Finite Group Databasebeta

← #32711

Subgroups — #32711

Order 256.

492 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 109 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#2848C212yesmaximal
2#2077C212yesmaximal, complemented
3#2075C211yesmaximal, characteristic, complemented
4#2036C212yesmaximal, complemented
5#2035C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#2001C211yesmaximal, characteristic, complemented
8#2001C212yesmaximal, complemented
9#1779C211yesmaximal, characteristic
10C24×C4C2211yescharacteristic
11C22×Q16C2211yescharacteristic, complemented
12C2×C22⋊Q8C2212yes—
13C2×C22⋊Q8C2212yes—
14C2×C22⋊Q8C2212yes—
15C2×#107C2211yescharacteristic, complemented
16C2×#106C2212yescomplemented
17C2×#103C2212yescomplemented
18C2×#103C2212yescomplemented
19C2×#103C2212yescomplemented
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2212yes—
24#385—22no—
25#385—22no—
26#385—22no—
27#380C2212yes—
28#380C2212yes—
29#380C2212yes—
30#379—22no—
31#340C2212yescomplemented
32#340C2211yescharacteristic, complemented
33#340C2211yescharacteristic, complemented
34#340C2212yescomplemented
35C25D411yescharacteristic
36C22×Q8C2312yes—
37C23×C4—42no—
38C23×C4—41no—
39C23×C4—42no—
40C23×C4—41no—
41C23×C4D411yescharacteristic
42C23×C4D412yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C2×Q16—42no—
46C2×Q16—41no—
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C22⋊Q8—42no—
51C22⋊Q8—42no—
52C22⋊Q8—42no—
53C22⋊Q8—42no—
54C22⋊Q8—42no—
55C22⋊Q8—42no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68#107—41no—
69#106—42no—
70#103—42no—
71#103—42no—
72#103—42no—
73C22⋊C8—41no—
74C22⋊C8—42no—
75C22⋊C8—42no—
76C22⋊C8—41no—
77#95—22no—
78#95—22no—
79#95—22no—
80C24—22no—
81C24—24no—
82C24—42no—
83C24—41no—
84C24—41no—
85C24C2×D411yescharacteristic
86C24C2×D412yes—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—41no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—48no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2411yescharacteristic
118Q16—44no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—21no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C22⋊C4—44no—
136C22⋊C4—44no—
137C22⋊C4—44no—
138C22⋊C4—44no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—21no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—21no—
151C23—48no—
152C23—44no—
153C23—44no—
154C23—48no—
155C23—44no—
156C23#13512yes—
157C23#13714yes—
158C23C22×D411yescharacteristic
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—48no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4H(5,𝔽2)12yes—
190C2×C4H(5,𝔽2)11yescharacteristic
191C2×C4H(5,𝔽2)12yes—
192C2×C4H(5,𝔽2)11yescharacteristic
193C8—42no—
194C8—41no—
195C8—41no—
196C22—24no—
197C22—24no—
198C22—22no—
199C22—24no—
200C22—24no—
201C22—416no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—42no—
206C22—42no—
207C22—41no—
208C22—42no—
209C22#53412yes—
210C22#53411yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×#13712yes—
213C22#57511yescharacteristic
214C4—41no—
215C4—42no—
216C4—82no—
217C4—82no—
218C4—82no—
219C4—82no—
220C4#56012yes—
221C4#56011yescharacteristic
222C4H(3,𝔽4)11yescharacteristic
223C2—42no—
224C2—44no—
225C2#281211yescharacteristic
226C2#319012yes—
227C2#319111yescharacteristic
228C2#319511yescharacteristic
229C2#319612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2077C2semidirect
3. #2075C2semidirect
4. #2036C2semidirect
5. #2035C2semidirect
6. #2035C2semidirect
7. #2001C2semidirect
8. #2001C2semidirect
11. C22×Q16C22semidirect
15. C2×#107C22semidirect
16. C2×#106C22semidirect
17. C2×#103C22semidirect
18. C2×#103C22semidirect
19. C2×#103C22semidirect
31. #340C22semidirect
32. #340C22semidirect
33. #340C22semidirect
34. #340C22semidirect

Maximal subgroups

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

subgroupindex
#28482
#20772
#20752
#20362
#20352
#20012
#17792