Finite Group Databasebeta

← #25921

Subgroups — #25921

Order 256.

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

normal subgroups only

This group has 486 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
1#3502C211yesmaximal, characteristic, complemented
2#3140C212yesmaximal
3#2545C211yesmaximal, characteristic
4#2028C212yesmaximal, complemented
5#2027C212yesmaximal
6#1986C212yesmaximal, complemented
7#1888C212yesmaximal
8#1886C212yesmaximal, complemented
9C4×#101C211yesmaximal, characteristic, complemented
10Q82—22no—
11#557—22no—
12#557—24no—
13#544—22no—
14#544C2212yes—
15#533—22no—
16#533C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#124C2211yescharacteristic
20#519C2212yes—
21C2×C4×Q8C2211yescharacteristic
22#489—22no—
23#489—22no—
24#488—24no—
25#419C2214yes—
26#418C2214yes—
27C2×#101C2212yes—
28C4×M16C2212yes—
29#396—22no—
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#388—22no—
33C4×C4⋊C4C2211yescharacteristic
34#343C2212yes—
35#339C2214yes—
36#339C2214yes—
37C22×Q8C2311yescharacteristic
38C22×Q8C2311yescharacteristic
39C22×D4C2311yescharacteristic
40C2×M16C2314yes—
41C4⋊Q8—22no—
42C4⋊Q8—22no—
43C4⋊Q8—28no—
44C4⋊Q8—24no—
45C4⋊Q8—24no—
46C4⋊Q8D412yes—
47C4⋊Q8D412yes—
48#125—24no—
49#125—28no—
50#125—24no—
51#125D414yes—
52#124—24no—
53#124—22no—
54#124—22no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C4×Q8—24no—
59C4×Q8—24no—
60C4×Q8—24no—
61C4×Q8—24no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—22no—
66#117—24no—
67#117C2314yes—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4D412yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C42D412yes—
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#103—24no—
81#103—24no—
82#102—24no—
83#102—24no—
84#101—24no—
85#97—22no—
86C4×C8—22no—
87#95—21no—
88#95—22no—
89C24—21no—
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8#4114yes—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110M16—24no—
111M16—24no—
112C2×C8—24no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—28no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4C2×D414yes—
126C4⋊C4C2×D414yes—
127C22⋊C4—24no—
128C22⋊C4—44no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C42—28no—
134C42—22no—
135C42—22no—
136C42—24no—
137C42—22no—
138C42—22no—
139C42C2×D412yes—
140C42C2×D412yes—
141C23—42no—
142C23—42no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23C22×D411yescharacteristic
147Q8—28no—
148Q8—44no—
149Q8—44no—
150Q8—82no—
151Q8—84no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4#12011yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12012yes—
173C2×C4#12414yes—
174C2×C4C22×D412yes—
175C2×C4C2×#4112yes—
176C2×C42−1+412yes—
177C8—44no—
178C22—21no—
179C22—21no—
180C22—82no—
181C22—82no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22C2×#12011yescharacteristic
186C22C2×#12411yescharacteristic
187C22#53611yescharacteristic
188C4—24no—
189C4—22no—
190C4—21no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—44no—
195C4—44no—
196C4—81no—
197C4—81no—
198C4#54814yes—
199C2—21no—
200C2—21no—
201C2—161no—
202C2#243611yescharacteristic
203C2#301811yescharacteristic
204C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3502C2semidirect
4. #2028C2semidirect
6. #1986C2semidirect
8. #1886C2semidirect
9. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
#35022
#31402
#25452
#20282
#20272
#19862
#18882
#18862
C4×#1012