Finite Group Databasebeta

← #25852

Subgroups — #25852

Order 256.

631 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 145 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#2434C212yesmaximal, complemented
3#2409C211yesmaximal, characteristic
4#2062C214yesmaximal, complemented
5#1996C212yesmaximal, complemented
6#1867C214yesmaximal
7C4×#99C211yesmaximal, characteristic
8C23×Q8C2211yescharacteristic
9#554—22no—
10#554—24no—
11D4×Q8—22no—
12#544—21no—
13#543—21no—
14#543C2212yes—
15#533—21no—
16C2×C4⋊Q8—22no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20#518C2212yes—
21C2×C4×D4C2211yescharacteristic
22C2×#99C2212yes—
23#395—21no—
24#395—24no—
25#393—21no—
26#391—22no—
27#391—21no—
28#386C2212yes—
29#385C2212yes—
30#385C2218yes—
31#380C2211yescharacteristic
32C4×C22⋊C4C2212yes—
33#342C2212yes—
34#342C2218yes—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×Q8—22no—
38C22×Q8—21no—
39C22×Q8—22no—
40C22×Q8—44no—
41C22×Q8D412yes—
42C22×D4C2311yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2312yes—
45C23×C4C2311yescharacteristic
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48C4⋊Q8—24no—
49C4⋊Q8—48no—
50#125—22no—
51#125—24no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64C22⋊Q8—44no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—24no—
70C4×D4—24no—
71#117—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—28no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D412yes—
79C2×C4⋊C4D414yes—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D411yescharacteristic
83C2×C22⋊C4C2312yes—
84C2×C22⋊C4C2311yescharacteristic
85C2×C22⋊C4C2314yes—
86C2×C22⋊C4C2314yes—
87C2×C42—22no—
88C2×C42D411yescharacteristic
89#99Q818yescomplemented
90#95—22no—
91#95—21no—
92#95—22no—
93#95—28no—
94#95—22no—
95C24C2×D411yescharacteristic
96C24C2411yescharacteristic
97C2×Q8—24no—
98C2×Q8—28no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—44no—
104C2×Q8—42no—
105C2×Q8—44no—
106C2×Q8—48no—
107C2×Q8—44no—
108C2×Q8—41no—
109C2×Q8—42no—
110C2×Q8—41no—
111C2×Q8—82no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4C2×Q814yes—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D414yes—
131C22×C4C2×D414yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D412yes—
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—24no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4—416no—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—22no—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—24no—
156C22⋊C4—44no—
157C22⋊C4C2×Q818yescomplemented
158C42—24no—
159C42—21no—
160C42—21no—
161C42—44no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23C4⋊Q812yescomplemented
169C23C22×D411yescharacteristic
170C23C22×D412yes—
171C23C22×Q812yes—
172C232−1+412yes—
173Q8—44no—
174Q8—48no—
175Q8—44no—
176Q8—44no—
177Q8—82no—
178Q8—82no—
179Q8—82no—
180D4—28no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—28no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—28no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4#12011yescharacteristic
211C2×C4#12012yes—
212C2×C4#12011yescharacteristic
213C2×C4C4⋊Q812yescomplemented
214C22—22no—
215C22—21no—
216C22—22no—
217C22—21no—
218C22—22no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22C2×#12011yescharacteristic
226C22C2×C4⋊Q811yescharacteristic
227C22#53611yescharacteristic
228C22D4×Q814yes—
229C4—22no—
230C4—21no—
231C4—22no—
232C4—48no—
233C4—44no—
234C4—41no—
235C4—44no—
236C4—44no—
237C4—41no—
238C4—82no—
239C4B(4,𝔽2)12yes—
240C2—24no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2#243411yescharacteristic
245C2C2×B(4,𝔽2)11yescharacteristic
246C2#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. #2434C2semidirect
4. #2062C2semidirect
5. #1996C2semidirect
89. #99Q8semidirect
157. C22⋊C4C2×Q8semidirect
168. C23C4⋊Q8semidirect
213. C2×C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#24342
#24092
#20622
#19962
#18672
C4×#992