Finite Group Databasebeta

← #29942

Subgroups — #29942

Order 256.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#2283C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2085C212yesmaximal, complemented
5#2077C212yesmaximal
6#2075C212yesmaximal
7#1987C211yesmaximal, characteristic, complemented
8#1987C211yesmaximal, characteristic, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C22⋊Q8—21no—
14C22×C42C2211yescharacteristic
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×C22⋊C8C2212yes—
20#391—21no—
21#389—21no—
22#389C2212yes—
23#384C2211yescharacteristic
24#383—21no—
25#382C2211yescharacteristic
26#382C2211yescharacteristic
27#381—21no—
28#380—21no—
29#380C2211yescharacteristic
30#380C2212yes—
31#380C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33C4×C4⋊C4C2211yescharacteristic
34C4×C22⋊C4—21no—
35#376—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C22⋊Q8—44no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#106—42no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77C22⋊C8—42no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88C24C2×D411yescharacteristic
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C42—22no—
142C42—22no—
143C42—21no—
144C42—21no—
145C42—42no—
146C42—42no—
147C42#4112yes—
148C42#4112yes—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23AGL(1,ℤ/8)12yes—
156C23C22×D411yescharacteristic
157Q8—84no—
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—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4#12312yes—
194C2×C4#12312yes—
195C2×C4#13512yes—
196C2×C4#13712yes—
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4H(5,𝔽2)11yescharacteristic
200C2×C42−1+411yescharacteristic
201C2×C42−1+411yescharacteristic
202C2×C42−1+411yescharacteristic
203C8—42no—
204C8—42no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22C2×#12311yescharacteristic
213C22#53611yescharacteristic
214C22#53711yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×AGL(1,ℤ/8)11yescharacteristic
217C22C2×#13711yescharacteristic
218C22#57511yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—81no—
225C4—81no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4#48514yes—
230C4#54111yescharacteristic
231C4#54111yescharacteristic
232C4#54211yescharacteristic
233C4#54211yescharacteristic
234C2—42no—
235C2#262511yescharacteristic
236C2C2×#48511yescharacteristic
237C2#309411yescharacteristic
238C2#323311yescharacteristic
239C2#323811yescharacteristic
240C2#324711yescharacteristic
241C2#325211yescharacteristic
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. #2101C2semidirect
4. #2085C2semidirect
7. #1987C2semidirect
8. #1987C2semidirect
9. #1948C2semidirect
10. #1947C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22832
#21012
#20852
#20772
#20752
#19872
#19482
#19472
#17972