Finite Group Databasebeta

← #32840

Subgroups — #32840

Order 256.

324 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 107 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#2862C211yesmaximal, characteristic, complemented
2#2860C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2092C212yesmaximal
5#2085C212yesmaximal, complemented
6#2073C212yesmaximal
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#2023C211yesmaximal, characteristic
10#2023C211yesmaximal, characteristic
11#1839C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#126—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×#102C2212yes—
18C2×#102C2212yes—
19#391—21no—
20#391—21no—
21#391—21no—
22#391—21no—
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#388—21no—
31#388—21no—
32#382C2212yes—
33#382C2212yes—
34C43C2211yescharacteristic
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×D4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48C4⋊Q8—42no—
49C4⋊Q8—42no—
50#126—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—42no—
75#106—42no—
76#102—42no—
77#102—42no—
78#95—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—21no—
85C24—21no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—21no—
95C2×D4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C42—22no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42—21no—
138C42—21no—
139C42—41no—
140C42—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23C22×D411yescharacteristic
149Q8—84no—
150Q8—84no—
151D4—42no—
152D4—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
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—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
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#13512yes—
185C2×C4AGL(1,ℤ/8)12yes—
186C2×C4H(5,𝔽2)12yes—
187C2×C42−1+412yes—
188C2×C42−1+411yescharacteristic
189C2×C42−1+411yescharacteristic
190C8—42no—
191C8—42no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22#53511yescharacteristic
200C22#53611yescharacteristic
201C22#53611yescharacteristic
202C22C2×#13511yescharacteristic
203C22C2×AGL(1,ℤ/8)11yescharacteristic
204C22#57511yescharacteristic
205C22#57511yescharacteristic
206C4—22no—
207C4—22no—
208C4—41no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—82no—
213C4—81no—
214C4—82no—
215C4—81no—
216C4—81no—
217C4#56312yes—
218C4#56312yes—
219C2—161no—
220C2#282111yescharacteristic
221C2#321111yescharacteristic
222C2#321311yescharacteristic
223C2#322811yescharacteristic
224C2#323311yescharacteristic
225C2#323511yescharacteristic
226C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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. #2862C2semidirect
3. #2101C2semidirect
5. #2085C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
11. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28622
#28602
#21012
#20922
#20852
#20732
#20582
#20572
#20232
#18392