Finite Group Databasebeta

← #16828346

Subgroups — #16828346

Order 3712.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D29C211yesmaximal, characteristic
2#14271639C211yesmaximal, characteristic, complemented
3#14271635C211yesmaximal, characteristic
4#14271266C211yesmaximal, characteristic, complemented
5#14271263C211yesmaximal, characteristic, complemented
6#14271261C211yesmaximal, characteristic, complemented
7#14271260C211yesmaximal, characteristic, complemented
8C4×#90773C211yesmaximal, characteristic, complemented
9SD16×Dic29C211yesmaximal, characteristic, complemented
10#14271250C211yesmaximal, characteristic, complemented
11#14271198C211yesmaximal, characteristic, complemented
12#14271170C211yesmaximal, characteristic, complemented
13#14271166C211yesmaximal, characteristic
14#14271117C211yesmaximal, characteristic, complemented
15C4×SD16×C29C211yesmaximal, characteristic, complemented
16C2×Q8×D29C2211yescharacteristic
17C2×D4×D29C2211yescharacteristic
18C2×SD16×C29C2211yescharacteristic
19C4×Q8×C29C2211yescharacteristic, complemented
20C4×D4×C29C2211yescharacteristic
21Q8×Dic29C2211yescharacteristic, complemented
22C2×#90784C2211yescharacteristic
23D4×Dic29C2211yescharacteristic
24C2×#90783C2211yescharacteristic
25C4×C22⋊D29—21no—
26C2×#90781—21no—
27SD16×D29C418yescomplemented
28C2×#90773C2211yescharacteristic
29C2×#90772C2211yescharacteristic
30C2×C8×D29C2211yescharacteristic
31#11745255C2211yescharacteristic
32#11745254—21no—
33#11745254C2211yescharacteristic
34#11745254C2211yescharacteristic
35#11745249—21no—
36#11745249C2211yescharacteristic, complemented
37#11745243—21no—
38C22⋊C4×D29—21no—
39C4×D116C2211yescharacteristic
40#11745234—21no—
41#11745234C2211yescharacteristic
42C4×Dic58—21no—
43C4×Dic58C2211yescharacteristic, complemented
44C29×#106C2211yescharacteristic, complemented
45C29×#103C2211yescharacteristic, complemented
46C29×#102C2211yescharacteristic
47C4×C8×C29C2211yescharacteristic, complemented
48#11745183C2211yescharacteristic, complemented
49#11745180C2211yescharacteristic
50#11745169C2211yescharacteristic
51#11745165C2211yescharacteristic, complemented
52#11745164C2211yescharacteristic, complemented
53#11745163C2211yescharacteristic, complemented
54C8×Dic29C2211yescharacteristic, complemented
55#11745158C2211yescharacteristic, complemented
56#11745157C2211yescharacteristic
57#11745155C2211yescharacteristic, complemented
58#11745155C2211yescharacteristic, complemented
59#11745151C2211yescharacteristic, complemented
60C23×D29—21no—
61C2×Q8×C29C2311yescharacteristic
62C2×D4×C29C2311yescharacteristic
63C22×C4×C29—21no—
64C2×C22⋊D29—21no—
65C22×Dic29—21no—
66Q8×D29—22no—
67Q8×D29C2×C414yescomplemented
68D4×D29—22no—
69D4×D29C2×C414yes—
70C2×D116C2311yescharacteristic
71C2×C4×D29—21no—
72C2×C4×D29—21no—
73C2×C4×D29—21no—
74C2×C4×D29C2311yescharacteristic
75C2×Dic58—21no—
76C2×Dic58C2311yescharacteristic
77SD16×C29C2×C414yescomplemented
78C2×C8×C29C2311yescharacteristic
79C2×C8×C29C2311yescharacteristic
80C4⋊C4×C29—21no—
81C4⋊C4×C29C2311yescharacteristic
82C4⋊C4×C29C2311yescharacteristic
83C22⋊C4×C29—21no—
84C42×C29—21no—
85C42×C29C2311yescharacteristic
86C22⋊Dic29—21no—
87#90784C2×C414yescomplemented
88#90783C2×C414yescomplemented
89#90781—21no—
90#90781—21no—
91#90781—21no—
92#90781—21no—
93#90781D411yescharacteristic
94#90781D411yescharacteristic
95C4⋊Dic29—21no—
96C4⋊Dic29C2311yescharacteristic
