Finite Group Databasebeta

← #16828743

Subgroups — #16828743

Order 3712.

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

normal subgroups only

This group has 806 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
1C22×#90809C211yesmaximal, characteristic, complemented
2C2×#11745276C212yesmaximal, complemented
3#121×D29C211yesmaximal, characteristic, complemented
4#14271311C212yesmaximal, complemented
5#14271307C212yesmaximal, complemented
6#14271302C211yesmaximal, characteristic, complemented
7#14271299C211yesmaximal, characteristic, complemented
8#14271296C211yesmaximal, characteristic, complemented
9#14271163C212yesmaximal, complemented
10#14271125C211yesmaximal, characteristic, complemented
11C29×#448C211yesmaximal, characteristic, complemented
12C22×Q8×C29C2211yescharacteristic, complemented
13C2×#90809—28no—
14C2×#90809—21no—
15C2×#90809—21no—
16C2×#90809C2212yes—
17C2×#90809C2212yescomplemented
18C2×D4×D29—21no—
19C2×D4×D29C2211yescharacteristic, complemented
20C22×D116—21no—
21C22×D116C2211yescharacteristic, complemented
22C22×C4×D29—21no—
23C22×C4×D29C2211yescharacteristic
24C2×SD16×C29C2212yescomplemented
25C29×#121C2211yescharacteristic
26C2×#90784C2212yescomplemented
27#11745301—21no—
28#11745300C2211yescharacteristic, complemented
29C2×#90782C2212yescomplemented
30#11745292C2211yescharacteristic
31#11745276—28no—
32C2×D232C2212yescomplemented
33C2×#90772C2212yescomplemented
34#11745257C2211yescharacteristic, complemented
35C4⋊C4×D29C2211yescharacteristic, complemented
36#11745246—21no—
37C22⋊C4×D29—21no—
38C29×#103C2212yescomplemented
39C29×C22⋊C8C2211yescharacteristic, complemented
40#11745183C2212yescomplemented
41C22⋊#6597C2211yescharacteristic, complemented
42#11745169C2212yescomplemented
43#11745168C2212yescomplemented
44#11745157C2212yescomplemented
45C23×D29—21no—
46C23×D29—21no—
47C23×D29D411yescharacteristic
48C2×Q8×C29—24no—
49C2×Q8×C29—21no—
50C2×Q8×C29C2312yes—
51C2×D4×C29—21no—
52C2×D4×C29C2311yescharacteristic
53C22×C4×C29—21no—
54C22×C4×C29C2311yescharacteristic
55C2×C22⋊D29—21no—
56C2×C22⋊D29—21no—
57C2×C22⋊D29—21no—
58C22×Dic29D411yescharacteristic
59#90809—28no—
60#90809—28no—
61#90809—24no—
62#90809—24no—
63#90809—24no—
64#90809D418yescomplemented
65D4×D29—24no—
66D4×D29—44no—
67C2×D116—28no—
68C2×D116—22no—
69C2×D116—22no—
70C2×D116—21no—
71C2×D116—24no—
72C2×D116—21no—
73C2×D116C2311yescharacteristic
74C2×D116C2312yes—
75C2×C4×D29—28no—
76C2×C4×D29—22no—
77C2×C4×D29—22no—
78C2×C4×D29—24no—
79C2×C4×D29—21no—
80C2×C4×D29D411yescharacteristic
81C2×C4×D29D411yescharacteristic
82C2×C4×D29C2311yescharacteristic
83C2×C4×D29C2311yescharacteristic
84SD16×C29—24no—
85C2×C8×C29C2312yes—
86C4⋊C4×C29C2311yescharacteristic
87C22⋊C4×C29—21no—
88C22⋊Dic29—21no—
89#90784—24no—
90#90782—24no—
91#90781—21no—
92#90781—21no—
93C4⋊Dic29C2311yescharacteristic
94#90779—21no—
95C2×#6597C2312yes—
96D232—24no—
97#90772—24no—
98C23×C29—21no—
99C23×C29C2×D411yescharacteristic
100C22×D29—24no—
101C22×D29—22no—
102C22×D29—22no—
103C22×D29—22no—
104C22×D29—21no—
105C22×D29—41no—
106C22×D29—42no—
107C22×D29—42no—
108C22×D29—42no—
