Finite Group Databasebeta

← #23986

Subgroups — #23986

Order 256.

713 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 123 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#3491C211yesmaximal, characteristic, complemented
2#2849C212yesmaximal, complemented
3#2280C211yesmaximal, characteristic, complemented
4#2030C211yesmaximal, characteristic, complemented
5#2029C211yesmaximal, characteristic, complemented
6#2022C211yesmaximal, characteristic, complemented
7#2021C211yesmaximal, characteristic
8#1899C212yesmaximal, complemented
9#1890C212yesmaximal, complemented
10#1889C212yesmaximal, complemented
11#1820C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13#552—24no—
14#551—24no—
15#543C2211yescharacteristic
16#539—22no—
17#534C2211yescharacteristic
18#529—22no—
19C2×#123C2212yes—
20C2×#123C2212yes—
21C2×#120C2212yes—
22#518C2212yes—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#99C2212yes—
26C2×#99C2212yes—
27#386—24no—
28#385—24no—
29#381C2212yes—
30#381C2211yescharacteristic
31#381C2211yescharacteristic
32#381C2212yes—
33#380—22no—
34#379C2212yes—
35#379C2211yescharacteristic
36#379C2211yescharacteristic
37C4×C22⋊C4—22no—
38#342C2212yes—
39#342C2211yescharacteristic
40#342C2211yescharacteristic
41#342C2212yes—
42#342C2212yes—
43C25C2311yescharacteristic
44C22×D4—22no—
45C22×D4C2312yes—
46C23×C4—22no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—22no—
50C23×C4—24no—
51C23×C4—42no—
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54#126—24no—
55#126—24no—
56#125—22no—
57#125—24no—
58#124—22no—
59#123—24no—
60#123—22no—
61#123—22no—
62#123—24no—
63#123—22no—
64#123—24no—
65#123—42no—
66C22⋊Q8—24no—
67C22⋊Q8—22no—
68C22⋊Q8—24no—
69#121—22no—
70#120—22no—
71#120—24no—
72#120—42no—
73C4×Q8—22no—
74C4×D4—22no—
75C4×D4—24no—
76C4×D4—24no—
77#117—24no—
78#117—24no—
79#117—22no—
80#117—42no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D412yescomplemented
84C2×C4⋊C4D414yescomplemented
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4D414yescomplemented
92C2×C22⋊C4C2311yescharacteristic
93C2×C22⋊C4C2311yescharacteristic
94C2×C22⋊C4C2312yes—
95C2×C22⋊C4C2312yes—
96C2×C22⋊C4C2311yescharacteristic
97C2×C22⋊C4C2311yescharacteristic
98C2×C22⋊C4C2312yes—
99C2×C42—21no—
100C2×C42D412yescomplemented
101#99—42no—
102#99—42no—
103#95—22no—
104#95—22no—
105#95—24no—
106#95—24no—
107C24—21no—
108C24—22no—
109C24—21no—
110C24—22no—
111C24—44no—
112C24C2×D412yes—
113C24#4112yes—
114C24C2411yescharacteristic
115C2×Q8—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—44no—
124C2×D4—48no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4C2×D412yes—
146C22×C4C2×D414yes—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D412yes—
150C22×C4C2×D412yes—
151C22×C4C2×D414yes—
152C22×C4#4112yes—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—24no—
160C4⋊C4—44no—
161C4⋊C4—42no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—44no—
177C22⋊C4—44no—
178C22⋊C4—44no—
179C22⋊C4—44no—
180C22⋊C4—48no—
181C42—24no—
182C42—22no—
183C42—22no—
184C42—42no—
185C23—24no—
186C23—28no—
187C23—22no—
188C23—24no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—44no—
193C23—42no—
194C23—42no—
195C23—44no—
196C23—48no—
197C23—44no—
198C23—44no—
199C23—41no—
200C23—41no—
201C23—42no—
202C23C22×D411yescharacteristic
203C23C22×D412yes—
204C23C2×#4111yescharacteristic
205C23C2×#4111yescharacteristic
206C23H(5,𝔽2)12yes—
207Q8—82no—
208D4—42no—
209D4—42no—
210D4—42no—
211D4—42no—
212D4—82no—
213D4—88no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—48no—
229C2×C4—48no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—84no—
245C2×C4#12012yes—
246C2×C4#12012yes—
247C22—22no—
248C22—24no—
249C22—28no—
250C22—24no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—44no—
255C22—44no—
256C22—48no—
257C22—42no—
258C22—42no—
259C22—44no—
260C22—42no—
261C22—44no—
262C22—82no—
263C22—82no—
264C22C2×#12011yescharacteristic
265C22#52511yescharacteristic
266C22#53511yescharacteristic
267C22#54612yes—
268C22#54612yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—44no—
275C4—42no—
276C4—81no—
277C4—81no—
278C4—82no—
279C4—82no—
280C2—21no—
281C2—22no—
282C2—22no—
283C2—21no—
284C2—44no—
285C2—82no—
286C2#240411yescharacteristic
287C2#302512yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#2021C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3491C2semidirect
2. #2849C2semidirect
3. #2280C2semidirect
4. #2030C2semidirect
5. #2029C2semidirect
6. #2022C2semidirect
8. #1899C2semidirect
9. #1890C2semidirect
10. #1889C2semidirect
11. #1820C2semidirect
83. C2×C4⋊C4D4semidirect
84. C2×C4⋊C4D4semidirect
91. C2×C22⋊C4D4semidirect
100. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
#34912
#28492
#22802
#20302
#20292
#20222
#20212
#18992
#18902
#18892
#18202