97C4⋊Dic29C2311yescharacteristic
98#90779—21no—
99#90779—21no—
100#90779D411yescharacteristic
101#90779D411yescharacteristic
102C4×Dic29—21no—
103C4×Dic29—21no—
104C4×Dic29—21no—
105C4×Dic29—21no—
106C4×Dic29—21no—
107C4×Dic29C2311yescharacteristic
108C2×#6597C2311yescharacteristic
109C2×#6597C2311yescharacteristic
110#90773C2×C414yescomplemented
111#90772—22no—
112C8×D29C2×C414yescomplemented
113C23×C29—21no—
114C22×D29—21no—
115C22×D29—24no—
116C22×D29—81no—
117C22×D29—81no—
118C22×D29C2×D411yescharacteristic
119Q8×C29—21no—
120Q8×C29C22×C412yes—
121D4×C29—21no—
122D4×C29C22×C412yes—
123C2×C4×C29—21no—
124C2×C4×C29—21no—
125C2×C4×C29—21no—
126C2×C4×C29—81no—
127C2×C4×C29C2×D411yescharacteristic
128C2×C4×C29C2×D411yescharacteristic
129C2×C4×C29C2411yescharacteristic
130C22⋊D29—24no—
131C2×Dic29—21no—
132C2×Dic29—21no—
133C2×Dic29—21no—
134C2×Dic29—81no—
135C2×Dic29C2×D411yescharacteristic
136C2×Dic29C2×D411yescharacteristic
137C2×Dic29C2×D411yescharacteristic
138D116—21no—
139D116C22×C412yes—
140C4×D29—24no—
141C4×D29—81no—
142C4×D29—81no—
143C4×D29C22×C412yes—
144C4×D29#4112yes—
145Dic58—21no—
146Dic58—24no—
147Dic58C22×C412yes—
148C8×C29—21no—
149C8×C29C22×C412yes—
150#6597—21no—
151#6597C22×C412yes—
152#2949—291nomaximal, Sylow
153C22×C29—22no—
154C22×C29—41no—
155C22×C29—41no—
156C22×C29C22×D411yescharacteristic
157C2×D29—21no—
158C2×D29—21no—
159C2×D29—22no—
160C2×D29—41no—
161C2×D29—41no—
162C2×D29—84no—
163C2×D29C4×D412yes—
164C4×C29—22no—
165C4×C29—21no—
166C4×C29—21no—
167C4×C29—41no—
168C4×C29—41no—
169C4×C29C23×C411yescharacteristic
170C4×C29C2×#4111yescharacteristic
171Dic29—22no—
172Dic29—21no—
173Dic29—21no—
174Dic29—41no—
175Dic29—41no—
176Dic29C4×D412yes—
177C22×SD16—291no—
178#519—291no—
179#518—291no—
180#440—291no—
181#440—291no—
182#440—291no—
183#440—291no—
184C4×SD16—291no—
185C4×SD16—291no—
186C4×SD16—291no—
187C4×SD16D2911yescharacteristic, complemented
188#427—291no—
189#419—291no—
190#418—291no—
191#405—291no—
192C2×C29—42no—
193C2×C29C2×C4×D411yescharacteristic
194C2×C29#57711yescharacteristic
195C2×C29#57711yescharacteristic
196D29—22no—
197D29—42no—
198C22×Q8—291no—
199C22×D4—291no—
200C2×SD16—291no—
201C2×SD16—291no—
202C2×SD16—298no—
203C2×SD16—291no—
204C2×SD16C2×D2911yescharacteristic
205C2×M16—291no—
206C22×C8—291no—
207C4×Q8—291no—
208C4×Q8—291no—
209C4×Q8—291no—
210C4×Q8—581no—
211C4×Q8—581no—
212C4×Q8C2×D2911yescharacteristic, complemented
213C4×D4—291no—
214C4×D4—291no—
215C4×D4—291no—
216C4×D4—581no—
217C4×D4—581no—
218C4×D4C2×D2911yescharacteristic
219#117—291no—
220#117—291no—
221#117—291no—
222#117—581no—
223#117—581no—
224C2×C22⋊C4—581no—
225C2×C22⋊C4—581no—
226#107—291no—
227#107—291no—
228#106—291no—
229#106C2×D2911yescharacteristic, complemented
230#103—291no—
231#103—291no—
232#103—291no—
233#103C2×D2911yescharacteristic, complemented
234#102—291no—
235#102—291no—
236#102—291no—
237#102C2×D2911yescharacteristic
238#97—291no—
239#97—291no—
240C4×C8—291no—
241C4×C8C2×D2911yescharacteristic, complemented
242C29#294911yescharacteristic, complemented, Sylow
243C24—581no—
244C2×Q8—294no—
245C2×Q8—291no—
246C2×Q8—581no—
247C2×Q8—582no—
248C2×Q8C22×D2911yescharacteristic
249C2×D4—294no—
250C2×D4—291no—
251C2×D4—582no—
252C2×D4—581no—
253C2×D4C22×D2911yescharacteristic
254C22×C4—21no—
255C22×C4—291no—
256C22×C4—581no—
257C22×C4—581no—
258C22×C4—581no—
259C22×C4—581no—
260SD16—294no—
