Finite Group Databasebeta

← #29986

Subgroups — #29986

Order 256.

359 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 127 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#2859C211yesmaximal, characteristic, complemented
2#2293C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal
5#2059C212yesmaximal, complemented
6#2025C212yesmaximal
7#1948C211yesmaximal, characteristic, complemented
8#1947C211yesmaximal, characteristic, complemented
9#1831C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic
11#1802C211yesmaximal, characteristic
12C2×#125C2212yes—
13C2×C4×D4C2211yescharacteristic
14C2×#117—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×#102C2212yes—
20C2×#102C2212yes—
21#390C2211yescharacteristic
22#390C2211yescharacteristic
23#389—21no—
24#389—21no—
25#389—21no—
26#389—21no—
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#388—21no—
31#384C2211yescharacteristic
32#384C2211yescharacteristic
33#382C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—21no—
36C43C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—42no—
48#125—42no—
49C4×D4—22no—
50C4×D4Q814yes—
51#117—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—21no—
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82#102—42no—
83#102—42no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89C24—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4C2×Q812yes—
96C22×C4—21no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4C2×Q812yes—
135C22⋊C4—24no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C42—22no—
141C42—22no—
142C42—21no—
143C42—21no—
144C42—21no—
145C42—21no—
146C42—41no—
147C42—41no—
148C42—42no—
149C42C2×Q812yes—
150C42#4112yes—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23C22×D411yescharacteristic
159D4—22no—
160D4—22no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—48no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4C22⋊Q812yes—
194C2×C4C22⋊Q812yes—
195C2×C4#13512yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4C22×Q811yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4H(5,𝔽2)11yescharacteristic
200C2×C4H(5,𝔽2)11yescharacteristic
201C2×C42−1+411yescharacteristic
202C2×C42−1+411yescharacteristic
203C8—42no—
204C8—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22C2×C22⋊Q811yescharacteristic
213C22#53511yescharacteristic
214C22#53611yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×AGL(1,ℤ/8)11yescharacteristic
217C22#57511yescharacteristic
218C22#57511yescharacteristic
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—82no—
226C4—82no—
227C4—81no—
228C4—81no—
229C4#47814yes—
230C4#54211yescharacteristic
231C4#54211yescharacteristic
232C4#54411yescharacteristic
233C4#54411yescharacteristic
234C2—82no—
235C2#262811yescharacteristic
236C2C2×#47811yescharacteristic
237C2#308011yescharacteristic
238C2#321211yescharacteristic
239C2#321411yescharacteristic
240C2#323311yescharacteristic
241C2#323511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2859C2semidirect
3. #2101C2semidirect
5. #2059C2semidirect
7. #1948C2semidirect
8. #1947C2semidirect
9. #1831C2semidirect

Maximal subgroups

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

subgroupindex
#28592
#22932
#21012
#20922
#20592
#20252
#19482
#19472
#18312
#18022