Finite Group Databasebeta

← #9372

Subgroups — #9372

Order 256.

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

normal subgroups only

This group has 195 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#3235C211yesmaximal, characteristic
2#2985C211yesmaximal, characteristic, complemented
3#1552C211yesmaximal, characteristic, complemented
4#1536C211yesmaximal, characteristic
5#1480C211yesmaximal, characteristic
6#1290C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#501C412yes—
11#490C412yes—
12#482—22no—
13#476—21no—
14#474—21no—
15#446—22no—
16#446—21no—
17C8×Q8—21no—
18#433—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24#332—21no—
25#332—21no—
26#329—22no—
27D4⋊C8—21no—
28D4⋊C8—21no—
29C2×#41—21no—
30C22×C8—21no—
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8D411yescharacteristic
33C4⋊Q8D411yescharacteristic
34#127C2×C411yescharacteristic
35#125C2×C412yes—
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—
55C4⋊C8—21no—
56C4⋊C8—21no—
57#102—22no—
58#102—22no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63#97—22no—
64#97—21no—
65#97C2×C412yes—
66#97D412yes—
67C4×C8—22no—
68C4×C8D411yescharacteristic
69C4×C8D411yescharacteristic
70#41—81no—
71#41—81no—
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—22no—
86C2×C8—21no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—81no—
90C4⋊C4—22no—
91C4⋊C4—22no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4C22⋊C412yes—
97C4⋊C4C22⋊C412yes—
98C22⋊C4—22no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C42—22no—
102C42C22×C411yescharacteristic
103C42C2×D411yescharacteristic
104C42C2×D411yescharacteristic
105C42C2×D411yescharacteristic
106C23—41no—
107C23#12311yescharacteristic
108Q8—41no—
109Q8—41no—
110Q8—81no—
111Q8—81no—
112D4—42no—
113D4—42no—
114D4—81no—
115D4—82no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—81no—
124C2×C4—81no—
125C2×C4C2×C22⋊C411yescharacteristic
126C2×C4C4×D411yescharacteristic
127C2×C4C4×D411yescharacteristic
128C2×C4#12011yescharacteristic
129C2×C4#12111yescharacteristic
130C2×C4#12111yescharacteristic
131C8—41no—
132C8—42no—
133C8—41no—
134C8—42no—
135C8—81no—
136C8—81no—
137C22—21no—
138C22—21no—
139C22—21no—
140C22—81no—
141C22—81no—
142C22—81no—
143C22#38611yescharacteristic
144C4—21no—
145C4—21no—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—42no—
150C4—42no—
151C4—81no—
152C4#47111yescharacteristic
153C4#47211yescharacteristic
154C2—41no—
155C2—161no—
156C2#188111yescharacteristic
157C2#190711yescharacteristic
158C2#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. #1552C2semidirect
6. #1290C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32352
#29852
#15522
#15362
#14802
#12902
#12872