Finite Group Databasebeta

← #18006207

Subgroups — #18006207

Order 4736.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924562C211yesmaximal, characteristic, complemented
2C2×#11924560C211yesmaximal, characteristic, complemented
3#14558180C211yesmaximal, characteristic, complemented
4#14558177C211yesmaximal, characteristic, complemented
5#14557916C211yesmaximal, characteristic, complemented
6#14557778C211yesmaximal, characteristic, complemented
7#14557777C211yesmaximal, characteristic, complemented
8#14557747C211yesmaximal, characteristic, complemented
9#14557746C211yesmaximal, characteristic, complemented
10#14557697C211yesmaximal, characteristic, complemented
11#14557696C211yesmaximal, characteristic, complemented
12#14557665C211yesmaximal, characteristic, complemented
13#14557613C211yesmaximal, characteristic, complemented
14#14557608C211yesmaximal, characteristic, complemented
15C37×#447C211yesmaximal, characteristic, complemented
16C2×#10596962C2211yescharacteristic
17C2×#10596964C2211yescharacteristic
18C2×D4×D37C2211yescharacteristic, complemented
19#11924562—24no—
20#11924560—24no—
21C2×Dic148C2211yescharacteristic, complemented
22C2×#10596930C2211yescharacteristic, complemented
23C2×#10596940C2211yescharacteristic, complemented
24C2×#10596940C2211yescharacteristic, complemented
25C2×#10596940C2211yescharacteristic, complemented
26C2×#10596939C2211yescharacteristic
27C2×#10596939C2211yescharacteristic
28C2×#10596934C2211yescharacteristic, complemented
29C2×#10596929C2211yescharacteristic, complemented
30#11924523C2211yescharacteristic, complemented
31#11924523C2211yescharacteristic, complemented
32#11924518—21no—
33#11924517—21no—
34#11924516C2211yescharacteristic, complemented
35#11924515C2211yescharacteristic, complemented
36#11924508—21no—
37#11924507—21no—
38#11924507C2211yescharacteristic
39#11924505—21no—
40#11924504—21no—
41D4×Dic37C2211yescharacteristic, complemented
42#11924498C2211yescharacteristic, complemented
43C4×C22⋊D37C2211yescharacteristic
44C2×C22⋊Dic37—21no—
45C22⋊C4×D37—21no—
46#11924469C2211yescharacteristic, complemented
47#11924465C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924459C2211yescharacteristic, complemented
50#11924458C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53C8×Dic37C2211yescharacteristic, complemented
54C22×D4×C37C2211yescharacteristic, complemented
55C2×D8×C37C2211yescharacteristic
56C2×D8×C37C2211yescharacteristic
57#121×C37C2211yescharacteristic
58#102×C37C2211yescharacteristic, complemented
59#102×C37C2211yescharacteristic, complemented
60C22⋊C8×C37C2211yescharacteristic, complemented
61C24×C37—21no—
62C23×D37—21no—
63C2×D4×C37—22no—
64C2×D4×C37—21no—
65C2×D4×C37—21no—
66C2×D4×C37—41no—
67C2×D4×C37C2311yescharacteristic
68C2×D4×C37C2311yescharacteristic
69C2×D4×C37C2311yescharacteristic
70C22×C4×C37C2311yescharacteristic
71C2×C22⋊D37—21no—
72C2×C22⋊D37—21no—
73C2×C22⋊D37—21no—
74C2×C22⋊D37—21no—
75C2×C22⋊D37D411yescharacteristic
76C22×Dic37—21no—
77C22×Dic37—21no—
78#10596964—22no—
79#10596964—41no—
80D4×D37—22no—
81D4×D37—41no—
82#10596962—22no—
83#10596962—41no—
84C2×D148C2311yescharacteristic
85C2×C4×D37—21no—
86C2×C4×D37—21no—
87C2×C4×D37C2311yescharacteristic
88C2×Dic74C2311yescharacteristic
89C2×Dic74C2311yescharacteristic
90D8×C37—22no—
91D8×C37—41no—
92C2×C8×C37C2311yescharacteristic
93C2×C8×C37C2311yescharacteristic
94C4⋊C4×C37C2311yescharacteristic
95C22⋊C4×C37—21no—
96C22⋊Dic37—21no—
97C22⋊Dic37—21no—
98C22⋊Dic37—21no—
