Finite Group Databasebeta

← #16828981

Subgroups — #16828981

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745277C211yesmaximal, characteristic, complemented
2C2×#11745273C211yesmaximal, characteristic
3#124×D29C211yesmaximal, characteristic, complemented
4C4×#90807C211yesmaximal, characteristic, complemented
5#14271388C211yesmaximal, characteristic, complemented
6#14271386C211yesmaximal, characteristic, complemented
7#14271384C211yesmaximal, characteristic, complemented
8#14271382C211yesmaximal, characteristic, complemented
9#14271380C211yesmaximal, characteristic, complemented
10#14271378C211yesmaximal, characteristic, complemented
11#14271376C211yesmaximal, characteristic, complemented
12#14271185C211yesmaximal, characteristic, complemented
13#14271170C211yesmaximal, characteristic, complemented
14#14271166C211yesmaximal, characteristic, complemented
15C29×#463C211yesmaximal, characteristic, complemented
16C2×Q8×D29C2211yescharacteristic
17C2×#90807C2211yescharacteristic
18C2×D4×D29C2211yescharacteristic
19C2×SD16×C29C2211yescharacteristic, complemented
20C2×D8×C29C2211yescharacteristic
21C29×#124C2211yescharacteristic
22C4×D4×C29C2211yescharacteristic, complemented
23#11745309C2211yescharacteristic
24C2×#90785C2211yescharacteristic, complemented
25#11745298C2211yescharacteristic
26D4×Dic29C2211yescharacteristic, complemented
27C2×#90783C2211yescharacteristic
28C2×#90783C2211yescharacteristic, complemented
29C2×#90782C2211yescharacteristic
30C4×C22⋊D29—21no—
31C2×C4×Dic29—21no—
32#11745277—24no—
33#11745273—24no—
34C2×Dic116C2211yescharacteristic, complemented
35C2×#90773C2211yescharacteristic
36C2×#90772C2211yescharacteristic
37C2×#90772C2211yescharacteristic
38#11745254C2211yescharacteristic
39#11745249C2211yescharacteristic, complemented
40#11745247—21no—
41#11745247—21no—
42#11745243—21no—
43C22⋊C4×D29—21no—
44C22⋊C4×D29—21no—
45#11745239—21no—
46#11745237C2211yescharacteristic
47C42×D29C2211yescharacteristic
48C4×Dic58C2211yescharacteristic, complemented
49C29×C4⋊C8C2211yescharacteristic, complemented
50C29×#103C2211yescharacteristic, complemented
51C29×#102C2211yescharacteristic
52#11745183C2211yescharacteristic, complemented
53#11745180C2211yescharacteristic
54#11745169C2211yescharacteristic
55#11745164C2211yescharacteristic, complemented
56#11745162C2211yescharacteristic, complemented
57#11745162C2211yescharacteristic, complemented
58#11745158C2211yescharacteristic, complemented
59#11745157C2211yescharacteristic
60C4⋊#6597C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×Q8×C29C2311yescharacteristic
63C2×D4×C29C2311yescharacteristic
64C2×D4×C29C2311yescharacteristic
65C22×C4×C29—21no—
66C2×C22⋊D29—21no—
67C2×C22⋊D29—21no—
68C22×Dic29—21no—
69C22×Dic29—21no—
70Q8×D29—22no—
71Q8×D29—22no—
72#90807—22no—
73#90807D414yes—
74D4×D29—22no—
75D4×D29—22no—
76C2×D116C2311yescharacteristic
77C2×C4×D29—21no—
78C2×C4×D29D411yescharacteristic
79C2×C4×D29D411yescharacteristic
80C2×C4×D29C2311yescharacteristic
81C2×Dic58—21no—
82C2×Dic58C2311yescharacteristic
83C2×Dic58C2311yescharacteristic
84SD16×C29—22no—
85D8×C29—22no—
86C2×C8×C29C2311yescharacteristic
87C2×C8×C29C2311yescharacteristic
88C4⋊C4×C29C2311yescharacteristic
89C22⋊C4×C29—21no—
90C22⋊C4×C29—21no—
91C22⋊C4×C29—21no—
92C42×C29C2311yescharacteristic
93C22⋊Dic29—21no—
94C22⋊Dic29—21no—
95C22⋊Dic29—21no—
96#90785—22no—
97#90783—22no—
98#90783—22no—
99#90782—22no—
100#90781—21no—
101#90781—21no—
