Finite Group Databasebeta

← #23911

Subgroups — #23911

Order 256.

869 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 147 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#3486C211yesmaximal, characteristic
2C22×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#456C212yesmaximal, complemented
4#3004C212yesmaximal, complemented
5#2005C214yesmaximal, complemented
6#1896C212yesmaximal, complemented
7#1886C212yesmaximal, complemented
8#1784C211yesmaximal, characteristic, complemented
9C2×2−1+4C2211yescharacteristic
10C23×Q8C2211yescharacteristic
11C23×Q8C2211yescharacteristic
12C2×#137C2214yes—
13D4×Q8—24no—
14#548—24no—
15#536—22no—
16#536C2212yes—
17C2×C4⋊Q8—22no—
18C2×C4⋊Q8—28no—
19C2×C4⋊Q8C2212yes—
20C2×C4⋊Q8C2212yes—
21#525—22no—
22#525C2212yes—
23#519C2211yescharacteristic
24C22×C4⋊C4—22no—
25C22×C42C2211yescharacteristic
26#456—28no—
27#451—24no—
28#448—24no—
29C2×#104C2214yes—
30#419C2214yes—
31C2×#101C2212yes—
32C2×#101C2212yes—
33#407C2212yes—
34#339C2214yes—
352−1+4—24no—
362−1+4—42no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41C2312yes—
40C22×Q8—24no—
41C22×Q8—24no—
42C22×Q8—22no—
43C22×Q8—21no—
44C22×Q8—24no—
45C22×Q8—24no—
46C22×Q8—42no—
47C22×Q8—42no—
48C22×Q8D412yes—
49C22×Q8C2312yes—
50C22×Q8C2311yescharacteristic
51C22×Q8C2311yescharacteristic
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56#137—28no—
57#137—44no—
58C2×Q16—24no—
59C2×SD16—24no—
60C2×M16C2314yes—
61C4⋊Q8—216no—
62C4⋊Q8—24no—
63C4⋊Q8—24no—
64C4⋊Q8—24no—
65C4⋊Q8—22no—
66C4⋊Q8—24no—
67C4⋊Q8D418yescomplemented
68#124—24no—
69#124—22no—
70#123—24no—
71#123—22no—
72C22⋊Q8—24no—
73C22⋊Q8—28no—
74C22⋊Q8—24no—
75C22⋊Q8D414yes—
76#121—24no—
77#121D414yes—
78#120—22no—
79C4×Q8—24no—
80C4×D4—28no—
81C4×D4—24no—
82#117—22no—
83#117C2312yes—
84C2×C4⋊C4—216no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C42—21no—
88C2×C42—24no—
89C2×C42C2312yes—
90#104—28no—
91#103—28no—
92#101—24no—
93#101—42no—
94C22⋊C8—24no—
95C24C2×D411yescharacteristic
96#41—24no—
97#41—24no—
98#41—44no—
99#41—48no—
100#41—44no—
101C2×Q8—22no—
102C2×Q8—24no—
103C2×Q8—24no—
104C2×Q8—24no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—24no—
112C2×Q8—24no—
113C2×Q8—42no—
114C2×Q8—416no—
115C2×Q8—416no—
116C2×Q8—42no—
117C2×Q8—48no—
118C2×Q8—42no—
119C2×Q8C2×D412yes—
120C2×Q8C2×D414yes—
121C2×Q8C2×D414yes—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4C2×D412yes—
126C22×C4—24no—
127C22×C4—28no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D412yes—
143C22×C4C2×D414yes—
144C22×C4C2411yescharacteristic
145Q16—48no—
146SD16—48no—
147M16—24no—
148M16—24no—
149C2×C8—24no—
150C4⋊C4—216no—
151C4⋊C4—24no—
152C4⋊C4—28no—
153C4⋊C4—216no—
154C4⋊C4—24no—
155C4⋊C4C2×D414yes—
156C22⋊C4—28no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—24no—
160C42—24no—
161C42—24no—
162C42—22no—
163C42—22no—
164C42C2×D414yes—
165C23—21no—
166C23—21no—
167C23—24no—
168C23—22no—
169C23#12012yes—
170C23C22×D411yescharacteristic
171Q8—28no—
172Q8—28no—
173Q8—24no—
174Q8—48no—
175Q8—48no—
176Q8—44no—
177Q8—44no—
178Q8—44no—
179D4—24no—
180D4—44no—
181D4—42no—
182D4—42no—
183D4—48no—
184C2×C4—216no—
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—48no—
200C2×C4—48no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4#12014yes—
205C2×C4#12012yes—
206C2×C4C22×D412yes—
207C2×C4C22×D412yes—
208C2×C4H(5,𝔽2)12yes—
209C8—44no—
210C22—24no—
211C22—24no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—44no—
217C22#45614yescomplemented
218C22C2×#12011yescharacteristic
219C22C2×#12011yescharacteristic
220C22#53411yescharacteristic
221C4—28no—
222C4—22no—
223C4—24no—
224C4—42no—
225C4—42no—
226C4—44no—
227C4—44no—
228C4D4214yes—
229C2—21no—
230C2—21no—
231C2—24no—
232C2—82no—
233C2#240311yescharacteristic
234C2C2×#45612yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3486C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Q8C2semidirect
3. C2×#456C2semidirect
4. #3004C2semidirect
5. #2005C2semidirect
6. #1896C2semidirect
7. #1886C2semidirect
8. #1784C2semidirect
67. C4⋊Q8D4semidirect
217. C22#456semidirect

Maximal subgroups

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

subgroupindex
#34862
C22×C4⋊Q82
C2×#4562
#30042
#20052
#18962
#18862
#17842