Finite Group Databasebeta

← #29664

Subgroups — #29664

Order 256.

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

normal subgroups only

This group has 566 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×#462C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic
4#2024C212yesmaximal, complemented
5#2016C212yesmaximal, complemented
6#2001C214yesmaximal
7#1803C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic
9C22×Q16C2214yes—
10#533—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17C2×C22⋊Q8C2212yes—
18C2×C4×Q8—21no—
19C2×C4×Q8C2212yes—
20C2×#117—21no—
21C22×C42C2211yescharacteristic
22#462—28no—
23C2×C4⋊C8C2212yes—
24C2×#103C2214yes—
25C2×#103C2214yes—
26C2×C22⋊C8C2212yes—
27#391C2212yes—
28#389—21no—
29#389—21no—
30#384C2211yescharacteristic
31#380—21no—
32#380—21no—
33#340C2214yes—
34#336C2212yes—
35C22×Q8—21no—
36C22×Q8C2312yes—
37C22×Q8C2312yes—
38C23×C4D411yescharacteristic
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C2×Q16—28no—
42C2×Q16—44no—
43C2×Q16—44no—
44C22×C8C2314yes—
45C4⋊Q8—24no—
46C4⋊Q8—24no—
47C4⋊Q8—24no—
48#125—28no—
49C22⋊Q8—28no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C22⋊Q8—44no—
53C4×Q8—28no—
54C4×Q8—24no—
55C4×Q8D418yes—
56#117—28no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—22no—
72C2×C42—24no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C4⋊C8—24no—
78#103—28no—
79#103—44no—
80C22⋊C8—42no—
81#95—22no—
82#95—22no—
83C24C2×D411yescharacteristic
84C2×Q8—24no—
85C2×Q8—22no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—42no—
96C2×Q8C2×D414yes—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—41no—
110C22×C4Q1614yescomplemented
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—416no—
118C2×C8—24no—
119C2×C8—24no—
120C2×C8—24no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—28no—
124C4⋊C4—24no—
125C4⋊C4—28no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4C2×D414yes—
134C22⋊C4—28no—
135C22⋊C4—44no—
136C42—22no—
137C42—22no—
138C42—28no—
139C42—22no—
140C42C2×D414yes—
141C23—21no—
142C23—21no—
143C23—21no—
144C23—21no—
145C23—21no—
146C23—21no—
147C23C2×Q1612yes—
148C23C22×D411yescharacteristic
149Q8—24no—
150Q8—24no—
151Q8—48no—
152Q8—42no—
153Q8—42no—
154Q8—44no—
155Q8—44no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—28no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
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—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4#12012yes—
182C2×C4#12012yes—
183C2×C4C2×Q1612yes—
184C2×C4C2×Q1612yes—
185C2×C4#13712yes—
186C2×C4#13712yes—
187C2×C4C22×D412yes—
188C2×C4H(5,𝔽2)11yescharacteristic
189C2×C4H(5,𝔽2)12yes—
190C2×C42−1+411yescharacteristic
191C8—44no—
192C22—22no—
193C22—22no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C2×#12011yescharacteristic
200C22#53511yescharacteristic
201C22#53711yescharacteristic
202C22C22×Q1611yescharacteristic
203C22C2×#13711yescharacteristic
204C22C2×#13711yescharacteristic
205C22#57511yescharacteristic
206C4—22no—
207C4—22no—
208C4—22no—
209C4—41no—
210C4—44no—
211C4—44no—
212C4—82no—
213C4—82no—
214C4#45114yes—
215C4#53512yes—
216C4#53712yes—
217C2—42no—
218C2#261511yescharacteristic
219C2C2×#45111yescharacteristic
220C2#300711yescharacteristic
221C2#320911yescharacteristic
222C2#321611yescharacteristic
223C2#324911yescharacteristic
224C2#325511yescharacteristic
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×#462C2semidirect
4. #2024C2semidirect
5. #2016C2semidirect
7. #1803C2semidirect
110. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4622
#28572
#20242
#20162
#20012
#18032
#17972