Finite Group Databasebeta

← #18006128

Subgroups — #18006128

Order 4736.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924564C211yesmaximal, characteristic, complemented
2C2×#11924559C211yesmaximal, characteristic, complemented
3#14558185C211yesmaximal, characteristic, complemented
4#14558174C211yesmaximal, characteristic, complemented
5#14557926C211yesmaximal, characteristic, complemented
6#14557763C211yesmaximal, characteristic, complemented
7#14557751C211yesmaximal, characteristic, complemented
8#14557737C211yesmaximal, characteristic, complemented
9#14557698C211yesmaximal, characteristic, complemented
10#14557693C211yesmaximal, characteristic, complemented
11#14557683C211yesmaximal, characteristic, complemented
12#14557681C211yesmaximal, characteristic, complemented
13#14557644C211yesmaximal, characteristic, complemented
14#14557602C211yesmaximal, characteristic, complemented
15C37×#470C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic
17C2×#10596966C2211yescharacteristic, complemented
18C2×#10596962C2211yescharacteristic
19C22×D148C2211yescharacteristic
20#11924564—24no—
21#11924559—24no—
22C2×#10596942C2211yescharacteristic, complemented
23C2×#10596930C2211yescharacteristic
24C2×#10596930C2211yescharacteristic
25C2×#10596941C2211yescharacteristic
26C2×#10596941C2211yescharacteristic
27C2×#10596941C2211yescharacteristic, complemented
28C2×D296C2211yescharacteristic
29C2×D296C2211yescharacteristic
30C2×C8×D37C2211yescharacteristic, complemented
31#11924534C2211yescharacteristic, complemented
32#11924525C2211yescharacteristic
33#11924521—21no—
34#11924521—21no—
35#11924518—21no—
36#11924516C2211yescharacteristic, complemented
37#11924511—21no—
38#11924511C2211yescharacteristic
39#11924509—21no—
40#11924506—21no—
41#11924498C2211yescharacteristic
42#11924498C2211yescharacteristic, complemented
43C4×C22⋊D37C2211yescharacteristic
44C4⋊C4×D37—21no—
45C22⋊C4×D37—21no—
46#11924465C2211yescharacteristic, complemented
47#11924459C2211yescharacteristic, complemented
48#11924456C2211yescharacteristic
49#11924456C2211yescharacteristic
50#11924456C2211yescharacteristic, complemented
51C22⋊#64436C2211yescharacteristic
52#11924448C2211yescharacteristic, complemented
53#11924446C2211yescharacteristic, complemented
54C2×Q16×C37C2211yescharacteristic, complemented
55C2×M16×C37C2211yescharacteristic
56C22⋊Q8×C37C2211yescharacteristic
57C22⋊Q8×C37C2211yescharacteristic
58#106×C37C2211yescharacteristic, complemented
59#103×C37C2211yescharacteristic, complemented
60#103×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C2×Q8×C37C2311yescharacteristic
63C2×Q8×C37C2311yescharacteristic
64C22×C4×C37C2311yescharacteristic
65C2×C22⋊D37—21no—
66C2×C22⋊D37D411yescharacteristic
67#10596966—22no—
68#10596966—22no—
69#10596966—41no—
70#10596966—41no—
71#10596962—22no—
72#10596962—41no—
73C2×D148—22no—
74C2×D148—21no—
75C2×D148—21no—
76C2×D148—21no—
77C2×D148—21no—
78C2×D148—41no—
79C2×D148C2311yescharacteristic
80C2×D148C2311yescharacteristic
81C2×D148C2311yescharacteristic
82C2×C4×D37—21no—
83C2×C4×D37—21no—
84C2×C4×D37—21no—
85C2×C4×D37—21no—
86C2×C4×D37—21no—
87C2×C4×D37C2311yescharacteristic
88C2×Dic74C2311yescharacteristic
89Q16×C37—22no—
90M16×C37—22no—
91C2×C8×C37C2311yescharacteristic
92C2×C8×C37C2311yescharacteristic
93C4⋊C4×C37—21no—
94C4⋊C4×C37—21no—
95C4⋊C4×C37C2311yescharacteristic
96C4⋊C4×C37C2311yescharacteristic
97C22⋊C4×C37—21no—
98C22⋊C4×C37—21no—
99C22⋊Dic37D411yescharacteristic
100#10596942—41no—
101#10596941—22no—
102#10596941—22no—
