Finite Group Databasebeta

← #23618

Subgroups — #23618

Order 256.

659 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 179 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×#428C211yesmaximal, characteristic, complemented
2C2×#423C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic
4C2×#418C211yesmaximal, characteristic
5#2296C211yesmaximal, characteristic, complemented
6C2×#390C211yesmaximal, characteristic
7#2067C214yesmaximal, complemented
8#2065C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4—22no—
17C22×C42C2211yescharacteristic
18#428—24no—
19C2×#107C2212yescomplemented
20C2×#106C2212yescomplemented
21#423—24no—
22C2×C4⋊C8C2214yescomplemented
23#418—24no—
24#418—24no—
25C2×#102C2212yes—
26C2×#102C2214yes—
27C2×#102C2212yes—
28#390—21no—
29#390—21no—
30#390C2214yes—
31#389C2212yescomplemented
32#389C2212yescomplemented
33#385—22no—
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36#340C2214yescomplemented
37C25—21no—
38C22×D4—22no—
39C22×D4—22no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—22no—
43C22×D4—22no—
44C22×D4Q814yes—
45C22×D4C2312yes—
46C23×C4—21no—
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C2×M16—22no—
52C2×M16—22no—
53C2×M16—22no—
54C2×M16D414yes—
55C22×C8C2312yes—
56C22×C8C2312yes—
57#117—22no—
58#117—22no—
59#117—22no—
60#117D414yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—48no—
75C2×C42—21no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2312yes—
82#107—24no—
83#106—24no—
84C4⋊C8—28no—
85#102—28no—
86#102—24no—
87#102—24no—
88#95—22no—
89#95—22no—
90C24—24no—
91C24—22no—
92C24—44no—
93C24—44no—
94C24—42no—
95C24—42no—
96C24C2×D411yescharacteristic
97C2×D4—28no—
98C2×D4—28no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—44no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—44no—
109C2×D4C2×Q814yes—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—24no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—42no—
117C22×C4—44no—
118C22×C4—42no—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×Q812yes—
125C22×C4#4112yes—
126C22×C4#4112yes—
127C22×C4#4112yes—
128C22×C4C2411yescharacteristic
129M16—28no—
130M16—44no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C2×D414yes—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4C2×D412yes—
147C4⋊C4C2×D412yes—
148C22⋊C4—24no—
149C22⋊C4—48no—
150C22⋊C4—42no—
151C22⋊C4—48no—
152C42—22no—
153C42—22no—
154C42—24no—
155C42—22no—
156C42—22no—
157C42—42no—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—416no—
162C23—42no—
163C23—42no—
164C23—44no—
165C23—44no—
166C23—48no—
167C23—84no—
168C23—84no—
169C23—88no—
170C23#12112yes—
171C23#12112yes—
172C23C22⋊Q812yes—
173C23C22×D411yescharacteristic
174D4—24no—
175D4—24no—
176D4—42no—
177D4—42no—
178D4—48no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—84no—
195C2×C4#12114yes—
196C2×C4#12112yes—
197C2×C4#12112yes—
198C2×C4C22⋊Q814yes—
199C2×C4C22⋊Q814yes—
200C2×C4C22⋊Q812yes—
201C2×C4#12614yes—
202C2×C4AGL(1,ℤ/8)14yes—
203C2×C4C22×D411yescharacteristic
204C2×C4C22×D411yescharacteristic
205C2×C4C22×Q811yescharacteristic
206C2×C4C2×#4111yescharacteristic
207C2×C4C2×#4111yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C8—24no—
210C8—42no—
211C22—24no—
212C22—22no—
213C22—22no—
214C22—24no—
215C22—48no—
216C22—44no—
217C22—816no—
218C22—82no—
219C22—82no—
220C22—84no—
221C22#39614yes—
222C22C2×#12111yescharacteristic
223C22C2×#12111yescharacteristic
224C22C2×C22⋊Q811yescharacteristic
225C22C2×AGL(1,ℤ/8)12yes—
226C22#57512yes—
227C4—24no—
228C4—42no—
229C4—42no—
230C4—42no—
231C4—82no—
232C4—82no—
233C4#39614yes—
234C4C2×#12111yescharacteristic
235C4C2×C22⋊Q811yescharacteristic
236C4C2×C22⋊Q811yescharacteristic
237C4C2×#12611yescharacteristic
238C2—24no—
239C2—84no—
240C2C2×#39611yescharacteristic
241C2#303811yescharacteristic
242C2#304011yescharacteristic
243C2#305512yes—
244C2#307712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
1. C2×#428C2semidirect
2. C2×#423C2semidirect
5. #2296C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1736C2semidirect
19. C2×#107C22semidirect
20. C2×#106C22semidirect
22. C2×C4⋊C8C22semidirect
31. #389C22semidirect
32. #389C22semidirect
36. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C2×#4232
C2×#4182
#22962
C2×#3902
#20672
#20652
#17362