Finite Group Databasebeta

← #9159

Subgroups — #9159

Order 256.

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

normal subgroups only

This group has 226 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#3116C211yesmaximal, characteristic
2#2970C211yesmaximal, characteristic
3#1759C211yesmaximal, characteristic
4#1526C211yesmaximal, characteristic, complemented
5#1506C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#485—21no—
11#477C411yescharacteristic, complemented
12#476C411yescharacteristic, complemented
13#462C411yescharacteristic, complemented
14#461C411yescharacteristic, complemented
15#452—21no—
16#446—21no—
17#441—21no—
18#440—21no—
19#435—22no—
20C8×D4—21no—
21#432—21no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24C2×#104—21no—
25C4×M16C2211yescharacteristic
26C4×M16C2211yescharacteristic
27C2×#97—21no—
28#376C2211yescharacteristic
29#339—21no—
30#330—21no—
31#330—21no—
32#328—22no—
33C2×#41D411yescharacteristic
34C2×Q16—21no—
35C2×SD16—21no—
36C2×M16—21no—
37C2×M16—21no—
38C22×C8—22no—
39C4⋊Q8C2×C411yescharacteristic
40C4⋊Q8C2×C411yescharacteristic
41#125—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8C2×C411yescharacteristic
47C4×Q8D411yescharacteristic
48C4×D4—21no—
49C4×D4—21no—
50C4×D4C2×C411yescharacteristic
51C4×D4D411yescharacteristic
52#117—21no—
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#103—21no—
67#103—21no—
68#102—21no—
69C22⋊C8—22no—
70C22⋊C8—21no—
71C22⋊C8D412yes—
72#97—22no—
73#97—21no—
74#97—21no—
75#97—21no—
76#97D411yescharacteristic
77#97D411yescharacteristic
78C4×C8—21no—
79C4×C8—21no—
80#95—21no—
81#95—21no—
82#41—22no—
83#41—41no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8C22⋊C411yescharacteristic
87C2×D4—21no—
88C2×D4C22⋊C411yescharacteristic
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4111yescharacteristic
96Q16—42no—
97SD16—42no—
98M16—22no—
99M16—42no—
100M16—41no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—44no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4C22⋊C411yescharacteristic
117C4⋊C4C22⋊C411yescharacteristic
118C22⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—21no—
121C22⋊C4—41no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—22no—
126C42—41no—
127C42—41no—
128C42C22×C411yescharacteristic
129C42C2×D411yescharacteristic
130C42C2×D411yescharacteristic
131C42#4111yescharacteristic
132C23—21no—
133C23#12111yescharacteristic
134Q8—21no—
135Q8—21no—
136Q8—41no—
137Q8—41no—
138Q8—81no—
139D4—21no—
140D4—42no—
141D4#10412yes—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4C2×C22⋊C411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4C4×D411yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12311yescharacteristic
163C8—22no—
164C8—42no—
165C8—41no—
166C8—42no—
167C8—41no—
168C8—41no—
169C22—21no—
170C22—22no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22#38611yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—82no—
185C4—81no—
186C4—81no—
187C4C2×#10411yescharacteristic
188C4#42111yescharacteristic
189C2—41no—
190C2—42no—
191C2#187111yescharacteristic
192C2#190211yescharacteristic
193C2#190611yescharacteristic
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
4. #1526C2semidirect
5. #1506C2semidirect
6. #1300C2semidirect
7. #1291C2semidirect
11. #477C4semidirect
12. #476C4semidirect
13. #462C4semidirect
14. #461C4semidirect

Maximal subgroups

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

subgroupindex
#31162
#29702
#17592
#15262
#15062
#13002
#12912