Finite Group Databasebeta

← #31112

Subgroups — #31112

Order 256.

325 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#3099C212yesmaximal
2#3084C212yesmaximal, complemented
3#2090C212yesmaximal, complemented
4#2079C212yesmaximal, complemented
5#2060C211yesmaximal, characteristic, complemented
6#2039C211yesmaximal, characteristic
7#1937C212yesmaximal, complemented
8#1859C212yesmaximal, complemented
9#1844C211yesmaximal, characteristic, complemented
10#544C2211yescharacteristic
11#537C2211yescharacteristic
12#520C2212yes—
13#485—22no—
14#483—22no—
15#482—22no—
16#481—22no—
17#479—22no—
18#478—22no—
19#476—22no—
20#475—22no—
21#428C2212yes—
22#428C2212yes—
23C2×#106C2212yes—
24C2×#106C2212yes—
25#424C2212yes—
26C2×#104C2212yes—
27#417C2212yes—
28#417C2212yes—
29#416C2212yes—
30#408C2212yes—
31C2×C4×C8C2211yescharacteristic
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34#343C2211yescharacteristic
35#343C2212yes—
36#343C2211yescharacteristic
37#341C2212yes—
38#339C2212yes—
39C2×#41C2312yes—
40C2×M16C2312yes—
41C2×M16C2312yes—
42C22×C8C2312yes—
43C4⋊Q8—22no—
44#125—22no—
45C22⋊Q8—22no—
46C22⋊Q8—41no—
47#121—22no—
48#121—41no—
49C4×Q8—22no—
50C4×D4—22no—
51C4×D4—42no—
52#117—22no—
53#117C2312yes—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C42C2311yescharacteristic
61#107—22no—
62#107—22no—
63#106—22no—
64#106—22no—
65#106—22no—
66#106—22no—
67#106—22no—
68#106—22no—
69C4⋊C8—22no—
70C4⋊C8—22no—
71#104—24no—
72#103—22no—
73#103—22no—
74#103D412yes—
75#103D412yes—
76#102—22no—
77#102—22no—
78#102D412yes—
79#102D412yes—
80C22⋊C8—22no—
81C22⋊C8—22no—
82C4×C8—22no—
83#41—22no—
84#41—22no—
85#41—22no—
86#41—22no—
87C2×Q8—41no—
88C2×Q8C2×D412yes—
89C2×D4—22no—
90C2×D4—41no—
91C2×D4C2×D412yes—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4C2411yescharacteristic
99M16—22no—
100M16—22no—
101M16—22no—
102M16—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8C2×D412yes—
108C2×C8C2×D412yes—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—42no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—42no—
117C4⋊C4—82no—
118C4⋊C4—82no—
119C4⋊C4C2×D412yes—
120C4⋊C4C2×D412yes—
121C22⋊C4—22no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C42—22no—
125C42—21no—
126C42—42no—
127C42#4112yes—
128C23—22no—
129C23H(5,𝔽2)11yescharacteristic
130Q8—22no—
131Q8—22no—
132Q8—161no—
133D4—22no—
134D4—22no—
135D4—42no—
136D4—42no—
137D4—161no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—162no—
153C2×C4C22×D412yes—
154C2×C4C2×#4111yescharacteristic
155C2×C42−1+411yescharacteristic
156C2×C42−1+412yes—
157C8—22no—
158C8—22no—
159C8—42no—
160C8—42no—
161C22—21no—
162C22—21no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#53911yescharacteristic
167C22D4211yescharacteristic
168C22#56311yescharacteristic
169C4—24no—
170C4—42no—
171C4—42no—
172C4—42no—
173C4—82no—
174C4—81no—
175C4—81no—
176C4#53612yes—
177C4#54712yes—
178C2—21no—
179C2—21no—
180C2—82no—
181C2#268011yescharacteristic
182C2#329111yescharacteristic
183C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3084C2semidirect
3. #2090C2semidirect
4. #2079C2semidirect
5. #2060C2semidirect
7. #1937C2semidirect
8. #1859C2semidirect
9. #1844C2semidirect

Maximal subgroups

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

subgroupindex
#30992
#30842
#20902
#20792
#20602
#20392
#19372
#18592
#18442