Finite Group Databasebeta

← #24502

Subgroups — #24502

Order 256.

496 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 141 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×#451C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic
3#2855C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#1969C212yesmaximal, complemented
8#1933C212yesmaximal, complemented
9#1872C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C42C2211yescharacteristic
17#451—24no—
18#431—24no—
19C2×#107C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C4×C8C2212yes—
28#391—21no—
29#391—21no—
30#391C2211yescharacteristic
31#391C2212yes—
32#391C2211yescharacteristic
33#391C2212yes—
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#387—21no—
37#387—21no—
38#385—21no—
39#381—21no—
40#380C2211yescharacteristic
41C2×#95—21no—
42C2×#95—21no—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×Q8C2311yescharacteristic
51C22×Q8C2312yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×Q16—24no—
58C2×Q16—24no—
59C2×Q16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—42no—
82C4⋊C8—24no—
83#103—24no—
84#103—24no—
85#103—42no—
86#103—42no—
87C22⋊C8—22no—
88C22⋊C8D414yes—
89C4×C8—24no—
90#95—22no—
91#95—22no—
92#95—22no—
93#95—22no—
94#95—21no—
95#95—21no—
96#95—21no—
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—21no—
106C2×Q8—22no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—48no—
113C2×Q8—48no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8#4114yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4Q1614yescomplemented
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4#4112yes—
144C22×C4C2411yescharacteristic
145Q16—48no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8C2×D414yes—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—22no—
165C42—22no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#12112yes—
175C23C2×Q1612yes—
176C23#13512yes—
177C23C22×D411yescharacteristic
178Q8—24no—
179Q8—24no—
180Q8—48no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4#12112yes—
221C2×C4C2×Q1612yes—
222C2×C4C2×Q1612yes—
223C2×C4#13712yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C4C2×#4111yescharacteristic
227C2×C4H(5,𝔽2)12yes—
228C8—24no—
229C8—42no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22#46714yes—
237C22C2×#12111yescharacteristic
238C22#52511yescharacteristic
239C22#53511yescharacteristic
240C22C22×Q1611yescharacteristic
241C22C2×#13511yescharacteristic
242C22C2×#13711yescharacteristic
243C22#57511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—44no—
249C4—41no—
250C4—81no—
251C4—81no—
252C4—81no—
253C4—82no—
254C4—81no—
255C4#54612yes—
256C4#55112yes—
257C2—24no—
258C2#241111yescharacteristic
259C2C2×#46711yescharacteristic
260C2#305611yescharacteristic
261C2#310411yescharacteristic
262C2#310811yescharacteristic
263C2#320911yescharacteristic
264C2#321111yescharacteristic
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
5. #2068C2semidirect
6. #2056C2semidirect
7. #1969C2semidirect
8. #1933C2semidirect
136. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
C2×#4312
#28552
C2×#3912
#20682
#20562
#19692
#19332
#18722
#18522
#17872