Finite Group Databasebeta

← #9164

Subgroups — #9164

Order 256.

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

normal subgroups only

This group has 229 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#3112C211yesmaximal, characteristic
2#2984C211yesmaximal, characteristic, complemented
3#1763C211yesmaximal, characteristic, complemented
4#1530C211yesmaximal, characteristic
5#1502C211yesmaximal, characteristic
6#1302C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#480—21no—
11#477C411yescharacteristic, complemented
12#477C411yescharacteristic, complemented
13#460C411yescharacteristic, complemented
14#460C411yescharacteristic, complemented
15#448—21no—
16#446—21no—
17#446—22no—
18C8×Q8—21no—
19#440—21no—
20#440—21no—
21#432—21no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431—21no—
25#431C2211yescharacteristic
26C2×#103—21no—
27C2×#97C2211yescharacteristic
28C4×C4⋊C4C2211yescharacteristic
29#340—21no—
30#329—22no—
31#327—21no—
32#327—21no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C2×SD16—21no—
36C2×SD16—21no—
37C22×C8—21no—
38C4⋊Q8C2×C411yescharacteristic
39#127C2×C411yescharacteristic
40#124—21no—
41#121—21no—
42#121—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8C2×C411yescharacteristic
46C4×Q8C2×C411yescharacteristic
47C4×Q8D411yescharacteristic
48C4×Q8D411yescharacteristic
49C4×D4—21no—
50C4×D4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D411yescharacteristic
54C2×C42—21no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#107—21no—
58#107—21no—
59C4⋊C8—22no—
60C4⋊C8—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C411yescharacteristic
64C4⋊C8C2×C411yescharacteristic
65#103—24no—
66#103—21no—
67#103—21no—
68#102—21no—
69#102—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8D411yescharacteristic
73C22⋊C8D411yescharacteristic
74#97—22no—
75#97—21no—
76#97—21no—
77#97—21no—
78#97D412yes—
79C4×C8—22no—
80C4×C8—21no—
81#95—21no—
82#41—81no—
83#41—81no—
84C2×Q8—21no—
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—41no—
88C2×Q8C22⋊C411yescharacteristic
89C2×Q8C22⋊C411yescharacteristic
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4#4111yescharacteristic
101SD16—42no—
102SD16—42no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—41no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—41no—
120C4⋊C4C22⋊C411yescharacteristic
121C4⋊C4C22⋊C411yescharacteristic
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42—21no—
128C42—81no—
129C42C22×C411yescharacteristic
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42#4111yescharacteristic
133C23—41no—
134C23#12111yescharacteristic
135Q8—21no—
136Q8—21no—
137Q8—21no—
138Q8—42no—
139Q8—81no—
140Q8#10412yes—
141D4—41no—
142D4—41no—
143D4—81no—
144D4—81no—
145D4—81no—
146D4—81no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—82no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4C4×D411yescharacteristic
166C2×C4C4×D411yescharacteristic
167C2×C4#12011yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12311yescharacteristic
170C8—41no—
171C8—41no—
172C8—42no—
173C8—42no—
174C8—41no—
175C8—41no—
176C22—21no—
177C22—21no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22#38611yescharacteristic
182C22#47111yescharacteristic
183C22#47211yescharacteristic
184C4—21no—
185C4—22no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—82no—
194C4—81no—
195C4C2×#10411yescharacteristic
196C4#42111yescharacteristic
197C2—21no—
198C2—21no—
199C2—161no—
200C2#187111yescharacteristic
201C2#190511yescharacteristic
202C2#190711yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
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. #2984C2semidirect
3. #1763C2semidirect
6. #1302C2semidirect
7. #1285C2semidirect
11. #477C4semidirect
12. #477C4semidirect
13. #460C4semidirect
14. #460C4semidirect

Maximal subgroups

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

subgroupindex
#31122
#29842
#17632
#15302
#15022
#13022
#12852