Finite Group Databasebeta

← #25609

Subgroups — #25609

Order 256.

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

normal subgroups only

This group has 526 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
2C2×#494C211yesmaximal, characteristic, complemented
3C2×#491C211yesmaximal, characteristic, complemented
4#2621C211yesmaximal, characteristic
5#2001C214yesmaximal
6#1991C211yesmaximal, characteristic, complemented
7#1960C211yesmaximal, characteristic, complemented
8#1878C214yesmaximal, complemented
9#1750C211yesmaximal, characteristic
10C22×Q16C2214yes—
11#541—24no—
12#541—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—22no—
16C2×#125—22no—
17C2×#125C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2212yes—
20C2×C4×Q8C2212yes—
21C2×#117C2211yescharacteristic
22#494—24no—
23#491—24no—
24C2×#103C2214yes—
25C2×#103C2214yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—22no—
30#397—22no—
31#391C2212yes—
32#382C2211yescharacteristic
33#340C2214yes—
34#336C2212yes—
35C22×Q8C2312yes—
36C22×Q8C2312yes—
37C23×C4C2311yescharacteristic
38C2×Q16—28no—
39C2×Q16—28no—
40C2×Q16—44no—
41C22×C8C2312yes—
42C22×C8C2312yes—
43C4⋊Q8—22no—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8D414yes—
47#126—216no—
48#125—216no—
49#125—22no—
50#125D414yes—
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53C22⋊Q8—44no—
54C4×Q8—24no—
55C4×Q8—24no—
56#117—22no—
57#117—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—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⋊C4D412yes—
70C2×C42—22no—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#103—28no—
74#103—44no—
75C22⋊C8—24no—
76#97—22no—
77C4×C8—22no—
78#95—22no—
79#95—24no—
80C24C2×D411yescharacteristic
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—24no—
87C2×Q8—24no—
88C2×Q8C2×D414yes—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D412yes—
100C22×C4C2411yescharacteristic
101Q16—416no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8#4114yes—
108C4⋊C4—28no—
109C4⋊C4—28no—
110C4⋊C4—28no—
111C4⋊C4—28no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—44no—
116C4⋊C4—44no—
117C4⋊C4C2×D414yes—
118C22⋊C4—28no—
119C22⋊C4—24no—
120C22⋊C4—44no—
121C42—28no—
122C42—21no—
123C42—21no—
124C42C2×D412yes—
125C42C2×D412yes—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—24no—
130C23C22×D411yescharacteristic
131Q8—28no—
132Q8—44no—
133Q8—44no—
134Q8—44no—
135C2×C4—24no—
136C2×C4—22no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—24no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4#12014yes—
154C2×C4C22×D411yescharacteristic
155C2×C4C22×D411yescharacteristic
156C2×C4C2×#4112yes—
157C2×C4H(5,𝔽2)12yes—
158C8—24no—
159C8—42no—
160C22—42no—
161C22—41no—
162C22—42no—
163C22—42no—
164C22C2×#12011yescharacteristic
165C22#53211yescharacteristic
166C22#53511yescharacteristic
167C22#57814yes—
168C4—24no—
169C4—41no—
170C4—44no—
171C4—42no—
172C4—44no—
173C4—82no—
174C4—82no—
175C4#54612yes—
176C4#55012yes—
177C2—81no—
178C2#243111yescharacteristic
179C2#301112yes—
180C2#317712yes—
181C2#322312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
2. C2×#494C2semidirect
3. C2×#491C2semidirect
6. #1991C2semidirect
7. #1960C2semidirect
8. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4942
C2×#4912
#26212
#20012
#19912
#19602
#18782
#17502