Finite Group Databasebeta

← #24353

Subgroups — #24353

Order 256.

539 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 135 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#3501C211yesmaximal, characteristic
2#2627C212yesmaximal, complemented
3#2322C211yesmaximal, characteristic
4#2062C212yesmaximal, complemented
5#2030C212yesmaximal, complemented
6#1996C212yesmaximal, complemented
7#1890C212yesmaximal
8#1867C212yesmaximal
9#1799C211yesmaximal, characteristic
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544—21no—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×#123C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20#518C2212yes—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#99C2212yes—
24#400—22no—
25#398—24no—
26#395—22no—
27#389—21no—
28#388—21no—
29#387—21no—
30#386C2211yescharacteristic
31#385—21no—
32#385C2211yescharacteristic
33#385C2214yes—
34#381C2211yescharacteristic
35#381C2214yes—
36#380C2212yes—
37C4×C22⋊C4C2211yescharacteristic
38#342C2212yes—
39#342C2214yes—
40#342C2214yes—
41C22×Q8—21no—
42C22×Q8D412yes—
43C22×D4C2311yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50C4⋊Q8—24no—
51#125—22no—
52#125—24no—
53#123—22no—
54#123Q814yescomplemented
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—24no—
72C4×D4—24no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D414yes—
86C2×C4⋊C4D412yes—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4C2311yescharacteristic
92C2×C22⋊C4C2314yes—
93C2×C22⋊C4C2311yescharacteristic
94C2×C22⋊C4C2314yes—
95C2×C22⋊C4C2311yescharacteristic
96C2×C42—21no—
97C2×C42D411yescharacteristic
98#99—24no—
99#95—21no—
100#95—22no—
101#95—21no—
102#95—21no—
103#95—24no—
104#95—24no—
105#95—21no—
106#95—24no—
107#95—21no—
108C24C2×D411yescharacteristic
109C24C2411yescharacteristic
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—82no—
115C2×Q8—82no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4C2×Q812yes—
119C2×D4#4112yes—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—24no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D414yes—
139C22×C4C2×D414yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×Q812yescomplemented
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C4⋊C4—81no—
163C4⋊C4—84no—
164C22⋊C4—24no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—24no—
169C22⋊C4—21no—
170C22⋊C4—22no—
171C22⋊C4—24no—
172C22⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—81no—
175C22⋊C4C2×Q812yescomplemented
176C42—24no—
177C42—21no—
178C42—21no—
179C42—81no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23C22⋊Q812yes—
187C23C22×D411yescharacteristic
188C23C22×D412yes—
189C23C22×Q811yescharacteristic
190C23C2×#4111yescharacteristic
191C232−1+412yes—
192Q8—44no—
193Q8—82no—
194Q8—82no—
195Q8—82no—
196D4—24no—
197D4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—44no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4#12011yescharacteristic
235C2×C4#12011yescharacteristic
236C2×C4#12012yes—
237C2×C4C22⋊Q812yes—
238C22—22no—
239C22—21no—
240C22—22no—
241C22—21no—
242C22—22no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—81no—
250C22C2×#12011yescharacteristic
251C22C2×C22⋊Q811yescharacteristic
252C22#53611yescharacteristic
253C22#54812yes—
254C22D4×Q812yes—
255C4—22no—
256C4—21no—
257C4—21no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—41no—
262C4—44no—
263C4—44no—
264C4—41no—
265C4—84no—
266C4—81no—
267C4—81no—
268C2—21no—
269C2—22no—
270C2—21no—
271C2—22no—
272C2—42no—
273C2#240911yescharacteristic
274C2#302611yescharacteristic
275C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radicalC22×D4C23
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
2. #2627C2semidirect
4. #2062C2semidirect
5. #2030C2semidirect
6. #1996C2semidirect
54. #123Q8semidirect
145. C22×C4C2×Q8semidirect
175. C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#26272
#23222
#20622
#20302
#19962
#18902
#18672
#17992