Finite Group Databasebeta

← #16828795

Subgroups — #16828795

Order 3712.

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

normal subgroups only

This group has 764 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
1C2×#11745359C211yesmaximal, characteristic, complemented
2C2×#11745312C211yesmaximal, characteristic, complemented
3#14271741C211yesmaximal, characteristic, complemented
4#14271322C212yesmaximal, complemented
5#14271312C212yesmaximal, complemented
6#14271305C212yesmaximal, complemented
7#14271295C212yesmaximal, complemented
8#14271165C211yesmaximal, characteristic, complemented
9C2×#11745180C211yesmaximal, characteristic, complemented
10#14271131C211yesmaximal, characteristic, complemented
11C2×C29×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
12C2×#41×C29C2211yescharacteristic, complemented
13C22×D4×C29C2211yescharacteristic
14C22×C22⋊D29—22no—
15#11745359—24no—
16#11745359—24no—
17C2×#90807—21no—
18C2×#90807C2212yes—
19C2×D4×D29—21no—
20C2×D4×D29C2212yes—
21C2×#90805C2211yescharacteristic
22C2×#90805C2211yescharacteristic
23C29×AGL(1,ℤ/8)—24no—
24C2×SD16×C29C2212yescomplemented
25C2×D8×C29C2212yescomplemented
26C2×M16×C29C2211yescharacteristic, complemented
27#11745312—24no—
28#11745308C2212yescomplemented
29C2×#90785C2211yescharacteristic, complemented
30C2×#90784C2211yescharacteristic, complemented
31#11745301—22no—
32#11745300C2212yescomplemented
33C2×#90783C2211yescharacteristic, complemented
34C2×#90782C2211yescharacteristic, complemented
35#11745292—21no—
36C2×C4⋊Dic29C2211yescharacteristic, complemented
37#11745286—21no—
38C22×#6597C2211yescharacteristic, complemented
39#11745180C2212yescomplemented
40#11745180C2212yescomplemented
41#11745169C2212yescomplemented
42#11745168C2212yescomplemented
43#11745168C2212yescomplemented
44#11745164C2212yescomplemented
45C24×C29—21no—
46C23×D29—22no—
47#41×C29—22no—
48#41×C29—22no—
49C2×Q8×C29C2311yescharacteristic
50C2×D4×C29—21no—
51C2×D4×C29—21no—
52C2×D4×C29—22no—
53C2×D4×C29D414yescomplemented
54C2×D4×C29C2311yescharacteristic
55C2×D4×C29C2312yes—
56C22×C4×C29—21no—
57C22×C4×C29C2311yescharacteristic
58C2×C22⋊D29—216no—
59C2×C22⋊D29—21no—
60C2×C22⋊D29—22no—
61C2×C22⋊D29—22no—
62C2×C22⋊D29—22no—
63C2×C22⋊D29D412yes—
64C22×Dic29—22no—
65C22×Dic29—21no—
66#90807—28no—
67#90807—24no—
68#90807—24no—
69D4×D29—28no—
70D4×D29—24no—
71D4×D29—24no—
72#90805—22no—
73#90805—22no—
74#90805—22no—
75#90805D414yescomplemented
76C2×D116D411yescharacteristic
77C2×D116C2311yescharacteristic
78C2×C4×D29—21no—
79C2×C4×D29C2312yes—
80C2×Dic58D411yescharacteristic
81C2×Dic58C2311yescharacteristic
82SD16×C29—24no—
83D8×C29—24no—
84M16×C29—22no—
85C2×C8×C29C2312yes—
86C22⋊Dic29—22no—
87#90785—22no—
88#90784—22no—
89#90783—22no—
90#90782—22no—
91#90781—22no—
92C4⋊Dic29—21no—
93C4⋊Dic29C2312yes—
94#90779—22no—
95C2×#6597—22no—
96C2×#6597C2311yescharacteristic
97C2×#6597C2311yescharacteristic
98C23×C29—22no—
99C23×C29—24no—
100C23×C29—21no—