102#90781—21no—
103#90781—21no—
104#90781—21no—
105C4⋊Dic29C2311yescharacteristic
106#90779—21no—
107#90779—21no—
108C4×Dic29—21no—
109C4×Dic29—21no—
110C4×Dic29—21no—
111C4×Dic29D411yescharacteristic
112C4×Dic29D411yescharacteristic
113C4×Dic29C2311yescharacteristic
114C2×#6597C2311yescharacteristic
115C2×#6597C2311yescharacteristic
116Dic116—22no—
117#90773—22no—
118#90772—22no—
119#90772—22no—
120C23×C29—21no—
121C23×C29—21no—
122C22×D29—21no—
123C22×D29—42no—
124C22×D29—42no—
125C22×D29—42no—
126C22×D29C2×D411yescharacteristic
127Q8×C29—21no—
128Q8×C29—21no—
129D4×C29—21no—
130D4×C29—21no—
131D4×C29—21no—
132D4×C29C2×D412yes—
133C2×C4×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29—42no—
136C2×C4×C29C2×D411yescharacteristic
137C2×C4×C29C2×D411yescharacteristic
138C2×C4×C29C2411yescharacteristic
139C22⋊D29—24no—
140C22⋊D29—42no—
141C2×Dic29—21no—
142C2×Dic29—24no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—42no—
146C2×Dic29C2×D411yescharacteristic
147C2×Dic29C2×D411yescharacteristic
148C2×Dic29C2×D411yescharacteristic
149D116—21no—
150D116—21no—
151C4×D29—41no—
152C4×D29—41no—
153C4×D29—42no—
154C4×D29C2×D412yes—
155C4×D29#4112yes—
156Dic58—21no—
157Dic58—21no—
158Dic58—21no—
159Dic58—42no—
160Dic58C2×D412yes—
161C8×C29—21no—
162C8×C29—21no—
163#6597—21no—
164#6597—21no—
165#3040—291nomaximal, Sylow
166C22×C29—22no—
167C22×C29—41no—
168C22×C29—41no—
169C22×C29—41no—
170C22×C29—41no—
171C22×C29—41no—
172C22×C29C22×D411yescharacteristic
173C2×D29—21no—
174C2×D29—21no—
175C2×D29—41no—
176C2×D29—41no—
177C2×D29—41no—
178C2×D29—84no—
179C2×D29#12112yes—
180C4×C29—21no—
181C4×C29—21no—
182C4×C29—41no—
183C4×C29—41no—
184C4×C29C22×D411yescharacteristic
185C4×C29C2×#4111yescharacteristic
186Dic29—22no—
187Dic29—21no—
188Dic29—21no—
189Dic29—41no—
190Dic29—41no—
191Dic29#12112yes—
192C2×#137—291no—
193C2×AGL(1,ℤ/8)—291no—
194C2×#124—291no—
195C4×#41—291no—
196#464—291no—
197#464—291no—
198#464—291no—
199#464—291no—
200#463—291no—
201#463—291no—
202#463—291no—
203#463D2911yescharacteristic, complemented
204#423—291no—
205#419—291no—
206#418—291no—
207C2×C29—42no—
208C2×C29—81no—
209C2×C29C2×#12111yescharacteristic
210C2×C29#57511yescharacteristic
211C2×C29#57511yescharacteristic
212D29—22no—
213D29—81no—
214C2×#41—291no—
215C22×Q8—291no—
216C22×D4—291no—
217#137—584no—
218AGL(1,ℤ/8)—584no—
219C2×Q16—291no—
220C2×Q16—291no—
221C2×SD16—291no—
222C2×SD16—291no—
223C2×SD16—291no—
224C2×SD16C2×D2911yescharacteristic, complemented
225C2×D8—291no—
226C2×D8C2×D2911yescharacteristic
227C2×M16—291no—
228C2×M16—291no—
229#124—291no—
230#124—291no—
231#124—291no—
232#124—581no—
233#124—581no—
234#124C2×D2911yescharacteristic
235C4×Q8—291no—
236C4×Q8—291no—
237C4×D4—291no—
238C4×D4—581no—
239C4×D4—581no—
240C4×D4C2×D2911yescharacteristic, complemented
241#117—291no—
242#117—581no—
243C2×C22⋊C4—581no—
244C2×C22⋊C4—581no—
245C2×C42—291no—
246C2×C42—581no—
247C4⋊C8—291no—
248C4⋊C8—291no—
249C4⋊C8—291no—
250C4⋊C8C2×D2911yescharacteristic, complemented
251#103—291no—
252#103—291no—
253#103—291no—
254#103C2×D2911yescharacteristic, complemented
255#102—291no—
256#102—291no—
257#102—291no—
258#102C2×D2911yescharacteristic
259C29#304011yescharacteristic, complemented, Sylow
260C24—581no—
261#41—294no—
262#41—582no—
263C2×Q8—291no—
264C2×Q8—291no—
265C2×Q8—581no—
266C2×Q8—582no—
267C2×Q8—582no—
268C2×Q8C22×D2911yescharacteristic
269C2×D4—291no—
270C2×D4—582no—
271C2×D4—582no—
