Finite Group Databasebeta

← #18875

Subgroups — #18875

Order 256.

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

normal subgroups only

This group has 493 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
1#3508C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic, complemented
3#3208C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1641C212yesmaximal, complemented
6#1640C212yesmaximal, complemented
7#1631C212yesmaximal, complemented
8#1630C212yesmaximal, complemented
9C2×#331C211yesmaximal, characteristic, complemented
10#1481C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12#559—21no—
13#559C2212yes—
14#550—21no—
15#550C2212yes—
16#546—22no—
17#546—22no—
18#540—22no—
19#535—21no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#532—21no—
23#532—21no—
24C2×#127C2211yescharacteristic
25#518—21no—
26C4×#41C2211yescharacteristic
27#492C2212yes—
28#486C2212yes—
29#479—21no—
30#478—21no—
31#477—21no—
32#477C2211yescharacteristic, complemented
33#477C2212yescomplemented
34#476C2211yescharacteristic, complemented
35#475C2211yescharacteristic, complemented
36#474C2211yescharacteristic, complemented
37#467—22no—
38#462—21no—
39#462C2212yescomplemented
40#460—22no—
41#450—22no—
42#431C2211yescharacteristic
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#331C2212yes—
48#331C2212yes—
49Q8⋊C8C2212yescomplemented
50Q8⋊C8C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—22no—
54C2×#41—22no—
55C2×#41D411yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4D412yes—
59C2×Q16—22no—
60C2×SD16—24no—
61C22×C8—21no—
62C22×C8—21no—
63C4⋊Q8D411yescharacteristic
64C4⋊Q8D411yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#127—22no—
68#127—21no—
69#127—21no—
70#127D412yes—
71#127C2312yes—
72#125—21no—
73#125—21no—
74#124—22no—
75#124—22no—
76#124—22no—
77#123—24no—
78#123—24no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8D412yes—
83#121—22no—
84#121—22no—
85#121—22no—
86#121—24no—
87#121—24no—
88#121—22no—
89#120—24no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8D411yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×Q8C2312yes—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4D411yescharacteristic
105C4×D4C2311yescharacteristic
106#117—21no—
107#117—21no—
108#117—21no—
109#117D411yescharacteristic
110C2×C22⋊C4—24no—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113#107—21no—
114#107—21no—
115#107—22no—
116#106—21no—
117#106—21no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8C2312yes—
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124#103—21no—
125#103—21no—
126#103—22no—
127#103—22no—
128#102—21no—
129#102—21no—
130#102—24no—
131C22⋊C8—21no—
132C4×C8C2312yes—
133C24—22no—
134#41—22no—
135#41—22no—
136#41—82no—
137#41—82no—
138#41—84no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×D411yescharacteristic
143C2×Q8C2×D412yes—
144C2×D4—21no—
145C2×D4—24no—
146C2×D4—24no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4—82no—
157C2×D4C2×D411yescharacteristic
158C2×D4C2×D414yes—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167Q16—82no—
168SD16—84no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—22no—
173C2×C8—42no—
174C2×C8—81no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D411yescharacteristic
190C4⋊C4C2×D411yescharacteristic
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—21no—
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—41no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—22no—
204C42—41no—
205C42C2×D411yescharacteristic
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2411yescharacteristic
209C23—24no—
210C23—21no—
211C23—44no—
212C23—82no—
213C23—82no—
214C23#12011yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—42no—
221Q8—41no—
222Q8—41no—
223D4—21no—
224D4—21no—
225D4—44no—
226D4—44no—
227D4—42no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—42no—
232D4—44no—
233D4—82no—
234D4—84no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4#12011yescharacteristic
253C2×C4#12011yescharacteristic
254C2×C4#12011yescharacteristic
255C2×C4#13512yes—
256C2×C4#13712yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C8—42no—
261C8—41no—
262C8—41no—
263C8—42no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—44no—
268C22—41no—
269C22—44no—
270C22—41no—
271C22—44no—
272C22—164no—
273C22#45212yes—
274C22C2×#12011yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—42no—
281C4—41no—
282C4—42no—
283C4—81no—
284C4—81no—
285C4#45212yes—
286C4C2×#13511yescharacteristic
287C4C2×#13711yescharacteristic
288C4#57611yescharacteristic
289C4#57711yescharacteristic
290C2—22no—
291C2—81no—
292C2—84no—
293C2C2×#45211yescharacteristic
294C2#301211yescharacteristic
295C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3508C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3249C2semidirect
3. #3208C2semidirect
4. #3076C2semidirect
5. #1641C2semidirect
6. #1640C2semidirect
7. #1631C2semidirect
8. #1630C2semidirect
9. C2×#331C2semidirect
10. #1481C2semidirect
11. #1477C2semidirect
32. #477C22semidirect
33. #477C22semidirect
34. #476C22semidirect
35. #475C22semidirect
36. #474C22semidirect
39. #462C22semidirect
49. Q8⋊C8C22semidirect
50. Q8⋊C8C22semidirect
51. Q8⋊C8C22semidirect
52. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#32492
#32082
#30762
#16412
#16402
#16312
#16302
C2×#3312
#14812
#14772