101C23×C29—81no—
102C23×C29—81no—
103C23×C29C2×D411yescharacteristic
104C22×D29—21no—
105C22×D29—28no—
106C22×D29—82no—
107C22×D29—82no—
108C22×D29C2×D412yes—
109Q8×C29—21no—
110Q8×C29—21no—
111D4×C29—24no—
112D4×C29—22no—
113D4×C29—21no—
114D4×C29—21no—
115D4×C29—42no—
116D4×C29C2×D414yes—
117C2×C4×C29—21no—
118C2×C4×C29—42no—
119C2×C4×C29C2×D411yescharacteristic
120C2×C4×C29C2×D412yes—
121C2×C4×C29C2×D412yes—
122C2×C4×C29C2411yescharacteristic
123C22⋊D29—28no—
124C22⋊D29—28no—
125C22⋊D29—24no—
126C22⋊D29—24no—
127C22⋊D29—82no—
128C2×Dic29—28no—
129C2×Dic29—21no—
130C2×Dic29—22no—
131C2×Dic29—81no—
132C2×Dic29C2×D412yes—
133D116—22no—
134D116—21no—
135D116C2×D412yes—
136C4×D29—24no—
137C4×D29—22no—
138C4×D29—22no—
139Dic58—22no—
140Dic58—21no—
141Dic58C2×D412yes—
142C8×C29—22no—
143#6597—21no—
144#6597—21no—
145#3012—291nomaximal, Sylow
146C22×C29—21no—
147C22×C29—21no—
148C22×C29—24no—
149C22×C29—42no—
150C22×C29—42no—
151C22×C29—41no—
152C22×C29—41no—
153C22×C29—41no—
154C22×C29—84no—
155C22×C29#12012yes—
156C22×C29C22×D411yescharacteristic
157C2×D29—22no—
158C2×D29—22no—
159C2×D29—22no—
160C2×D29—22no—
161C2×D29—41no—
162C2×D29—41no—
163C2×D29—88no—
164C4×C29—41no—
165C4×C29#12012yes—
166C4×C29C22×D411yescharacteristic
167C4×C29C22×D411yescharacteristic
168Dic29—22no—
169Dic29—22no—
170Dic29—42no—
171C2×H(5,𝔽2)—291no—
172C2×AGL(1,ℤ/8)D2911yescharacteristic, complemented
173C2×#135—291no—
174#537—291no—
175#452—292no—
176#450—292no—
177#449—292no—
178#447—292no—
179#417—291no—
180C2×#102—291no—
181#408—291no—
182C2×C29—22no—
183C2×C29—44no—
184C2×C29—81no—
185C2×C29C2×#12011yescharacteristic
186C2×C29#57611yescharacteristic
187C2×C29#57711yescharacteristic
188D29—42no—
189D29—42no—
190H(5,𝔽2)—584no—
191H(5,𝔽2)—584no—
192C2×#41—291no—
193C2×#41—292no—
194C2×#41—291no—
195C2×#41—581no—
196C2×#41C2×D2911yescharacteristic, complemented
197C22×D4—292no—
198C22×D4—581no—
199C22×D4—582no—
200C22×D4C2×D2911yescharacteristic
201AGL(1,ℤ/8)—24no—
202#135—584no—
203C2×Q16—291no—
204C2×SD16—291no—
205C2×SD16—291no—
206C2×SD16C2×D2912yescomplemented
207C2×D8—291no—
208C2×D8C2×D2912yescomplemented
209C2×M16C2×D2911yescharacteristic, complemented
210C22×C8—291no—
211C22⋊Q8—292no—
212C22⋊Q8—581no—
213#121—292no—
214#121—581no—
215#121—582no—
216C2×C4⋊C4—291no—
217#103—292no—
218#102—292no—
219#102—292no—
220#102—292no—
221C22⋊C8—292no—
222C22⋊C8—292no—
223C29#301211yescharacteristic, complemented, Sylow
224C24—21no—
225C24—582no—
226#41—22no—
227#41—22no—
228#41—294no—
229#41—584no—
230#41—588no—
231#41—584no—
232#41—582no—
233#41—582no—
234#41—582no—
235C2×Q8—291no—
236C2×Q8—291no—
237C2×Q8C22×D2911yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—292no—
242C2×D4—291no—
243C2×D4—291no—
244C2×D4—582no—
245C2×D4—584no—
246C2×D4—588no—
247C2×D4—582no—
248C2×D4—5816no—
249C2×D4—581no—
250C2×D4—584no—
251C2×D4—582no—
252C2×D4C22⋊D2914yescomplemented
253C2×D4C22×D2911yescharacteristic
254C2×D4C22×D2912yes—
255C22×C4—21no—
