Finite Group Databasebeta

← #25600

Subgroups — #25600

Order 256.

579 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 155 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×#494C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic
5#2001C214yesmaximal
6#1990C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1878C214yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic
10C22×Q16C2214yes—
11#533—24no—
12#533—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2212yes—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×#117—21no—
23C22×C42C2211yescharacteristic
24#494—24no—
25#487—24no—
26C2×#103C2214yes—
27C2×#103C2214yes—
28C2×C22⋊C8C2212yes—
29C2×C4×C8C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#391C2212yes—
32#389—21no—
33#389—21no—
34#384C2211yescharacteristic
35#380—21no—
36#380—21no—
37#340C2214yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2312yes—
41C22×Q8C2312yes—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×Q16—28no—
46C2×Q16—28no—
47C2×Q16—44no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55C4⋊Q8D414yes—
56#125—28no—
57C22⋊Q8—28no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—44no—
61C4×Q8—28no—
62C4×Q8—24no—
63C4×Q8—24no—
64#117—28no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#103—28no—
87#103—44no—
88C22⋊C8—24no—
89C4×C8—22no—
90C4×C8—22no—
91#95—22no—
92#95—22no—
93C24C2×D411yescharacteristic
94C2×Q8—24no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—24no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—24no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8C2×D414yes—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4Q1614yescomplemented
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125Q16—416no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8#4114yes—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4C2×D414yes—
145C22⋊C4—28no—
146C22⋊C4—44no—
147C42—28no—
148C42—22no—
149C42—22no—
150C42—21no—
151C42—21no—
152C42C2×D412yes—
153C42C2×D412yes—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—21no—
159C23—21no—
160C23C2×Q1612yes—
161C23C22×D411yescharacteristic
162Q8—28no—
163Q8—44no—
164Q8—48no—
165Q8—44no—
166Q8—44no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4#12012yes—
192C2×C4#12012yes—
193C2×C4C2×Q1612yes—
194C2×C4C2×Q1612yes—
195C2×C4#13512yes—
196C2×C4#13712yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4112yes—
201C2×C4H(5,𝔽2)12yes—
202C8—24no—
203C8—42no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22C2×#12011yescharacteristic
212C22#53211yescharacteristic
213C22#53511yescharacteristic
214C22C22×Q1611yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×#13711yescharacteristic
217C22C2×#13711yescharacteristic
218C4—22no—
219C4—41no—
220C4—44no—
221C4—41no—
222C4—44no—
223C4—82no—
224C4—82no—
225C4#45114yes—
226C4#45214yes—
227C4#54612yes—
228C4#55012yes—
229C2—42no—
230C2#243111yescharacteristic
231C2C2×#45111yescharacteristic
232C2C2×#45211yescharacteristic
233C2#317011yescharacteristic
234C2#318511yescharacteristic
235C2#320911yescharacteristic
236C2#321011yescharacteristic
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×#494C2semidirect
3. C2×#487C2semidirect
6. #1990C2semidirect
7. #1956C2semidirect
8. #1878C2semidirect
118. 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×#4942
C2×#4872
#28572
#20012
#19902
#19562
#18782
C4×C22⋊C82