Finite Group Databasebeta

← #9329

Subgroups — #9329

Order 256.

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

normal subgroups only

This group has 265 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#3109C211yesmaximal, characteristic
2#2319C211yesmaximal, characteristic, complemented
3#1980C211yesmaximal, characteristic, complemented
4#1505C211yesmaximal, characteristic, complemented
5#1490C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#481—21no—
11#476C412yescomplemented
12#459C412yescomplemented
13#447—21no—
14#442—22no—
15#432C2211yescharacteristic
16#432C2211yescharacteristic
17C2×#104—22no—
18C4×M16C2211yescharacteristic
19#390—21no—
20#389—21no—
21#388—22no—
22#381—21no—
23#380—21no—
24C4×C4⋊C4C2211yescharacteristic
25#376C2211yescharacteristic
26#327—21no—
27#327—21no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C2×D8—22no—
33C2×M16—21no—
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#124—21no—
37#121—21no—
38#121—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4C2×C412yes—
44C4×D4D412yes—
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#106—22no—
55C4⋊C8—21no—
56C4⋊C8C2×C412yes—
57#104—44no—
58#102—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—
72#41—42no—
73#41—42no—
74C2×Q8—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—42no—
80C2×D4—41no—
81C2×D4C22⋊C412yes—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—41no—
89C22×C4—82no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93D8—44no—
94M16—22no—
95M16—41no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—42no—
103C4⋊C4—82no—
104C4⋊C4C22⋊C412yes—
105C22⋊C4—22no—
106C22⋊C4—22no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—82no—
110C22⋊C4—82no—
111C42—42no—
112C42—42no—
113C42C22×C411yescharacteristic
114C42C2×D411yescharacteristic
115C42#4111yescharacteristic
116C42#4111yescharacteristic
117C23—22no—
118C23—42no—
119C23—42no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23#12011yescharacteristic
124Q8—42no—
125D4—22no—
126D4—22no—
127D4—42no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—81no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4C2×C22⋊C411yescharacteristic
153C2×C4C4×D411yescharacteristic
154C2×C4C4×D411yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12311yescharacteristic
158C8—42no—
159C8—42no—
160C8—81no—
161C22—21no—
162C22—21no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—82no—
167C22—82no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#38611yescharacteristic
172C22#45311yescharacteristic
173C22#45411yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—81no—
181C4—82no—
182C4—81no—
183C4—81no—
184C4#42112yes—
185C2—21no—
186C2—21no—
187C2—82no—
188C2—161no—
189C2#188311yescharacteristic
190C2#189711yescharacteristic
191C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3109C2
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. #2319C2semidirect
3. #1980C2semidirect
4. #1505C2semidirect
5. #1490C2semidirect
6. #1300C2semidirect
7. #1285C2semidirect
11. #476C4semidirect
12. #459C4semidirect

Maximal subgroups

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

subgroupindex
#31092
#23192
#19802
#15052
#14902
#13002
#12852