Finite Group Databasebeta

← #9284

Subgroups — #9284

Order 256.

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

normal subgroups only

This group has 198 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#3228C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic, complemented
3#1531C211yesmaximal, characteristic, complemented
4#1530C211yesmaximal, characteristic
5#1496C211yesmaximal, characteristic
6#1287C211yesmaximal, characteristic, complemented
7#1275C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#500C411yescharacteristic
11#498C411yescharacteristic
12#486C411yescharacteristic
13#486C411yescharacteristic
14#482—22no—
15#478—22no—
16#446—22no—
17#446—21no—
18C8×Q8—21no—
19#433—22no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24C2×#97C2211yescharacteristic
25#330—22no—
26#329—22no—
27D4⋊C8—21no—
28D4⋊C8—21no—
29C2×#41—21no—
30C22×C8—21no—
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C411yescharacteristic
34#127C2×C411yescharacteristic
35#125D412yes—
36#124—21no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39#121—21no—
40#121—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×D4—21no—
44C4×D4—21no—
45#117D412yes—
46C2×C42C2311yescharacteristic
47#107—22no—
48#106—22no—
49C4⋊C8—22no—
50C4⋊C8—22no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55#102—22no—
56#102—22no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61#97—22no—
62#97—21no—
63#97—21no—
64#97—21no—
65#97D412yes—
66C4×C8—22no—
67C4×C8C2×C411yescharacteristic
68C4×C8C2×C411yescharacteristic
69C4×C8D411yescharacteristic
70C4×C8D411yescharacteristic
71#41—82no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—41no—
79C22×C4—41no—
80C22×C4C2×D411yescharacteristic
81C22×C4#4111yescharacteristic
82C22×C4#4111yescharacteristic
83C2×C8—22no—
84C2×C8—21no—
85C2×C8—21no—
86C2×C8—21no—
87C2×C8—22no—
88C2×C8—21no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—22no—
95C4⋊C4—22no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4C22⋊C412yes—
99C4⋊C4C22⋊C412yes—
100C22⋊C4—22no—
101C22⋊C4—41no—
102C22⋊C4—41no—
103C42—22no—
104C42C22×C411yescharacteristic
105C42C2×D411yescharacteristic
106C42C2×D411yescharacteristic
107C42C2×D411yescharacteristic
108C23—41no—
109C23#12311yescharacteristic
110Q8—41no—
111Q8—41no—
112Q8—81no—
113Q8—81no—
114D4—42no—
115D4—41no—
116D4—41no—
117D4—81no—
118D4—82no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—82no—
127C2×C4C2×C22⋊C411yescharacteristic
128C2×C4C4×D411yescharacteristic
129C2×C4C4×D411yescharacteristic
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C8—41no—
134C8—42no—
135C8—41no—
136C8—41no—
137C8—41no—
138C8—42no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22—21no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#38611yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—21no—
151C4—21no—
152C4—42no—
153C4—42no—
154C4—81no—
155C4#44311yescharacteristic
156C4#44411yescharacteristic
157C2—41no—
158C2—161no—
159C2#187611yescharacteristic
160C2#190311yescharacteristic
161C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ 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. #2985C2semidirect
3. #1531C2semidirect
6. #1287C2semidirect
7. #1275C2semidirect

Maximal subgroups

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

subgroupindex
#32282
#29852
#15312
#15302
#14962
#12872
#12752