Finite Group Databasebeta

← #24065

Subgroups — #24065

Order 256.

536 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#533C211yesmaximal, characteristic
2C2×#485C212yesmaximal
3#2296C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#1873C212yesmaximal
9#1865C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic
12#533—28no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125—21no—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8C2212yes—
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×#117—21no—
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#485—28no—
25C2×#107C2212yes—
26C2×#106C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×C22⋊C8C2212yes—
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#385—21no—
37#385—21no—
38C4×C4⋊C4—21no—
39C4×C4⋊C4—21no—
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8C2312yes—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—24no—
53C4⋊Q8—24no—
54#125—28no—
55C22⋊Q8—28no—
56C22⋊Q8—24no—
57C22⋊Q8D418yes—
58C4×Q8—28no—
59C4×Q8—24no—
60C4×Q8—24no—
61#117—28no—
62#117—22no—
63#117—22no—
64#117—24no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42—24no—
85C2×C42—21no—
86C2×C42D411yescharacteristic
87C2×C42D411yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90#107—24no—
91#106—24no—
92#103—24no—
93#103—24no—
94#103—24no—
95#103—24no—
96C22⋊C8—24no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24C2×D411yescharacteristic
102C2×Q8—24no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8C2×D414yes—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D414yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—28no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4C2×D414yes—
154C4⋊C4#4112yes—
155C4⋊C4#4112yes—
156C22⋊C4—24no—
157C22⋊C4—28no—
158C22⋊C4—42no—
159C42—24no—
160C42—28no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—42no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23#12012yes—
173C23C22×D411yescharacteristic
174Q8—24no—
175Q8—24no—
176Q8—48no—
177Q8—42no—
178Q8—42no—
179C2×C4—28no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4#12012yes—
213C2×C4#13512yes—
214C2×C4#13712yes—
215C2×C4#13712yes—
216C2×C4C22×D412yes—
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C42−1+412yes—
220C8—42no—
221C8—42no—
222C22—22no—
223C22—22no—
224C22—22no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22C2×#12011yescharacteristic
230C22#52511yescharacteristic
231C22#53611yescharacteristic
232C22C2×#13511yescharacteristic
233C22C2×#13711yescharacteristic
234C22C2×#13711yescharacteristic
235C22#57511yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—44no—
243C4—41no—
244C4—44no—
245C4—81no—
246C4—81no—
247C4#54712yes—
248C4#54712yes—
249C2—42no—
250C2#240611yescharacteristic
251C2#300211yescharacteristic
252C2#300411yescharacteristic
253C2#310711yescharacteristic
254C2#311611yescharacteristic
255C2#323111yescharacteristic
256C2#323711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2058C2semidirect
5. #2057C2semidirect
6. #2043C2semidirect
7. #2043C2semidirect
9. #1865C2semidirect
10. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4852
#22962
#20582
#20572
#20432
#18732
#18652
#18632
#18172