Finite Group Databasebeta

← #9327

Subgroups — #9327

Order 256.

246 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 61 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#3110C211yesmaximal, characteristic
2#2319C211yesmaximal, characteristic
3#1986C211yesmaximal, characteristic, complemented
4#1506C211yesmaximal, characteristic, complemented
5#1490C211yesmaximal, characteristic
6#1300C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#484—21no—
11#474C412yescomplemented
12#461C412yescomplemented
13#450—21no—
14#440—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C4×M16C2211yescharacteristic
18#390—21no—
19#389—21no—
20#388—22no—
21#381—21no—
22#380—21no—
23C4×C4⋊C4C2211yescharacteristic
24#376C2211yescharacteristic
25#339—22no—
26#328—21no—
27#328—21no—
28D4⋊C8—22no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C2×SD16—22no—
32C2×M16—21no—
33C4⋊Q8C2×C411yescharacteristic
34C4⋊Q8C2×C411yescharacteristic
35#125—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4C2×C412yes—
43C4×D4D412yes—
44#117—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4D411yescharacteristic
48C2×C22⋊C4—21no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C42—21no—
52C2×C42D412yes—
53C2×C42C2311yescharacteristic
54#107—22no—
55C4⋊C8—21no—
56C4⋊C8C2×C412yes—
57#103—22no—
58#102—22no—
59C22⋊C8—21no—
60C22⋊C8—21no—
61#97—21no—
62#97—21no—
63#97—21no—
64C4×C8—21no—
65#95—22no—
66#95—21no—
67#95—21no—
68#95D411yescharacteristic
69#95D411yescharacteristic
70C24—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×D4—21no—
74C2×D4—42no—
75C2×D4C22⋊C412yes—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—82no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4C2×D411yescharacteristic
86SD16—44no—
87M16—22no—
88M16—41no—
89C2×C8—22no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—42no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—82no—
101C4⋊C4C22⋊C412yes—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—41no—
105C22⋊C4—82no—
106C22⋊C4—82no—
107C42—42no—
108C42—42no—
109C42—41no—
110C42C22×C411yescharacteristic
111C42C2×D411yescharacteristic
112C42#4111yescharacteristic
113C42#4111yescharacteristic
114C23—22no—
115C23—42no—
116C23—42no—
117C23—41no—
118C23—41no—
119C23#12011yescharacteristic
120Q8—42no—
121Q8—81no—
122D4—22no—
123D4—22no—
124D4—42no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—42no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4C2×C22⋊C411yescharacteristic
145C2×C4C4×D411yescharacteristic
146C2×C4C4×D411yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12311yescharacteristic
150C8—42no—
151C8—42no—
152C8—81no—
153C22—21no—
154C22—21no—
155C22—42no—
156C22—42no—
157C22—42no—
158C22—82no—
159C22—82no—
160C22#38611yescharacteristic
161C22#45511yescharacteristic
162C22#45611yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—42no—
169C4—81no—
170C4—82no—
171C4—81no—
172C4—81no—
173C4—81no—
174C4#42112yes—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#188211yescharacteristic
179C2#189711yescharacteristic
180C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
3. #1986C2semidirect
4. #1506C2semidirect
6. #1300C2semidirect
7. #1285C2semidirect
11. #474C4semidirect
12. #461C4semidirect

Maximal subgroups

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

subgroupindex
#31102
#23192
#19862
#15062
#14902
#13002
#12852