Finite Group Databasebeta

← #10200

Subgroups — #10200

Order 256.

275 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 59 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#3245C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic, complemented
4#1981C211yesmaximal, characteristic, complemented
5#1980C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic, complemented
7#1283C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#498—21no—
12#489—22no—
13#487—21no—
14#486—21no—
15#486—21no—
16#481—22no—
17#476—22no—
18#474—21no—
19#474—21no—
20#466—22no—
21#459—21no—
22#459—21no—
23#431C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—22no—
26#417—22no—
27C2×#103—22no—
28C4×M16C2211yescharacteristic
29C4×C4⋊C4C2211yescharacteristic
30#327—22no—
31C2×#41—21no—
32C2×#41—21no—
33C22×Q8—21no—
34C2×D8—22no—
35C2×M16D412yes—
36C22×C8D412yes—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D412yes—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46#124—21no—
47#121—21no—
48#121—21no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#107—22no—
61#106—24no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65#104—44no—
66#103—24no—
67#103—44no—
68#102—24no—
69#102—22no—
70#102—22no—
71#102—24no—
72C22⋊C8D412yes—
73#97—21no—
74C4×C8—21no—
75C4×C8—21no—
76#95—21no—
77#41—42no—
78#41—42no—
79#41—42no—
80#41—42no—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8—41no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C2×D4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102D8—82no—
103M16—22no—
104M16—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—42no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C42—42no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C23—41no—
129C23—41no—
130C23#12711yescharacteristic
131Q8—44no—
132Q8—42no—
133Q8—42no—
134Q8—81no—
135Q8—81no—
136D4—42no—
137D4—42no—
138D4—41no—
139D4—41no—
140D4—41no—
141D4—41no—
142D4—42no—
143D4—81no—
144D4—81no—
145D4—81no—
146D4—81no—
147D4—81no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—81no—
156C2×C4—82no—
157C2×C4—81no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4#12011yescharacteristic
161C2×C4#12011yescharacteristic
162C2×C4#12011yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#13512yes—
167C2×C4AGL(1,ℤ/8)12yes—
168C8—42no—
169C8—42no—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22#39211yescharacteristic
180C22#49511yescharacteristic
181C22#49611yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—82no—
186C4—82no—
187C4—81no—
188C4—81no—
189C4#44912yes—
190C4#45311yescharacteristic
191C4#45411yescharacteristic
192C4#46312yes—
193C2—21no—
194C2—21no—
195C2—161no—
196C2—161no—
197C2#200611yescharacteristic
198C2#200911yescharacteristic
199C2#201211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3245C2
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
2. #3227C2semidirect
3. #3141C2semidirect
4. #1981C2semidirect
5. #1980C2semidirect
6. #1502C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
#32452
#32272
#31412
#19812
#19802
#15022
#12832