Finite Group Databasebeta

← #15683079

Subgroups — #15683079

Order 2944.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10625120C211yesmaximal, characteristic, complemented
2C2×#10625119C211yesmaximal, characteristic, complemented
3#13092981C211yesmaximal, characteristic, complemented
4#13092668C212yesmaximal, complemented
5#13092665C212yesmaximal, complemented
6#13092651C212yesmaximal, complemented
7#13092648C212yesmaximal, complemented
8#13092457C211yesmaximal, characteristic, complemented
9#13092413C211yesmaximal, characteristic, complemented
10C2×#10624991C211yesmaximal, characteristic, complemented
11C23×#537C211yesmaximal, characteristic, complemented
12C2×#41×C23C2211yescharacteristic
13C2×#41×C23C2211yescharacteristic
14C22×D92C2211yescharacteristic
15C23×C22⋊Q8—21no—
16C23×C22⋊Q8C2212yescomplemented
17C23×#121—21no—
18C23×#121—22no—
19C23×#121C2212yescomplemented
20C2×C4⋊C4×C23C2211yescharacteristic
21#10625120—24no—
22#10625119—24no—
23#10625118—21no—
24C2×#67274C2212yes—
25#10625111—21no—
26#10625108—22no—
27C2×#67272C2212yes—
28#10625101C2212yes—
29#10625101C2212yes—
30#10625097C2211yescharacteristic
31C2×#67267C2211yescharacteristic
32C2×#67267C2212yescomplemented
33C22×C23⋊C8C2211yescharacteristic
34C22×C23⋊C8C2212yescomplemented
35#10625017C2211yescharacteristic, complemented
36#10625017C2211yescharacteristic, complemented
37#10625014C2211yescharacteristic, complemented
38#10625014C2211yescharacteristic, complemented
39#10624991C2212yes—
40#10624991C2212yes—
41#10624990C2212yescomplemented
42#10624989C2212yescomplemented
43C23×D23—21no—
44#41×C23—22no—
45#41×C23—22no—
46#41×C23—22no—
47#41×C23—22no—
48C2×Q8×C23D411yescharacteristic
49C2×Q8×C23C2311yescharacteristic
50C2×D4×C23—21no—
51C2×D4×C23—22no—
52C2×D4×C23D412yes—
53C2×D4×C23D411yescharacteristic
54C2×D4×C23C2311yescharacteristic
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23C2312yes—
58C22×C4×C23C2311yescharacteristic
59C2×C22⋊D23—22no—
60C2×C22⋊D23—22no—
61C2×D92—22no—
62C2×D92—22no—
63C2×D92—21no—
64C2×D92C2312yes—
65C4⋊C4×C23—21no—
66C4⋊C4×C23—22no—
67C4⋊C4×C23C2312yes—
68C22⋊C4×C23—22no—
69C22⋊C4×C23—22no—
70C22⋊Dic23—21no—
71#67274—24no—
72#67272—24no—
73(C2×D23)⋊C4—22no—
74(C2×D23)⋊C4—22no—
75C4⋊Dic23C2311yescharacteristic
76C4⋊Dic23C2311yescharacteristic
77C4×Dic23—21no—
78#67267—24no—
79#67267D414yescomplemented
80C2×C23⋊C8—22no—
81C2×C23⋊C8—24no—
82C2×C23⋊C8C2311yescharacteristic
83C2×C23⋊C8C2311yescharacteristic
84C2×C23⋊C8C2312yes—
85C23×C23—21no—
86C23×C23C2×D412yes—
87C23×C23C2×D411yescharacteristic
88C22×D23—22no—
89C22×D23—42no—
90C22×D23—81no—
91C22×D23—81no—
92Q8×C23—22no—
93Q8×C23—21no—
94Q8×C23—21no—
95D4×C23—24no—
96D4×C23—22no—
97D4×C23—21no—
98D4×C23—21no—
99D4×C23—42no—
100D4×C23—82no—
101C2×C4×C23—22no—
102C2×C4×C23—22no—
103C2×C4×C23—22no—
104C2×C4×C23—21no—
105C2×C4×C23—42no—
106C2×C4×C23—81no—
107C2×C4×C23C2×D412yes—
108C2×C4×C23C2×D412yes—
109C2×C4×C23C2×D411yescharacteristic
110C2×C4×C23#4112yes—
111C2×C4×C23C2411yescharacteristic
112C22⋊D23—82no—
113C22⋊D23—82no—
114C2×Dic23—21no—
115C2×Dic23—21no—
116D92—22no—
117D92—22no—
118D92—42no—
119C23⋊C8—21no—
120C23⋊C8—21no—
121C23⋊C8C2×D414yes—
122#3062—231nomaximal, Sylow
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—22no—
126C22×C23—22no—
127C22×C23—22no—
128C22×C23—41no—
129C22×C23—41no—
130C22×C23—41no—
131C22×C23#12112yes—
132C22×C23C22×D411yescharacteristic
133C2×D23—42no—
134C2×D23—42no—
135C2×D23—42no—
136C2×D23—84no—
137C4×C23—22no—
138C4×C23—41no—
139C4×C23—42no—
140C4×C23#12112yes—
141C4×C23C22×D411yescharacteristic
142C4×C23C2×#4111yescharacteristic
143Dic23—41no—
144Dic23—41no—
145C2×AGL(1,ℤ/8)—231no—
146C2×#131—231no—
147#537D2311yescharacteristic, complemented
148#535—231no—
149#469—232no—
150#468—232no—
