Finite Group Databasebeta

← #15916

Subgroups — #15916

Order 256.

693 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 163 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×#399C211yesmaximal, characteristic
2C2×#381C211yesmaximal, characteristic
3#1443C212yesmaximal, complemented
4#1442C212yesmaximal, complemented
5#1442C212yesmaximal, complemented
6#1440C212yesmaximal
7#1437C211yesmaximal, characteristic
8#1436C212yesmaximal, complemented
9C22⋊#95C212yesmaximal
10C24×C4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2211yescharacteristic
13C22×C22⋊C4C2211yescharacteristic
14C22×C22⋊C4C2212yes—
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#399—24no—
19#381—24no—
20C2×#95C2211yescharacteristic
21C2×#95C2212yes—
22C2×#95C2212yes—
23C2×#95C2212yes—
24C2×#95C2212yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2212yes—
33C25C2311yescharacteristic
34C23×C4—22no—
35C23×C4—24no—
36C23×C4—24no—
37C23×C4—41no—
38C23×C4—41no—
39C23×C4C2312yes—
40C23×C4C2312yes—
41C23×C4C2312yes—
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—42no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—41no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—24no—
55C2×C22⋊C4—24no—
56C2×C22⋊C4—24no—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—24no—
59C2×C22⋊C4—41no—
60C2×C22⋊C4—41no—
61C2×C22⋊C4—42no—
62C2×C22⋊C4—42no—
63C2×C22⋊C4D414yes—
64C2×C42—41no—
65C2×C42—42no—
66#95—24no—
67#95—24no—
68#95—22no—
69#95—24no—
70#95—24no—
71#95—24no—
72#95—24no—
73#95—24no—
74#95—42no—
75#95—42no—
76#95—41no—
77#95—42no—
78#95—42no—
79C24—22no—
80C24—22no—
81C24—22no—
82C24—22no—
83C24—21no—
84C24—21no—
85C24—22no—
86C24C2×D412yes—
87C24#4114yes—
88C24C2411yescharacteristic
89C22×C4—28no—
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—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
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×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—48no—
138C22×C4—48no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4#4114yes—
144C22×C4#4114yes—
145C22×C4#4114yes—
146C4⋊C4—48no—
147C4⋊C4—44no—
148C22⋊C4—44no—
149C22⋊C4—48no—
150C22⋊C4—48no—
151C22⋊C4—44no—
152C42—44no—
153C23—22no—
154C23—24no—
155C23—22no—
156C23—24no—
157C23—24no—
158C23—24no—
159C23—24no—
160C23—24no—
161C23—24no—
162C23—24no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23#12414yes—
174C23C22×D411yescharacteristic
175C23C2×#4112yes—
176C23C2×#4112yes—
177C23C2×#4112yes—
178C23C2×#4112yes—
179C23H(5,𝔽2)11yescharacteristic
180C23H(5,𝔽2)11yescharacteristic
181C23H(5,𝔽2)12yes—
182C232−1+412yes—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—416no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
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—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C22—24no—
237C22—28no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—44no—
245C22#52511yescharacteristic
246C22C2×#12411yescharacteristic
247C22#52912yes—
248C22#52912yes—
249C22#53411yescharacteristic
250C22#53912yes—
251C22#53912yes—
252C22#54012yes—
253C22#54212yes—
254C22#54612yes—
255C22#54712yes—
256C22#55112yes—
257C22#55112yes—
258C22#55112yes—
259C22#55212yes—
260C22#55212yes—
261C22#55312yes—
262C22#55911yescharacteristic
263C22#55911yescharacteristic
264C22#56011yescharacteristic
265C22#56311yescharacteristic
266C4—44no—
267C4—42no—
268C4—42no—
269C4—44no—
270C4—44no—
271C4—82no—
272C4—82no—
273C4—82no—
274C2—44no—
275C2#247012yes—
276C2#248812yes—
277C2#251612yes—
278C2#256111yescharacteristic
279C2#257512yes—
280C2#270911yescharacteristic
281C2#272411yescharacteristic
282C2#273812yes—
283C2#274612yes—
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
3. #1443C2semidirect
4. #1442C2semidirect
5. #1442C2semidirect
8. #1436C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3812
#14432
#14422
#14402
#14372
#14362
C22⋊#952