Finite Group Databasebeta

← #24522

Subgroups — #24522

Order 256.

424 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 117 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#3095C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic
3#2924C211yesmaximal, characteristic
4C2×#419C211yesmaximal, characteristic
5#2064C212yesmaximal, complemented
6#2047C212yesmaximal, complemented
7#1979C212yesmaximal, complemented
8#1930C212yesmaximal, complemented
9#1901C211yesmaximal, characteristic
10#1855C211yesmaximal, characteristic
11#1784C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×M16C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#485—21no—
18#485—21no—
19#484—21no—
20#483—21no—
21#451—21no—
22#451—21no—
23#451—22no—
24#433—22no—
25#432—21no—
26#431—21no—
27C2×#108C2212yes—
28#428C2211yescharacteristic
29#428C2211yescharacteristic
30#424C2212yes—
31#419—21no—
32#419—21no—
33#419C2212yes—
34#419C2211yescharacteristic
35#419C2211yescharacteristic
36#419C2212yes—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103C2212yes—
40C2×#101C2212yes—
41C2×#101C2212yes—
42C2×#101C2212yes—
43#407C2211yescharacteristic
44#407C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#405C2212yes—
47#343C2212yes—
48#341C2212yes—
49C22×Q8—21no—
50C22×Q8—24no—
51C22×Q8—21no—
52C22×Q8—21no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—42no—
56C22×Q8C2311yescharacteristic
57C22×Q8C2312yes—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C2×Q16—24no—
61C2×Q16—22no—
62C2×Q16—22no—
63C2×Q16—42no—
64C2×M16—21no—
65C2×M16—42no—
66C2×M16C2312yes—
67C2×M16C2311yescharacteristic
68C2×M16C2312yes—
69C2×M16C2311yescharacteristic
70C22×C8—21no—
71C22×C8C2312yes—
72C22⋊Q8—21no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76#117—21no—
77#117—21no—
78#117D411yescharacteristic
79#117D411yescharacteristic
80#117C2311yescharacteristic
81#117C2311yescharacteristic
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—21no—
89#108—42no—
90#107—21no—
91#107—21no—
92#106—21no—
93#106—21no—
94C4⋊C8—22no—
95C4⋊C8—22no—
96#103—22no—
97#103—22no—
98#103—22no—
99#103—22no—
100#103—22no—
101#103—22no—
102#103—21no—
103#103—21no—
104#103—21no—
105#103—21no—
106#101—24no—
107#101—42no—
108#101—42no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8D411yescharacteristic
114C22⋊C8D411yescharacteristic
115C22⋊C8D412yes—
116#97—22no—
117C4×C8—22no—
118C24#4111yescharacteristic
119C2×Q8—22no—
120C2×Q8—24no—
121C2×Q8—24no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8#4114yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—44no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4#4111yescharacteristic
153C22×C4C2411yescharacteristic
154Q16—44no—
155Q16—44no—
156M16—22no—
157M16—22no—
158M16—22no—
159M16—22no—
160M16—22no—
161M16—22no—
162M16—42no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—42no—
170C2×C8C2×D412yes—
171C2×C8C2×D412yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4C2×D411yescharacteristic
179C4⋊C4C2×D411yescharacteristic
180C4⋊C4C2×D411yescharacteristic
181C4⋊C4C2×D411yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23#12111yescharacteristic
196C23#12111yescharacteristic
197C23C2×#4111yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—44no—
201Q8—44no—
202Q8—42no—
203Q8—42no—
204Q8—41no—
205Q8—41no—
206Q8—42no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#12111yescharacteristic
230C2×C4#12111yescharacteristic
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C2×#4112yes—
234C2×C4H(5,𝔽2)12yes—
235C8—22no—
236C8—22no—
237C8—41no—
238C8—42no—
239C8—41no—
240C8—42no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—44no—
249C22#47312yes—
250C22C2×#12111yescharacteristic
251C22#55112yes—
252C4—21no—
253C4—21no—
254C4—44no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—81no—
260C4—81no—
261C4#52511yescharacteristic
262C4#53511yescharacteristic
263C4#54612yes—
264C2—22no—
265C2—22no—
266C2—41no—
267C2#241111yescharacteristic
268C2C2×#47311yescharacteristic
269C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC24#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2064C2semidirect
6. #2047C2semidirect
7. #1979C2semidirect
8. #1930C2semidirect

Maximal subgroups

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

subgroupindex
#30952
C2×#4512
#29242
C2×#4192
#20642
#20472
#19792
#19302
#19012
#18552
#17842