Finite Group Databasebeta

← #25701

Subgroups — #25701

Order 256.

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

normal subgroups only

This group has 519 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×#498C211yesmaximal, characteristic, complemented
3C2×#477C211yesmaximal, characteristic, complemented
4#2563C211yesmaximal, characteristic
5#2035C214yesmaximal, complemented
6#1988C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic
8#1854C214yesmaximal, complemented
9C4×#103C211yesmaximal, characteristic, complemented
10#556—24no—
11#556—24no—
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#126—22no—
15C2×#124—21no—
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8—21no—
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2212yes—
21#498—24no—
22#477—24no—
23C2×#107C2212yes—
24C2×#106C2214yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×C22⋊C8C2214yes—
29C2×C4×C8C2211yescharacteristic
30#393—22no—
31#390—21no—
32#382—21no—
33#380C2214yes—
34C4×C4⋊C4C2211yescharacteristic
35#340C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8C2311yescharacteristic
40C22×D4D411yescharacteristic
41C23×C4C2312yes—
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8D414yes—
48#126—216no—
49#124—28no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C4×Q8—28no—
53C4×Q8—22no—
54C4×Q8D414yes—
55C4×D4—22no—
56C4×D4—22no—
57#117—24no—
58#117—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4D412yes—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—28no—
78C4⋊C8—22no—
79#103—24no—
80#103—24no—
81C22⋊C8—28no—
82C4×C8—22no—
83#95—21no—
84#95—24no—
85C24C2×D412yes—
86C2×Q8—24no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—82no—
94C2×Q8C2×D412yes—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
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#4114yes—
116C4⋊C4—28no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4C2×D414yes—
128C4⋊C4C2×D412yes—
129C22⋊C4—28no—
130C22⋊C4—28no—
131C22⋊C4—24no—
132C22⋊C4—84no—
133C42—28no—
134C42—24no—
135C42—42no—
136C42C2×D412yes—
137C42C2×D412yes—
138C23—22no—
139C23—24no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23C22×D411yescharacteristic
144Q8—22no—
145Q8—22no—
146Q8—42no—
147Q8—48no—
148Q8—84no—
149D4—44no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—44no—
168C2×C4#12012yes—
169C2×C4#12012yes—
170C2×C4#13512yes—
171C2×C4#13712yes—
172C2×C4C22×D411yescharacteristic
173C2×C4C22×D411yescharacteristic
174C2×C4C2×#4112yes—
175C2×C42−1+412yes—
176C8—24no—
177C8—42no—
178C22—44no—
179C22—44no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22C2×#12011yescharacteristic
184C22#54712yes—
185C22C2×#13511yescharacteristic
186C22C2×#13711yescharacteristic
187C22#57712yes—
188C4—22no—
189C4—42no—
190C4—42no—
191C4—44no—
192C4—42no—
193C4—82no—
194C4—82no—
195C4#45214yes—
196C4#53211yescharacteristic
197C4#53611yescharacteristic
198C4#54812yes—
199C2—82no—
200C2#243211yescharacteristic
201C2C2×#45211yescharacteristic
202C2#301011yescharacteristic
203C2#334012yes—
204C2#335612yes—
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×#498C2semidirect
3. C2×#477C2semidirect
5. #2035C2semidirect
6. #1988C2semidirect
8. #1854C2semidirect
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×#4982
C2×#4772
#25632
#20352
#19882
#19542
#18542
C4×#1032