Finite Group Databasebeta

← #29691

Subgroups — #29691

Order 256.

534 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 145 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×#533C211yesmaximal, characteristic
2C2×#477C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic
4#2057C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2036C214yesmaximal
7#1803C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9#533—28no—
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8—21no—
18C2×C4×Q8C2212yes—
19C2×#117—21no—
20C22×C42C2211yescharacteristic
21#477—28no—
22C2×#107C2214yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2214yes—
25C2×#103C2214yes—
26C2×C22⋊C8C2212yes—
27#391—21no—
28#389—21no—
29#389C2212yes—
30#384C2211yescharacteristic
31#380—21no—
32#380C2212yes—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×Q8C2312yes—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8C2314yes—
42C4⋊Q8—24no—
43C4⋊Q8—24no—
44C4⋊Q8—24no—
45#125—28no—
46C22⋊Q8—28no—
47C22⋊Q8—24no—
48C22⋊Q8—24no—
49C22⋊Q8—44no—
50C4×Q8—28no—
51C4×Q8—24no—
52C4×Q8D418yes—
53#117—28no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C42—22no—
69C2×C42—24no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#107—28no—
75C4⋊C8—24no—
76#103—28no—
77#103—44no—
78C22⋊C8—42no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8C2×D414yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—41no—
107C22×C4Q1614yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2411yescharacteristic
114C2×C8—24no—
115C2×C8—24no—
116C2×C8—24no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—28no—
120C4⋊C4—24no—
121C4⋊C4—28no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—42no—
130C4⋊C4C2×D414yes—
131C22⋊C4—28no—
132C22⋊C4—44no—
133C42—22no—
134C42—22no—
135C42—28no—
136C42—22no—
137C42C2×D414yes—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23C2×Q1612yes—
145C23C22×D411yescharacteristic
146Q8—24no—
147Q8—24no—
148Q8—48no—
149Q8—42no—
150Q8—42no—
151Q8—84no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—28no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4#12012yes—
178C2×C4#12012yes—
179C2×C4C2×Q1612yes—
180C2×C4C2×Q1612yes—
181C2×C4AGL(1,ℤ/8)12yes—
182C2×C4#13712yes—
183C2×C4C22×D412yes—
184C2×C4H(5,𝔽2)11yescharacteristic
185C2×C42−1+411yescharacteristic
186C2×C42−1+412yes—
187C8—44no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22C2×#12011yescharacteristic
196C22#53611yescharacteristic
197C22#53711yescharacteristic
198C22C22×Q1611yescharacteristic
199C22C2×AGL(1,ℤ/8)11yescharacteristic
200C22C2×#13711yescharacteristic
201C22#57511yescharacteristic
202C4—22no—
203C4—22no—
204C4—22no—
205C4—41no—
206C4—44no—
207C4—44no—
208C4—82no—
209C4—82no—
210C4#45114yes—
211C4#53612yes—
212C4#53712yes—
213C2—42no—
214C2#261611yescharacteristic
215C2C2×#45111yescharacteristic
216C2#300611yescharacteristic
217C2#323011yescharacteristic
218C2#323711yescharacteristic
219C2#324911yescharacteristic
220C2#325411yescharacteristic
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
2. C2×#477C2semidirect
4. #2057C2semidirect
5. #2043C2semidirect
7. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4772
#28572
#20572
#20432
#20362
#18032
#17972