Finite Group Databasebeta

← #28207

Subgroups — #28207

Order 256.

462 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 129 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×#478C212yesmaximal, complemented
2C4×#127C211yesmaximal, characteristic
3C4×#125C211yesmaximal, characteristic, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic
7#2041C211yesmaximal, characteristic
8#1926C212yesmaximal, complemented
9#1925C212yesmaximal, complemented
10#1831C211yesmaximal, characteristic, complemented
11#1804C212yesmaximal, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×D4—21no—
15C2×C4×D4—21no—
16C2×C4×D4C2212yes—
17#478—28no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25#390—21no—
26#390—21no—
27#390C2212yes—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4C2312yes—
47C23×C4—22no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50#127—24no—
51#125—22no—
52#125D414yescomplemented
53C4×D4—24no—
54C4×D4—24no—
55C4×D4—44no—
56C4×D4—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#107—24no—
81#106—24no—
82C4⋊C8—24no—
83#102—24no—
84#102—24no—
85#102—24no—
86#102—24no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93C24—22no—
94C2×D4—24no—
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4#4114yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4C2×D412yes—
140C4⋊C4C2×D412yes—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C42—22no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—41no—
150C42—42no—
151C42—42no—
152C42C2×D412yes—
153C42#4112yes—
154C23—24no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23C22×D411yescharacteristic
162D4—24no—
163D4—24no—
164D4—48no—
165D4—48no—
166D4—42no—
167D4—42no—
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—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—88no—
197C2×C4#12112yes—
198C2×C4#12112yes—
199C2×C4#13512yes—
200C2×C4AGL(1,ℤ/8)12yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4112yes—
204C2×C4C2×#4111yescharacteristic
205C2×C42−1+412yes—
206C8—42no—
207C8—42no—
208C22—44no—
209C22—44no—
210C22—44no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—82no—
215C22C2×#12111yescharacteristic
216C22#52511yescharacteristic
217C22#53611yescharacteristic
218C22C2×#13511yescharacteristic
219C22C2×AGL(1,ℤ/8)11yescharacteristic
220C22C2×AGL(1,ℤ/8)11yescharacteristic
221C22#57511yescharacteristic
222C4—22no—
223C4—22no—
224C4—22no—
225C4—22no—
226C4—22no—
227C4—41no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—81no—
232C4—82no—
233C4#54812yes—
234C4#55212yes—
235C2—84no—
236C2#253611yescharacteristic
237C2#303511yescharacteristic
238C2#303811yescharacteristic
239C2#310511yescharacteristic
240C2#311411yescharacteristic
241C2#322811yescharacteristic
242C2#323411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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. C2×#478C2semidirect
3. C4×#125C2semidirect
4. #2084C2semidirect
5. #2083C2semidirect
8. #1926C2semidirect
9. #1925C2semidirect
10. #1831C2semidirect
11. #1804C2semidirect
52. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C4×#1272
C4×#1252
#20842
#20832
#20412
#19262
#19252
#18312
#18042