272C2×D4—581no—
273C2×D4—581no—
274C2×D4C22×D2911yescharacteristic
275C2×D4C22×D2911yescharacteristic
276C22×C4—21no—
277C22×C4—291no—
278C22×C4—291no—
279C22×C4—291no—
280C22×C4—581no—
281C22×C4—581no—
282C22×C4—581no—
283Q16—582no—
284Q16—582no—
285SD16—22no—
286SD16—582no—
287SD16—582no—
288SD16—582no—
289D8—22no—
290D8—582no—
291M16—582no—
292M16—582no—
293C2×C8—291no—
294C2×C8—291no—
295C2×C8C22×D2911yescharacteristic
296C2×C8C22×D2911yescharacteristic
297C4⋊C4—291no—
298C4⋊C4—581no—
299C4⋊C4—581no—
300C4⋊C4C22×D2911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—581no—
305C22⋊C4—581no—
306C22⋊C4—581no—
307C22⋊C4—581no—
308C22⋊C4—581no—
309C22⋊C4—581no—
310C22⋊C4—581no—
311C22⋊C4—581no—
312C42—291no—
313C42—291no—
314C42—291no—
315C42—581no—
316C42—581no—
317C42—581no—
318C42C22×D2911yescharacteristic
319C23—21no—
320C23—21no—
321C23—291no—
322C23—581no—
323C23—1162no—
324C23—1162no—
325C23—1162no—
326Q8—21no—
327Q8—21no—
328Q8—292no—
329Q8—581no—
330Q8—581no—
331Q8—581no—
332Q8—1162no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—584no—
337D4—581no—
338D4—581no—
339D4—1162no—
340D4D4×D2912yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—291no—
345C2×C4—292no—
346C2×C4—292no—
347C2×C4—291no—
348C2×C4—291no—
349C2×C4—584no—
350C2×C4—581no—
351C2×C4—581no—
352C2×C4—581no—
353C2×C4—1161no—
354C2×C4—1161no—
355C2×C4—1162no—
356C2×C4—1162no—
357C2×C4D4×D2911yescharacteristic
358C2×C4D4×D2911yescharacteristic
359C2×C4C23×D2911yescharacteristic
360C8—21no—
361C8—21no—
362C8—581no—
363C8—581no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—292no—
371C22—581no—
372C22—581no—
373C22—1161no—
374C22—1161no—
375C22—1161no—
376C22—2324no—
377C22C2×D4×D2911yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—292no—
383C4—582no—
384C4—581no—
385C4—581no—
386C4—1161no—
387C4—1161no—
388C4C2×D4×D2911yescharacteristic
389C4#41×D2911yescharacteristic
390C2—42no—
391C2—81no—
392C2—582no—
393C2—2321no—
394C2#121×D2911yescharacteristic
395C2#1427195911yescharacteristic
396C2#1427196011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#463C2
socleC22×C29C22×D4
prime radicalC2×#11745273C2
hypercenter#463D29
hypocenterC29#3040
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#11745277C2semidirect
3. #124×D29C2semidirect
4. C4×#90807C2semidirect
5. #14271388C2semidirect
6. #14271386C2semidirect
7. #14271384C2semidirect
8. #14271382C2semidirect
9. #14271380C2semidirect
10. #14271378C2semidirect
11. #14271376C2semidirect
12. #14271185C2semidirect
13. #14271170C2semidirect
14. #14271166C2semidirect
15. C29×#463C2semidirect
19. C2×SD16×C29C22semidirect
22. C4×D4×C29C22semidirect
24. C2×#90785C22semidirect
26. D4×Dic29C22semidirect
28. C2×#90783C22semidirect
34. C2×Dic116C22semidirect
39. #11745249C22semidirect
48. C4×Dic58C22semidirect
49. C29×C4⋊C8C22semidirect
50. C29×#103C22semidirect
52. #11745183C22semidirect
55. #11745164C22semidirect
56. #11745162C22semidirect
57. #11745162C22semidirect
58. #11745158C22semidirect
60. C4⋊#6597C22semidirect
203. #463D29semidirect
224. C2×SD16C2×D29semidirect
240. C4×D4C2×D29semidirect
250. C4⋊C8C2×D29semidirect
254. #103C2×D29semidirect
259. C29#3040semidirect

Maximal subgroups

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

subgroupindex
C2×#117452772
C2×#117452732
#124×D292
C4×#908072
#142713882
#142713862
#142713842
#142713822
#142713802
#142713782
#142713762
#142711852
#142711702
#142711662
C29×#4632
#304029