Finite Group Databasebeta

← #24573

Subgroups — #24573

Order 256.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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, complemented
2#3006C211yesmaximal, characteristic
3C2×#417C211yesmaximal, characteristic, complemented
4#2880C211yesmaximal, characteristic, complemented
5#2079C212yesmaximal, complemented
6#2051C212yesmaximal, complemented
7#1907C212yesmaximal, complemented
8#1900C211yesmaximal, characteristic, complemented
9#1855C211yesmaximal, characteristic, complemented
10#1821C211yesmaximal, characteristic, complemented
11#1815C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C2×AGL(1,ℤ/8)C2211yescharacteristic
14C2×AGL(1,ℤ/8)C2211yescharacteristic
15C2×#131C2212yes—
16C22×M16C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#481—21no—
20#481—21no—
21#481—22no—
22#449—21no—
23#449—21no—
24#448—21no—
25#447—21no—
26C2×#108C2212yescomplemented
27C2×#106C2212yescomplemented
28#417—21no—
29#417—21no—
30#417C2212yes—
31#417C2211yescharacteristic
32#417C2212yes—
33#417C2211yescharacteristic
34C2×#103—21no—
35C2×#102—21no—
36C2×#102C2212yescomplemented
37C2×#100C2212yescomplemented
38#408—22no—
39#408C2212yes—
40#407—21no—
41#407C2211yescharacteristic
42#407C2211yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8C2211yescharacteristic
45#343C2212yescomplemented
46#343C2212yescomplemented
47#343C2212yescomplemented
48#341C2212yescomplemented
49C2×#41—21no—
50C2×#41—21no—
51C2×#41—24no—
52C2×#41—42no—
53C2×#41D411yescharacteristic
54C2×#41D411yescharacteristic
55C2×#41C2311yescharacteristic
56C2×#41C2311yescharacteristic
57C22×Q8D411yescharacteristic
58C22×D4—21no—
59C22×D4—21no—
60C22×D4D411yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65AGL(1,ℤ/8)—24no—
66AGL(1,ℤ/8)—42no—
67C2×SD16—21no—
68C2×SD16—21no—
69C2×D8—21no—
70C2×D8—21no—
71#131—24no—
72#131—42no—
73C2×M16—21no—
74C2×M16—22no—
75C2×M16—42no—
76C2×M16C2312yes—
77C2×M16C2312yes—
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C22×C8—21no—
81C22×C8—22no—
82C22×C8C2312yes—
83#121—21no—
84#121—21no—
85#121—21no—
86#121—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95#108—42no—
96#106—22no—
97#106—22no—
98#103—22no—
99#103—22no—
100#103—21no—
101#103—21no—
102#102—22no—
103#102—22no—
104#102—21no—
105#102—21no—
106#102—22no—
107#102—22no—
108#100—42no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8—22no—
114C22⋊C8—22no—
115C22⋊C8D411yescharacteristic
116C22⋊C8D411yescharacteristic
117C22⋊C8D412yes—
118C24—21no—
119C24—21no—
120C24#4111yescharacteristic
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—44no—
127#41—41no—
128#41—41no—
129#41—44no—
130#41#4114yes—
131C2×Q8—21no—
132C2×Q8—22no—
133C2×Q8—41no—
134C2×Q8C2×D411yescharacteristic
135C2×Q8C2×D411yescharacteristic
136C2×D4—21no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—41no—
147C2×D4—44no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—82no—
151C2×D4C2×D411yescharacteristic
152C2×D4C2×D411yescharacteristic
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—44no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—82no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4#4111yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—42no—
173SD16—42no—
174D8—42no—
175D8—42no—
176M16—22no—
177M16—22no—
178M16—22no—
179M16—22no—
180M16—22no—
181M16—22no—
182M16—44no—
183M16—42no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8C2×D412yes—
193C2×C8C2×D412yes—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—21no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23—41no—
210C23—42no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#12111yescharacteristic
215C23#12111yescharacteristic
216C23C2×#4111yescharacteristic
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—42no—
221Q8#12312yes—
222D4—24no—
223D4—21no—
224D4—21no—
225D4—21no—
226D4—42no—
227D4—42no—
228D4—42no—
229D4—42no—
230D4—41no—
231D4—41no—
232D4—42no—
233D4—82no—
234D4—82no—
235D4#12312yes—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4#12111yescharacteristic
257C2×C4#12111yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4C2×#4112yes—
261C2×C4H(5,𝔽2)12yes—
262C8—22no—
263C8—22no—
264C8—41no—
265C8—42no—
266C8—41no—
267C8—42no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—84no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—162no—
287C22#47112yes—
288C22C2×#12111yescharacteristic
289C22#55112yes—
290C4—21no—
291C4—21no—
292C4—22no—
293C4—41no—
294C4—81no—
295C4—82no—
296C4C2×#12311yescharacteristic
297C4#53411yescharacteristic
298C4#54612yes—
299C2—22no—
300C2—22no—
301C2—41no—
302C2—42no—
303C2—81no—
304C2—161no—
305C2#241211yescharacteristic
306C2C2×#47111yescharacteristic
307C2#306811yescharacteristic
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. C2×#481C2semidirect
3. C2×#417C2semidirect
4. #2880C2semidirect
5. #2079C2semidirect
6. #2051C2semidirect
7. #1907C2semidirect
8. #1900C2semidirect
9. #1855C2semidirect
10. #1821C2semidirect
11. #1815C2semidirect
26. C2×#108C22semidirect
27. C2×#106C22semidirect
36. C2×#102C22semidirect
37. C2×#100C22semidirect
45. #343C22semidirect
46. #343C22semidirect
47. #343C22semidirect
48. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
#30062
C2×#4172
#28802
#20792
#20512
#19072
#19002
#18552
#18212
#18152