Finite Group Databasebeta

← #17106

Subgroups — #17106

Order 256.

719 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 167 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×#397C212yesmaximal
3C2×#395C212yesmaximal, complemented
4C2×#393C211yesmaximal, characteristic
5#1444C212yesmaximal, complemented
6#1443C212yesmaximal, complemented
7#1442C212yesmaximal, complemented
8#1440C211yesmaximal, characteristic
9#1439C212yesmaximal
10C23×Q8C2211yescharacteristic
11C22×C4⋊C4C2212yes—
12C22×C4⋊C4C2212yes—
13C22×C4⋊C4C2212yes—
14C22×C4⋊C4C2211yescharacteristic
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20#399—24no—
21#397—28no—
22#395—28no—
23#393—24no—
24C2×#95C2211yescharacteristic
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2212yes—
28C2×#95C2212yes—
29C2×#95C2212yes—
30C2×#95C2211yescharacteristic
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C25C2311yescharacteristic
36C22×Q8—22no—
37C22×Q8—24no—
38C22×Q8—41no—
39C22×Q8—42no—
40C22×Q8—41no—
41C23×C4C2312yes—
42C23×C4C2312yes—
43C23×C4C2312yes—
44C23×C4C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2311yescharacteristic
48C23×C4C2312yes—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—41no—
68C2×C22⋊C4—41no—
69C2×C22⋊C4—41no—
70C2×C22⋊C4—41no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4D414yes—
74#95—24no—
75#95—22no—
76#95—24no—
77#95—24no—
78#95—24no—
79#95—24no—
80#95—22no—
81#95—24no—
82#95—24no—
83#95—42no—
84#95D418yes—
85C24—22no—
86C24—22no—
87C24—21no—
88C24—21no—
89C24—22no—
90C24—22no—
91C24—22no—
92C24—22no—
93C24C2×D412yes—
94C24C2411yescharacteristic
95C2×Q8—48no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—48no—
99C2×Q8—44no—
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—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
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—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4C2×D412yes—
147C22×C4C2×D414yes—
148C22×C4C2×D414yes—
149C22×C4C2×D412yes—
150C22×C4C2×D414yes—
151C22×C4#4114yes—
152C4⋊C4—48no—
153C4⋊C4—48no—
154C4⋊C4—44no—
155C4⋊C4—48no—
156C22⋊C4—44no—
157C22⋊C4—44no—
158C22⋊C4—44no—
159C22⋊C4—48no—
160C22⋊C4—48no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—24no—
165C23—24no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—41no—
173C23—42no—
174C23—42no—
175C23—41no—
176C23—41no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—41no—
182C23C22×D411yescharacteristic
183C23C22×D412yes—
184C23C2×#4112yes—
185C23H(5,𝔽2)11yescharacteristic
186C23H(5,𝔽2)11yescharacteristic
187C23H(5,𝔽2)12yes—
188C232−1+412yes—
189C232−1+412yes—
190C232−1+412yes—
191Q8—416no—
192C2×C4—22no—
193C2×C4—22no—
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—42no—
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—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
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—
242C2×C4—42no—
243C2×C4—42no—
244C22—42no—
245C22—42no—
246C22—44no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22#53411yescharacteristic
254C22#53512yes—
255C22#53611yescharacteristic
256C22#53611yescharacteristic
257C22#53612yes—
258C22#53612yes—
259C22#53912yes—
260C22#54212yes—
261C22#54212yes—
262C22D4211yescharacteristic
263C22D4212yes—
264C22#54712yes—
265C22#54712yes—
266C22#54812yes—
267C22#55111yescharacteristic
268C22#56011yescharacteristic
269C22#56312yes—
270C22#56312yes—
271C22#56311yescharacteristic
272C22#56312yes—
273C22#56312yes—
274C4—42no—
275C4—44no—
276C4—44no—
277C4—44no—
278C4—42no—
279C4—44no—
280C4—82no—
281C4—82no—
282C2—82no—
283C2#266911yescharacteristic
284C2#267412yes—
285C2#267412yes—
286C2#268012yes—
287C2#271711yescharacteristic
288C2#272312yes—
289C2#281711yescharacteristic
290C2#282112yes—
291C2#283812yes—
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. C2×#395C2semidirect
5. #1444C2semidirect
6. #1443C2semidirect
7. #1442C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3992
C2×#3972
C2×#3952
C2×#3932
#14442
#14432
#14422
#14402
#14392