Finite Group Databasebeta

← #12326

Subgroups — #12326

Order 256.

195 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 41 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#3273C211yesmaximal, characteristic
2#2251C211yesmaximal, characteristic, complemented
3#2208C211yesmaximal, characteristic, complemented
4#1689C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1333C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#546—21no—
10C2×Q32—21no—
11C2×SD32—21no—
12C2×SD32—21no—
13#495C2211yescharacteristic
14#494C2211yescharacteristic, complemented
15#474—21no—
16#468—21no—
17#463—21no—
18#452—21no—
19#447—21no—
20C4×D8C2211yescharacteristic
21#436C2211yescharacteristic
22C4⋊C16C2211yescharacteristic, complemented
23#358—21no—
24#346C2211yescharacteristic, complemented
25#334C2211yescharacteristic, complemented
26#332—21no—
27C2×#41—21no—
28C22×D4—21no—
29AGL(1,ℤ/8)—42no—
30AGL(1,ℤ/8)—42no—
31C2×Q16—21no—
32C2×Q16—21no—
33C2×Q16D411yescharacteristic
34C2×SD16—21no—
35C2×SD16—21no—
36C2×D8—21no—
37C2×D8D411yescharacteristic
38C2×D8D411yescharacteristic
39C2×M16—21no—
40C4⋊Q8—21no—
41#124—21no—
42#123—41no—
43C22⋊Q8—21no—
44#121—21no—
45#121—41no—
46#120—41no—
47C4×D4—21no—
48C4×D4D411yescharacteristic
49C2×C22⋊C4—41no—
50Q32—42no—
51SD32—42no—
52SD32—42no—
53C2×C16—21no—
54C2×C16—21no—
55#107—41no—
56#107D411yescharacteristic
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59#103—21no—
60#102—21no—
61#102—21no—
62#102—41no—
63C22⋊C8—21no—
64C4×C8C2311yescharacteristic
65C24—41no—
66#41—42no—
67#41—81no—
68C2×Q8—21no—
69C2×Q8—41no—
70C2×D4—21no—
71C2×D4—21no—
72C2×D4—41no—
73C2×D4—42no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—81no—
77C2×D4D811yescharacteristic
78C22×C4—21no—
79C22×C4—41no—
80C22×C4—41no—
81Q16—41no—
82Q16—41no—
83Q16—41no—
84Q16—41no—
85Q16—41no—
86SD16—42no—
87SD16—41no—
88SD16—41no—
89D8—21no—
90D8—41no—
91D8—42no—
92D8D812yes—
93M16—41no—
94M16—41no—
95C2×C8—21no—
96C2×C8—41no—
97C2×C8C2×D411yescharacteristic
98C2×C8C2×D411yescharacteristic
99C4⋊C4—21no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4D811yescharacteristic
104C22⋊C4—21no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C42C2×D411yescharacteristic
111C16—22no—
112C16—41no—
113C23—21no—
114C23—41no—
115C23—41no—
116C23—82no—
117C23—82no—
118C23—161no—
119Q8—41no—
120Q8—41no—
121Q8—81no—
122Q8—81no—
123D4—22no—
124D4—41no—
125D4—41no—
126D4—41no—
127D4—41no—
128D4—82no—
129D4—82no—
130D4—161no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—82no—
137C2×C4—81no—
138C2×C4—161no—
139C2×C4#12011yescharacteristic
140C2×C4C2×D811yescharacteristic
141C2×C4AGL(1,ℤ/8)11yescharacteristic
142C8—21no—
143C8—41no—
144C8—81no—
145C8C2×D811yescharacteristic
146C8AGL(1,ℤ/8)11yescharacteristic
147C22—41no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22—164no—
157C22#44711yescharacteristic
158C4—21no—
159C4—21no—
160C4—41no—
161C4—81no—
162C4—81no—
163C4—161no—
164C4#44711yescharacteristic
165C4#45311yescharacteristic
166C2—81no—
167C2—82no—
168C2—161no—
169C2#161911yescharacteristic
170C2#219111yescharacteristic
171C2#219411yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3273C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2251C2semidirect
3. #2208C2semidirect
4. #1689C2semidirect
5. #1360C2semidirect
6. #1333C2semidirect
7. #1330C2semidirect
14. #494C22semidirect
22. C4⋊C16C22semidirect
24. #346C22semidirect
25. #334C22semidirect

Maximal subgroups

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

subgroupindex
#32732
#22512
#22082
#16892
#13602
#13332
#13302