Finite Group Databasebeta

← #27172116

Subgroups — #27172116

Order 12928.

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

normal subgroups only

This group has 324 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×#15692234C211yesmaximal, characteristic
2C4×C22⋊Dic101C211yesmaximal, characteristic
3#20264943C214yesmaximal
4#380×C101C211yesmaximal, characteristic
5C22×Dic202C411yescharacteristic, complemented
6#15692234—21no—
7#15692234—21no—
8#15692234—21no—
9#15692234—21no—
10C2×C4⋊Dic101C411yescharacteristic, complemented
11C2×#13108053C412yescomplemented
12C2×C22⋊Dic101C2212yes—
13C2×C22⋊Dic101C2212yes—
14C2×C4×Dic101—21no—
15#15692162—21no—
16C23×C4×C101C2211yescharacteristic
17C2×C4⋊C4×C101—21no—
18C2×C22⋊C4×C101C2212yes—
19#95×C101—21no—
20C2×Dic202—21no—
21C2×Dic202—21no—
22C2×Dic202—42no—
23C22×Dic101—22no—
24C22×Dic101C2×C412yes—
25C22×Dic101C2×C412yes—
26C4⋊Dic101—21no—
27C4⋊Dic101—21no—
28#13108053—22no—
29#13108053—22no—
30C22⋊Dic101—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33C22⋊Dic101Q814yescomplemented
34C4×Dic101—42no—
35C24×C101C2311yescharacteristic
36C22×C4×C101—22no—
37C22×C4×C101—42no—
38C22×C4×C101C2×C411yescharacteristic
39C22×C4×C101C2×C411yescharacteristic
40C22×C4×C101D411yescharacteristic
41C22×C4×C101D411yescharacteristic
42C22×C4×C101D411yescharacteristic
43C22×C4×C101D411yescharacteristic
44C4⋊C4×C101—42no—
45C22⋊C4×C101—24no—
46C23×C101—41no—
47C23×C101—41no—
48C23×C101C22×C411yescharacteristic
49C23×C101C2×D411yescharacteristic
50C23×C101C2×D411yescharacteristic
51C23×C101C2×Q812yes—
52C23×C101#4112yes—
53C2×C4×C101—22no—
54C2×C4×C101—22no—
55C2×C4×C101—22no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—42no—
60C2×C4×C101—42no—
61C2×C4×C101—42no—
62C2×C4×C101C22⋊C411yescharacteristic
63C2×C4×C101C22⋊C411yescharacteristic
64C2×C4×C101C22⋊C411yescharacteristic
65C2×C4×C101C22⋊C411yescharacteristic
66C2×Dic101—22no—
67C2×Dic101—22no—
68C2×Dic101—24no—
69C2×Dic101—42no—
70C2×Dic101—42no—
71C2×Dic101—42no—
72C2×Dic101—42no—
73Dic202—42no—
74Dic202—42no—
75C22×C101—21no—
76C22×C101—21no—
77C22×C101—21no—
78C22×C101—21no—
79C22×C101—21no—
80C22×C101—21no—
81C22×C101—42no—
82C22×C101—42no—
83C22×C101C2×C22⋊C411yescharacteristic
84C22×C101C4×Q812yes—
85C22×C101C22⋊Q812yes—
86C22×C101#12411yescharacteristic
87C22×C101C4⋊Q811yescharacteristic
88C4×C101—21no—
89C4×C101—21no—
90C4×C101—21no—
91C4×C101—82no—
92C4×C101#9912yes—
93Dic101—44no—
94Dic101—42no—
95Dic101—42no—
96C2×C101—22no—
97C2×C101—21no—
98C2×C101—22no—
99C2×C101—21no—
100C2×C101#39111yescharacteristic
101C2×C101C2×#9911yescharacteristic
102C2×C101#41011yescharacteristic
103#1996—1011nomaximal, Sylow
104C101#199611yescharacteristic, complemented, Sylow
105C2×C22⋊Q8—1011no—
106#380D10111yescharacteristic
107C4×C22⋊C4—1011no—
108#342—1014no—
109C22×Q8—1011no—
110C23×C4C2×D10111yescharacteristic
111C22⋊Q8—2021no—
112C22⋊Q8—2021no—
113C22⋊Q8—2021no—
114C22⋊Q8—2021no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—1012no—
117C2×C4⋊C4—1011no—
118C2×C22⋊C4—1012no—
119C2×C22⋊C4—1012no—
120C2×C22⋊C4C2×D10112yes—
121C2×C42—2021no—
122#95—21no—
123#95—2021no—
124C24C22×D10111yescharacteristic
125C2×Q8—4042no—
126C2×Q8—2021no—
127C2×Q8—2021no—
128C22×C4—22no—
129C22×C4—42no—
130C22×C4—1012no—
131C22×C4—1012no—
132C22×C4—2022no—
133C22×C4C4×D10111yescharacteristic
134C22×C4C4×D10111yescharacteristic
135C22×C4D40411yescharacteristic
136C22×C4D40411yescharacteristic
137C22×C4C22⋊D10111yescharacteristic
138C22×C4C22⋊D10111yescharacteristic
139C4⋊C4—42no—
140C4⋊C4—2022no—
141C4⋊C4—2022no—
142C4⋊C4—2021no—
143C4⋊C4—2021no—
144C22⋊C4—24no—
145C22⋊C4—1014no—
146C22⋊C4—2022no—
147C22⋊C4—2022no—
148C22⋊C4—2022no—
149C42—4042no—
150C23—41no—
151C23—41no—
152C23C2×C4×D10111yescharacteristic
153C23C2×C22⋊D10111yescharacteristic
154C23C2×D40411yescharacteristic
155C23Q8×D10112yes—
156C23#1310807312yes—
157Q8—4042no—
158Q8—4042no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—4042no—
169C2×C4—4042no—
170C2×C4—4042no—
171C2×C4—4042no—
172C2×C4—2022no—
173C2×C4—2022no—
174C2×C4—2024no—
175C2×C4#1310805111yescharacteristic
176C2×C4#1310805111yescharacteristic
177C2×C4#1310805111yescharacteristic
178C2×C4#1310805111yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22C2×#1310805111yescharacteristic
188C22#1569222012yes—
189C22#1569223512yes—
190C22#1569224211yescharacteristic
191C22#1569225311yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—82no—
196C4—4044no—
197C4—4042no—
198C4—4042no—
199C4#1569216912yes—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2#2026508211yescharacteristic
205C2C2×#1569216911yescharacteristic
206C2#2026515011yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#380×C101C2
socleC22×C101C2×C22⋊C4
prime radical, cosocleC24×C101C23
hypercenter#380D101
hypocenterC101#1996
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
5. C22×Dic202C4semidirect
10. C2×C4⋊Dic101C4semidirect
11. C2×#13108053C4semidirect
33. C22⋊Dic101Q8semidirect
104. C101#1996semidirect

Maximal subgroups

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

subgroupindex
C2×#156922342
C4×C22⋊Dic1012
#202649432
#380×C1012
#1996101