Finite Group Databasebeta

← #10021

Subgroups — #10021

Order 256.

246 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 49 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#3142C211yesmaximal, characteristic
2#3112C211yesmaximal, characteristic
3#2634C211yesmaximal, characteristic, complemented
4#1534C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic
6#1491C211yesmaximal, characteristic, complemented
7#1288C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×#125C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#491—21no—
12#490—21no—
13#488—22no—
14#480—21no—
15#477—21no—
16#460—21no—
17#448—21no—
18#440—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#417—21no—
22#417—21no—
23C4×M16C2211yescharacteristic
24#400—21no—
25#400—21no—
26#398—21no—
27#395—21no—
28#387—22no—
29#383C2211yescharacteristic
30#330—22no—
31#327—22no—
32Q8⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C2×SD16—22no—
36C2×M16—21no—
37C4⋊Q8D411yescharacteristic
38#127D411yescharacteristic
39#125—21no—
40#125—21no—
41#125—21no—
42#125—21no—
43#124—21no—
44#121—21no—
45#121—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×D4—21no—
51C4×D4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61#107—22no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64#103—22no—
65#103—22no—
66#103—22no—
67#102—22no—
68#102—22no—
69#102—22no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72#97—21no—
73#97—21no—
74#97D412yes—
75C4×C8D412yes—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—21no—
80#95—21no—
81#95D412yes—
82#41—44no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—41no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100SD16—44no—
101M16—42no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—44no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—41no—
123C42—41no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C23—41no—
129C23#12011yescharacteristic
130Q8—22no—
131Q8—22no—
132Q8—41no—
133Q8—41no—
134Q8—42no—
135D4—42no—
136D4—42no—
137D4—81no—
138D4—82no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—82no—
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12711yescharacteristic
162C8—42no—
163C8—42no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22#39211yescharacteristic
171C22#45412yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—82no—
180C4—81no—
181C4—81no—
182C4—81no—
183C4—81no—
184C2—22no—
185C2—161no—
186C2#200311yescharacteristic
187C2#200911yescharacteristic
188C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2634C2semidirect
4. #1534C2semidirect
6. #1491C2semidirect
7. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#31422
#31122
#26342
#15342
#15142
#14912
#12882