Finite Group Databasebeta

← #29959

Subgroups — #29959

Order 256.

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

normal subgroups only

This group has 479 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#2628C211yesmaximal, characteristic, complemented
2#2309C211yesmaximal, characteristic
3#2092C212yesmaximal, complemented
4#2067C212yesmaximal, complemented
5#2065C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1854C212yesmaximal, complemented
8#1798C211yesmaximal, characteristic, complemented
9#1786C212yesmaximal
10C23×D4C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic, complemented
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#107C2212yescomplemented
15C2×#106C2212yescomplemented
16C2×C4⋊C8C2211yescharacteristic, complemented
17C2×C4⋊C8C2211yescharacteristic, complemented
18C2×#102C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2212yes—
23#400—21no—
24#399—21no—
25#395—21no—
26#393—21no—
27#390—21no—
28#390—21no—
29#390C2212yes—
30#390C2212yes—
31#389C2212yescomplemented
32#382C2211yescharacteristic, complemented
33#380C2212yes—
34#379—21no—
35#379—21no—
36#340C2212yescomplemented
37#340C2212yescomplemented
38#336C2212yescomplemented
39C25—21no—
40C22×Q8—21no—
41C22×D4—21no—
42C22×D4—22no—
43C22×D4—41no—
44C22×D4—41no—
45C22×D4—41no—
46C22×D4—41no—
47C22×D4Q814yescomplemented
48C22×D4C2312yes—
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—41no—
53C4⋊Q8—41no—
54#117—41no—
55#117—41no—
56#117—41no—
57#117—41no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4D411yescharacteristic
72C2×C22⋊C4D411yescharacteristic
73C2×C42—21no—
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—42no—
77#106—42no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80#102—42no—
81#102—42no—
82#102—42no—
83#102—42no—
84C22⋊C8—24no—
85#95—22no—
86#95—21no—
87#95—21no—
88#95—22no—
89C24—22no—
90C24—24no—
91C24—42no—
92C24—42no—
93C24—82no—
94C24—82no—
95C24C2×D411yescharacteristic
96C2×Q8—42no—
97C2×Q8—82no—
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—42no—
107C2×D4—42no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4C2×Q814yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×Q812yes—
124C22×C4#4112yes—
125C22×C4C2411yescharacteristic
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—82no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C22⋊C4—82no—
148C22⋊C4—82no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—42no—
155C23—21no—
156C23—21no—
157C23—22no—
158C23—22no—
159C23—24no—
160C23—48no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—88no—
168C23—84no—
169C23—84no—
170C23—88no—
171C23C22⋊Q812yes—
172C23C22×D411yescharacteristic
173Q8—84no—
174D4—24no—
175D4—24no—
176D4—48no—
177D4—42no—
178D4—42no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4C22⋊Q812yes—
203C2×C4C22×Q811yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4H(5,𝔽2)11yescharacteristic
206C2×C4H(5,𝔽2)11yescharacteristic
207C2×C4H(5,𝔽2)11yescharacteristic
208C2×C4H(5,𝔽2)11yescharacteristic
209C8—42no—
210C8—42no—
211C22—24no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—816no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22C2×C22⋊Q811yescharacteristic
225C22#53411yescharacteristic
226C22#53511yescharacteristic
227C22#57612yes—
228C22#57712yes—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—82no—
235C4—82no—
236C4—82no—
237C4#53811yescharacteristic
238C4#53811yescharacteristic
239C4#54311yescharacteristic
240C4#54311yescharacteristic
241C2—42no—
242C2—84no—
243C2#262611yescharacteristic
244C2#308112yes—
245C2#319911yescharacteristic
246C2#320011yescharacteristic
247C2#322412yes—
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. #2628C2semidirect
3. #2092C2semidirect
4. #2067C2semidirect
5. #2065C2semidirect
6. #2023C2semidirect
7. #1854C2semidirect
8. #1798C2semidirect
11. C2×C4⋊Q8C22semidirect
14. C2×#107C22semidirect
15. C2×#106C22semidirect
16. C2×C4⋊C8C22semidirect
17. C2×C4⋊C8C22semidirect
31. #389C22semidirect
32. #382C22semidirect
36. #340C22semidirect
37. #340C22semidirect
38. #336C22semidirect
47. C22×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#26282
#23092
#20922
#20672
#20652
#20232
#18542
#17982
#17862