Finite Group Databasebeta

← #25854

Subgroups — #25854

Order 256.

547 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 143 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#2583C212yesmaximal, complemented
3#2579C211yesmaximal, characteristic
4#2062C212yesmaximal, complemented
5#2062C212yesmaximal, complemented
6#1996C212yesmaximal, complemented
7#1867C212yesmaximal
8#1867C212yesmaximal
9C4×#99C211yesmaximal, characteristic
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544—21no—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19#518C2212yes—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#99C2212yes—
23#398—21no—
24#398—22no—
25#396—21no—
26#395—22no—
27#389—24no—
28#387—21no—
29#386C2211yescharacteristic
30#386C2211yescharacteristic
31#385—21no—
32#385C2211yescharacteristic
33#385C2211yescharacteristic
34#385C2214yes—
35#385C2214yes—
36#380C2211yescharacteristic
37C4×C22⋊C4C2212yes—
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—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
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—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D414yes—
85C2×C4⋊C4D412yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4D411yescharacteristic
90C2×C22⋊C4C2311yescharacteristic
91C2×C22⋊C4C2311yescharacteristic
92C2×C22⋊C4C2311yescharacteristic
93C2×C22⋊C4C2314yes—
94C2×C22⋊C4C2314yes—
95C2×C42—22no—
96C2×C42D411yescharacteristic
97#99Q818yescomplemented
98#95—22no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—24no—
103#95—24no—
104#95—21no—
105#95—21no—
106C24C2×D411yescharacteristic
107C24C2411yescharacteristic
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—82no—
113C2×Q8—82no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4C2×Q814yes—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—81no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D414yes—
135C22×C4C2×D414yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C4⋊C4—22no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—24no—
152C4⋊C4—42no—
153C4⋊C4—48no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—81no—
158C4⋊C4—84no—
159C4⋊C4—81no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—24no—
164C22⋊C4—24no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—81no—
170C22⋊C4C2×Q818yescomplemented
171C42—24no—
172C42—21no—
173C42—21no—
174C42—82no—
175C23—21no—
176C23—21no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23C4⋊Q812yes—
182C23C22×D411yescharacteristic
183C23C22×D412yes—
184C23C22×Q812yes—
185C232−1+412yes—
186Q8—44no—
187Q8—82no—
188Q8—82no—
189Q8—82no—
190D4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—28no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—44no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—44no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#12012yes—
230C2×C4C4⋊Q812yes—
231C22—22no—
232C22—21no—
233C22—22no—
234C22—21no—
235C22—22no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22C2×#12011yescharacteristic
243C22C2×C4⋊Q811yescharacteristic
244C22#53611yescharacteristic
245C22D4×Q814yes—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—48no—
251C4—42no—
252C4—44no—
253C4—41no—
254C4—41no—
255C4—44no—
256C4—81no—
257C4—81no—
258C4—81no—
259C2—24no—
260C2—21no—
261C2—21no—
262C2—42no—
263C2#243411yescharacteristic
264C2#302511yescharacteristic
265C2#302611yescharacteristic
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. #2583C2semidirect
4. #2062C2semidirect
5. #2062C2semidirect
6. #1996C2semidirect
97. #99Q8semidirect
170. C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#35012
#25832
#25792
#20622
#19962
#18672
C4×#992