103#10596941—41no—
104#10596938—21no—
105#10596938—21no—
106#10596938—21no—
107#10596938—21no—
108#10596938—21no—
109#10596938—21no—
110#10596938D411yescharacteristic
111C4⋊Dic37—21no—
112#10596936—21no—
113#10596936—21no—
114#10596936D411yescharacteristic
115C4×Dic37C2311yescharacteristic
116C2×#64436C2311yescharacteristic
117C2×#64436C2311yescharacteristic
118D296—22no—
119D296D414yescomplemented
120#10596930—22no—
121#10596930—22no—
122C8×D37—22no—
123C23×C37C2×D411yescharacteristic
124C22×D37—21no—
125C22×D37—21no—
126C22×D37—21no—
127C22×D37—42no—
128C22×D37—42no—
129C22×D37—81no—
130C22×D37C2×D411yescharacteristic
131Q8×C37—21no—
132Q8×C37—21no—
133Q8×C37—21no—
134Q8×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—21no—
140C2×C4×C37—21no—
141C2×C4×C37C2×D411yescharacteristic
142C2×C4×C37C2411yescharacteristic
143C22⋊D37—41no—
144C22⋊D37—42no—
145C2×Dic37—21no—
146C2×Dic37C2×D411yescharacteristic
147C2×Dic37C2×D411yescharacteristic
148C2×Dic37C2×D411yescharacteristic
149D148—21no—
150D148—21no—
151D148—21no—
152D148—21no—
153D148—42no—
154D148—42no—
155D148—42no—
156D148—81no—
157D148C2×D412yes—
158D148C2×D412yes—
159C4×D37—21no—
160C4×D37—21no—
161C4×D37—42no—
162C4×D37—42no—
163C4×D37—42no—
164C4×D37—42no—
165C4×D37—81no—
166Dic74—21no—
167Dic74—21no—
168C8×C37—21no—
169C8×C37C2×D412yes—
170#64436—21no—
171#64436—21no—
172C22×C37—21no—
173C22×C37—21no—
174C22×C37—21no—
175C22×C37C22×D411yescharacteristic
176C2×D37—21no—
177C2×D37—22no—
178C2×D37—22no—
179C2×D37—21no—
180C2×D37—21no—
181C2×D37—41no—
182C2×D37—41no—
183C2×D37—41no—
184C2×D37—41no—
185C2×D37—41no—
186C2×D37—41no—
187C2×D37—41no—
188C2×D37—84no—
189C4×C37—21no—
190C4×C37—41no—
191C4×C37—41no—
192C4×C37—41no—
193C4×C37—41no—
194C4×C37C22×D411yescharacteristic
195C4×C37H(5,𝔽2)11yescharacteristic
196Dic37—21no—
197Dic37—21no—
198Dic37—21no—
199Dic37—41no—
200#3267—371nomaximal, Sylow
201C2×C37—41no—
202C2×C37D4211yescharacteristic
203C2×C37#57511yescharacteristic
204C2×C37#57711yescharacteristic
205D37—41no—
206D37—42no—
207D37—42no—
208D37—81no—
209C2×AGL(1,ℤ/8)—371no—
210C2×#135—371no—
211#547—371no—
212#546—371no—
213#497—371no—
214#470D3711yescharacteristic, complemented
215#465—371no—
216#463—371no—
217#461—371no—
218#452—371no—
219#452—371no—
220#450—371no—
221#449—371no—
222#442—371no—
223#435—371no—
224C37#326711yescharacteristic, complemented, Sylow
225C2×#41—371no—
226C2×#41—371no—
227C2×#41—371no—
228C22×D4—371no—
229AGL(1,ℤ/8)—744no—
230#135—744no—
231C2×Q16—371no—
232C2×Q16C2×D3711yescharacteristic, complemented
233C2×SD16—371no—
234C2×SD16—371no—
235C2×SD16—371no—
236C2×SD16—371no—
237C2×SD16—371no—
238C2×D8—371no—
239C2×D8—371no—
240C2×M16C2×D3711yescharacteristic
241C22×C8—371no—
242C4⋊Q8—371no—
243#124—371no—
244#123—741no—
245#123—741no—
246#123—741no—
247C22⋊Q8—371no—
248C22⋊Q8C2×D3711yescharacteristic
249C22⋊Q8C2×D3711yescharacteristic
250#121—371no—
251#121—741no—
252#121—741no—
253#120—741no—
254C4×D4—371no—
255C4×D4—371no—
256C4×D4—371no—
257C2×C4⋊C4—741no—
258C2×C22⋊C4—741no—
259#106C2×D3711yescharacteristic, complemented
260C4⋊C8—371no—
261#103—371no—
262#103C2×D3711yescharacteristic, complemented
263#103C2×D3711yescharacteristic, complemented
264#102—371no—
265#102—371no—
266#102—371no—
267C22⋊C8—371no—
268C22⋊C8—371no—
269#97—371no—
270C24—741no—
271#41—742no—
272#41—742no—
273#41—742no—
274#41—1481no—
275#41—1481no—
276#41—1481no—
277C2×Q8—371no—
278C2×Q8C22×D3711yescharacteristic
279C2×Q8C22×D3711yescharacteristic
280C2×D4—371no—
281C2×D4—371no—
