Finite Group Databasebeta

← #9278

Subgroups — #9278

Order 256.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#3187C211yesmaximal, characteristic
2#2368C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1520C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10#483C412yescomplemented
11#481—21no—
12#465—22no—
13#450C412yescomplemented
14#448—21no—
15C2×#99—21no—
16C2×#99—21no—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#391—21no—
20#387—21no—
21#387—21no—
22#385—21no—
23#380C2211yescharacteristic
24C4×C4⋊C4—21no—
25C4×C22⋊C4—21no—
26C4×C22⋊C4C2211yescharacteristic
27C2×#95C2211yescharacteristic
28#328—22no—
29#327—21no—
30#327—21no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4C2×C411yescharacteristic
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C2×SD16—22no—
37C22×C8—21no—
38C22×C8—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8C2×C412yes—
42C22⋊Q8D412yes—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4C2×C411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4D411yescharacteristic
57C2×C22⋊C4D411yescharacteristic
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61#106—22no—
62#103—22no—
63#102—22no—
64#99—42no—
65#99—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8C2×C412yes—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95D411yescharacteristic
78#95D411yescharacteristic
79C24—21no—
80C24C2×D411yescharacteristic
81C2×Q8—21no—
82C2×Q8—81no—
83C2×Q8C22⋊C412yes—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—81no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—81no—
105C22×C4—81no—
106C22×C4C22×C411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4#4111yescharacteristic
111C22×C4#4111yescharacteristic
112SD16—82no—
113C2×C8—22no—
114C2×C8—41no—
115C2×C8—81no—
116C2×C8—81no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—42no—
123C4⋊C4—81no—
124C4⋊C4—81no—
125C4⋊C4C22⋊C412yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C22⋊C4SD1612yescomplemented
136C42—42no—
137C42—81no—
138C42—81no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—81no—
145C23—81no—
146C23C4×D411yescharacteristic
147C23#12011yescharacteristic
148C23#12111yescharacteristic
149C23C2×SD1611yescharacteristic
150C23#13511yescharacteristic
151C23AGL(1,ℤ/8)11yescharacteristic
152C23#13711yescharacteristic
153Q8—42no—
154Q8—81no—
155D4—42no—
156D4—81no—
157D4—81no—
158D4—81no—
159D4—81no—
160D4—81no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4C2×C22⋊C411yescharacteristic
194C2×C4C4×D411yescharacteristic
195C2×C4#12111yescharacteristic
196C2×C4#12311yescharacteristic
197C8—42no—
198C8—81no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—161no—
213C22—161no—
214C22#38611yescharacteristic
215C22C4×SD1611yescharacteristic
216C22#44011yescharacteristic
217C22#44811yescharacteristic
218C22#44911yescharacteristic
219C22#46011yescharacteristic
220C22#46411yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—42no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—161no—
233C2—21no—
234C2—21no—
235C2—21no—
236C2—21no—
237C2—21no—
238C2—21no—
239C2—161no—
240C2#188011yescharacteristic
241C2#190611yescharacteristic
242C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2368C2semidirect
4. #1520C2semidirect
6. #1294C2semidirect
7. #1282C2semidirect
10. #483C4semidirect
13. #450C4semidirect
135. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#31872
#23682
#19642
#15202
C2×#3272
#12942
#12822