99C22⋊Dic37—21no—
100C22⋊Dic37—21no—
101C22⋊Dic37—21no—
102C22⋊Dic37—21no—
103C22⋊Dic37D411yescharacteristic
104#10596940—22no—
105#10596940—22no—
106#10596940—22no—
107#10596939—22no—
108#10596939D414yescomplemented
109#10596938—21no—
110#10596938—21no—
111#10596938D411yescharacteristic
112C4⋊Dic37—21no—
113C4⋊Dic37C2311yescharacteristic
114#10596936—21no—
115#10596936D411yescharacteristic
116C4×Dic37C2311yescharacteristic
117#10596934—22no—
118C2×#64436C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120Dic148—41no—
121#10596930—22no—
122#10596929—22no—
123C23×C37—21no—
124C23×C37—21no—
125C23×C37—21no—
126C23×C37—42no—
127C23×C37—42no—
128C23×C37—81no—
129C23×C37C2×D411yescharacteristic
130C22×D37—21no—
131C22×D37—42no—
132C22×D37—42no—
133C22×D37—81no—
134C22×D37C2×D411yescharacteristic
135D4×C37—21no—
136D4×C37—21no—
137D4×C37—21no—
138D4×C37—21no—
139D4×C37—21no—
140D4×C37—42no—
141D4×C37—81no—
142D4×C37C2×D412yes—
143C2×C4×C37—21no—
144C2×C4×C37—21no—
145C2×C4×C37—21no—
146C2×C4×C37C2×D411yescharacteristic
147C2×C4×C37C2411yescharacteristic
148C22⋊D37—41no—
149C22⋊D37—42no—
150C22⋊D37—42no—
151C22⋊D37—42no—
152C22⋊D37—81no—
153C2×Dic37—21no—
154C2×Dic37—21no—
155C2×Dic37—21no—
156C2×Dic37—42no—
157C2×Dic37—81no—
158C2×Dic37C2×D411yescharacteristic
159C2×Dic37C2×D411yescharacteristic
160C2×Dic37C2×D411yescharacteristic
161D148—21no—
162D148C2×D412yes—
163C4×D37—21no—
164C4×D37—21no—
165C4×D37—42no—
166C4×D37—81no—
167Dic74—21no—
168Dic74—21no—
169Dic74—21no—
170Dic74—21no—
171C8×C37—21no—
172C8×C37—21no—
173#64436—21no—
174#64436C2×D412yes—
175C22×C37—22no—
176C22×C37—21no—
177C22×C37—21no—
178C22×C37—21no—
179C22×C37—41no—
180C22×C37—41no—
181C22×C37—41no—
182C22×C37—41no—
183C22×C37—41no—
184C22×C37—41no—
185C22×C37—41no—
186C22×C37—41no—
187C22×C37—84no—
188C22×C37C22×D411yescharacteristic
189C2×D37—21no—
190C2×D37—22no—
191C2×D37—21no—
192C2×D37—21no—
193C2×D37—41no—
194C2×D37—41no—
195C2×D37—84no—
196C4×C37—21no—
197C4×C37—41no—
198C4×C37C22×D411yescharacteristic
199C4×C37H(5,𝔽2)11yescharacteristic
200Dic37—21no—
201Dic37—21no—
202Dic37—21no—
203Dic37—41no—
204Dic37—41no—
205Dic37—41no—
206#3273—371nomaximal, Sylow
207C2×C37—41no—
208C2×C37—42no—
209C2×C37—81no—
210C2×C37—81no—
211C2×C37D4211yescharacteristic
212C2×C37#57511yescharacteristic
213C2×C37#57611yescharacteristic
214D37—41no—
215D37—42no—
216C2×AGL(1,ℤ/8)—371no—
217C2×AGL(1,ℤ/8)—371no—
218#546—371no—
219#546—371no—
220#495—371no—
221#468—371no—
222#468—371no—
223#463—371no—
224#463—371no—
225#452—371no—
226#452—371no—
227#447—371no—
228#447D3711yescharacteristic, complemented
229C4×D8—371no—
230#436—371no—
231C37#327311yescharacteristic, complemented, Sylow
232C2×#41—371no—
233C2×#41—371no—
234C22×D4—371no—
235C22×D4C2×D3711yescharacteristic, complemented
236AGL(1,ℤ/8)—744no—
237AGL(1,ℤ/8)—744no—
238C2×Q16—371no—
239C2×SD16—371no—
240C2×SD16—371no—
241C2×SD16—371no—
242C2×SD16—371no—
243C2×D8—371no—
244C2×D8—371no—
245C2×D8C2×D3711yescharacteristic
246C2×D8C2×D3711yescharacteristic
247C2×M16—371no—
248C2×M16—371no—
249#124—371no—
250#124—371no—
251#123—741no—
252#123—741no—
253C22⋊Q8—371no—
254C22⋊Q8—371no—
255#121—371no—
256#121—741no—
257#121—741no—
258#121C2×D3711yescharacteristic
259#120—741no—
260#120—741no—
261C4×D4—371no—
262C4×D4—371no—
263C4×D4—371no—
264C2×C22⋊C4—741no—
265C2×C22⋊C4—741no—
266#107—371no—
267C4⋊C8—371no—
268#103—371no—
269#103—371no—
270#102—371no—
271#102—371no—
272#102C2×D3711yescharacteristic, complemented