151#466—232no—
152#465—232no—
153#428—231no—
154#416—231no—
155C2×#102—231no—
156C2×C23—22no—
157C2×C23—42no—
158C2×C23—81no—
159C2×C23C2×#12111yescharacteristic
160C2×C23#57611yescharacteristic
161C2×C23#57711yescharacteristic
162D23—82no—
163C2×#41C2×D2311yescharacteristic
164C2×#41C2×D2311yescharacteristic
165C22×D4—231no—
166AGL(1,ℤ/8)—464no—
167C2×SD16—232no—
168C2×D8—232no—
169#131—464no—
170C2×M16—231no—
171C2×M16—232no—
172C22×C8—231no—
173C22×C8—232no—
174#127—461no—
175#124—461no—
176C22⋊Q8—21no—
177C22⋊Q8C2×D2312yescomplemented
178#121—21no—
179#121—22no—
180#121—232no—
181#121—232no—
182#121C2×D2312yescomplemented
183#120—462no—
184#117—231no—
185C2×C4⋊C4C2×D2311yescharacteristic
186#107—232no—
187#106—232no—
188#103—231no—
189#103—231no—
190#102—231no—
191#102—231no—
192#102—232no—
193#102—232no—
194C23#306211yescharacteristic, complemented, Sylow
195C24—461no—
196#41—22no—
197#41—22no—
198#41—22no—
199#41—22no—
200C2×Q8C22⋊D2311yescharacteristic
201C2×Q8C22×D2311yescharacteristic
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—232no—
205C2×D4—462no—
206C2×D4—462no—
207C2×D4—462no—
208C2×D4—462no—
209C2×D4—461no—
210C2×D4C22⋊D2312yes—
211C2×D4C22⋊D2311yescharacteristic
212C2×D4C22×D2311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4C22×D2312yes—
216C22×C4C22×D2311yescharacteristic
217SD16—464no—
218D8—464no—
219M16—234no—
220M16—464no—
221C2×C8—232no—
222C2×C8—231no—
223C2×C8—231no—
224C2×C8—464no—
225C2×C8—462no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—231no—
229C4⋊C4—231no—
230C4⋊C4C22×D2312yes—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—461no—
234C22⋊C4—462no—
235C22⋊C4—462no—
236C42—461no—
237C23—21no—
238C23—462no—
239C23—922no—
240C23—1841no—
241C23—1841no—
242C23C2×C22⋊D2312yes—
243C23C2×C22⋊D2311yescharacteristic
244Q8—22no—
245Q8—21no—
246Q8—21no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—21no—
251D4—42no—
252D4—82no—
253D4—462no—
254D4—462no—
255D4—922no—
256D4—1842no—
257D4—1842no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—461no—
265C2×C4—461no—
266C2×C4D4×D2312yes—
267C2×C4#6729012yes—
268C2×C4C2×C22⋊D2312yes—
269C2×C4C2×C22⋊D2311yescharacteristic
270C2×C4C23×D2311yescharacteristic
271C8—234no—
272C8—461no—
273C8—461no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—922no—
283C22—922no—
284C22—922no—
285C22—1844no—
286C22#1062511012yes—
287C22C22×C22⋊D2311yescharacteristic
288C4—22no—
289C4—41no—
290C4—42no—
291C4—921no—
292C4—921no—
293C4#1062511012yes—
294C4C2×D4×D2311yescharacteristic
295C4C2×#6729011yescharacteristic
296C2—22no—
297C2—42no—
298C2—81no—
299C2—1842no—
300C2C2×#1062511011yescharacteristic
301C2#1309322011yescharacteristic
302C2#1309322511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#537C2
socleC22×C23C22×D4
hypercenter#537D23
hypocenterC23#3062
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#10625120C2semidirect
2. C2×#10625119C2semidirect
3. #13092981C2semidirect
4. #13092668C2semidirect
5. #13092665C2semidirect
6. #13092651C2semidirect
7. #13092648C2semidirect
8. #13092457C2semidirect
9. #13092413C2semidirect
10. C2×#10624991C2semidirect
11. C23×#537C2semidirect
16. C23×C22⋊Q8C22semidirect
19. C23×#121C22semidirect
32. C2×#67267C22semidirect
34. C22×C23⋊C8C22semidirect
35. #10625017C22semidirect
36. #10625017C22semidirect
37. #10625014C22semidirect
38. #10625014C22semidirect
41. #10624990C22semidirect
42. #10624989C22semidirect
79. #67267D4semidirect
147. #537D23semidirect
177. C22⋊Q8C2×D23semidirect
182. #121C2×D23semidirect
194. C23#3062semidirect

Maximal subgroups

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

subgroupindex
C2×#106251202
C2×#106251192
#130929812
#130926682
#130926652
#130926512
#130926482
#130924572
#130924132
C2×#106249912
C23×#5372
#306223