Finite Group Databasebeta

← #9243

Subgroups — #9243

Order 256.

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

normal subgroups only

This group has 257 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
1C2×#463C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#1527C211yesmaximal, characteristic, complemented
4#1490C211yesmaximal, characteristic, complemented
5#1469C211yesmaximal, characteristic, complemented
6#1290C211yesmaximal, characteristic, complemented
7#1283C211yesmaximal, characteristic, complemented
8C22×SD16C411yescharacteristic
9C22×D8C411yescharacteristic
10C2×#124C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#463—21no—
13#463—21no—
14#463—21no—
15#463—21no—
16#436—21no—
17#435—22no—
18C8×D4—21no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#431C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C411yescharacteristic
24C2×#102C411yescharacteristic
25#407—21no—
26C2×C22⋊C8—21no—
27C4×M16C2211yescharacteristic
28#329—22no—
29D4⋊C8—21no—
30D4⋊C8—21no—
31#323—21no—
32#323—21no—
33C22×Q8C2×C411yescharacteristic
34C22×D4C2×C411yescharacteristic
35C22×D4C2×C411yescharacteristic
36C23×C4—21no—
37C2×SD16—21no—
38C2×SD16—21no—
39C2×SD16—42no—
40C2×D8—21no—
41C2×D8—21no—
42C2×D8—42no—
43C2×M16—22no—
44C2×M16—21no—
45C22×C8—22no—
46C22×C8C2×C411yescharacteristic
47C22×C8C2×C411yescharacteristic
48#124—21no—
49#124—21no—
50#124—21no—
51#124—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4D411yescharacteristic
55C4×D4D411yescharacteristic
56C4×D4D411yescharacteristic
57C4×D4D411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C42C2311yescharacteristic
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71#102—21no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D411yescharacteristic
77C22⋊C8D411yescharacteristic
78#97—21no—
79#97—21no—
80#97D412yes—
81C4×C8—21no—
82C4×C8—21no—
83C24—21no—
84C24—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—41no—
88C2×Q8—41no—
89C2×D4—22no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4C22⋊C411yescharacteristic
97C2×D4C22⋊C411yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—82no—
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4111yescharacteristic
108SD16—42no—
109SD16—42no—
110D8—42no—
111D8—42no—
112M16—22no—
113M16—41no—
114M16—82no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—82no—
125C4⋊C4—21no—
126C4⋊C4C22⋊C411yescharacteristic
127C4⋊C4C22⋊C411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42#4111yescharacteristic
140C23—21no—
141C23—21no—
142C23—42no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—81no—
149C23—81no—
150C23—81no—
151C23C4×D411yescharacteristic
152Q8—41no—
153Q8—41no—
154Q8—81no—
155D4—21no—
156D4—21no—
157D4—21no—
158D4—42no—
159D4—41no—
160D4—41no—
161D4—81no—
162D4#10012yes—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4C2×C22⋊C411yescharacteristic
178C2×C4C4×D411yescharacteristic
179C2×C4#12011yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12311yescharacteristic
183C8—41no—
184C8—42no—
185C8—41no—
186C8—42no—
187C8—41no—
188C8—41no—
189C8—81no—
190C22—21no—
191C22—21no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—82no—
199C22—82no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—161no—
204C22—161no—
205C22#38611yescharacteristic
206C22#44311yescharacteristic
207C22#44411yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—81no—
216C4C2×#10011yescharacteristic
217C4#41311yescharacteristic
218C2—21no—
219C2—21no—
220C2—42no—
221C2—81no—
222C2—161no—
223C2#187611yescharacteristic
224C2#190011yescharacteristic
225C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D8C4
LCSG ▷ C22×C4 ▷ 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. #2974C2semidirect
3. #1527C2semidirect
4. #1490C2semidirect
5. #1469C2semidirect
6. #1290C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
#29742
#15272
#14902
#14692
#12902
#12832