Finite Group Databasebeta

← #16777286

Subgroups — #16777286

Order 3328.

286 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 66 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#13131065C211yesmaximal, characteristic
2#13130572C211yesmaximal, characteristic
3#13116445C211yesmaximal, characteristic
4C22×C8×D13C2211yescharacteristic
5C4×C22⋊#261C411yescharacteristic
6D13⋊(C22⋊C8)C411yescharacteristic
7C2×#88106C412yescomplemented
8C2×#88100—21no—
9C2×#88038—21no—
10(C2×#261)⋊C8—21no—
11#11718259—21no—
12#11718259—21no—
13C13×C22⋊C16C411yescharacteristic, complemented
14C22⋊#6090C411yescharacteristic, complemented
15#11718099—21no—
16C22×C4×D13C2×C411yescharacteristic
17C2×C22⋊#261C812yes—
18C2×C4×#261—21no—
19C2×D13⋊C8—21no—
20C22×C8×C13C2×C411yescharacteristic
21C22×#1055C2×C411yescharacteristic
22C2×C8×D13—42no—
23C2×C8×D13D411yescharacteristic
24C2×C8×D13D411yescharacteristic
25C2×C8×D13D411yescharacteristic
26C2×C8×D13Q811yescharacteristic
27C22⋊#1057C812yes—
28(C2×#261)⋊C4—21no—
29#88111—21no—
30#88106—42no—
31C2×#6092—22no—
32#88100—42no—
33C2×C13×C16—21no—
34C2×#6090—21no—
35#88038—42no—
36#7201—131nomaximal, Sylow
37C23×D13C2×C811yescharacteristic
38C22×#261—22no—
39C22×C4×C13C4211yescharacteristic
40C22×Dic13C2×C811yescharacteristic
41C2×C4×D13—41no—
42C2×C4×D13—41no—
43C2×C4×D13C22⋊C411yescharacteristic
44C2×C4×D13C4⋊C411yescharacteristic
45C2×C4×D13M1611yescharacteristic
46C2×C4×D13M1611yescharacteristic
47C22⋊#261—42no—
48C2×#1057—22no—
49C4×#261—41no—
50C4×#261—41no—
51D13⋊C8—41no—
52D13⋊C8—41no—
53C2×C8×C13—41no—
54C2×C8×C13C22⋊C411yescharacteristic
55C2×C8×C13C4⋊C411yescharacteristic
56C2×#1055—22no—
57C2×#1055C22⋊C411yescharacteristic
58C2×#1055C4⋊C411yescharacteristic
59C8×D13—22no—
60C8×D13—22no—
61C8×D13—41no—
62C8×D13—41no—
63#6092—42no—
64C13×C16—41no—
65#6090—41no—
66#2112—131no—
67#2112—131no—
68#1752—131no—
69C23×C13C4×C811yescharacteristic
70C22×D13—22no—
71C22×D13—41no—
72C22×D13—41no—
73C22×D13C22⋊C811yescharacteristic
74C22×D13C4⋊C811yescharacteristic
75C2×#261—42no—
76C2×#261—42no—
77C2×#261—42no—
78C2×C4×C13—21no—
79C2×C4×C13—21no—
80C2×C4×C13#9511yescharacteristic
81C2×C4×C13#9711yescharacteristic
82C2×Dic13—41no—
83C2×Dic13C22⋊C811yescharacteristic
84C2×Dic13C4⋊C811yescharacteristic
85C4×D13—21no—
86C4×D13—21no—
87C4×D13—21no—
88C4×D13—21no—
89C4×D13—41no—
90C4×D13—41no—
91#1057—42no—
92C8×C13—21no—
93C8×C13—21no—
94#1055—21no—
95#1055#9912yes—
96C23×C8—131no—
97C2×M32—132no—
98C2×M32—261no—
99C2×C22⋊C8—131no—
100C2×#97—261no—
101C4×C22⋊C4—131no—
102C22⋊C16—131no—
103C22⋊C16—261no—
104C22⋊C16—261no—
105C22⋊C16—261no—
106C22⋊C16#26111yescharacteristic, complemented
107#336—261no—
108C22×C13—21no—
109C22×C13—21no—
110C22×C13—21no—
111C22×C13#33611yescharacteristic
