Finite Group Databasebeta

← #9320

Subgroups — #9320

Order 256.

245 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 69 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#3039C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic
3#1988C211yesmaximal, characteristic, complemented
4#1806C211yesmaximal, characteristic
5#1546C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1292C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C4×#41C2211yescharacteristic
11#462—21no—
12#462C412yescomplemented
13#461—21no—
14#461C412yescomplemented
15#446—21no—
16#436—21no—
17#435—22no—
18#432—21no—
19#431C2211yescharacteristic
20#423—21no—
21#423C2211yescharacteristic
22#423C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#104—21no—
25#419—21no—
26C4×C4⋊C4C2211yescharacteristic
27#340—22no—
28#339—21no—
29#332—21no—
30#332—21no—
31C2×#41D411yescharacteristic
32C22×Q8—21no—
33#137—42no—
34#137—42no—
35C2×Q16—22no—
36C2×SD16—22no—
37C2×M16—21no—
38C2×M16—22no—
39C2×M16—21no—
40C22×C8D412yes—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8C2×C412yes—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4C2×C411yescharacteristic
50C4×D4D411yescharacteristic
51#117—21no—
52#117D411yescharacteristic
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59C4⋊C8—22no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8C2×C412yes—
66C4⋊C8D411yescharacteristic
67C4⋊C8D411yescharacteristic
68#104—42no—
69#103—22no—
70#103—22no—
71C22⋊C8—22no—
72C22⋊C8—21no—
73#97—22no—
74C4×C8—21no—
75#95—21no—
76#41—21no—
77#41—21no—
78#41—21no—
79#41—21no—
80C2×Q8—22no—
81C2×Q8—41no—
82C2×Q8—41no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×Q8C22⋊C411yescharacteristic
87C2×D4—21no—
88C2×D4C22⋊C411yescharacteristic
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97Q16—44no—
98SD16—44no—
99M16—41no—
100M16—41no—
101M16—41no—
102M16—41no—
103M16—82no—
104C2×C8—22no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—42no—
109C2×C8—41no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4C22⋊C411yescharacteristic
119C4⋊C4C22⋊C411yescharacteristic
120C22⋊C4—21no—
121C22⋊C4—21no—
122C22⋊C4—21no—
123C42—21no—
124C42—21no—
125C42—21no—
126C42—41no—
127C42—41no—
128C42C22×C411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42#4111yescharacteristic
132C23—21no—
133C23#12111yescharacteristic
134Q8—21no—
135Q8—21no—
136Q8—42no—
137Q8—42no—
138Q8—81no—
139Q8—81no—
140D4—21no—
141D4—21no—
142D4—41no—
143D4—41no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4C2×C22⋊C411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4C4×D411yescharacteristic
164C2×C4#12011yescharacteristic
165C2×C4#12111yescharacteristic
166C2×C4#12311yescharacteristic
167C2×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C4#13711yescharacteristic
171C8—42no—
172C8—42no—
173C8—41no—
174C8—41no—
175C8—81no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22#38611yescharacteristic
182C22#46811yescharacteristic
183C22#46911yescharacteristic
184C4—21no—
185C4—21no—
186C4—21no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—82no—
191C4—82no—
192C4#41711yescharacteristic
193C4#41811yescharacteristic
194C4#45511yescharacteristic
195C4#45611yescharacteristic
196C4#46811yescharacteristic
197C4#46911yescharacteristic
198C2—21no—
199C2—21no—
200C2—81no—
201C2#188211yescharacteristic
202C2#189511yescharacteristic
203C2#190211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
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. #1988C2semidirect
5. #1546C2semidirect
6. #1303C2semidirect
7. #1292C2semidirect
12. #462C4semidirect
14. #461C4semidirect

Maximal subgroups

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

subgroupindex
#30392
#29892
#19882
#18062
#15462
#13032
#12922