Finite Group Databasebeta

← #9312

Subgroups — #9312

Order 256.

246 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 63 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#2875C211yesmaximal, characteristic
3#1531C211yesmaximal, characteristic, complemented
4#1488C211yesmaximal, characteristic
5#1485C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8C2×#131—21no—
9#533C2211yescharacteristic
10C4×#41C2211yescharacteristic
11#485—21no—
12#477C411yescharacteristic
13#476C411yescharacteristic
14#462C411yescharacteristic
15#461C411yescharacteristic
16#452—21no—
17#441—21no—
18#440—21no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22#405—21no—
23#405—22no—
24C4×M16—21no—
25C4×M16C2211yescharacteristic
26C2×#97—21no—
27C2×#97C2211yescharacteristic
28#330—21no—
29#330—21no—
30Q8⋊C8—21no—
31Q8⋊C8—22no—
32D4⋊C8—21no—
33C2×#41D411yescharacteristic
34C2×Q16—21no—
35C2×SD16—21no—
36#131—24no—
37#131—24no—
38C2×M16—22no—
39C2×M16—21no—
40C22×C8—22no—
41C22×C8—21no—
42C4⋊Q8C2×C411yescharacteristic
43C4⋊Q8C2×C411yescharacteristic
44#125—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8C2×C411yescharacteristic
50C4×Q8D411yescharacteristic
51C4×D4—21no—
52C4×D4—21no—
53C4×D4C2×C411yescharacteristic
54C4×D4D411yescharacteristic
55#117—21no—
56#117—21no—
57#117D411yescharacteristic
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#107—21no—
61#106—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8C2×C411yescharacteristic
65C4⋊C8C2×C411yescharacteristic
66#103—21no—
67#103—21no—
68#103—21no—
69#102—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73#97—22no—
74#97—21no—
75#97—21no—
76#97—21no—
77#97—21no—
78#97D411yescharacteristic
79#97D411yescharacteristic
80C4×C8—22no—
81C4×C8—21no—
82C4×C8D412yes—
83#41—22no—
84#41—22no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8C22⋊C411yescharacteristic
88C2×D4—21no—
89C2×D4C22⋊C411yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4C2×D411yescharacteristic
94C22×C4#4111yescharacteristic
95C22×C4#4111yescharacteristic
96Q16—42no—
97SD16—42no—
98M16—28no—
99M16—22no—
100M16—22no—
101C2×C8—28no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—41no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4C22⋊C411yescharacteristic
120C4⋊C4C22⋊C411yescharacteristic
121C22⋊C4—21no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—41no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—41no—
129C42C22×C411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C23—21no—
134C23#12311yescharacteristic
135Q8—21no—
136Q8—42no—
137Q8—41no—
138Q8—41no—
139Q8#10412yes—
140D4—24no—
141D4—21no—
142D4#10412yes—
143C2×C4—24no—
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—81no—
154C2×C4—81no—
155C2×C4C2×C22⋊C411yescharacteristic
156C2×C4C4×D411yescharacteristic
157C2×C4C4×D411yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12111yescharacteristic
161C8—22no—
162C8—22no—
163C8—24no—
164C8—41no—
165C8—41no—
166C8—81no—
167C8—81no—
168C22—21no—
169C22—21no—
170C22—22no—
171C22—21no—
172C22—41no—
173C22—41no—
174C22#38611yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—22no—
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—81no—
184C4—81no—
185C4C2×#10411yescharacteristic
186C4C2×#10411yescharacteristic
187C2—41no—
188C2—42no—
189C2#188111yescharacteristic
190C2#189611yescharacteristic
191C2#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. #1531C2semidirect
5. #1485C2semidirect
6. #1302C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#31162
#28752
#15312
#14882
#14852
#13022
#12912