Finite Group Databasebeta

← #29865

Subgroups — #29865

Order 256.

387 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 127 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×#476C211yesmaximal, characteristic
2C2×#474C211yesmaximal, characteristic
3#2862C211yesmaximal, characteristic, complemented
4#2293C211yesmaximal, characteristic
5#2085C214yesmaximal, complemented
6#2073C214yesmaximal
7#1988C211yesmaximal, characteristic, complemented
8#1987C211yesmaximal, characteristic, complemented
9#1831C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×D4C2211yescharacteristic
13C2×#117—21no—
14#476—24no—
15#474—24no—
16C2×#107C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2211yescharacteristic
20C2×#102C2212yes—
21C2×#102C2212yes—
22#391—21no—
23#391—21no—
24#391—22no—
25#390C2212yes—
26#389C2212yes—
27#388—22no—
28#382C2214yes—
29C4×C4⋊C4C2211yescharacteristic
30C4×C22⋊C4—21no—
31C43C2211yescharacteristic
32#340C2214yes—
33#340C2214yes—
34#336C2214yes—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×D4C2311yescharacteristic
38C23×C4—21no—
39C22×C8C2312yes—
40C22×C8C2312yes—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C4⋊Q8—22no—
44C4⋊Q8—22no—
45C4×D4—22no—
46C4×D4D414yes—
47#117—44no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4C2312yes—
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C4⋊C4C2311yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C42—21no—
59C2×C42D411yescharacteristic
60C2×C42D411yescharacteristic
61C2×C42D412yes—
62C2×C42C2312yes—
63C2×C42C2311yescharacteristic
64#107—24no—
65#106—24no—
66C4⋊C8—22no—
67C4⋊C8—22no—
68#102—24no—
69#102—24no—
70#95—22no—
71#95—21no—
72#95—24no—
73#95—24no—
74C24—21no—
75C2×Q8—44no—
76C2×Q8—42no—
77C2×Q8—42no—
78C2×Q8—42no—
79C2×Q8—42no—
80C2×D4—21no—
81C2×D4—22no—
82C2×D4—22no—
83C2×D4C2×D412yes—
84C22×C4—21no—
85C22×C4—24no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4C2×D412yes—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—24no—
99C2×C8—24no—
100C4⋊C4—21no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—44no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4C2×D412yes—
111C22⋊C4—24no—
112C22⋊C4—44no—
113C22⋊C4—44no—
114C42—22no—
115C42—22no—
116C42—22no—
117C42—22no—
118C42—42no—
119C42—42no—
120C42C2×D412yes—
121C42#4112yes—
122C23—22no—
123C23—42no—
124C23—41no—
125C23—41no—
126C23—42no—
127C23C22×D411yescharacteristic
128Q8—84no—
129Q8—84no—
130D4—22no—
131D4—22no—
132D4—42no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—48no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—44no—
152C2×C4#12112yes—
153C2×C4#12112yes—
154C2×C4#13512yes—
155C2×C4AGL(1,ℤ/8)12yes—
156C2×C4C22×D411yescharacteristic
157C2×C4C2×#4111yescharacteristic
158C2×C42−1+412yes—
159C2×C42−1+412yes—
160C8—42no—
161C8—42no—
162C22—42no—
163C22—42no—
164C22—42no—
165C22—82no—
166C22—82no—
167C22C2×#12111yescharacteristic
168C22#53612yes—
169C22C2×#13511yescharacteristic
170C22C2×AGL(1,ℤ/8)11yescharacteristic
171C22#57512yes—
172C4—22no—
173C4—42no—
174C4—42no—
175C4—42no—
176C4—82no—
177C4—82no—
178C4—82no—
179C4#46314yes—
180C4#53611yescharacteristic
181C4#53611yescharacteristic
182C4#54112yes—
183C2—82no—
184C2#262111yescharacteristic
185C2C2×#46311yescharacteristic
186C2#304011yescharacteristic
187C2#323312yes—
188C2#323512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2862C2semidirect
5. #2085C2semidirect
7. #1988C2semidirect
8. #1987C2semidirect
9. #1831C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4742
#28622
#22932
#20852
#20732
#19882
#19872
#18312