Finite Group Databasebeta

← #27172297

Subgroups — #27172297

Order 12928.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#20265803C211yesmaximal, characteristic, complemented
2#20265032C212yesmaximal
3#20265032C211yesmaximal, characteristic
4#20264943C212yesmaximal, complemented
5#342×C101C211yesmaximal, characteristic, complemented
6C24×D101C2211yescharacteristic
7#15692234—22no—
8#15692232—24no—
9C2×C22⋊Dic101C2211yescharacteristic
10C2×C22⋊Dic101C2212yes—
11C2×#13108051—24no—
12C2×#13108051—22no—
13C2×C22⋊C4×C101C2212yes—
14C2×C22⋊C4×C101C2211yescharacteristic
15C23×D101—22no—
16C23×D101—21no—
17C23×D101—21no—
18C23×D101—22no—
19C23×D101—44no—
20C23×D101Q812yescomplemented
21C2×Dic202—22no—
22C22×Dic101—21no—
23C22×Dic101D414yes—
24C4⋊Dic101—22no—
25#13108053—24no—
26C22⋊Dic101—22no—
27C22⋊Dic101—22no—
28C22⋊Dic101—22no—
29#13108051—48no—
30#13108051—44no—
31C24×C101C2311yescharacteristic
32C22×C4×C101—22no—
33C22×C4×C101D412yes—
34C22⋊C4×C101—22no—
35C22⋊C4×C101—22no—
36C22⋊C4×C101—22no—
37C23×C101—42no—
38C23×C101C2×D412yes—
39C23×C101C2×D411yescharacteristic
40C23×C101C2×Q811yescharacteristic
41C23×C101#4111yescharacteristic
42C23×C101#4112yes—
43C22×D101—28no—
44C22×D101—24no—
45C22×D101—24no—
46C22×D101—22no—
47C22×D101—44no—
48C22×D101—44no—
49C22×D101—48no—
50C22×D101—44no—
51C22×D101—42no—
52C22×D101—41no—
53C22×D101—42no—
54C22×D101—41no—
55C2×C4×C101—22no—
56C2×C4×C101—42no—
57C2×C4×C101—44no—
58C2×C4×C101—42no—
59C2×Dic101—24no—
60C2×Dic101—44no—
61C2×Dic101—42no—
62C2×Dic101—41no—
63Dic202—82no—
64C22×C101—24no—
65C22×C101—22no—
66C22×C101—41no—
67C22×C101—41no—
68C22×C101—42no—
69C22×C101#12011yescharacteristic
70C22×C101C22⋊Q812yes—
71C22×C101C22⋊Q811yescharacteristic
72C22×C101#12412yes—
73C22×C101#12411yescharacteristic
74C2×D101—28no—
75C2×D101—24no—
76C2×D101—44no—
77C2×D101—48no—
78C2×D101—44no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—44no—
82C4×C101—42no—
83C4×C101—82no—
84Dic101—44no—
85Dic101—81no—
86C2×C101—21no—
87C2×C101—22no—
88C2×C101—22no—
89C2×C101—21no—
90C2×C101#39311yescharacteristic
91C2×C101B(4,𝔽2)12yes—
92D101—44no—
93#2029—1011nomaximal, Sylow
94C101#202911yescharacteristic, complemented, Sylow
95#543—1011no—
96#379—1012no—
97#379—1011no—
98#342—1012no—
99#342D10111yescharacteristic, complemented
100C25—1011no—
101C22⋊Q8—2024no—
102C22⋊Q8—2022no—
103C2×C22⋊C4—1012no—
104C2×C22⋊C4—1011no—
105C2×C22⋊C4—2024no—
106C2×C22⋊C4—2022no—
107C2×C22⋊C4C2×D10112yes—
108C2×C22⋊C4C2×D10111yescharacteristic
109C24—1012no—
110C24—4044no—
111C24—2021no—
112C24—2022no—
113C24—2022no—
114C24—2021no—
115C24C22×D10111yescharacteristic
116C2×Q8—2022no—
117C22×C4—22no—
118C22×C4—1014no—
119C22×C4—2021no—
120C22×C4D40412yes—
121C4⋊C4—2024no—
122C4⋊C4—2022no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—4044no—
127C22⋊C4—4048no—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C22⋊C4—2022no—
131C23—42no—
132C23—4044no—
133C23—4042no—
134C23—4044no—
135C23—4048no—
136C23—4044no—
137C23—4041no—
138C23—4042no—
139C23—4041no—
140C23—2028no—
141C23—2024no—
142C23—2024no—
143C23—2022no—
144C23D4×D10112yes—
145C23C2×D40411yescharacteristic
146C23Q8×D10111yescharacteristic
147C23#1310807311yescharacteristic
148C23#1310807412yes—
149Q8—8082no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—4042no—
155C2×C4—4041no—
156C2×C4—4044no—
157C2×C4—2024no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—4048no—
164C22—4044no—
165C22—4044no—
166C22—4042no—
167C22—4042no—
168C22—4044no—
169C22—2028no—
170C22—2024no—
171C22#1569222511yescharacteristic
172C22#1569223312yes—
173C22#1569223511yescharacteristic
174C22#1569224112yes—
175C22#1569224311yescharacteristic
176C4—42no—
177C4—82no—
178C4—8081no—
179C4—4044no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—4044no—
185C2#2026509011yescharacteristic
186C2#2026536512yes—
subgroupG/H
centerC22#15692225
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101#120
prime radicalC24×D101C22
hypercenter#342D101
hypocenterC101#2029
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #20265803C2semidirect
4. #20264943C2semidirect
5. #342×C101C2semidirect
20. C23×D101Q8semidirect
94. C101#2029semidirect
99. #342D101semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202650322
#202649432
#342×C1012
#2029101