Finite Group Databasebeta

← #10202

Subgroups — #10202

Order 256.

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

normal subgroups only

This group has 256 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#3246C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#1986C211yesmaximal, characteristic, complemented
5#1981C211yesmaximal, characteristic, complemented
6#1502C211yesmaximal, characteristic
7#1283C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#500—21no—
12#489—22no—
13#487—21no—
14#487—21no—
15#486—21no—
16#484—22no—
17#477—22no—
18#476—21no—
19#475—21no—
20#465—22no—
21#461—21no—
22#460—21no—
23#431C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25#417—22no—
26C2×#103—22no—
27C4×M16C2211yescharacteristic
28C4×C4⋊C4C2211yescharacteristic
29#339—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C2×SD16—22no—
34C2×M16D412yes—
35C22×C8D412yes—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8D412yes—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#125—21no—
45#124—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×D4—21no—
53C4×D4—21no—
54#117—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#107—24no—
61#106—22no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65#103—24no—
66#103—22no—
67#103—24no—
68#103—44no—
69#102—24no—
70#102—22no—
71C22⋊C8D412yes—
72#97—21no—
73C4×C8—21no—
74C4×C8—21no—
75#95—21no—
76#41—42no—
77#41—42no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—22no—
82C2×Q8—42no—
83C2×Q8—42no—
84C2×Q8—41no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95SD16—82no—
96M16—22no—
97M16—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—42no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C42—41no—
120C42—42no—
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C23—41no—
126C23#12711yescharacteristic
127Q8—42no—
128Q8—44no—
129Q8—42no—
130Q8—41no—
131Q8—41no—
132Q8—81no—
133Q8—81no—
134D4—42no—
135D4—41no—
136D4—41no—
137D4—81no—
138D4—81no—
139D4—81no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—81no—
149C2×C4—82no—
150C2×C4—81no—
151C2×C4—82no—
152C2×C4#12011yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#13512yes—
159C2×C4#13712yes—
160C8—42no—
161C8—42no—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22#39211yescharacteristic
169C22#49511yescharacteristic
170C22#49711yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—82no—
175C4—82no—
176C4—81no—
177C4—81no—
178C4—81no—
179C4#45212yes—
180C4#45311yescharacteristic
181C4#45411yescharacteristic
182C4#46412yes—
183C2—21no—
184C2—21no—
185C2—161no—
186C2#200611yescharacteristic
187C2#200911yescharacteristic
188C2#201511yescharacteristic
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
2. #3230C2semidirect
4. #1986C2semidirect
5. #1981C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
#32462
#32302
#31412
#19862
#19812
#15022
#12832