Finite Group Databasebeta

← #25570

Subgroups — #25570

Order 256.

563 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 139 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×#556C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4#2542C211yesmaximal, characteristic
5#2001C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1873C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10C22×Q16C2212yes—
11#556—24no—
12#556—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#124—21no—
17C2×C22⋊Q8C2212yes—
18C2×C22⋊Q8C2214yes—
19C2×C4×Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2212yes—
23#487—24no—
24#462—24no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2214yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×C22⋊C8C2214yes—
30C2×C4×C8C2211yescharacteristic
31#399—22no—
32#390—21no—
33#384—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8C2312yes—
39C22×Q8C2311yescharacteristic
40C22×D4D411yescharacteristic
41C23×C4C2312yes—
42C2×Q16—24no—
43C2×Q16—44no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8D414yes—
50#126—216no—
51#124—28no—
52C22⋊Q8—24no—
53C22⋊Q8—24no—
54C22⋊Q8—28no—
55C22⋊Q8—44no—
56C4×Q8—28no—
57C4×Q8—22no—
58C4×Q8D414yes—
59C4×D4—22no—
60C4×D4—22no—
61#117—24no—
62#117—24no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—24no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4D412yes—
77C2×C42—22no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81C4⋊C8—22no—
82#103—24no—
83#103—28no—
84#103—24no—
85C22⋊C8—44no—
86C4×C8—22no—
87#95—21no—
88C24C2×D412yes—
89C2×Q8—24no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—24no—
95C2×Q8—22no—
96C2×Q8—24no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8C2×D412yes—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116Q16—48no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—24no—
122C4⋊C4—28no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—48no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4C2×D412yes—
133C4⋊C4C2×D414yes—
134C4⋊C4#4114yes—
135C22⋊C4—28no—
136C22⋊C4—28no—
137C22⋊C4—24no—
138C22⋊C4—48no—
139C42—28no—
140C42—24no—
141C42—42no—
142C42C2×D412yes—
143C42C2×D412yes—
144C23—22no—
145C23—24no—
146C23—22no—
147C23—22no—
148C23—24no—
149C23C22×D411yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—42no—
153Q8—48no—
154Q8—42no—
155Q8—42no—
156Q8—44no—
157D4—44no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—24no—
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—44no—
174C2×C4—42no—
175C2×C4—44no—
176C2×C4#12012yes—
177C2×C4#12012yes—
178C2×C4#13512yes—
179C2×C4#13712yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C22×D411yescharacteristic
182C2×C4C2×#4112yes—
183C2×C4H(5,𝔽2)12yes—
184C8—42no—
185C8—42no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—44no—
191C22C2×#12011yescharacteristic
192C22#54612yes—
193C22C2×#13511yescharacteristic
194C22C2×#13711yescharacteristic
195C22#57812yes—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—44no—
202C4—82no—
203C4#45214yes—
204C4#53211yescharacteristic
205C4#53511yescharacteristic
206C4#55012yes—
207C2—82no—
208C2#243111yescharacteristic
209C2C2×#45211yescharacteristic
210C2#301111yescharacteristic
211C2#330212yes—
212C2#332812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#487C2semidirect
3. C2×#462C2semidirect
5. #2001C2semidirect
6. #1988C2semidirect
8. #1873C2semidirect
9. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4872
C2×#4622
#25422
#20012
#19882
#19542
#18732
C4×#1032