Finite Group Databasebeta

← #18340

Subgroups — #18340

Order 256.

438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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#3458C211yesmaximal, characteristic
2#3236C212yesmaximal, complemented
3#2611C211yesmaximal, characteristic
4#1617C212yesmaximal, complemented
5#1614C212yesmaximal
6#1611C212yesmaximal, complemented
7#1608C212yesmaximal
8#1506C212yesmaximal, complemented
9#1464C211yesmaximal, characteristic
10#544—22no—
11#544—22no—
12#544—21no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—24no—
17#533—21no—
18#533C2212yes—
19C2×#125C2211yescharacteristic
20C2×#117—21no—
21C2×#117C2211yescharacteristic
22#501—22no—
23#491—22no—
24#484C2214yes—
25#483C2214yes—
26#479—24no—
27#432C2212yes—
28#400C2211yescharacteristic
29#399C2211yescharacteristic
30#398C2211yescharacteristic
31#397C2211yescharacteristic
32#389—21no—
33#388—21no—
34#387—21no—
35#385—21no—
36#376C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C23×C4D411yescharacteristic
42C4⋊Q8—24no—
43C4⋊Q8—21no—
44C4⋊Q8—21no—
45C4⋊Q8—24no—
46C4⋊Q8D412yes—
47C4⋊Q8D412yes—
48#125—24no—
49#125—22no—
50#125—24no—
51#125—22no—
52#125—22no—
53#125—22no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—24no—
61C22⋊Q8C2314yes—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8D414yes—
66#117—22no—
67#117—22no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4D411yescharacteristic
86C2×C22⋊C4C2311yescharacteristic
87C2×C42—22no—
88C2×C42—21no—
89C2×C42D411yescharacteristic
90C2×C42C2311yescharacteristic
91#107—24no—
92#106—24no—
93C4⋊C8—22no—
94#103—24no—
95#103—24no—
96C22⋊C8C2314yes—
97#97—22no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24C2×D411yescharacteristic
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8C2×D414yes—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2411yescharacteristic
126C2×C8—24no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4C2×D414yes—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C42—24no—
155C42—24no—
156C42—21no—
157C42—21no—
158C42—42no—
159C42C2×D412yes—
160C42C2×D412yes—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23#13712yes—
166C23C22×D411yescharacteristic
167Q8—44no—
168Q8—82no—
169Q8—82no—
170Q8—82no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4#12011yescharacteristic
200C2×C4#12011yescharacteristic
201C2×C4#12012yes—
202C2×C4#13712yes—
203C2×C4C22×D412yes—
204C8—44no—
205C22—21no—
206C22—21no—
207C22—22no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—81no—
212C22C2×#12011yescharacteristic
213C22C2×#13711yescharacteristic
214C22C2×#13711yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—44no—
223C4—41no—
224C4—44no—
225C4—81no—
226C4—81no—
227C4—81no—
228C2—21no—
229C2—21no—
230C2—42no—
231C2#300411yescharacteristic
232C2#301811yescharacteristic
233C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3236C2semidirect
4. #1617C2semidirect
6. #1611C2semidirect
8. #1506C2semidirect

Maximal subgroups

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

subgroupindex
#34582
#32362
#26112
#16172
#16142
#16112
#16082
#15062
#14642