Finite Group Databasebeta

← #32941

Subgroups — #32941

Order 256.

671 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#131C211yesmaximal, characteristic, complemented
2C2×#471C212yesmaximal, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4#3007C211yesmaximal, characteristic, complemented
5#3006C211yesmaximal, characteristic
6#1908C212yesmaximal, complemented
7#1907C212yesmaximal, complemented
8#1901C211yesmaximal, characteristic, complemented
9#1900C211yesmaximal, characteristic, complemented
10#1815C212yesmaximal, complemented
11#1810C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C22×SD16C2211yescharacteristic, complemented
16C2×#131—28no—
17C2×#131—21no—
18C2×#131—21no—
19C2×#131C2212yes—
20C2×#131C2212yes—
21C22×M16—21no—
22C22×M16C2211yescharacteristic
23C23×C8—21no—
24C23×C8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121C2211yescharacteristic
27#471—28no—
28#465—21no—
29#465—21no—
30#465—21no—
31#465—21no—
32#452—22no—
33#451—21no—
34#450—21no—
35#449—22no—
36#448—21no—
37#447—21no—
38C2×#108C2212yescomplemented
39C2×#106C2211yescharacteristic, complemented
40#417C2212yes—
41#417C2212yes—
42C2×#103C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#101C2212yescomplemented
45C2×#100C2212yescomplemented
46#407C2211yescharacteristic
47#407C2211yescharacteristic
48#343C2212yescomplemented
49#343C2212yescomplemented
50#341C2212yescomplemented
51C2×#41—24no—
52C2×#41—24no—
53C2×#41—21no—
54C2×#41—21no—
55C2×#41D412yes—
56C2×#41C2312yes—
57C22×Q8D411yescharacteristic
58C22×Q8C2311yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4D411yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65#137—24no—
66#137—24no—
67AGL(1,ℤ/8)—24no—
68AGL(1,ℤ/8)—24no—
69C2×Q16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—24no—
75C2×D8—22no—
76#131—28no—
77#131—28no—
78#131—24no—
79#131—24no—
80#131—24no—
81#131D418yescomplemented
82C2×M16—28no—
83C2×M16—21no—
84C2×M16—24no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16C2312yes—
88C2×M16C2312yes—
89C2×M16C2312yes—
90C22×C8—28no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—24no—
95C22×C8D411yescharacteristic
96C22×C8D411yescharacteristic
97C22×C8D411yescharacteristic
98C22×C8D411yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105#121—21no—
106#121—21no—
107#121—21no—
108#121—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114#108—24no—
115#106—21no—
116#106—21no—
117#103—22no—
118#103—22no—
119#103—21no—
120#103—21no—
121#102—22no—
122#102—22no—
123#102—21no—
124#102—21no—
125#101—24no—
126#100—24no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130C22⋊C8—21no—
131C24—21no—
132C24—21no—
133C24#4111yescharacteristic
134#41—24no—
135#41—28no—
136#41—24no—
137#41—24no—
138#41—24no—
139#41—22no—
140#41—22no—
141#41C2×D414yes—
142#41#4114yes—
143C2×Q8—24no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8C2×D412yes—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—82no—
162C2×D4C2×D412yes—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4#4111yescharacteristic
178C22×C4C2411yescharacteristic
179Q16—44no—
180SD16—44no—
181SD16—44no—
182SD16—42no—
183SD16—42no—
184D8—44no—
185M16—24no—
186M16—28no—
187M16—22no—
188M16—28no—
189M16—22no—
190M16—22no—
191M16—22no—
192M16—22no—
193M16C2×D414yes—
194C2×C8—24no—
195C2×C8—28no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—28no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8C2×D411yescharacteristic
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D411yescharacteristic
209C2×C8C2×D414yes—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C23—24no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—42no—
229C23—81no—
230C23—81no—
231C23#12111yescharacteristic
232C23#12111yescharacteristic
233C23C2×#4111yescharacteristic
234Q8—24no—
235Q8—22no—
236Q8—41no—
237Q8—42no—
238Q8—41no—
239Q8#12114yes—
240D4—28no—
241D4—24no—
242D4—24no—
243D4—24no—
244D4—22no—
245D4—41no—
246D4—42no—
247D4—41no—
248D4—82no—
249D4—82no—
250D4#12114yes—
251C2×C4—24no—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#12111yescharacteristic
270C2×C4#12111yescharacteristic
271C2×C4C22×D412yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C2×C4C2×#4112yes—
275C8—21no—
276C8—24no—
277C8—21no—
278C8—42no—
279C8—42no—
280C8#12012yes—
281C8#12012yes—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—24no—
288C22—21no—
289C22—21no—
290C22—44no—
291C22—48no—
292C22—42no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—162no—
298C22#47112yescomplemented
299C22C2×#12111yescharacteristic
300C22#52512yes—
301C4—21no—
302C4—24no—
303C4—21no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4C2×#12011yescharacteristic
308C4C2×#12112yes—
309C4#52511yescharacteristic
310C2—22no—
311C2—22no—
312C2—41no—
313C2—44no—
314C2—161no—
315C2#284711yescharacteristic
316C2C2×#47111yescharacteristic
317C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#131C2semidirect
2. C2×#471C2semidirect
3. C2×#465C2semidirect
4. #3007C2semidirect
6. #1908C2semidirect
7. #1907C2semidirect
8. #1901C2semidirect
9. #1900C2semidirect
10. #1815C2semidirect
11. #1810C2semidirect
15. C22×SD16C22semidirect
38. C2×#108C22semidirect
39. C2×#106C22semidirect
42. C2×#103C22semidirect
43. C2×#102C22semidirect
44. C2×#101C22semidirect
45. C2×#100C22semidirect
48. #343C22semidirect
49. #343C22semidirect
50. #341C22semidirect
81. #131D4semidirect
298. C22#471semidirect

Maximal subgroups

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

subgroupindex
C22×#1312
C2×#4712
C2×#4652
#30072
#30062
#19082
#19072
#19012
#19002
#18152
#18102