Finite Group Databasebeta

← #15730

Subgroups — #15730

Order 256.

737 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 179 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×#385C211yesmaximal, characteristic
3#1441C212yesmaximal, complemented
4#1441C212yesmaximal, complemented
5#1441C212yesmaximal, complemented
6#1439C212yesmaximal
7#1437C211yesmaximal, characteristic
8#1436C212yesmaximal, complemented
9C22⋊#95C212yesmaximal
10C24×C4C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C22⋊C4C2211yescharacteristic
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C22⋊C4C2212yes—
19C22×C42C2211yescharacteristic
20#399—24no—
21#385—24no—
22C2×#95C2211yescharacteristic
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—41no—
37C23×C4—41no—
38C23×C4—42no—
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—41no—
64C2×C22⋊C4—41no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4D414yes—
69C2×C22⋊C4Q818yes—
70C2×C42—41no—
71C2×C42—42no—
72#95—24no—
73#95—22no—
74#95—24no—
75#95—24no—
76#95—41no—
77#95—42no—
78#95—42no—
79#95—42no—
80#95—42no—
81#95—42no—
82#95—42no—
83C24—22no—
84C24—22no—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—22no—
90C24C2×D412yes—
91C24C2×Q814yes—
92C24C2411yescharacteristic
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—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
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—22no—
138C22×C4—44no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2×Q814yes—
146C22×C4C2×Q814yes—
147C22×C4#4114yes—
148C22×C4#4114yes—
149C22×C4#4114yes—
150C4⋊C4—48no—
151C4⋊C4—48no—
152C4⋊C4—48no—
153C4⋊C4—44no—
154C22⋊C4—28no—
155C22⋊C4—44no—
156C22⋊C4—48no—
157C22⋊C4—48no—
158C42—44no—
159C23—22no—
160C23—24no—
161C23—22no—
162C23—24no—
163C23—24no—
164C23—24no—
165C23—24no—
166C23—24no—
167C23—24no—
168C23—24no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—41no—
174C23—42no—
175C23—42no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23C4⋊Q814yes—
180C23C22×D411yescharacteristic
181C23C22×Q812yes—
182C23C2×#4112yes—
183C23C2×#4112yes—
184C23C2×#4112yes—
185C23H(5,𝔽2)11yescharacteristic
186C23H(5,𝔽2)11yescharacteristic
187C23H(5,𝔽2)12yes—
188C23H(5,𝔽2)12yes—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—416no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
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—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C22—24no—
243C22—28no—
244C22—44no—
245C22—42no—
246C22—42no—
247C22—44no—
248C22—44no—
249C22—44no—
250C22—44no—
251C22#52511yescharacteristic
252C22#52912yes—
253C22C2×C4⋊Q811yescharacteristic
254C22#53312yes—
255C22#53411yescharacteristic
256C22#53812yes—
257C22#54012yes—
258C22#54012yes—
259C22#54312yes—
260C22#54612yes—
261C22#54612yes—
262C22#55112yes—
263C22#55111yescharacteristic
264C22#55312yes—
265C22#55412yes—
266C22#55412yes—
267C22#55412yes—
268C22#55412yes—
269C22#55911yescharacteristic
270C22#56011yescharacteristic
271C22#56011yescharacteristic
272C4—42no—
273C4—44no—
274C4—42no—
275C4—44no—
276C4—44no—
277C4—44no—
278C4—44no—
279C4—82no—
280C2—44no—
281C2#245712yes—
282C2#252212yes—
283C2#254912yes—
284C2#258311yescharacteristic
285C2#258912yes—
286C2#269711yescharacteristic
287C2#270911yescharacteristic
288C2#274812yes—
289C2#278012yes—
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. #1441C2semidirect
4. #1441C2semidirect
5. #1441C2semidirect
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×#3852
#14412
#14392
#14372
#14362
C22⋊#952