Finite Group Databasebeta

← #9323

Subgroups — #9323

Order 256.

306 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 65 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#2275C211yesmaximal, characteristic
3#1986C211yesmaximal, characteristic, complemented
4#1506C211yesmaximal, characteristic, complemented
5#1490C211yesmaximal, characteristic
6#1300C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4—21no—
10C2×C4×D4C2211yescharacteristic
11#484—21no—
12#474C412yescomplemented
13#461C412yescomplemented
14#450—21no—
15#440—22no—
16#432C2211yescharacteristic
17#432C2211yescharacteristic
18C4×M16C2211yescharacteristic
19C4×C4⋊C4—21no—
20C4×C4⋊C4C2211yescharacteristic
21C4×C22⋊C4—22no—
22C4×C22⋊C4—21no—
23#376C2211yescharacteristic
24C2×#95—21no—
25#339—22no—
26#328—21no—
27#328—21no—
28D4⋊C8—22no—
29C22×D4D411yescharacteristic
30C23×C4—22no—
31C23×C4—21no—
32C2×SD16—22no—
33C2×M16—21no—
34C4⋊Q8C2×C411yescharacteristic, complemented
35C4⋊Q8C2×C411yescharacteristic, complemented
36#125—21no—
37C22⋊Q8—21no—
38C22⋊Q8—21no—
39C4×Q8—21no—
40C4×Q8—21no—
41C4×D4—28no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4C2×C412yes—
45C4×D4D412yes—
46#117—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C22⋊C4—22no—
50C2×C22⋊C4—21no—
51C2×C42—21no—
52C2×C42—22no—
53C2×C42D412yes—
54C2×C42C2311yescharacteristic
55#107—22no—
56C4⋊C8—21no—
57C4⋊C8C2×C412yescomplemented
58#103—22no—
59#102—22no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62#97—21no—
63#97—21no—
64#97—21no—
65C4×C8—21no—
66#95—22no—
67#95—21no—
68#95—21no—
69#95D411yescharacteristic
70#95D411yescharacteristic
71C24—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×D4—24no—
75C2×D4—21no—
76C2×D4C22⋊C412yes—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—28no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—82no—
85C22×C4—82no—
86C22×C4—82no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90SD16—44no—
91M16—22no—
92M16—41no—
93C2×C8—22no—
94C2×C8—41no—
95C2×C8—41no—
96C4⋊C4—24no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4C22⋊C412yes—
105C22⋊C4—28no—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—41no—
109C42—24no—
110C42—42no—
111C42—41no—
112C42—82no—
113C42C22×C411yescharacteristic
114C42C2×D411yescharacteristic
115C42#4111yescharacteristic
116C42#4111yescharacteristic
117C23—24no—
118C23—22no—
119C23—42no—
120C23—41no—
121C23—41no—
122C23#12011yescharacteristic
123Q8—42no—
124Q8—81no—
125D4—24no—
126D4—22no—
127D4#10414yes—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—88no—
148C2×C4C2×C22⋊C411yescharacteristic
149C2×C4C4×D411yescharacteristic
150C2×C4C4×D411yescharacteristic
151C2×C4#12111yescharacteristic
152C2×C4#12111yescharacteristic
153C2×C4#12311yescharacteristic
154C8—42no—
155C8—42no—
156C8—81no—
157C22—21no—
158C22—21no—
159C22—24no—
160C22—44no—
161C22—44no—
162C22—42no—
163C22—42no—
164C22#38611yescharacteristic
165C22#45511yescharacteristic
166C22#45611yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—42no—
173C4—44no—
174C4—42no—
175C4—42no—
176C4—81no—
177C4—81no—
178C4C2×#10412yes—
179C2—21no—
180C2—21no—
181C2—44no—
182C2#188211yescharacteristic
183C2#189611yescharacteristic
184C2#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
12. #474C4semidirect
13. #461C4semidirect
34. C4⋊Q8C2×C4semidirect
35. C4⋊Q8C2×C4semidirect
57. C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31102
#22752
#19862
#15062
#14902
#13002
#12852