yescharacteristic
57C2×C4×C101#4112yes—
58C22⋊D101—41no—
59C22⋊D101—41no—
60C22⋊D101—84no—
61C2×Dic101—22no—
62C2×Dic101—21no—
63D404—22no—
64C4×D101—42no—
65Dic202—22no—
66#11717579—42no—
67C22×C101—24no—
68C22×C101—41no—
69C22×C101—44no—
70C22×C101—44no—
71C22×C101—82no—
72C22×C101C2×C22⋊C411yescharacteristic
73C22×C101#12011yescharacteristic
74C22×C101#12311yescharacteristic
75C2×D101—42no—
76C2×D101—41no—
77C4×C101—44no—
78C4×C101C4×D412yes—
79C4×C101#12112yes—
80Dic101—42no—
81Dic101—41no—
82C2×C101—21no—
83C2×C101—21no—
84C2×C101—21no—
85C2×C101—44no—
86C2×C101#38611yescharacteristic
87D101—81no—
88#1911—1011nomaximal, Sylow
89C101#191111yescharacteristic, complemented, Sylow
90#535—1011no—
91#518D10111yescharacteristic, complemented
92#421—1012no—
93C2×#100—1011no—
94H(3,ℤ/4)—1012no—
95C2×#41—1011no—
96C22×D4C2×D10111yescharacteristic
97C2×M16—1012no—
98#127—1012no—
99#124—1012no—
100#121—2022no—
101#120—2022no—
102C4×D4—24no—
103#117—1011no—
104#117C2×D10112yes—
105C2×C22⋊C4—22no—
106#104—2024no—
107#100—2022no—
108C24C22⋊D10112yes—
109#41—2022no—
110#41—2022no—
111C2×Q8—1011no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—1011no—
115C2×D4—2024no—
116C2×D4—2021no—
117C2×D4C4×D10112yes—
118C2×D4D40412yes—
119C22×C4—24no—
120C22×C4—2021no—
121C22×C4C22×D10111yescharacteristic
122M16—2022no—
123M16—2022no—
124C2×C8—2022no—
125C4⋊C4—22no—
126C4⋊C4—2021no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—2024no—
130C22⋊C4—2021no—
131C42—1012no—
132C42C22⋊D10114yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—2021no—
137C23#1310805114yes—
138C23C2×C22⋊D10111yescharacteristic
139Q8—2022no—
140D4—24no—
141D4—24no—
142D4—8084no—
143D4—4041no—
144D4—4041no—
145D4—2022no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—4042no—
149C2×C4—2022no—
150C2×C4—2021no—
151C2×C4C2×C4×D10111yescharacteristic
152C2×C4C2×C22⋊D10112yes—
153C2×C4C2×D40411yescharacteristic
154C2×C4#1310807412yes—
155C8—4042no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—4042no—
162C22—4041no—
163C22C2×#1310805111yescharacteristic
164C22#1569222411yescharacteristic
165C22#1569223811yescharacteristic
166C4—44no—
167C4—4042no—
168C4—4041no—
169C4C4×C22⋊D10112yes—
170C4#1569222912yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—8081no—
176C2#2026505611yescharacteristic
subgroupG/H
centerC2#20265056
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#518×C101C2
socleC2×C101#386
prime radical#20265744C2
cosocleC22×C4×C101C23
hypercenter#518D101
hypocenterC101#1911
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20265195C2semidirect
3. C2×#15692172C2semidirect
4. #20264933C2semidirect
5. #518×C101C2semidirect
8. #15692251C4semidirect
9. #15692242C4semidirect
21. C2×Dic202C2×C4semidirect
22. C2×D404C2×C4semidirect
32. C4×Dic101C2×C4semidirect
43. C42×C101D4semidirect
89. C101#1911semidirect
91. #518D101semidirect
132. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657442
#202651952
C2×#156921722
#202649332
#518×C1012
#1911101