Finite Group Databasebeta

← #9045

Subgroups — #9045

Order 256.

339 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 77 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×#481C211yesmaximal, characteristic
2#2325C211yesmaximal, characteristic
3#1824C211yesmaximal, characteristic, complemented
4C2×#327C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1294C211yesmaximal, characteristic, complemented
7#1280C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×C4×D4—21no—
10C22×C4⋊C4C2211yescharacteristic
11#481—22no—
12#481C414yescomplemented
13C2×#106—21no—
14C2×#102—21no—
15C2×C22⋊C8C2211yescharacteristic
16C2×C22⋊C8C2211yescharacteristic
17#388—21no—
18#387—21no—
19#386—21no—
20#386—21no—
21#385—21no—
22#380C2211yescharacteristic
23C2×#95—21no—
24C2×#95—21no—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27#340—21no—
28#340—21no—
29#327—22no—
30C22×D4—21no—
31C22×D4D411yescharacteristic
32C23×C4—21no—
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—21no—
39C22×C8—21no—
40#121—21no—
41#121—21no—
42#121C2×C412yes—
43#121D412yes—
44C4×D4—42no—
45C4×D4—42no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×C412yescomplemented
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C42—21no—
62#106—24no—
63#102—24no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C22⋊C8C2×C412yescomplemented
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—42no—
76#95—42no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—21no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4C22⋊C412yes—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—42no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—81no—
119C22×C4—81no—
120C22×C4SD1612yes—
121C22×C4C22×C411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4#4111yescharacteristic
126C22×C4#4111yescharacteristic
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—41no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4C22⋊C412yes—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—42no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C22⋊C4—81no—
148C22⋊C4—81no—
149C42—42no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—21no—
154C23—22no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23C4×D411yescharacteristic
162C23#12011yescharacteristic
163C23#12111yescharacteristic
164C23C2×SD1611yescharacteristic
165C23#13511yescharacteristic
166C23AGL(1,ℤ/8)11yescharacteristic
167C23#13711yescharacteristic
168D4—42no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—81no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
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—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—84no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4C4×D411yescharacteristic
218C2×C4#12111yescharacteristic
219C2×C4#12311yescharacteristic
220C8—42no—
221C8—81no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22#38611yescharacteristic
242C22C4×SD1611yescharacteristic
243C22#44011yescharacteristic
244C22#45011yescharacteristic
245C22#45211yescharacteristic
246C22#45311yescharacteristic
247C22#45411yescharacteristic
248C22#45811yescharacteristic
249C22#45811yescharacteristic
250C22#46111yescharacteristic
251C22#46311yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—82no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4—82no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#161311yescharacteristic
269C2#161511yescharacteristic
270C2#161611yescharacteristic
271C2#161811yescharacteristic
272C2#187711yescharacteristic
273C2#188311yescharacteristic
274C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1824C2semidirect
5. #1445C2semidirect
6. #1294C2semidirect
7. #1280C2semidirect
12. #481C4semidirect
52. C2×C4⋊C4C2×C4semidirect
67. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
#23252
#18242
C2×#3272
#14452
#12942
#12802