Finite Group Databasebeta

← #22228

Subgroups — #22228

Order 256.

387 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×#488C211yesmaximal, characteristic
2#2330C211yesmaximal, characteristic, complemented
3#2326C211yesmaximal, characteristic
4#1992C211yesmaximal, characteristic, complemented
5#1959C211yesmaximal, characteristic
6#1928C212yesmaximal, complemented
7#1925C212yesmaximal
8#1863C212yesmaximal, complemented
9#1862C212yesmaximal
10#1845C211yesmaximal, characteristic, complemented
11#1746C211yesmaximal, characteristic, complemented
12C2×#125C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#488C418yescomplemented
17C2×#107C2212yes—
18C2×#107C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#97C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#389—22no—
28#389C2212yes—
29#388—22no—
30#387—22no—
31#385—22no—
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34C4×C4⋊C4C2212yes—
35#376C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39#336C2212yes—
40C22×Q8C2311yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—22no—
46#125C2×C414yescomplemented
47#124—22no—
48#124C2×C414yes—
49C4×Q8—42no—
50C4×D4—42no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C42—21no—
64C2×C42—22no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#107—42no—
69#107—42no—
70#103—42no—
71#103C2×C418yescomplemented
72#102—42no—
73#102C2×C418yes—
74#97—22no—
75C4×C8C2×C414yescomplemented
76#95—22no—
77#95—22no—
78#95—22no—
79#95D412yes—
80#95D412yes—
81C24—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8C22×C412yes—
86C2×D4—21no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4C22×C412yes—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—42no—
96C22×C4—82no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D412yes—
101C22×C4C2411yescharacteristic
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8C22×C414yes—
108C4⋊C4—28no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C4⋊C4—82no—
118C4⋊C4C22×C414yes—
119C22⋊C4—28no—
120C22⋊C4—42no—
121C22⋊C4—82no—
122C42—22no—
123C42—22no—
124C42—42no—
125C42—82no—
126C42C22×C412yes—
127C42#4112yes—
128C23—22no—
129C23—41no—
130C23—41no—
131C23—42no—
132C23—42no—
133C23C22×D411yescharacteristic
134Q8—24no—
135Q8—42no—
136D4—24no—
137D4—42no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—24no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4C4×D414yes—
162C2×C4C23×C411yescharacteristic
163C2×C4C2×#4111yescharacteristic
164C2×C4H(5,𝔽2)12yes—
165C2×C42−1+412yes—
166C8—24no—
167C8—42no—
168C22—42no—
169C22—44no—
170C22—81no—
171C22—81no—
172C22—82no—
173C22C2×C4×D411yescharacteristic
174C22#53511yescharacteristic
175C22#53611yescharacteristic
176C22#57612yes—
177C22#57812yes—
178C4—24no—
179C4—42no—
180C4—44no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4—81no—
185C4—82no—
186C4—81no—
187C4#52012yes—
188C4#54212yes—
189C2—82no—
190C2#232611yescharacteristic
191C2#295112yes—
192C2#322011yescharacteristic
193C2#322311yescharacteristic
194C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2330C2semidirect
4. #1992C2semidirect
6. #1928C2semidirect
8. #1863C2semidirect
10. #1845C2semidirect
11. #1746C2semidirect
16. #488C4semidirect
46. #125C2×C4semidirect
71. #103C2×C4semidirect
75. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
#23302
#23262
#19922
#19592
#19282
#19252
#18632
#18622
#18452
#17462