Finite Group Databasebeta

← #25607

Subgroups — #25607

Order 256.

576 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 131 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×#541C211yesmaximal, characteristic, complemented
2C2×#492C211yesmaximal, characteristic, complemented
3C2×#490C211yesmaximal, characteristic, complemented
4#2619C211yesmaximal, characteristic
5#2000C214yesmaximal, complemented
6#1991C211yesmaximal, characteristic, complemented
7#1960C211yesmaximal, characteristic, complemented
8#1880C214yesmaximal, complemented
9#1750C211yesmaximal, characteristic, complemented
10C22×SD16C2214yes—
11#541—24no—
12#541—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#127C2211yescharacteristic
15C2×#126—22no—
16C2×#125—22no—
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19C2×#121C2212yes—
20C2×C4×Q8C2212yes—
21C2×#117C2211yescharacteristic
22#492—24no—
23#490—24no—
24C2×#103C2214yes—
25C2×#102C2214yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#394—22no—
30#392—22no—
31#390C2212yes—
32#382C2211yescharacteristic
33#340C2214yes—
34#336C2212yes—
35C22×Q8C2312yes—
36C22×D4—22no—
37C22×D4—22no—
38C22×D4C2312yes—
39C23×C4C2311yescharacteristic
40C2×SD16—28no—
41C2×SD16—28no—
42C2×SD16—44no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8D414yescomplemented
47#127—22no—
48#127—22no—
49#126—216no—
50#125—216no—
51#125—22no—
52#125D414yescomplemented
53C22⋊Q8—24no—
54C22⋊Q8—24no—
55#121—44no—
56C4×Q8—24no—
57C4×Q8—24no—
58#117—22no—
59#117—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D412yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4D412yes—
71C2×C42—22no—
72C2×C42C2311yescharacteristic
73C2×C42C2311yescharacteristic
74#103—44no—
75#102—28no—
76C22⋊C8—24no—
77#97—22no—
78C4×C8—22no—
79#95—22no—
80C24—22no—
81C24C2×D411yescharacteristic
82C2×Q8—22no—
83C2×Q8—24no—
84C2×Q8—24no—
85C2×Q8C2×D414yes—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—24no—
89C2×D4—24no—
90C2×D4—44no—
91C2×D4—44no—
92C2×D4—84no—
93C2×D4—84no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2411yescharacteristic
105SD16—416no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8#4114yes—
112C4⋊C4—28no—
113C4⋊C4—28no—
114C4⋊C4—28no—
115C4⋊C4—28no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4C2×D414yes—
120C22⋊C4—28no—
121C22⋊C4—24no—
122C22⋊C4—44no—
123C22⋊C4—84no—
124C42—28no—
125C42—21no—
126C42—21no—
127C42C2×D412yes—
128C42C2×D412yes—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—24no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—44no—
137C23—44no—
138C23C22×D411yescharacteristic
139Q8—28no—
140Q8—44no—
141D4—44no—
142D4—44no—
143D4—88no—
144D4—88no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—24no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4#12014yes—
161C2×C4C22×D411yescharacteristic
162C2×C4C22×D411yescharacteristic
163C2×C4C2×#4112yes—
164C2×C4H(5,𝔽2)12yes—
165C8—24no—
166C8—42no—
167C22—42no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—84no—
172C22—82no—
173C22—82no—
174C22—82no—
175C22—84no—
176C22C2×#12011yescharacteristic
177C22#53211yescharacteristic
178C22#53511yescharacteristic
179C22#57714yes—
180C4—24no—
181C4—41no—
182C4—44no—
183C4—42no—
184C4—44no—
185C4—82no—
186C4#54612yes—
187C4#55012yes—
188C2—81no—
189C2—162no—
190C2#243111yescharacteristic
191C2#301012yes—
192C2#317412yes—
193C2#322212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#541C2semidirect
2. C2×#492C2semidirect
3. C2×#490C2semidirect
5. #2000C2semidirect
6. #1991C2semidirect
7. #1960C2semidirect
8. #1880C2semidirect
9. #1750C2semidirect
46. C4⋊Q8D4semidirect
52. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4922
C2×#4902
#26192
#20002
#19912
#19602
#18802
#17502