261SD16—294no—
262SD16—294no—
263SD16C4×D2914yescomplemented
264M16—582no—
265C2×C8—291no—
266C2×C8—294no—
267C2×C8—291no—
268C2×C8C22×D2911yescharacteristic
269C2×C8C22×D2911yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—291no—
272C4⋊C4—291no—
273C4⋊C4—291no—
274C4⋊C4—291no—
275C4⋊C4—581no—
276C4⋊C4—581no—
277C4⋊C4—581no—
278C4⋊C4C22×D2911yescharacteristic
279C4⋊C4C22×D2911yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—291no—
282C22⋊C4—291no—
283C22⋊C4—581no—
284C22⋊C4—581no—
285C22⋊C4—581no—
286C22⋊C4—581no—
287C22⋊C4—581no—
288C42—21no—
289C42—291no—
290C42—581no—
291C42—581no—
292C42—581no—
293C42—581no—
294C42—581no—
295C42C22×D2911yescharacteristic
296C23—21no—
297C23—291no—
298C23—584no—
299C23—581no—
300C23—2321no—
301C23—2321no—
302Q8—21no—
303Q8—292no—
304Q8—584no—
305Q8—581no—
306Q8C2×C4×D2912yes—
307D4—21no—
308D4—292no—
309D4—584no—
310D4—581no—
311D4C2×C4×D2912yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—291no—
317C2×C4—292no—
318C2×C4—292no—
319C2×C4—291no—
320C2×C4—291no—
321C2×C4—584no—
322C2×C4—581no—
323C2×C4—581no—
324C2×C4—581no—
325C2×C4—2321no—
326C2×C4—2321no—
327C2×C4—2321no—
328C2×C4D4×D2911yescharacteristic
329C2×C4D4×D2911yescharacteristic
330C2×C4C23×D2911yescharacteristic
331C8—21no—
332C8—292no—
333C8—581no—
334C8C2×C4×D2912yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—292no—
339C22—581no—
340C22—581no—
341C22—582no—
342C22—1161no—
343C22—1161no—
344C22—2324no—
345C22C2×D4×D2911yescharacteristic
346C4—22no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—292no—
352C4—582no—
353C4—581no—
354C4—581no—
355C4—1161no—
356C4—1161no—
357C4C22×C4×D2911yescharacteristic
358C4#41×D2911yescharacteristic
359C2—42no—
360C2—582no—
361C2—1162no—
362C2C4×D4×D2911yescharacteristic
363C2#1427196911yescharacteristic
364C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutatorC4×C29C23×C4
FrattiniC2×C4C23×D29
FittingC4×SD16×C29C2
socleC22×C29C22×D4
prime radicalC2×D4×D29C22
cosocleC2×C4×C29C24
hypercenterC4×SD16D29
hypocenterC29#2949
LCSG ▷ C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #14271639C2semidirect
4. #14271266C2semidirect
5. #14271263C2semidirect
6. #14271261C2semidirect
7. #14271260C2semidirect
8. C4×#90773C2semidirect
9. SD16×Dic29C2semidirect
10. #14271250C2semidirect
11. #14271198C2semidirect
12. #14271170C2semidirect
14. #14271117C2semidirect
15. C4×SD16×C29C2semidirect
19. C4×Q8×C29C22semidirect
21. Q8×Dic29C22semidirect
27. SD16×D29C4semidirect
36. #11745249C22semidirect
43. C4×Dic58C22semidirect
44. C29×#106C22semidirect
45. C29×#103C22semidirect
47. C4×C8×C29C22semidirect
48. #11745183C22semidirect
51. #11745165C22semidirect
52. #11745164C22semidirect
53. #11745163C22semidirect
54. C8×Dic29C22semidirect
55. #11745158C22semidirect
57. #11745155C22semidirect
58. #11745155C22semidirect
59. #11745151C22semidirect
67. Q8×D29C2×C4semidirect
77. SD16×C29C2×C4semidirect
87. #90784C2×C4semidirect
88. #90783C2×C4semidirect
110. #90773C2×C4semidirect
112. C8×D29C2×C4semidirect
187. C4×SD16D29semidirect
212. C4×Q8C2×D29semidirect
229. #106C2×D29semidirect
233. #103C2×D29semidirect
241. C4×C8C2×D29semidirect
242. C29#2949semidirect
263. SD16C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D292
#142716392
#142716352
#142712662
#142712632
#142712612
#142712602
C4×#907732
SD16×Dic292
#142712502
#142711982
#142711702
#142711662
#142711172
C4×SD16×C292
#294929