Finite Group Databasebeta

← #9162

Subgroups — #9162

Order 256.

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

normal subgroups only

This group has 236 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#3115C211yesmaximal, characteristic
2#2971C211yesmaximal, characteristic, complemented
3#1759C211yesmaximal, characteristic, complemented
4#1525C211yesmaximal, characteristic, complemented
5#1505C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#482—21no—
11#475C411yescharacteristic, complemented
12#474C411yescharacteristic, complemented
13#460C411yescharacteristic, complemented
14#459C411yescharacteristic, complemented
15#449—21no—
16C8×Q8—21no—
17#442—21no—
18#440—21no—
19#436—21no—
20#436—22no—
21#432—21no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24C2×#104—21no—
25C4×M16—21no—
26C4×M16C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#376C2211yescharacteristic
29#339—21no—
30#329—21no—
31#329—21no—
32#327—22no—
33C2×#41—21no—
34C2×#41D411yescharacteristic
35C2×SD16—21no—
36C2×D8—21no—
37C2×M16—22no—
38C2×M16—21no—
39C2×M16—21no—
40C4⋊Q8C2×C411yescharacteristic
41#127C2×C411yescharacteristic
42#124—21no—
43#121—21no—
44#121—21no—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4C2×C411yescharacteristic
52C4×D4D411yescharacteristic
53#117—21no—
54#117D411yescharacteristic
55C2×C42—21no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#107—21no—
59#106—21no—
60C4⋊C8—22no—
61C4⋊C8—21no—
62C4⋊C8C2×C411yescharacteristic
63C4⋊C8C2×C411yescharacteristic
64#104—24no—
65#103—21no—
66#102—21no—
67#102—21no—
68#102—21no—
69C22⋊C8—22no—
70C22⋊C8—21no—
71C22⋊C8D412yes—
72#97—21no—
73#97—21no—
74#97—21no—
75#97D411yescharacteristic
76#97D411yescharacteristic
77C4×C8—22no—
78C4×C8—21no—
79C4×C8—21no—
80#95—21no—
81#95—21no—
82#41—22no—
83#41—41no—
84#41—82no—
85C2×Q8—21no—
86C2×Q8C22⋊C411yescharacteristic
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—41no—
92C2×D4C22⋊C411yescharacteristic
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4#4111yescharacteristic
101SD16—42no—
102D8—42no—
103M16—22no—
104M16—42no—
105M16—44no—
106M16—41no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—22no—
111C2×C8—21no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4C22⋊C411yescharacteristic
118C4⋊C4C22⋊C411yescharacteristic
119C22⋊C4—21no—
120C22⋊C4—21no—
121C22⋊C4—21no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—22no—
128C42—41no—
129C42C22×C411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—21no—
134C23—41no—
135C23#12111yescharacteristic
136Q8—21no—
137Q8—81no—
138Q8#10412yes—
139D4—21no—
140D4—21no—
141D4—42no—
142D4—41no—
143D4—41no—
144D4—81no—
145D4—81no—
146D4—82no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—82no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4C2×C22⋊C411yescharacteristic
163C2×C4C4×D411yescharacteristic
164C2×C4C4×D411yescharacteristic
165C2×C4#12011yescharacteristic
166C2×C4#12111yescharacteristic
167C2×C4#12311yescharacteristic
168C8—22no—
169C8—42no—
170C8—41no—
171C8—42no—
172C8—41no—
173C8—41no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22#38611yescharacteristic
184C4—21no—
185C4—22no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—41no—
191C4—41no—
192C4—82no—
193C4—81no—
194C4C2×#10411yescharacteristic
195C4#42111yescharacteristic
196C2—41no—
197C2—81no—
198C2—161no—
199C2#187111yescharacteristic
200C2#190311yescharacteristic
201C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3115C2
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. #2971C2semidirect
3. #1759C2semidirect
4. #1525C2semidirect
5. #1505C2semidirect
6. #1300C2semidirect
7. #1291C2semidirect
11. #475C4semidirect
12. #474C4semidirect
13. #460C4semidirect
14. #459C4semidirect

Maximal subgroups

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

subgroupindex
#31152
#29712
#17592
#15252
#15052
#13002
#12912