Finite Group Databasebeta

← #29910

Subgroups — #29910

Order 256.

384 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 115 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#2854C211yesmaximal, characteristic
2#2285C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2071C211yesmaximal, characteristic, complemented
8#2071C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal
10#1854C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C2×#117C2212yes—
15C22×C42C2211yescharacteristic
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×#102C2212yes—
19C2×#102C2212yes—
20C2×C22⋊C8C2212yes—
21#390C2211yescharacteristic
22#390C2211yescharacteristic
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#388—21no—
26#388—21no—
27#386—21no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#382C2212yes—
31#380—21no—
32#380—21no—
33#380C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4—21no—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×D4—21no—
44C22×D4C2311yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50#123—44no—
51#121—24no—
52#117—42no—
53#117—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
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—22no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—24no—
73C2×C42—22no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#107—42no—
79#106—42no—
80#102—42no—
81#102—42no—
82C22⋊C8—24no—
83#95—22no—
84#95—22no—
85#95—21no—
86#95—21no—
87#95—21no—
88C24—21no—
89C24C2×D411yescharacteristic
90C2×D4—22no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4—42no—
95C2×D4—42no—
96C2×D4—42no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4#4114yes—
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—44no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23#12312yes—
164C23C22×D411yescharacteristic
165D4—42no—
166D4—42no—
167D4—84no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4#12312yes—
200C2×C4#13512yes—
201C2×C4AGL(1,ℤ/8)12yes—
202C2×C4C2×#4112yes—
203C2×C4H(5,𝔽2)11yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C42−1+412yes—
206C8—42no—
207C8—42no—
208C22—22no—
209C22—22no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22C2×#12311yescharacteristic
223C22#53511yescharacteristic
224C22#53611yescharacteristic
225C22C2×#13511yescharacteristic
226C22C2×AGL(1,ℤ/8)11yescharacteristic
227C22#57511yescharacteristic
228C22#57511yescharacteristic
229C4—22no—
230C4—22no—
231C4—22no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—82no—
236C4—82no—
237C4—81no—
238C4—81no—
239C4#53912yes—
240C4#53912yes—
241C2—42no—
242C2—161no—
243C2#262311yescharacteristic
244C2#309111yescharacteristic
245C2#309411yescharacteristic
246C2#321211yescharacteristic
247C2#321411yescharacteristic
248C2#322811yescharacteristic
249C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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
2. #2285C2semidirect
3. #2101C2semidirect
4. #2092C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
7. #2071C2semidirect
8. #2071C2semidirect
10. #1854C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
#28542
#22852
#21012
#20922
#20842
#20832
#20712
#18752
#18542
#18172