112C2×D13—21no—
113C2×D13—21no—
114C2×D13—21no—
115C2×D13—21no—
116C2×D13—21no—
117C2×D13—44no—
118C2×D13#34912yes—
119#261—82no—
120C4×C13—21no—
121C4×C13#34211yescharacteristic
122C4×C13#34411yescharacteristic
123Dic13—21no—
124Dic13—21no—
125C23×C4—131no—
126C22×C8—131no—
127C22×C8—131no—
128C22×C8—131no—
129C22×C8—131no—
130C22×C8—131no—
131C22×C8—261no—
132C22×C8—522no—
133C22×C8C2×#26111yescharacteristic
134C2×C22⋊C4—132no—
135C2×C42—261no—
136M32—522no—
137M32—522no—
138C2×C16—21no—
139C2×C16—262no—
140C2×C16—261no—
141C22⋊C8—132no—
142C22⋊C8—261no—
143#97—522no—
144#95—261no—
145C2×C13—41no—
146C2×C13#128211yescharacteristic
147C2×C13#137711yescharacteristic
148C2×C13#137811yescharacteristic
149D13—22no—
150D13—41no—
151C24—131no—
152C22×C4—131no—
153C22×C4—131no—
154C22×C4—131no—
155C22×C4—131no—
156C22×C4—131no—
157C22×C4—262no—
158C22×C4—521no—
159C22×C4—521no—
160C22×C4C4×#26111yescharacteristic
161C2×C8—41no—
162C2×C8—131no—
163C2×C8—131no—
164C2×C8—262no—
165C2×C8—262no—
166C2×C8—262no—
167C2×C8—262no—
168C2×C8—521no—
169C2×C8—521no—
170C2×C8—521no—
171C2×C8—521no—
172C2×C8C4⋊#26111yescharacteristic
173C2×C8C22⋊#26111yescharacteristic
174C22⋊C4—522no—
175C42—521no—
176C42—521no—
177C16—41no—
178C16—521no—
179C16—522no—
180C13#720111yescharacteristic, complemented, Sylow
181C23—131no—
182C23—131no—
183C23—262no—
184C23—521no—
185C23—521no—
186C23C8×#26111yescharacteristic
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—131no—
190C2×C4—131no—
191C2×C4—261no—
192C2×C4—261no—
193C2×C4—261no—
194C2×C4—261no—
195C2×C4—521no—
196C2×C4—521no—
197C2×C4—521no—
198C2×C4—522no—
199C2×C4—522no—
200C2×C4—522no—
201C2×C4#8810011yescharacteristic
202C2×C4(C2×#261)⋊C411yescharacteristic
203C8—21no—
204C8—21no—
205C8—132no—
206C8—261no—
207C8—522no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—132no—
212C22—261no—
213C22—261no—
214C22—261no—
215C22—261no—
216C22—261no—
217C22—524no—
218C22(C2×#261)⋊C811yescharacteristic
219C4—21no—
220C4—261no—
221C4—261no—
222C4—1042no—
223C4#1171823811yescharacteristic
224C4#1171827411yescharacteristic
225C2—41no—
226C2—262no—
227C2—521no—
228C2#1312967711yescharacteristic
229C2#1312988211yescharacteristic
230C2#1312988511yescharacteristic
subgroupG/H
centerC2×C4(C2×#261)⋊C4
commutatorC23×C13C4×C8
FrattiniC22×C8C2×#261
FittingC13×C22⋊C16C4
socleC22×C13#336
prime radicalC23×D13C2×C8
cosocleC22×C8×D13C22
hypercenterC22⋊C16#261
hypocenterC13#7201
LCSG ▷ C23×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
7. C2×#88106C4semidirect
13. C13×C22⋊C16C4semidirect
14. C22⋊#6090C4semidirect
106. C22⋊C16#261semidirect
180. C13#7201semidirect

Maximal subgroups

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

subgroupindex
#131310652
#131305722
#131164452
#720113