Finite Group Databasebeta

← #23610

Subgroups — #23610

Order 256.

551 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C22×#106C211yesmaximal, characteristic, complemented
2C2×#424C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic
4C2×#416C211yesmaximal, characteristic
5#2316C211yesmaximal, characteristic, complemented
6#2311C211yesmaximal, characteristic
7#2066C214yesmaximal, complemented
8#2065C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C23×C8C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C2×#106—24no—
18C2×#106C2214yes—
19#424—24no—
20C2×C4⋊C8C2214yes—
21#417—24no—
22#416—24no—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27#391C2212yes—
28#390C2212yes—
29#389—22no—
30#389C2214yes—
31#386—22no—
32#380—22no—
33#380—22no—
34#340C2214yes—
35C2×#41—22no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—22no—
40C2×#41Q814yes—
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C2311yescharacteristic
48C2×M16—22no—
49C2×M16—22no—
50C2×M16—22no—
51C22×C8—22no—
52C22×C8—22no—
53C22×C8D414yes—
54C22×C8C2312yes—
55C22×C8C2312yes—
56#117—22no—
57#117—22no—
58#117—22no—
59#117—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4D412yes—
67C2×C4⋊C4D414yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4D412yes—
74C2×C42—21no—
75C2×C42C2312yes—
76#106—28no—
77C4⋊C8—28no—
78#103—24no—
79#103—24no—
80#102—24no—
81#102—24no—
82#95—24no—
83#95—24no—
84C24—21no—
85C24C2×D411yescharacteristic
86#41—28no—
87#41—28no—
88#41—44no—
89#41—44no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8C2×Q812yes—
96C2×D4—24no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4C2×Q812yes—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4C2×D414yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×Q812yes—
121C22×C4#4112yes—
122C22×C4#4112yes—
123C22×C4#4112yes—
124C22×C4C2411yescharacteristic
125M16—44no—
126C2×C8—28no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C2×D414yes—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—84no—
142C4⋊C4C2×D414yes—
143C22⋊C4—24no—
144C22⋊C4—42no—
145C22⋊C4—84no—
146C42—22no—
147C42—22no—
148C42—42no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23#12112yes—
157C23#12112yes—
158C23C22⋊Q812yes—
159C23C22×D411yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—22no—
165D4—22no—
166D4—48no—
167D4—41no—
168D4—41no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—44no—
177C2×C4—48no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4—84no—
185C2×C4#12114yes—
186C2×C4#12112yes—
187C2×C4#12112yes—
188C2×C4C22⋊Q814yes—
189C2×C4C22⋊Q814yes—
190C2×C4C22⋊Q812yes—
191C2×C4#12614yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C2×C4C22×Q811yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4C2×#4111yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C8—24no—
199C8—42no—
200C22—24no—
201C22—22no—
202C22—22no—
203C22—24no—
204C22—44no—
205C22—42no—
206C22—81no—
207C22—81no—
208C22—82no—
209C22#39614yes—
210C22C2×#12111yescharacteristic
211C22C2×#12111yescharacteristic
212C22C2×C22⋊Q811yescharacteristic
213C22#57712yes—
214C22#57712yes—
215C4—42no—
216C4—44no—
217C4—44no—
218C4—82no—
219C4—82no—
220C4#39614yes—
221C4C2×#12111yescharacteristic
222C4C2×C22⋊Q811yescharacteristic
223C4C2×C22⋊Q811yescharacteristic
224C4C2×#12611yescharacteristic
225C2—24no—
226C2—82no—
227C2C2×#39611yescharacteristic
228C2#304211yescharacteristic
229C2#304311yescharacteristic
230C2#306012yes—
231C2#308412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#106C2semidirect
2. C2×#424C2semidirect
5. #2316C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1062
C2×#4242
C2×#4172
C2×#4162
#23162
#23112
#20662
#20652
#17362