Finite Group Databasebeta

← #26463

Subgroups — #26463

Order 256.

449 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 139 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
2#2552C211yesmaximal, characteristic, complemented
3#2366C211yesmaximal, characteristic
4#2065C212yesmaximal, complemented
5#2025C212yesmaximal, complemented
6#1982C211yesmaximal, characteristic
7#1957C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1862C212yesmaximal, complemented
11C4×#106C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2212yes—
14C2×C4×D4—21no—
15C2×C4×D4C2212yes—
16#478—28no—
17C2×#107C2212yes—
18C2×#106C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C4×C8C2211yescharacteristic
27#398—22no—
28#390—21no—
29#390C2212yes—
30#389—21no—
31#389C2212yes—
32#386—22no—
33#384—21no—
34#384C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×D4—21no—
43C22×D4—21no—
44C22×D4C2312yes—
45C23×C4—22no—
46C22×C8C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2311yescharacteristic
49#127—22no—
50#127Q814yes—
51#125—24no—
52#125—42no—
53C4×D4—24no—
54C4×D4—42no—
55C4×D4—44no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
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×C42C2312yes—
74C2×C42C2311yescharacteristic
75#107—24no—
76#106—22no—
77#106D414yescomplemented
78C4⋊C8—24no—
79C4⋊C8—22no—
80#102—24no—
81#102—24no—
82#102—24no—
83#102—42no—
84C4×C8—22no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95D412yes—
89C24—22no—
90C2×D4—24no—
91C2×D4—24no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—81no—
100C2×D4—81no—
101C2×D4C2×Q814yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—44no—
108C22×C4—44no—
109C22×C4—82no—
110C22×C4C2×D412yes—
111C22×C4C2×D412yes—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2411yescharacteristic
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8C2×D412yes—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C4⋊C4—82no—
138C4⋊C4C2×D412yes—
139C4⋊C4C2×D412yes—
140C4⋊C4#4112yes—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C22⋊C4—82no—
144C42—22no—
145C42—22no—
146C42—42no—
147C42—42no—
148C42C2×Q812yes—
149C42#4112yes—
150C23—24no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23C22×D411yescharacteristic
158D4—24no—
159D4—24no—
160D4—42no—
161D4—42no—
162D4—48no—
163D4—84no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—88no—
188C2×C4C22⋊Q812yes—
189C2×C4C22⋊Q812yes—
190C2×C4#13512yes—
191C2×C4AGL(1,ℤ/8)12yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C22×Q811yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4C2×#4111yescharacteristic
196C2×C4H(5,𝔽2)11yescharacteristic
197C2×C42−1+411yescharacteristic
198C8—22no—
199C8—42no—
200C8—41no—
201C22—44no—
202C22—44no—
203C22—44no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22C2×C22⋊Q811yescharacteristic
209C22D4211yescharacteristic
210C22#54811yescharacteristic
211C22C2×#13511yescharacteristic
212C22C2×AGL(1,ℤ/8)11yescharacteristic
213C22#57611yescharacteristic
214C22#57711yescharacteristic
215C4—22no—
216C4—42no—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—82no—
223C4#47814yescomplemented
224C4#53211yescharacteristic
225C4#54211yescharacteristic
226C4D4×Q811yescharacteristic
227C4#55411yescharacteristic
228C2—84no—
229C2#245211yescharacteristic
230C2C2×#47811yescharacteristic
231C2#308111yescharacteristic
232C2#327411yescharacteristic
233C2#328411yescharacteristic
234C2#336111yescharacteristic
235C2#336811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#478C2semidirect
2. #2552C2semidirect
4. #2065C2semidirect
5. #2025C2semidirect
8. #1947C2semidirect
9. #1939C2semidirect
10. #1862C2semidirect
11. C4×#106C2semidirect
77. #106D4semidirect
223. C4#478semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
#25522
#23662
#20652
#20252
#19822
#19572
#19472
#19392
#18622
C4×#1062