Finite Group Databasebeta

← #9523

Subgroups — #9523

Order 256.

371 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 59 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#3006C211yesmaximal, characteristic
2#2333C211yesmaximal, characteristic, complemented
3#1789C211yesmaximal, characteristic, complemented
4#1782C211yesmaximal, characteristic, complemented
5#1510C211yesmaximal, characteristic, complemented
6#1306C211yesmaximal, characteristic, complemented
7#1284C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C2×AGL(1,ℤ/8)—21no—
10C2×#121C2211yescharacteristic
11#449—22no—
12#448C412yescomplemented
13#447C412yescomplemented
14C2×#104—21no—
15C2×#104—21no—
16#417—21no—
17C2×#99—21no—
18C2×#99—21no—
19#407C2211yescharacteristic
20#407C2211yescharacteristic
21#390—21no—
22#388—21no—
23#387—21no—
24#386—21no—
25#386—21no—
26#381C2211yescharacteristic
27#381C2211yescharacteristic
28C4×C22⋊C4—21no—
29C4×C22⋊C4C2211yescharacteristic
30#327—22no—
31C2×#41—21no—
32C2×#41—21no—
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—42no—
36C2×#41—42no—
37C22×Q8C2×C411yescharacteristic, complemented
38C22×D4—21no—
39C22×D4—21no—
40C22×D4C2×C411yescharacteristic, complemented
41C22×D4D411yescharacteristic
42C23×C4—21no—
43C23×C4—21no—
44C23×C4C2311yescharacteristic
45AGL(1,ℤ/8)—42no—
46AGL(1,ℤ/8)—42no—
47C2×SD16—22no—
48C2×D8—22no—
49C2×M16—21no—
50C2×M16—21no—
51#121—21no—
52#121—21no—
53#121C2×C412yes—
54#121D412yes—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4D411yescharacteristic
64C2×C22⋊C4D411yescharacteristic
65C2×C42—21no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68#104—42no—
69#104—42no—
70#103—22no—
71#102—22no—
72#99—42no—
73#99—42no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8C2×C412yescomplemented
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85C24—21no—
86C24—21no—
87C24C2×D411yescharacteristic
88#41—22no—
89#41—22no—
90#41—22no—
91#41—22no—
92#41—41no—
93#41—41no—
94#41—41no—
95#41—41no—
96#41—42no—
97#41—42no—
98#41—42no—
99#41—42no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—42no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—41no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—81no—
118C2×D4—81no—
119C2×D4C22⋊C412yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4C22×C411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4#4111yescharacteristic
141C22×C4#4111yescharacteristic
142SD16—44no—
143D8—44no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—41no—
148C2×C8—22no—
149C2×C8—41no—
150C4⋊C4—21no—
151C4⋊C4—81no—
152C4⋊C4C22⋊C412yes—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C42—41no—
164C42—41no—
165C42—81no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—81no—
181C23—81no—
182C23C4×D411yescharacteristic
183C23#12011yescharacteristic
184C23#12111yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—41no—
188Q8—41no—
189D4—22no—
190D4—22no—
191D4—42no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—81no—
199D4—82no—
200D4—82no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—84no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4C2×C22⋊C411yescharacteristic
236C2×C4C4×D411yescharacteristic
237C2×C4#12111yescharacteristic
238C2×C4#12311yescharacteristic
239C8—42no—
240C8—81no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—81no—
262C22—81no—
263C22#38611yescharacteristic
264C22B(4,𝔽2)11yescharacteristic
265C22B(4,𝔽2)11yescharacteristic
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—161no—
277C2—21no—
278C2—21no—
279C2—41no—
280C2—41no—
281C2—82no—
282C2—82no—
283C2#188911yescharacteristic
284C2#190411yescharacteristic
285C2#190611yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3006C2
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. #2333C2semidirect
3. #1789C2semidirect
4. #1782C2semidirect
5. #1510C2semidirect
6. #1306C2semidirect
7. #1284C2semidirect
12. #448C4semidirect
13. #447C4semidirect
37. C22×Q8C2×C4semidirect
40. C22×D4C2×C4semidirect
77. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#30062
#23332
#17892
#17822
#15102
#13062
#12842