109C22×D29—42no—
110C22×D29—42no—
111C22×D29C2×D411yescharacteristic
112C22×D29C2×D411yescharacteristic
113Q8×C29—24no—
114Q8×C29—22no—
115Q8×C29C2×D414yescomplemented
116D4×C29—22no—
117D4×C29—42no—
118C2×C4×C29—24no—
119C2×C4×C29—22no—
120C2×C4×C29—22no—
121C2×C4×C29—21no—
122C2×C4×C29C2×D411yescharacteristic
123C2×C4×C29C2411yescharacteristic
124C22⋊D29—42no—
125C22⋊D29—42no—
126C22⋊D29—42no—
127C2×Dic29—22no—
128C2×Dic29—21no—
129C2×Dic29C2×D411yescharacteristic
130C2×Dic29C2×D411yescharacteristic
131D116—28no—
132D116—24no—
133D116—28no—
134D116—22no—
135D116—22no—
136D116—42no—
137D116C2×D414yescomplemented
138C4×D29—28no—
139C4×D29—22no—
140C4×D29—22no—
141C4×D29—28no—
142C4×D29—22no—
143C4×D29—42no—
144C4×D29C2×D414yes—
145C8×C29—22no—
146#6597—22no—
147#3006—291nomaximal, Sylow
148C22×C29—21no—
149C22×C29—21no—
150C22×C29—41no—
151C22×C29—41no—
152C22×C29—41no—
153C22×C29AGL(1,ℤ/8)12yescomplemented
154C22×C29C22×D411yescharacteristic
155C2×D29—21no—
156C2×D29—21no—
157C2×D29—21no—
158C2×D29—24no—
159C2×D29—21no—
160C2×D29—21no—
161C2×D29—44no—
162C2×D29—48no—
163C2×D29—42no—
164C2×D29—42no—
165C2×D29—41no—
166C2×D29—41no—
167C2×D29—41no—
168C2×D29—84no—
169C2×D29#12012yes—
170C4×C29—24no—
171C4×C29—21no—
172C4×C29—41no—
173C4×C29C22×D412yes—
174Dic29—21no—
175Dic29—41no—
176Dic29#12012yes—
177C22×#41—291no—
178C2×AGL(1,ℤ/8)—292no—
179C2×#121—291no—
180#449—292no—
181#449—292no—
182#448—291no—
183#448D2911yescharacteristic, complemented
184#447—291no—
185#447—291no—
186#417—292no—
187#407—291no—
188C2×C29—22no—
189C2×C29—81no—
190C2×C29C2×#12011yescharacteristic
191C2×C29C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C29#57511yescharacteristic
193D29—22no—
194D29—41no—
195D29—44no—
196D29—81no—
197C2×#41—292no—
198C2×#41—292no—
199C2×#41—581no—
200C2×#41—581no—
201C2×#41—588no—
202C22×Q8C2×D2911yescharacteristic, complemented
203C22×D4—291no—
204C22×D4—291no—
205C22×D4—581no—
206C22×D4—581no—
207C23×C4—291no—
208C23×C4—581no—
209AGL(1,ℤ/8)—588no—
210C2×SD16—292no—
211C2×SD16C2×D2912yescomplemented
212C2×D8—292no—
213C2×D8—292no—
214C2×M16—292no—
215#121—291no—
216#121—291no—
217#121—291no—
218#121—581no—
219#121—581no—
220#121C2×D2911yescharacteristic
221C2×C4⋊C4—291no—
222C2×C22⋊C4—581no—
223#103—292no—
224#103C2×D2912yescomplemented
225#102—292no—
226#102—292no—
227C22⋊C8—291no—
228C22⋊C8—292no—
229C22⋊C8C2×D2911yescharacteristic, complemented
230C29#300611yescharacteristic, complemented, Sylow
231C24—291no—
232C24—581no—
233C24—581no—
234#41—298no—
235#41—584no—
236#41—584no—
237#41—588no—
238#41—588no—
239#41—584no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D2912yes—
243C2×D4—21no—
244C2×D4—292no—
245C2×D4—291no—
246C2×D4—584no—
247C2×D4—582no—
248C2×D4—582no—
249C2×D4—588no—
250C2×D4—581no—
251C2×D4—584no—
252C2×D4—581no—
253C2×D4—581no—
254C2×D4—581no—
255C2×D4—581no—
256C2×D4—1164no—
257C2×D4C22×D2911yescharacteristic
258C22×C4—21no—
259C22×C4—291no—
260C22×C4—291no—
261C22×C4—291no—
262C22×C4—291no—
263C22×C4—291no—
264C22×C4—584no—
265C22×C4—582no—
266C22×C4—582no—
267C22×C4—588no—
268C22×C4—581no—
269C22×C4C22×D2911yescharacteristic