282C2×D4—371no—
283C2×D4—371no—
284C2×D4—742no—
285C2×D4—741no—
286C2×D4—741no—
287C2×D4—741no—
288C2×D4—741no—
289C2×D4—741no—
290C2×D4—1481no—
291C22×C4—371no—
292C22×C4—741no—
293C22×C4—741no—
294C22×C4—741no—
295C22×C4—741no—
296C22×C4—741no—
297C22×C4C22×D3711yescharacteristic
298Q16—22no—
299Q16—1481no—
300SD16—742no—
301SD16—742no—
302SD16—742no—
303SD16—742no—
304SD16—1481no—
305D8—374no—
306D8—742no—
307M16—22no—
308C2×C8—371no—
309C2×C8—371no—
310C2×C8—742no—
311C2×C8C22×D3711yescharacteristic
312C2×C8C22×D3711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—371no—
316C4⋊C4—741no—
317C4⋊C4—741no—
318C4⋊C4—741no—
319C4⋊C4C22×D3711yescharacteristic
320C4⋊C4C22×D3711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—371no—
324C22⋊C4—371no—
325C22⋊C4—741no—
326C22⋊C4—741no—
327C22⋊C4—741no—
328C22⋊C4—741no—
329C22⋊C4—741no—
330C22⋊C4—741no—
331C42—371no—
332C23—371no—
333C23—2961no—
334C23—741no—
335C23—741no—
336C23—741no—
337C23—1482no—
338C23—1482no—
339C23D4×D3711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—741no—
345Q8—741no—
346D4—372no—
347D4—372no—
348D4—2961no—
349D4—741no—
350D4—741no—
351D4—741no—
352D4—741no—
353D4—1482no—
354D4—1482no—
355D4—1482no—
356D4—1481no—
357D4—1482no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—371no—
365C2×C4—371no—
366C2×C4—371no—
367C2×C4—2961no—
368C2×C4—741no—
369C2×C4—741no—
370C2×C4—741no—
371C2×C4—1482no—
372C2×C4—1482no—
373C2×C4—1482no—
374C2×C4—1482no—
375C2×C4D4×D3711yescharacteristic
376C2×C4C23×D3711yescharacteristic
377C8—21no—
378C8—741no—
379C8—741no—
380C8D4×D3712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—2964no—
385C22—741no—
386C22—741no—
387C22—741no—
388C22—742no—
389C22—742no—
390C22—1481no—
391C22—1481no—
392C22—1481no—
393C22—1481no—
394C22—1481no—
395C22—1481no—
396C22—1481no—
397C22C2×D4×D3711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—741no—
404C4—741no—
405C4—741no—
406C4—1481no—
407C4C2×D4×D3711yescharacteristic
408C4#1192458211yescharacteristic
409C2—41no—
410C2—2961no—
411C2—1482no—
412C2—1481no—
413C2—1482no—
414C2#1455816711yescharacteristic
415C2#1455833211yescharacteristic
416C2#1455834211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#470C2
socleC22×C37C22×D4
hypercenter#470D37
hypocenterC37#3267
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#11924564C2semidirect
2. C2×#11924559C2semidirect
3. #14558185C2semidirect
4. #14558174C2semidirect
5. #14557926C2semidirect
6. #14557763C2semidirect
7. #14557751C2semidirect
8. #14557737C2semidirect
9. #14557698C2semidirect
10. #14557693C2semidirect
11. #14557683C2semidirect
12. #14557681C2semidirect
13. #14557644C2semidirect
14. #14557602C2semidirect
15. C37×#470C2semidirect
17. C2×#10596966C22semidirect
22. C2×#10596942C22semidirect
27. C2×#10596941C22semidirect
30. C2×C8×D37C22semidirect
31. #11924534C22semidirect
36. #11924516C22semidirect
42. #11924498C22semidirect
46. #11924465C22semidirect
47. #11924459C22semidirect
50. #11924456C22semidirect
52. #11924448C22semidirect
53. #11924446C22semidirect
54. C2×Q16×C37C22semidirect
58. #106×C37C22semidirect
59. #103×C37C22semidirect
60. #103×C37C22semidirect
119. D296D4semidirect
214. #470D37semidirect
224. C37#3267semidirect
232. C2×Q16C2×D37semidirect
259. #106C2×D37semidirect
262. #103C2×D37semidirect
263. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C2×#119245592
#145581852
#145581742
#145579262
#145577632
#145577512
#145577372
#145576982
#145576932
#145576832
#145576812
#145576442
#145576022
C37×#4702
#326737