Finite Group Databasebeta

← #9209

Subgroups — #9209

Order 256.

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

normal subgroups only

This group has 228 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#3076C211yesmaximal, characteristic
2#2965C211yesmaximal, characteristic
3#1806C211yesmaximal, characteristic
4#1503C211yesmaximal, characteristic, complemented
5#1493C211yesmaximal, characteristic, complemented
6#1301C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#479—21no—
11#478—21no—
12#477C411yescharacteristic, complemented
13#476C411yescharacteristic, complemented
14#475C411yescharacteristic, complemented
15#474C411yescharacteristic, complemented
16#446—21no—
17#436—21no—
18#435—22no—
19#431—21no—
20#431C2211yescharacteristic
21#427—21no—
22#423C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#104—21no—
25#416—21no—
26C4×M16—21no—
27C2×C4×C8C2211yescharacteristic
28C4×C4⋊C4C2211yescharacteristic
29#339—21no—
30#328—22no—
31Q8⋊C8—21no—
32Q8⋊C8—21no—
33C2×#41D411yescharacteristic
34C2×M16—22no—
35C2×M16—21no—
36C22×C8—21no—
37C22×C8—21no—
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8C2×C411yescharacteristic
40#125—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4C2×C411yescharacteristic
50C4×D4D411yescharacteristic
51#117—21no—
52#117—21no—
53#117D411yescharacteristic
54C2×C4⋊C4—21no—
55C2×C42—21no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#107—21no—
59#107—21no—
60#106—21no—
61#106—21no—
62C4⋊C8—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8C2×C411yescharacteristic
68C4⋊C8C2×C411yescharacteristic
69#104—42no—
70#103—21no—
71#103—21no—
72#102—21no—
73#102—21no—
74C22⋊C8—22no—
75C22⋊C8D412yes—
76#97—22no—
77C4×C8—21no—
78C4×C8—21no—
79C4×C8D411yescharacteristic
80C4×C8D411yescharacteristic
81#95—21no—
82#41—21no—
83#41—21no—
84#41—21no—
85#41—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8C22⋊C411yescharacteristic
89C2×D4—21no—
90C2×D4C22⋊C411yescharacteristic
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—41no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4#4111yescharacteristic
98M16—44no—
99M16—41no—
100M16—41no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—42no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4C22⋊C411yescharacteristic
121C4⋊C4C22⋊C411yescharacteristic
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—41no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42C22×C411yescharacteristic
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42#4111yescharacteristic
136C23—21no—
137C23#12111yescharacteristic
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—81no—
143D4—21no—
144D4—21no—
145D4—41no—
146D4—41no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4C2×C22⋊C411yescharacteristic
164C2×C4C4×D411yescharacteristic
165C2×C4C4×D411yescharacteristic
166C2×C4#12011yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12311yescharacteristic
169C2×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C4#13711yescharacteristic
171C2×C4#13711yescharacteristic
172C2×C4#13711yescharacteristic
173C8—22no—
174C8—41no—
175C8—42no—
176C8—41no—
177C8—41no—
178C8—81no—
179C22—21no—
180C22—21no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22#38611yescharacteristic
185C22#46811yescharacteristic
186C22#47011yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—82no—
194C4—81no—
195C4—81no—
196C4#41711yescharacteristic
197C4#41911yescharacteristic
198C4#44311yescharacteristic
199C4#44411yescharacteristic
200C4#46811yescharacteristic
201C4#47011yescharacteristic
202C2—21no—
203C2—21no—
204C2—81no—
205C2#187611yescharacteristic
206C2#189411yescharacteristic
207C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #1503C2semidirect
5. #1493C2semidirect
6. #1301C2semidirect
7. #1279C2semidirect
12. #477C4semidirect
13. #476C4semidirect
14. #475C4semidirect
15. #474C4semidirect

Maximal subgroups

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

subgroupindex
#30762
#29652
#18062
#15032
#14932
#13012
#12792