Finite Group Databasebeta

← #27171545

Subgroups — #27171545

Order 12928.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15692236C211yesmaximal, characteristic
2C2×#15692170C211yesmaximal, characteristic
3#20265038C212yesmaximal, complemented
4#20264944C212yesmaximal, complemented
5#518×C101C211yesmaximal, characteristic
6C23×Dic101C2211yescharacteristic
7#15692236—22no—
8#15692236C414yescomplemented
9C2×#13108053—21no—
10C2×C22⋊Dic101—21no—
11C2×C22⋊Dic101C2212yes—
12C2×C22⋊Dic101C2211yescharacteristic
13#15692170—22no—
14#15692162—24no—
15C22×D4×C101C2211yescharacteristic
16C4×D4×C101—24no—
17#117×C101—21no—
18C2×C22⋊C4×C101—21no—
19C2×C22⋊C4×C101C2212yes—
20C22×Dic101—22no—
21C22×Dic101—21no—
22C22×Dic101—21no—
23C22×Dic101—21no—
24C22×Dic101—22no—
25C22×Dic101—22no—
26C22×Dic101C2×C412yescomplemented
27#13108053—24no—
28C22⋊Dic101—24no—
29C22⋊Dic101—21no—
30C22⋊Dic101—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101C2×C412yescomplemented
33C24×C101D411yescharacteristic
34C24×C101C2311yescharacteristic
35C2×D4×C101—22no—
36C2×D4×C101—22no—
37C2×D4×C101C2×C412yes—
38C2×D4×C101D412yes—
39C22×C4×C101—22no—
40C22×C4×C101D411yescharacteristic
41C22×C4×C101D414yes—
42C4⋊C4×C101—22no—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C22⋊C4×C101—22no—
46C42×C101—22no—
47C23×C101—21no—
48C23×C101—21no—
49C23×C101—42no—
50C23×C101—42no—
51C23×C101C22⋊C412yes—
52C23×C101C22×C411yescharacteristic
53C23×C101C2×D411yescharacteristic
54C23×C101C2×D412yes—
55C23×C101C2×D411yescharacteristic
56C23×C101#4112yes—
57D4×C101—24no—
58D4×C101—24no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—24no—
62C2×C4×C101—44no—
63C2×C4×C101—82no—
64C2×C4×C101C22⋊C412yes—
65C2×Dic101—22no—
66C2×Dic101—22no—
67C2×Dic101—24no—
68C2×Dic101—42no—
69C2×Dic101—41no—
70C2×Dic101—42no—
71C2×Dic101—41no—
72C2×Dic101—44no—
73C2×Dic101—42no—
74C22×C101—24no—
75C22×C101—22no—
76C22×C101—22no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101—42no—
82C22×C101—81no—
83C22×C101C2×C22⋊C411yescharacteristic
84C22×C101C4×D412yes—
85C22×C101#12011yescharacteristic
86C22×C101#12112yes—
87C22×C101#12311yescharacteristic
88C4×C101—22no—
89C4×C101—44no—
90Dic101—42no—
91Dic101—42no—
92Dic101—82no—
93C2×C101—21no—
94C2×C101—22no—
95C2×C101—21no—
96C2×C101—22no—
97C2×C101—42no—
98C2×C101#38611yescharacteristic
99C2×C101#45812yes—
100#1890—1011nomaximal, Sylow
101C101#189011yescharacteristic, complemented, Sylow
102C2×#123—1011no—
103#518D10111yescharacteristic
104C2×#99—1011no—
105#381—1012no—
106#342—1012no—
107C22×D4C2×D10111yescharacteristic
108C23×C4—1011no—
109#123—1014no—
110#123—2022no—
111C4×D4—24no—
112#117—21no—
113C2×C4⋊C4—2021no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—1011no—
116C2×C22⋊C4—1012no—
117C2×C22⋊C4—2021no—
118C2×C22⋊C4C2×D10112yes—
119#99—2022no—
120#95—2024no—
121C24C22⋊D10111yescharacteristic
122C24C22×D10111yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D10112yes—
126C2×D4D40412yes—
127C22×C4—22no—
128C22×C4—1012no—
129C22×C4—2022no—
130C22×C4—2022no—
131C22×C4—2021no—
132C22×C4—2021no—
133C22×C4—2021no—
134C22×C4—2022no—
135C22×C4C22⋊D10111yescharacteristic
136C22×C4C22⋊D10114yes—
137C4⋊C4—22no—
138C4⋊C4—2024no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—1012no—
143C22⋊C4—2022no—
144C22⋊C4—2022no—
145C22⋊C4—2024no—
146C22⋊C4—2021no—
147C42—22no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23#1310805112yes—
153C23C2×C4×D10111yescharacteristic
154C23C2×C22⋊D10111yescharacteristic
155C23C2×C22⋊D10112yes—
156C23C2×D40411yescharacteristic
157C23#1310807412yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—82no—
165C2×C4—4044no—
166C2×C4—4042no—
167C2×C4—4042no—
168C2×C4—4041no—
169C2×C4—4042no—
170C2×C4—4041no—
171C2×C4—2022no—
172C2×C4—2022no—
173C2×C4—2024no—
174C2×C4#1310805112yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×#1310805111yescharacteristic
185C22C4×C22⋊D10112yes—
186C22#1569222411yescharacteristic
187C22#1569222912yes—
188C22#1569223811yescharacteristic
189C4—22no—
190C4—44no—
191C4—8082no—
192C4—4042no—
193C4—4042no—
194C2—22no—
195C2—21no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#2026505611yescharacteristic
200C2#2026536712yes—
subgroupG/H
centerC22#15692224
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#518×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
cosocleC24×C101C23
hypercenter#518D101
hypocenterC101#1890
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #20265038C2semidirect
4. #20264944C2semidirect
8. #15692236C4semidirect
26. C22×Dic101C2×C4semidirect
32. C22⋊Dic101C2×C4semidirect
101. C101#1890semidirect

Maximal subgroups

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

subgroupindex
C2×#156922362
C2×#156921702
#202650382
#202649442
#518×C1012
#1890101