Finite Group Databasebeta

← #23612

Subgroups — #23612

Order 256.

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

normal subgroups only

This group has 607 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
1C2×#427C211yesmaximal, characteristic, complemented
2C2×#424C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic
5#2317C211yesmaximal, characteristic, complemented
6#2309C211yesmaximal, characteristic
7#2067C214yesmaximal, complemented
8#2065C214yesmaximal, complemented
9#1736C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C23×C8C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17#427—24no—
18C2×#107C2212yescomplemented
19C2×#106C2212yescomplemented
20#424—24no—
21C2×C4⋊C8C2212yescomplemented
22C2×C4⋊C8C2212yescomplemented
23#418—24no—
24C2×#102—24no—
25C2×#102C2212yes—
26C2×#102C2214yes—
27C2×#102C2212yes—
28#390—21no—
29#390—21no—
30#390C2214yes—
31#389C2214yescomplemented
32#385—22no—
33#384—21no—
34#382—21no—
35#379—22no—
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×C4C2311yescharacteristic
48C2×M16—22no—
49C2×M16—22no—
50C2×M16—22no—
51C22×C8—22no—
52C22×C8—22no—
53C22×C8D414yes—
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57#117—22no—
58#117—22no—
59#117—22no—
60#117—22no—
61#117—22no—
62#117D414yes—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4D411yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4D412yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42C2312yes—
80#107—24no—
81#106—24no—
82C4⋊C8—24no—
83C4⋊C8—24no—
84#102—24no—
85#102—24no—
86#102—28no—
87#95—24no—
88C24—22no—
89C24—24no—
90C24—42no—
91C24—42no—
92C24—82no—
93C24—82no—
94C24C2×D411yescharacteristic
95C2×D4—28no—
96C2×D4—28no—
97C2×D4—24no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—44no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—44no—
107C2×D4C2×Q814yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—82no—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2×Q812yes—
119C22×C4#4112yes—
120C22×C4#4112yes—
121C22×C4#4112yes—
122C22×C4C2411yescharacteristic
123M16—44no—
124C2×C8—28no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8C2×D412yes—
133C2×C8C2×D412yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—84no—
143C4⋊C4C2×D414yes—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—42no—
147C22⋊C4—84no—
148C22⋊C4—84no—
149C42—24no—
150C42—22no—
151C42—22no—
152C42—42no—
153C23—22no—
154C23—22no—
155C23—24no—
156C23—42no—
157C23—42no—
158C23—44no—
159C23—44no—
160C23—48no—
161C23—88no—
162C23—88no—
163C23—84no—
164C23—84no—
165C23#12112yes—
166C23#12112yes—
167C23C22⋊Q812yes—
168C23C22×D411yescharacteristic
169D4—24no—
170D4—24no—
171D4—42no—
172D4—42no—
173D4—48no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—84no—
187C2×C4#12114yes—
188C2×C4#12112yes—
189C2×C4#12112yes—
190C2×C4C22⋊Q814yes—
191C2×C4C22⋊Q814yes—
192C2×C4C22⋊Q812yes—
193C2×C4#12614yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C22×Q811yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C8—22no—
201C8—22no—
202C8—42no—
203C22—24no—
204C22—24no—
205C22—22no—
206C22—22no—
207C22—48no—
208C22—44no—
209C22—816no—
210C22—84no—
211C22—82no—
212C22—82no—
213C22#39614yes—
214C22C2×#12111yescharacteristic
215C22C2×#12111yescharacteristic
216C22C2×C22⋊Q811yescharacteristic
217C22#57612yes—
218C22#57712yes—
219C4—44no—
220C4—42no—
221C4—42no—
222C4—82no—
223C4—82no—
224C4#39614yes—
225C4C2×#12111yescharacteristic
226C4C2×C22⋊Q811yescharacteristic
227C4C2×C22⋊Q811yescharacteristic
228C4C2×#12611yescharacteristic
229C2—24no—
230C2—84no—
231C2C2×#39611yescharacteristic
232C2#304512yes—
233C2#305711yescharacteristic
234C2#305811yescharacteristic
235C2#308112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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. C2×#427C2semidirect
2. C2×#424C2semidirect
5. #2317C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1736C2semidirect
18. C2×#107C22semidirect
19. C2×#106C22semidirect
21. C2×C4⋊C8C22semidirect
22. C2×C4⋊C8C22semidirect
31. #389C22semidirect
36. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4272
C2×#4242
C2×#4182
C22×#1022
#23172
#23092
#20672
#20652
#17362