Finite Group Databasebeta

← #26583

Subgroups — #26583

Order 256.

363 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 111 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#3098C212yesmaximal
2#2947C212yesmaximal, complemented
3#2095C211yesmaximal, characteristic, complemented
4#2052C212yesmaximal, complemented
5#2007C211yesmaximal, characteristic
6#1952C212yesmaximal, complemented
7#1929C212yesmaximal, complemented
8#1859C212yesmaximal, complemented
9#1857C211yesmaximal, characteristic, complemented
10C2×#137C2212yes—
11#537C2211yescharacteristic
12#520C2212yes—
13#519C2211yescharacteristic
14#485—22no—
15#484—22no—
16#482—22no—
17#480—22no—
18#441—22no—
19#440—22no—
20C4×Q16—22no—
21C4×SD16—22no—
22C2×#108C2212yes—
23#428C2212yes—
24#428C2212yes—
25C2×#107C2212yes—
26C2×#104C2212yes—
27#417C2212yes—
28#416C2212yes—
29C2×#103C2212yes—
30C2×#101C2211yescharacteristic
31#408C2212yes—
32#405C2212yes—
33#391C2211yescharacteristic
34#384C2211yescharacteristic
35#343C2212yes—
36#343C2211yescharacteristic
37#339C2212yes—
38#336C2211yescharacteristic
39C2×#41C2312yes—
40C22×Q8C2311yescharacteristic
41#137—24no—
42#137—24no—
43C2×Q16—22no—
44C2×SD16—22no—
45C2×M16C2312yes—
46C2×M16C2312yes—
47C22×C8C2312yes—
48C22⋊Q8—22no—
49#121—22no—
50#121—41no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×D4—22no—
55C4×D4—42no—
56#117—22no—
57#117—41no—
58#117C2312yes—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#108—24no—
65#107—22no—
66#107—22no—
67#107—22no—
68#107—22no—
69#106—22no—
70#106—22no—
71#104—24no—
72#103—22no—
73#103—22no—
74#103—22no—
75#103D412yes—
76#103D412yes—
77#102—22no—
78#102D412yes—
79#102D412yes—
80#101—22no—
81C22⋊C8—22no—
82C22⋊C8—22no—
83#97—22no—
84C4×C8—22no—
85#95—41no—
86#41—22no—
87#41—22no—
88#41—22no—
89#41—22no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8C2×D412yes—
94C2×Q8#4112yes—
95C2×D4—22no—
96C2×D4—41no—
97C2×D4C2×D412yes—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4C2411yescharacteristic
105Q16—28no—
106SD16—28no—
107M16—22no—
108M16—22no—
109M16—22no—
110M16#4114yes—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8C2×D412yes—
116C2×C8C2×D412yes—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—82no—
125C4⋊C4C2×D412yes—
126C4⋊C4C2×D412yes—
127C22⋊C4—22no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C42—22no—
131C42—22no—
132C42—42no—
133C42—42no—
134C23—22no—
135C23H(5,𝔽2)11yescharacteristic
136Q8—24no—
137Q8—22no—
138Q8—22no—
139Q8—42no—
140D4—22no—
141D4—22no—
142D4—42no—
143D4—42no—
144D4—161no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—82no—
159C2×C4—81no—
160C2×C4—162no—
161C2×C4C22×D412yes—
162C2×C4C2×#4111yescharacteristic
163C2×C4C2×#4112yes—
164C2×C42−1+411yescharacteristic
165C8—24no—
166C8—22no—
167C8—22no—
168C8—42no—
169C22—21no—
170C22—21no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22#53911yescharacteristic
175C22D4211yescharacteristic
176C22#55911yescharacteristic
177C4—42no—
178C4—42no—
179C4—42no—
180C4—42no—
181C4—42no—
182C4—82no—
183C4—81no—
184C4#52512yes—
185C4#54712yes—
186C2—21no—
187C2—21no—
188C2—82no—
189C2#245511yescharacteristic
190C2#328811yescharacteristic
191C2#329311yescharacteristic
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. #2947C2semidirect
3. #2095C2semidirect
4. #2052C2semidirect
6. #1952C2semidirect
7. #1929C2semidirect
8. #1859C2semidirect
9. #1857C2semidirect

Maximal subgroups

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

subgroupindex
#30982
#29472
#20952
#20522
#20072
#19522
#19292
#18592
#18572