Finite Group Databasebeta

← #29636

Subgroups — #29636

Order 256.

751 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 119 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#3449C211yesmaximal, characteristic, complemented
2#2482C211yesmaximal, characteristic, complemented
3#2404C212yesmaximal, complemented
4#2044C214yesmaximal, complemented
5#2021C214yesmaximal, complemented
6#1820C211yesmaximal, characteristic, complemented
7#1782C212yesmaximal, complemented
8C22×#41C2212yes—
9#537—21no—
10#537—21no—
11#536—22no—
12#535—22no—
13#534C2214yes—
14#532—22no—
15#525—24no—
16C2×#120C2214yes—
17C2×#117C2211yescharacteristic
18C22×C22⋊C4C2211yescharacteristic
19C2×#99C2218yes—
20#399—21no—
21#397—21no—
22#393—22no—
23#392—22no—
24#388—22no—
25#386—24no—
26#386C2214yes—
27#385C2212yes—
28#381C2212yes—
29#381C2211yescharacteristic
30#379C2212yes—
31#342C2214yes—
32C25C2311yescharacteristic
33C2×#41—22no—
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41—48no—
38C22×Q8D412yescomplemented
39C22×D4—24no—
40C22×D4D412yescomplemented
41C22×D4C2314yes—
42C23×C4D414yescomplemented
43C23×C4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C4⋊Q8—22no—
46#127—22no—
47#124—24no—
48#123—28no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51#121—24no—
52#121—24no—
53#121—24no—
54#121—24no—
55#120—28no—
56#120—48no—
57C4×D4—24no—
58C4×D4—24no—
59#117—22no—
60#117—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D411yescharacteristic, complemented
63C2×C4⋊C4D411yescharacteristic, complemented
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4C2314yes—
73C2×C22⋊C4C2312yes—
74C2×C22⋊C4C2312yes—
75C2×C42D412yescomplemented
76#99—48no—
77#95—22no—
78#95—21no—
79#95—22no—
80#95—21no—
81#95—22no—
82#95—24no—
83C24—21no—
84C24—22no—
85C24—24no—
86C24—42no—
87C24—81no—
88C24C2×D414yes—
89C24C2411yescharacteristic
90#41—48no—
91#41—48no—
92#41—48no—
93#41—48no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—44no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—28no—
100C2×D4—28no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—44no—
106C2×D4—48no—
107C2×D4—48no—
108C2×D4—84no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—81no—
119C22×C4C2×D414yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C4⋊C4—24no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—82no—
131C22⋊C4—28no—
132C22⋊C4—28no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—28no—
136C22⋊C4—48no—
137C22⋊C4—44no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—42no—
141C22⋊C4—82no—
142C22⋊C4—84no—
143C42—22no—
144C42—22no—
145C23—22no—
146C23—24no—
147C23—24no—
148C23—24no—
149C23—24no—
150C23—41no—
151C23—41no—
152C23—44no—
153C23—44no—
154C23—42no—
155C23—44no—
156C23—41no—
157C23—42no—
158C23—41no—
159C23—44no—
160C23—42no—
161C23—88no—
162C23C22×D412yes—
163C23C22×D411yescharacteristic
164C23H(5,𝔽2)14yes—
165Q8—44no—
166Q8—44no—
167D4—44no—
168D4—44no—
169D4—48no—
170D4—48no—
171D4—84no—
172D4—84no—
173D4—816no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—48no—
184C2×C4—48no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4#12011yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#12012yes—
199C22—21no—
200C22—22no—
201C22—21no—
202C22—22no—
203C22—44no—
204C22—48no—
205C22—48no—
206C22—44no—
207C22—44no—
208C22—44no—
209C22—42no—
210C22—44no—
211C22—84no—
212C22—84no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22C2×#12011yescharacteristic
217C22#53414yes—
218C22#53512yes—
219C4—21no—
220C4—22no—
221C4—21no—
222C4—44no—
223C4—42no—
224C4—42no—
225C4—82no—
226C4—84no—
227C4—81no—
228C2—24no—
229C2—22no—
230C2—82no—
231C2—84no—
232C2#261311yescharacteristic
233C2#302511yescharacteristic
234C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
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
1. #3449C2semidirect
2. #2482C2semidirect
3. #2404C2semidirect
4. #2044C2semidirect
5. #2021C2semidirect
6. #1820C2semidirect
7. #1782C2semidirect
38. C22×Q8D4semidirect
40. C22×D4D4semidirect
42. C23×C4D4semidirect
62. C2×C4⋊C4D4semidirect
63. C2×C4⋊C4D4semidirect
75. C2×C42D4semidirect

Maximal subgroups

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

subgroupindex
#34492
#24822
#24042
#20442
#20212
#18202
#17822