Finite Group Databasebeta

← #29701

Subgroups — #29701

Order 256.

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

normal subgroups only

This group has 470 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×#541C211yesmaximal, characteristic
2C2×#479C212yesmaximal, complemented
3#2629C211yesmaximal, characteristic
4#2059C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2036C212yesmaximal
7#2035C212yesmaximal
8#1803C212yesmaximal, complemented
9#1798C211yesmaximal, characteristic
10#541—28no—
11C2×C4⋊Q8C2211yescharacteristic
12C2×#126—22no—
13C2×#125—22no—
14C2×#125C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8C2212yes—
18C2×#117C2211yescharacteristic
19#479—28no—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#398—22no—
30#396—22no—
31#389C2212yes—
32#384C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37C22×Q8C2312yes—
38C23×C4C2311yescharacteristic
39C22×C8C2312yes—
40C22×C8C2312yes—
41C4⋊Q8—24no—
42#126—216no—
43#125—216no—
44#125—24no—
45#125—24no—
46C22⋊Q8—24no—
47C22⋊Q8—24no—
48C4×Q8—24no—
49C4×Q8D418yes—
50#117—24no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D412yes—
65C2×C42—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#107—24no—
69#106—24no—
70C4⋊C8—24no—
71#103—24no—
72#103—24no—
73#103—42no—
74#103—42no—
75C22⋊C8—41no—
76C22⋊C8—41no—
77#95—22no—
78#95—22no—
79C24C2×D411yescharacteristic
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8C2×D414yes—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4C2×D412yes—
97C22×C4C2411yescharacteristic
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C4⋊C4—28no—
105C4⋊C4—28no—
106C4⋊C4—28no—
107C4⋊C4—28no—
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—44no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C4⋊C4C2×D414yes—
118C22⋊C4—28no—
119C22⋊C4—24no—
120C22⋊C4—82no—
121C42—28no—
122C42—22no—
123C42C2×D414yes—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—22no—
128C23—22no—
129C23C22×D411yescharacteristic
130Q8—24no—
131Q8—24no—
132Q8—42no—
133Q8—42no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4#12014yes—
157C2×C4C22×D412yes—
158C2×C4H(5,𝔽2)11yescharacteristic
159C2×C42−1+411yescharacteristic
160C2×C42−1+412yes—
161C8—42no—
162C8—42no—
163C22—41no—
164C22—42no—
165C22—41no—
166C22—41no—
167C22—42no—
168C22C2×#12011yescharacteristic
169C22#53611yescharacteristic
170C22#53711yescharacteristic
171C22#57712yes—
172C22#57812yes—
173C4—24no—
174C4—41no—
175C4—44no—
176C4—42no—
177C4—44no—
178C4—82no—
179C4—82no—
180C4#53612yes—
181C4#53712yes—
182C2—81no—
183C2#261611yescharacteristic
184C2#301312yes—
185C2#324212yes—
186C2#326011yescharacteristic
187C2#326111yescharacteristic
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×#479C2semidirect
4. #2059C2semidirect
5. #2043C2semidirect
8. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4792
#26292
#20592
#20432
#20362
#20352
#18032
#17982