Finite Group Databasebeta

← #10122

Subgroups — #10122

Order 256.

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

normal subgroups only

This group has 238 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#3234C211yesmaximal, characteristic, complemented
2#3214C211yesmaximal, characteristic
3#3142C211yesmaximal, characteristic, complemented
4#1496C211yesmaximal, characteristic, complemented
5#1484C211yesmaximal, characteristic, complemented
6#1470C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9#532C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#496—21no—
13#491—21no—
14#490—21no—
15#490—21no—
16#489—21no—
17#488—22no—
18#481—21no—
19#480—21no—
20#478—22no—
21#461—21no—
22#459—21no—
23#449—22no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27#417—21no—
28#417—21no—
29C4×M16C2211yescharacteristic
30D4⋊C8—22no—
31D4⋊C8—22no—
32#324—22no—
33C2×#41—21no—
34C2×#41—21no—
35C2×SD16—22no—
36C2×D8—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#127D411yescharacteristic
41#127D411yescharacteristic
42#125—21no—
43#125—21no—
44#125—21no—
45#125—21no—
46#124—21no—
47#124—21no—
48#121—21no—
49#121—21no—
50#121—21no—
51#121—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C42C2311yescharacteristic
61#107—22no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D412yes—
66#103—22no—
67#103—22no—
68#103—22no—
69#102—22no—
70#102—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8D412yes—
82#41—44no—
83#41—82no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—41no—
93C2×D4—41no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4#4111yescharacteristic
101SD16—82no—
102D8—82no—
103M16—42no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C23—41no—
129C23—41no—
130C23#12111yescharacteristic
131Q8—42no—
132Q8—42no—
133D4—41no—
134D4—42no—
135D4—41no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—82no—
141D4—82no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—81no—
153C2×C4#12011yescharacteristic
154C2×C4#12011yescharacteristic
155C2×C4#12011yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#12711yescharacteristic
159C8—42no—
160C8—42no—
161C8—42no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#39211yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—81no—
179C4—81no—
180C4—81no—
181C4—81no—
182C4#45412yes—
183C2—41no—
184C2—161no—
185C2—161no—
186C2#200311yescharacteristic
187C2#200811yescharacteristic
188C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3214C2
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
1. #3234C2semidirect
3. #3142C2semidirect
4. #1496C2semidirect
5. #1484C2semidirect
6. #1470C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#32142
#31422
#14962
#14842
#14702
#12912