Finite Group Databasebeta

← #25948

Subgroups — #25948

Order 256.

541 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 129 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#2503C212yesmaximal, complemented
3#2486C211yesmaximal, characteristic
4#2030C214yesmaximal, complemented
5#1996C212yesmaximal, complemented
6#1890C214yesmaximal
7C4×#99C211yesmaximal, characteristic
8#554—22no—
9#554—24no—
10D4×Q8—22no—
11#544—21no—
12#543—21no—
13#543C2212yes—
14#533—21no—
15C2×#123C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8C2212yes—
18#518C2212yes—
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#99C2212yes—
22#400—24no—
23#398—21no—
24#397—21no—
25#391—22no—
26#389—21no—
27#382—22no—
28#381C2218yes—
29#381C2212yes—
30#380—21no—
31#380C2211yescharacteristic
32C4×C22⋊C4C2212yes—
33#342C2218yes—
34#342C2212yes—
35C22×Q8—21no—
36C22×Q8D412yes—
37C22×D4C2311yescharacteristic
38C23×C4D411yescharacteristic
39C23×C4C2312yes—
40C23×C4C2311yescharacteristic
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—24no—
44#125—22no—
45#125—24no—
46#123—28no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—24no—
64C4×D4—24no—
65#117—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D412yes—
76C2×C4⋊C4D414yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4D411yescharacteristic
80C2×C22⋊C4C2312yes—
81C2×C22⋊C4C2314yes—
82C2×C22⋊C4C2311yescharacteristic
83C2×C22⋊C4C2314yes—
84C2×C42—22no—
85C2×C42D411yescharacteristic
86#99—24no—
87#95—22no—
88#95—22no—
89#95—21no—
90#95—28no—
91#95—22no—
92#95—28no—
93C24C2×D411yescharacteristic
94C24C2411yescharacteristic
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—44no—
100C2×Q8—82no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4#4114yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D414yes—
121C22×C4C2×D414yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C4⋊C4—22no—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—24no—
136C4⋊C4—42no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—41no—
140C4⋊C4—44no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—24no—
145C22⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—44no—
150C42—24no—
151C42—21no—
152C42—21no—
153C42—44no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23#12412yes—
161C23C22×D411yescharacteristic
162C23C22×D412yes—
163C23C2×#4112yes—
164C232−1+412yes—
165Q8—44no—
166Q8—82no—
167Q8—82no—
168Q8—82no—
169D4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—44no—
194C2×C4—48no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4#12011yescharacteristic
199C2×C4#12012yes—
200C2×C4#12011yescharacteristic
201C2×C4#12412yes—
202C22—22no—
203C22—21no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—41no—
212C22—81no—
213C22C2×#12011yescharacteristic
214C22C2×#12411yescharacteristic
215C22#53611yescharacteristic
216C22#54814yes—
217C4—22no—
218C4—21no—
219C4—22no—
220C4—41no—
221C4—44no—
222C4—44no—
223C4—41no—
224C4—82no—
225C4—84no—
226C4—82no—
227C4#45812yes—
228C2—21no—
229C2—24no—
230C2—21no—
231C2—42no—
232C2#243611yescharacteristic
233C2C2×#45811yescharacteristic
234C2#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. #2503C2semidirect
4. #2030C2semidirect
5. #1996C2semidirect

Maximal subgroups

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

subgroupindex
#35012
#25032
#24862
#20302
#19962
#18902
C4×#992