Finite Group Databasebeta

← #29753

Subgroups — #29753

Order 256.

458 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 111 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×#468C211yesmaximal, characteristic, complemented
2#3007C211yesmaximal, characteristic, complemented
3#3006C211yesmaximal, characteristic
4#2879C211yesmaximal, characteristic, complemented
5#2078C212yesmaximal, complemented
6#2052C212yesmaximal, complemented
7#2051C212yesmaximal, complemented
8#2019C212yesmaximal, complemented
9#1901C211yesmaximal, characteristic, complemented
10#1900C211yesmaximal, characteristic, complemented
11#1811C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×#137C2212yes—
14C2×AGL(1,ℤ/8)C2212yes—
15C22×SD16C2211yescharacteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20#468—22no—
21#468—22no—
22#452—22no—
23#451—21no—
24#450—21no—
25#449—22no—
26#448—21no—
27#447—21no—
28C2×#108C2212yescomplemented
29C2×#107C2211yescharacteristic, complemented
30#417C2212yes—
31#417C2212yes—
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2211yescharacteristic, complemented
34C2×#101C2212yescomplemented
35C2×#100C2212yescomplemented
36#408—22no—
37#408C2212yes—
38#408C2212yes—
39#407—22no—
40#407C2211yescharacteristic
41#407C2211yescharacteristic
42#343C2212yescomplemented
43#343C2212yescomplemented
44#341C2212yescomplemented
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—42no—
48C2×#41—42no—
49C2×#41D412yes—
50C2×#41C2312yes—
51C22×Q8D411yescharacteristic
52C22×Q8C2311yescharacteristic
53C22×D4—21no—
54C22×D4—21no—
55C22×D4D411yescharacteristic
56C22×D4C2311yescharacteristic
57C23×C4—21no—
58C23×C4C2311yescharacteristic
59#137—42no—
60#137—42no—
61AGL(1,ℤ/8)—42no—
62AGL(1,ℤ/8)—42no—
63C2×Q16—22no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16—42no—
68C2×D8—22no—
69C2×M16—21no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16D412yes—
73C2×M16D412yes—
74C2×M16C2312yes—
75C2×M16C2312yes—
76C2×M16C2312yes—
77C22×C8—21no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84#121—21no—
85#121—21no—
86#121—21no—
87#121—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93#108—42no—
94#107—22no—
95#103—22no—
96#103—22no—
97#103—22no—
98#102—22no—
99#102—22no—
100#102—22no—
101#101—42no—
102#100—42no—
103C22⋊C8—21no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8—24no—
108C22⋊C8—24no—
109C24—21no—
110C24—21no—
111C24#4111yescharacteristic
112#41—22no—
113#41—22no—
114#41—22no—
115#41—22no—
116#41—42no—
117#41—42no—
118#41—44no—
119#41—42no—
120#41—42no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—42no—
127C2×Q8C2×D412yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—42no—
136C2×D4—44no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—82no—
140C2×D4C2×D412yes—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4#4111yescharacteristic
156C22×C4C2411yescharacteristic
157Q16—44no—
158SD16—44no—
159SD16—44no—
160SD16—44no—
161D8—44no—
162M16—22no—
163M16—22no—
164M16—22no—
165M16—22no—
166M16—22no—
167M16—22no—
168M16—42no—
169M16—42no—
170M16—42no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8C2×D412yes—
178C2×C8C2×D412yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C23—22no—
190C23—22no—
191C23—41no—
192C23—41no—
193C23—42no—
194C23—41no—
195C23—42no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23—81no—
200C23#12111yescharacteristic
201C23#12111yescharacteristic
202C23C2×#4111yescharacteristic
203Q8—22no—
204Q8—22no—
205Q8—42no—
206Q8—41no—
207Q8—42no—
208Q8—41no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—44no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—41no—
217D4—41no—
218D4—82no—
219D4—82no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4#12111yescharacteristic
239C2×C4#12111yescharacteristic
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4H(5,𝔽2)12yes—
243C2×C4H(5,𝔽2)12yes—
244C8—22no—
245C8—22no—
246C8—42no—
247C8—42no—
248C8—42no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—82no—
261C22—84no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—162no—
266C22C2×#12111yescharacteristic
267C22#53812yes—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4#53411yescharacteristic
275C4#53412yes—
276C4#53511yescharacteristic
277C2—22no—
278C2—41no—
279C2—41no—
280C2—82no—
281C2—161no—
282C2#261711yescharacteristic
283C2#306811yescharacteristic
284C2#306911yescharacteristic
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×#468C2semidirect
2. #3007C2semidirect
4. #2879C2semidirect
5. #2078C2semidirect
6. #2052C2semidirect
7. #2051C2semidirect
8. #2019C2semidirect
9. #1901C2semidirect
10. #1900C2semidirect
11. #1811C2semidirect
15. C22×SD16C22semidirect
28. C2×#108C22semidirect
29. C2×#107C22semidirect
32. C2×#103C22semidirect
33. C2×#102C22semidirect
34. C2×#101C22semidirect
35. C2×#100C22semidirect
42. #343C22semidirect
43. #343C22semidirect
44. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
#30072
#30062
#28792
#20782
#20522
#20512
#20192
#19012
#19002
#18112