Finite Group Databasebeta

← #12327

Subgroups — #12327

Order 256.

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

normal subgroups only

This group has 184 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#3277C211yesmaximal, characteristic
2#2251C211yesmaximal, characteristic, complemented
3#2209C211yesmaximal, characteristic, complemented
4#1689C211yesmaximal, characteristic, complemented
5#1360C211yesmaximal, characteristic, complemented
6#1334C211yesmaximal, characteristic, complemented
7#1330C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#548—21no—
10C2×Q32—21no—
11C2×Q32—21no—
12C2×SD32—21no—
13#495C2211yescharacteristic
14#494C2211yescharacteristic
15#474—21no—
16#468—21no—
17#464—21no—
18#451—21no—
19#449—21no—
20C4×Q16C2211yescharacteristic
21#436C2211yescharacteristic
22C4⋊C16C2211yescharacteristic
23#358—21no—
24#346C2211yescharacteristic
25#334C2211yescharacteristic
26#332—21no—
27C2×#41—21no—
28C22×Q8—21no—
29#137—42no—
30#137—42no—
31C2×Q16—21no—
32C2×Q16—21no—
33C2×Q16—21no—
34C2×Q16D411yescharacteristic
35C2×Q16D411yescharacteristic
36C2×SD16—21no—
37C2×SD16—21no—
38C2×D8D411yescharacteristic
39C2×M16—21no—
40C4⋊Q8—21no—
41#124—21no—
42#124—41no—
43C22⋊Q8—21no—
44C22⋊Q8—41no—
45#121—21no—
46#121—41no—
47C4×Q8—21no—
48C4×D4—41no—
49C4×D4D411yescharacteristic
50Q32—42no—
51Q32—42no—
52SD32—42no—
53C2×C16—21no—
54C2×C16—21no—
55#107—41no—
56#107D411yescharacteristic
57C4⋊C8—21no—
58C4⋊C8D411yescharacteristic
59#103—21no—
60#103—21no—
61#102—21no—
62#102—41no—
63C22⋊C8—21no—
64C4×C8C2311yescharacteristic
65#41—42no—
66#41—81no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—41no—
70C2×Q8—42no—
71C2×Q8—41no—
72C2×Q8—81no—
73C2×D4—21no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4D811yescharacteristic
77C22×C4—21no—
78C22×C4—41no—
79C22×C4—41no—
80Q16—21no—
81Q16—41no—
82Q16—41no—
83Q16—41no—
84Q16—42no—
85Q16—41no—
86Q16—41no—
87Q16D812yes—
88SD16—42no—
89SD16—41no—
90SD16—41no—
91D8—41no—
92M16—41no—
93M16—41no—
94C2×C8—21no—
95C2×C8—41no—
96C2×C8C2×D411yescharacteristic
97C2×C8C2×D411yescharacteristic
98C4⋊C4—21no—
99C4⋊C4—41no—
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—
109C42—41no—
110C42C2×D411yescharacteristic
111C16—22no—
112C16—41no—
113C23—21no—
114C23—41no—
115Q8—22no—
116Q8—41no—
117Q8—41no—
118Q8—41no—
119Q8—82no—
120Q8—81no—
121Q8—81no—
122D4—41no—
123D4—41no—
124D4—41no—
125D4—82no—
126D4—82no—
127C2×C4—21no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—81no—
136C2×C4#12011yescharacteristic
137C2×C4C2×D811yescharacteristic
138C2×C4AGL(1,ℤ/8)11yescharacteristic
139C8—21no—
140C8—41no—
141C8—81no—
142C8C2×D811yescharacteristic
143C8AGL(1,ℤ/8)11yescharacteristic
144C22—41no—
145C22—41no—
146C22—41no—
147C22—81no—
148C22—81no—
149C22—81no—
150C22#44711yescharacteristic
151C4—21no—
152C4—21no—
153C4—41no—
154C4—42no—
155C4—81no—
156C4—81no—
157C4—161no—
158C4#44711yescharacteristic
159C4#45311yescharacteristic
160C2—81no—
161C2—161no—
162C2#161911yescharacteristic
163C2#219111yescharacteristic
164C2#219511yescharacteristic
subgroupG/H
center, socleC22#447
commutator, Frattini, cosocleC4×C8C23
prime radical#3277C2
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. #2209C2semidirect
4. #1689C2semidirect
5. #1360C2semidirect
6. #1334C2semidirect
7. #1330C2semidirect

Maximal subgroups

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

subgroupindex
#32772
#22512
#22092
#16892
#13602
#13342
#13302