Finite Group Databasebeta

← #25646

Subgroups — #25646

Order 256.

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

normal subgroups only

This group has 517 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×#476C211yesmaximal, characteristic
4#2567C211yesmaximal, characteristic, complemented
5#2042C214yesmaximal
6#1984C211yesmaximal, characteristic, complemented
7#1981C211yesmaximal, characteristic, complemented
8#1865C214yesmaximal, complemented
9C4×#102C211yesmaximal, characteristic
10#556—24no—
11#556—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#126—22no—
15C2×#124C2212yes—
16C2×C22⋊Q8—21no—
17C2×C4×Q8C2212yes—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117—21no—
21#487—24no—
22#476—24no—
23C2×#106C2212yes—
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2214yes—
26C2×#103C2214yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C4×C8C2211yescharacteristic
30#395—22no—
31#391—21no—
32#389C2214yes—
33#384—21no—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#336C2214yes—
37C22×Q8—21no—
38C22×Q8D411yescharacteristic
39C22×Q8C2312yes—
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8D414yes—
48#126—216no—
49#124—24no—
50#124—24no—
51C22⋊Q8—28no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—22no—
57C4×D4D414yes—
58#117—28no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#106—24no—
77C4⋊C8—22no—
78#103—28no—
79#103—28no—
80#102—24no—
81#102—24no—
82C4×C8—22no—
83#95—21no—
84#95—24no—
85C24—21no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—24no—
91C2×Q8—24no—
92C2×Q8—42no—
93C2×Q8—82no—
94C2×Q8C2×D414yes—
95C2×D4—21no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4C2×D412yes—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4C2×D412yes—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—24no—
115C2×C8—22no—
116C4⋊C4—28no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—24no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—84no—
127C4⋊C4C2×D412yes—
128C4⋊C4#4114yes—
129C22⋊C4—28no—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C42—28no—
133C42—22no—
134C42—22no—
135C42—42no—
136C42C2×D412yes—
137C42C2×D412yes—
138C23—22no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23C22×D411yescharacteristic
144Q8—28no—
145Q8—42no—
146Q8—42no—
147Q8—44no—
148Q8—84no—
149D4—22no—
150D4—22no—
151D4—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—48no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4#12012yes—
170C2×C4#12012yes—
171C2×C4#13512yes—
172C2×C4#13712yes—
173C2×C4C22×D411yescharacteristic
174C2×C4C22×D411yescharacteristic
175C2×C4C2×#4112yes—
176C2×C42−1+412yes—
177C8—42no—
178C8—42no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—82no—
183C22—82no—
184C22C2×#12011yescharacteristic
185C22#54812yes—
186C22C2×#13511yescharacteristic
187C22C2×#13711yescharacteristic
188C22#57712yes—
189C4—22no—
190C4—44no—
191C4—44no—
192C4—42no—
193C4—42no—
194C4—42no—
195C4—82no—
196C4#45214yes—
197C4#53211yescharacteristic
198C4#53611yescharacteristic
199C4#54712yes—
200C2—82no—
201C2#243211yescharacteristic
202C2C2×#45211yescharacteristic
203C2#301011yescharacteristic
204C2#336012yes—
205C2#336912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#487C2semidirect
4. #2567C2semidirect
6. #1984C2semidirect
7. #1981C2semidirect
8. #1865C2semidirect

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×#4762
#25672
#20422
#19842
#19812
#18652
C4×#1022