256C22×C4—292no—
257C22×C4—582no—
258C22×C4—581no—
259C22×C4—581no—
260C22×C4C22×D2911yescharacteristic
261Q16—582no—
262SD16—24no—
263SD16—582no—
264SD16—582no—
265D8—24no—
266D8—582no—
267M16—22no—
268C2×C8—291no—
269C2×C8—291no—
270C2×C8—582no—
271C2×C8C22×D2912yes—
272C4⋊C4—292no—
273C4⋊C4—582no—
274C4⋊C4—581no—
275C22⋊C4—582no—
276C22⋊C4—582no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—292no—
283C23—588no—
284C23—581no—
285C23—2322no—
286C23—2322no—
287C23D4×D2911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—292no—
291Q8—582no—
292Q8—581no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—292no—
299D4—588no—
300D4—588no—
301D4—584no—
302D4—584no—
303D4—582no—
304D4—581no—
305D4—2322no—
306D4C2×C22⋊D2914yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—292no—
310C2×C4—588no—
311C2×C4—581no—
312C2×C4—584no—
313C2×C4—582no—
314C2×C4—582no—
315C2×C4—582no—
316C2×C4—2321no—
317C2×C4D4×D2911yescharacteristic
318C2×C4D4×D2912yes—
319C2×C4C2×C22⋊D2912yes—
320C2×C4C23×D2911yescharacteristic
321C8—22no—
322C8—581no—
323C8—581no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—582no—
334C22—582no—
335C22—582no—
336C22—582no—
337C22—1161no—
338C22—1161no—
339C22—2328no—
340C22#1174529912yes—
341C22C2×D4×D2911yescharacteristic
342C4—41no—
343C4—582no—
344C4—582no—
345C4—1162no—
346C4#1174529912yes—
347C4C2×D4×D2911yescharacteristic
348C4C22×C22⋊D2911yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—1162no—
353C2—1162no—
354C2C2×#1174529911yescharacteristic
355C2#1427196711yescharacteristic
356C2#1427197311yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC2×C29×AGL(1,ℤ/8)C2
socleC22×C29C22×D4
hypercenterC2×AGL(1,ℤ/8)D29
hypocenterC29#3012
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11745359C2semidirect
2. C2×#11745312C2semidirect
3. #14271741C2semidirect
4. #14271322C2semidirect
5. #14271312C2semidirect
6. #14271305C2semidirect
7. #14271295C2semidirect
8. #14271165C2semidirect
9. C2×#11745180C2semidirect
10. #14271131C2semidirect
11. C2×C29×AGL(1,ℤ/8)C2semidirect
12. C2×#41×C29C22semidirect
24. C2×SD16×C29C22semidirect
25. C2×D8×C29C22semidirect
26. C2×M16×C29C22semidirect
28. #11745308C22semidirect
29. C2×#90785C22semidirect
30. C2×#90784C22semidirect
32. #11745300C22semidirect
33. C2×#90783C22semidirect
34. C2×#90782C22semidirect
36. C2×C4⋊Dic29C22semidirect
38. C22×#6597C22semidirect
39. #11745180C22semidirect
40. #11745180C22semidirect
41. #11745169C22semidirect
42. #11745168C22semidirect
43. #11745168C22semidirect
44. #11745164C22semidirect
53. C2×D4×C29D4semidirect
75. #90805D4semidirect
172. C2×AGL(1,ℤ/8)D29semidirect
196. C2×#41C2×D29semidirect
206. C2×SD16C2×D29semidirect
208. C2×D8C2×D29semidirect
209. C2×M16C2×D29semidirect
223. C29#3012semidirect
252. C2×D4C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117453592
C2×#117453122
#142717412
#142713222
#142713122
#142713052
#142712952
#142711652
C2×#117451802
#142711312
C2×C29×AGL(1,ℤ/8)2
#301229