Finite Group Databasebeta

← #29962

Subgroups — #29962

Order 256.

475 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 119 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#2629C211yesmaximal, characteristic, complemented
2#2309C211yesmaximal, characteristic
3#2094C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10#1798C211yesmaximal, characteristic, complemented
11#1786C212yesmaximal
12C23×D4C2211yescharacteristic
13C2×#125C2211yescharacteristic, complemented
14C2×#117C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#107C2212yescomplemented
17C2×#106C2212yescomplemented
18C2×C4⋊C8C2212yescomplemented
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#396—22no—
26#390—22no—
27#390C2211yescharacteristic
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#390C2211yescharacteristic
31#389C2212yescomplemented
32#384C2211yescharacteristic, complemented
33#380C2212yes—
34#379—22no—
35#340C2212yescomplemented
36#340C2212yescomplemented
37#336C2211yescharacteristic, complemented
38#336C2211yescharacteristic, complemented
39C25—21no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—42no—
43C22×D4—42no—
44C22×D4Q814yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#125—42no—
51#117—42no—
52#117—42no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D412yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D412yes—
68C2×C42—21no—
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—42no—
72#106—42no—
73C4⋊C8—42no—
74#102—42no—
75#102—42no—
76#102—42no—
77#102—42no—
78C22⋊C8—24no—
79#95—22no—
80#95—22no—
81C24—21no—
82C24—21no—
83C24—24no—
84C24—42no—
85C24—42no—
86C24—82no—
87C24—82no—
88C24C2×D411yescharacteristic
89C2×D4—24no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—48no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—48no—
100C2×D4C2×Q812yes—
101C2×D4C2×Q812yes—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×Q812yes—
112C22×C4#4112yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—44no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—82no—
133C22⋊C4—82no—
134C22⋊C4—82no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C42—22no—
138C42—22no—
139C42—42no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—22no—
146C23—48no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—88no—
156C23—88no—
157C23—84no—
158C23—84no—
159C23C22⋊Q812yes—
160C23C22×D411yescharacteristic
161D4—24no—
162D4—24no—
163D4—48no—
164D4—42no—
165D4—42no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4C22⋊Q812yes—
187C2×C4C22×Q811yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4H(5,𝔽2)12yes—
190C2×C4H(5,𝔽2)12yes—
191C8—42no—
192C8—42no—
193C22—24no—
194C22—22no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22—44no—
199C22—44no—
200C22—42no—
201C22—42no—
202C22—816no—
203C22—82no—
204C22—82no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22C2×C22⋊Q811yescharacteristic
210C22#53411yescharacteristic
211C22#53511yescharacteristic
212C22#57612yes—
213C22#57712yes—
214C4—22no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4#53812yes—
222C4#54312yes—
223C2—42no—
224C2—82no—
225C2—82no—
226C2#262611yescharacteristic
227C2#308112yes—
228C2#319812yes—
229C2#322011yescharacteristic
230C2#322211yescharacteristic
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. #2629C2semidirect
3. #2094C2semidirect
4. #2093C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2025C2semidirect
8. #2025C2semidirect
9. #1854C2semidirect
10. #1798C2semidirect
13. C2×#125C22semidirect
16. C2×#107C22semidirect
17. C2×#106C22semidirect
18. C2×C4⋊C8C22semidirect
31. #389C22semidirect
32. #384C22semidirect
35. #340C22semidirect
36. #340C22semidirect
37. #336C22semidirect
38. #336C22semidirect

Maximal subgroups

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

subgroupindex
#26292
#23092
#20942
#20932
#20672
#20652
#20252
#18542
#17982
#17862