Finite Group Databasebeta

← #30367

Subgroups — #30367

Order 256.

358 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 113 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#2857C211yesmaximal, characteristic
2#2855C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2075C212yesmaximal
6#2058C211yesmaximal, characteristic, complemented
7#2035C212yesmaximal
8#2024C211yesmaximal, characteristic, complemented
9#1943C212yesmaximal, complemented
10#1935C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C2×C22⋊Q8C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×C22⋊C8C2212yes—
21C2×#97C2212yes—
22#391—21no—
23#391C2211yescharacteristic
24#391C2211yescharacteristic
25#389—21no—
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2212yes—
29#387—21no—
30#387—21no—
31#385—21no—
32#384C2211yescharacteristic
33#381—21no—
34#380—21no—
35#380C2211yescharacteristic
36#380C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—42no—
51C22⋊Q8—42no—
52C22⋊Q8—44no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C22⋊C4—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—42no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—24no—
74#106—42no—
75#103—24no—
76#103—42no—
77C22⋊C8—24no—
78#97—42no—
79#95—22no—
80#95—21no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24C2×D411yescharacteristic
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—42no—
94C2×Q8—42no—
95C2×Q8—42no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—44no—
142C42—22no—
143C42—22no—
144C42—41no—
145C42—41no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23AGL(1,ℤ/8)12yes—
153C23C22×D411yescharacteristic
154Q8—42no—
155Q8—42no—
156Q8—84no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4#13712yes—
193C2×C4#13712yes—
194C2×C4C2×#4112yes—
195C2×C4H(5,𝔽2)12yes—
196C2×C42−1+411yescharacteristic
197C2×C42−1+411yescharacteristic
198C8—24no—
199C8—42no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22#53211yescharacteristic
208C22#53411yescharacteristic
209C22#53611yescharacteristic
210C22C2×AGL(1,ℤ/8)11yescharacteristic
211C22C2×#13711yescharacteristic
212C22C2×#13711yescharacteristic
213C22#57511yescharacteristic
214C4—22no—
215C4—22no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—81no—
220C4—82no—
221C4—81no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4#53912yes—
226C4#54212yes—
227C2—42no—
228C2#266311yescharacteristic
229C2#318011yescharacteristic
230C2#318511yescharacteristic
231C2#319311yescharacteristic
232C2#319611yescharacteristic
233C2#323711yescharacteristic
234C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2100C2semidirect
4. #2095C2semidirect
6. #2058C2semidirect
8. #2024C2semidirect
9. #1943C2semidirect
10. #1935C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#28552
#21002
#20952
#20752
#20582
#20352
#20242
#19432
#19352
#17492