Finite Group Databasebeta

← #10041

Subgroups — #10041

Order 256.

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

normal subgroups only

This group has 239 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#3216C211yesmaximal, characteristic
2#3112C211yesmaximal, characteristic
3#3104C211yesmaximal, characteristic, complemented
4#1537C211yesmaximal, characteristic, complemented
5#1496C211yesmaximal, characteristic
6#1467C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11#497—21no—
12#489—21no—
13#484—21no—
14#480—21no—
15#477—21no—
16#477—21no—
17#462—21no—
18#462—21no—
19#460—21no—
20#460—21no—
21#452—22no—
22#451—21no—
23#448—21no—
24#440—22no—
25C4×Q16—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#432C2211yescharacteristic
29#431C2211yescharacteristic
30#330—22no—
31D4⋊C8—22no—
32#324—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C2×Q16—22no—
36C2×Q16—22no—
37C2×SD16—22no—
38C2×SD16—22no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#127D411yescharacteristic
43#125—21no—
44#124—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47#121—21no—
48#121—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×D4—21no—
56C4×D4—21no—
57#117—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—22no—
63#107—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—22no—
69#103—22no—
70#103—22no—
71#103—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8D412yes—
78#97—21no—
79#97—21no—
80#97D412yes—
81C4×C8D412yes—
82#41—82no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—42no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—41no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4#4111yescharacteristic
99Q16—44no—
100Q16—82no—
101SD16—44no—
102SD16—82no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—41no—
123C42—41no—
124C42—41no—
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C23—41no—
130C23#12111yescharacteristic
131Q8—22no—
132Q8—22no—
133Q8—41no—
134Q8—41no—
135Q8—41no—
136Q8—42no—
137Q8—41no—
138Q8—42no—
139D4—42no—
140D4—42no—
141D4—42no—
142D4—82no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—82no—
154C2×C4—81no—
155C2×C4#12011yescharacteristic
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12711yescharacteristic
161C8—42no—
162C8—42no—
163C8—42no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#39211yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—42no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4#47312yes—
183C2—41no—
184C2—161no—
185C2#200811yescharacteristic
186C2#201011yescharacteristic
187C2#202011yescharacteristic
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. #3104C2semidirect
4. #1537C2semidirect
6. #1467C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32162
#31122
#31042
#15372
#14962
#14672
#12872