270SD16—24no—
271SD16—584no—
272D8—584no—
273D8—584no—
274M16—584no—
275C2×C8—292no—
276C2×C8C22×D2912yes—
277C4⋊C4—291no—
278C4⋊C4—581no—
279C4⋊C4C22×D2911yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—581no—
282C22⋊C4—581no—
283C22⋊C4—581no—
284C23—21no—
285C23—291no—
286C23—291no—
287C23—582no—
288C23—584no—
289C23—582no—
290C23—582no—
291C23—581no—
292C23—1161no—
293C23—1162no—
294C23—1162no—
295C23—1162no—
296C23—1162no—
297C23—1162no—
298C23D4×D2911yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D2914yescomplemented
302D4—22no—
303D4—42no—
304D4—294no—
305D4—588no—
306D4—588no—
307D4—584no—
308D4—582no—
309D4—582no—
310D4—1162no—
311D4—1162no—
312D4—1162no—
313D4—1162no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—291no—
319C2×C4—294no—
320C2×C4—291no—
321C2×C4—582no—
322C2×C4—588no—
323C2×C4—582no—
324C2×C4—588no—
325C2×C4—582no—
326C2×C4—582no—
327C2×C4—581no—
328C2×C4—1162no—
329C2×C4D4×D2911yescharacteristic
330C2×C4C23×D2911yescharacteristic
331C8—22no—
332C8—582no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—292no—
339C22—581no—
340C22—581no—
341C22—584no—
342C22—581no—
343C22—581no—
344C22—581no—
345C22—1164no—
346C22—1168no—
347C22—1162no—
348C22—1162no—
349C22—1161no—
350C22—1161no—
351C22—1161no—
352C22—2324no—
353C22#1174527612yescomplemented
354C22C2×D4×D2911yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—292no—
359C4—581no—
360C4—1161no—
361C4C2×D4×D2912yes—
362C2—22no—
363C2—81no—
364C2—582no—
365C2—1161no—
366C2—1164no—
367C2—2321no—
368C2#120×D2911yescharacteristic
369C2C2×#1174527611yescharacteristic
370C2#1427196211yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#448C2
socleC22×C29C22×D4
hypercenter#448D29
hypocenterC29#3006
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#90809C2semidirect
2. C2×#11745276C2semidirect
3. #121×D29C2semidirect
4. #14271311C2semidirect
5. #14271307C2semidirect
6. #14271302C2semidirect
7. #14271299C2semidirect
8. #14271296C2semidirect
9. #14271163C2semidirect
10. #14271125C2semidirect
11. C29×#448C2semidirect
12. C22×Q8×C29C22semidirect
17. C2×#90809C22semidirect
19. C2×D4×D29C22semidirect
21. C22×D116C22semidirect
24. C2×SD16×C29C22semidirect
26. C2×#90784C22semidirect
28. #11745300C22semidirect
29. C2×#90782C22semidirect
32. C2×D232C22semidirect
33. C2×#90772C22semidirect
34. #11745257C22semidirect
35. C4⋊C4×D29C22semidirect
38. C29×#103C22semidirect
39. C29×C22⋊C8C22semidirect
40. #11745183C22semidirect
41. C22⋊#6597C22semidirect
42. #11745169C22semidirect
43. #11745168C22semidirect
44. #11745157C22semidirect
64. #90809D4semidirect
115. Q8×C29C2×D4semidirect
137. D116C2×D4semidirect
153. C22×C29AGL(1,ℤ/8)semidirect
183. #448D29semidirect
202. C22×Q8C2×D29semidirect
211. C2×SD16C2×D29semidirect
224. #103C2×D29semidirect
229. C22⋊C8C2×D29semidirect
230. C29#3006semidirect
301. Q8D4×D29semidirect
353. C22#11745276semidirect

Maximal subgroups

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

subgroupindex
C22×#908092
C2×#117452762
#121×D292
#142713112
#142713072
#142713022
#142712992
#142712962
#142711632
#142711252
C29×#4482
#300629