Finite Group Databasebeta

← #16246

Subgroups — #16246

Order 256.

615 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 159 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#1445C212yesmaximal, complemented
2#1442C212yesmaximal, complemented
3#1440C211yesmaximal, characteristic
4#1439C212yesmaximal
5#1439C212yesmaximal
6#1437C211yesmaximal, characteristic
7#1437C211yesmaximal, characteristic
8#1436C212yesmaximal, complemented
9#1436C212yesmaximal, complemented
10C22×C4⋊C4C2212yes—
11C22×C4⋊C4C2212yes—
12C22×C22⋊C4C2212yes—
13C22×C22⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2211yescharacteristic
16C22×C22⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#95C2212yes—
20C2×#95C2212yes—
21C2×#95C2212yes—
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2211yescharacteristic
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C25C2311yescharacteristic
32C23×C4C2312yes—
33C23×C4C2312yes—
34C23×C4C2311yescharacteristic
35C23×C4C2311yescharacteristic
36C23×C4C2312yes—
37C23×C4C2312yes—
38C23×C4C2312yes—
39C23×C4C2312yes—
40C2×C4⋊C4—24no—
41C2×C4⋊C4—24no—
42C2×C4⋊C4—24no—
43C2×C4⋊C4—42no—
44C2×C4⋊C4—42no—
45C2×C4⋊C4—42no—
46C2×C22⋊C4—24no—
47C2×C22⋊C4—24no—
48C2×C22⋊C4—24no—
49C2×C22⋊C4—24no—
50C2×C22⋊C4—41no—
51C2×C22⋊C4—41no—
52C2×C22⋊C4—42no—
53C2×C22⋊C4—41no—
54C2×C22⋊C4—42no—
55C2×C22⋊C4—42no—
56C2×C22⋊C4—42no—
57C2×C22⋊C4—41no—
58C2×C22⋊C4Q814yes—
59C2×C42—42no—
60C2×C42—42no—
61C2×C42—41no—
62C2×C42—41no—
63#95—24no—
64#95—24no—
65#95—24no—
66#95—24no—
67#95—24no—
68#95—42no—
69#95—42no—
70#95—42no—
71#95—42no—
72#95—41no—
73#95—42no—
74#95—41no—
75C24—22no—
76C24—22no—
77C24—22no—
78C24—22no—
79C24—21no—
80C24—21no—
81C24—22no—
82C24—22no—
83C24C2×Q812yes—
84C24C2411yescharacteristic
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4C2×Q812yes—
132C22×C4C2×Q812yes—
133C22×C4#4114yes—
134C22×C4#4114yes—
135C22×C4#4114yes—
136C22×C4#4114yes—
137C4⋊C4—48no—
138C4⋊C4—48no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—48no—
142C22⋊C4—48no—
143C22⋊C4—44no—
144C42—44no—
145C42—44no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—24no—
150C23—24no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—41no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23C22×Q811yescharacteristic
168C23C2×#4112yes—
169C23C2×#4112yes—
170C23C2×#4112yes—
171C23C2×#4112yes—
172C23H(5,𝔽2)12yes—
173C23H(5,𝔽2)11yescharacteristic
174C23H(5,𝔽2)12yes—
175C23H(5,𝔽2)11yescharacteristic
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—82no—
237C22#52912yes—
238C22#52912yes—
239C22#53311yescharacteristic
240C22#53311yescharacteristic
241C22#53812yes—
242C22#53812yes—
243C22#54012yes—
244C22#54012yes—
245C22#54311yescharacteristic
246C22#55111yescharacteristic
247C22#55112yes—
248C22#55112yes—
249C22#55312yes—
250C22#55312yes—
251C22#55312yes—
252C22#55412yes—
253C22#55412yes—
254C22#55912yes—
255C22#55911yescharacteristic
256C22#56011yescharacteristic
257C22H(3,𝔽4)11yescharacteristic
258C4—44no—
259C4—42no—
260C4—44no—
261C4—44no—
262C4—44no—
263C4—42no—
264C4—82no—
265C4—82no—
266C2—82no—
267C2#251612yes—
268C2#252212yes—
269C2#258712yes—
270C2#259712yes—
271C2#260711yescharacteristic
272C2#274811yescharacteristic
273C2#275312yes—
274C2#279712yes—
275C2#280411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #1445C2semidirect
2. #1442C2semidirect
8. #1436C2semidirect
9. #1436C2semidirect

Maximal subgroups

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

subgroupindex
#14452
#14422
#14402
#14392
#14372
#14362