Finite Group Databasebeta

← #9213

Subgroups — #9213

Order 256.

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

normal subgroups only

This group has 225 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#3106C211yesmaximal, characteristic
2#2922C211yesmaximal, characteristic, complemented
3#1525C211yesmaximal, characteristic, complemented
4#1492C211yesmaximal, characteristic, complemented
5#1489C211yesmaximal, characteristic, complemented
6#1299C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#482—21no—
11#477C411yescharacteristic
12#476C411yescharacteristic
13#460C411yescharacteristic
14#459C411yescharacteristic
15#449—21no—
16C4×SD16—21no—
17C4×D8—21no—
18#433—22no—
19#431—22no—
20#431C2211yescharacteristic
21#424—21no—
22#423C2211yescharacteristic
23#408—21no—
24C4×M16C2211yescharacteristic
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#329—22no—
28Q8⋊C8—21no—
29D4⋊C8—21no—
30D4⋊C8—22no—
31C2×#41—21no—
32C2×#41D411yescharacteristic
33C2×SD16—21no—
34C2×D8—21no—
35C2×M16—21no—
36C2×M16—21no—
37C2×M16—21no—
38C22×C8—21no—
39C4⋊Q8C2×C411yescharacteristic
40#127C2×C411yescharacteristic
41#124—21no—
42#121—21no—
43#121—21no—
44C4×Q8C2×C411yescharacteristic
45C4×Q8D411yescharacteristic
46C4×D4—21no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4C2×C411yescharacteristic
51C4×D4D411yescharacteristic
52#117—21no—
53#117D411yescharacteristic
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56#107—21no—
57#106—21no—
58C4⋊C8—24no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8C2×C411yescharacteristic
62C4⋊C8C2×C411yescharacteristic
63#103—21no—
64#102—21no—
65#102—21no—
66#102—21no—
67C22⋊C8—24no—
68C22⋊C8—21no—
69#97—21no—
70#97D412yes—
71C4×C8—21no—
72C4×C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75C4×C8D412yes—
76#41—41no—
77#41—41no—
78#41—81no—
79#41—81no—
80C2×Q8—21no—
81C2×Q8C22⋊C411yescharacteristic
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C2×D4C22⋊C411yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4#4111yescharacteristic
94C22×C4#4111yescharacteristic
95SD16—42no—
96D8—42no—
97M16—22no—
98M16—41no—
99M16—41no—
100M16—41no—
101M16—81no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—41no—
108C2×C8—41no—
109C2×C8—41no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4C22⋊C411yescharacteristic
115C4⋊C4C22⋊C411yescharacteristic
116C22⋊C4—21no—
117C22⋊C4—21no—
118C22⋊C4—21no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42—21no—
122C42—21no—
123C42—21no—
124C42C22×C411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C23—21no—
129C23—41no—
130C23#12311yescharacteristic
131Q8—21no—
132Q8—21no—
133Q8—81no—
134D4—21no—
135D4—21no—
136D4—42no—
137D4—41no—
138D4—41no—
139D4—41no—
140D4—41no—
141D4—81no—
142D4—82no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4C2×C22⋊C411yescharacteristic
161C2×C4C4×D411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4#12011yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12111yescharacteristic
166C8—42no—
167C8—42no—
168C8—42no—
169C8—41no—
170C8—41no—
171C8—81no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—41no—
176C22—41no—
177C22—41no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22#38611yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—81no—
189C4#42111yescharacteristic
190C4#42111yescharacteristic
191C4#44311yescharacteristic
192C4#44411yescharacteristic
193C4#47111yescharacteristic
194C4#47211yescharacteristic
195C2—41no—
196C2—81no—
197C2—161no—
198C2#187611yescharacteristic
199C2#189711yescharacteristic
200C2#190711yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3106C2
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
2. #2922C2semidirect
3. #1525C2semidirect
4. #1492C2semidirect
5. #1489C2semidirect
6. #1299C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#31062
#29222
#15252
#14922
#14892
#12992
#12782