273#102C2×D3711yescharacteristic, complemented
274C22⋊C8—371no—
275C22⋊C8C2×D3711yescharacteristic, complemented
276C4×C8—371no—
277C24—21no—
278C24—741no—
279#41—742no—
280#41—742no—
281#41—1481no—
282#41—1481no—
283C2×Q8—371no—
284C2×Q8—371no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—371no—
290C2×D4—371no—
291C2×D4—741no—
292C2×D4—742no—
293C2×D4—741no—
294C2×D4—741no—
295C2×D4—741no—
296C2×D4—1481no—
297C2×D4C22×D3711yescharacteristic
298C2×D4C22×D3711yescharacteristic
299C2×D4C22×D3711yescharacteristic
300C22×C4—371no—
301C22×C4—741no—
302C22×C4—741no—
303C22×C4—741no—
304C22×C4—741no—
305C22×C4C22×D3711yescharacteristic
306Q16—1481no—
307SD16—742no—
308SD16—742no—
309SD16—742no—
310SD16—742no—
311D8—22no—
312D8—41no—
313D8—374no—
314D8—742no—
315M16—742no—
316M16—742no—
317C2×C8—371no—
318C2×C8—371no—
319C2×C8C22×D3711yescharacteristic
320C2×C8C22×D3711yescharacteristic
321C4⋊C4—371no—
322C4⋊C4—371no—
323C4⋊C4—741no—
324C4⋊C4—741no—
325C4⋊C4C22×D3711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—371no—
328C22⋊C4—371no—
329C22⋊C4—741no—
330C22⋊C4—741no—
331C22⋊C4—741no—
332C22⋊C4—741no—
333C22⋊C4—741no—
334C22⋊C4—741no—
335C22⋊C4—741no—
336C22⋊C4—741no—
337C22⋊C4—741no—
338C42—371no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—371no—
346C23—2961no—
347C23—741no—
348C23—1482no—
349C23—1482no—
350C23D4×D3711yescharacteristic
351Q8—741no—
352Q8—741no—
353Q8—741no—
354Q8—741no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—21no—
360D4—42no—
361D4—81no—
362D4—372no—
363D4—2961no—
364D4—741no—
365D4—1481no—
366D4—1482no—
367D4—1482no—
368D4—1482no—
369D4D4×D3712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—371no—
374C2×C4—371no—
375C2×C4—371no—
376C2×C4—2961no—
377C2×C4—2961no—
378C2×C4—741no—
379C2×C4—741no—
380C2×C4—741no—
381C2×C4—741no—
382C2×C4—741no—
383C2×C4—1482no—
384C2×C4—1482no—
385C2×C4D4×D3711yescharacteristic
386C2×C4C23×D3711yescharacteristic
387C8—21no—
388C8—21no—
389C8—372no—
390C8—741no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—2964no—
405C22—741no—
406C22—741no—
407C22—741no—
408C22—742no—
409C22—1481no—
410C22—1481no—
411C22C2×D4×D3711yescharacteristic
412C4—21no—
413C4—41no—
414C4—741no—
415C4—741no—
416C4—741no—
417C4—1481no—
418C4—1481no—
419C4—1481no—
420C4C2×D4×D3711yescharacteristic
421C4#1192458111yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—81no—
426C2—1481no—
427C2—1482no—
428C2#1455816511yescharacteristic
429C2#1455832911yescharacteristic
430C2#1455833411yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#447C2
socleC22×C37C22×D4
hypercenter#447D37
hypocenterC37#3273
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#11924562C2semidirect
2. C2×#11924560C2semidirect
3. #14558180C2semidirect
4. #14558177C2semidirect
5. #14557916C2semidirect
6. #14557778C2semidirect
7. #14557777C2semidirect
8. #14557747C2semidirect
9. #14557746C2semidirect
10. #14557697C2semidirect
11. #14557696C2semidirect
12. #14557665C2semidirect
13. #14557613C2semidirect
14. #14557608C2semidirect
15. C37×#447C2semidirect
18. C2×D4×D37C22semidirect
21. C2×Dic148C22semidirect
22. C2×#10596930C22semidirect
23. C2×#10596940C22semidirect
24. C2×#10596940C22semidirect
25. C2×#10596940C22semidirect
28. C2×#10596934C22semidirect
29. C2×#10596929C22semidirect
30. #11924523C22semidirect
31. #11924523C22semidirect
34. #11924516C22semidirect
35. #11924515C22semidirect
41. D4×Dic37C22semidirect
42. #11924498C22semidirect
46. #11924469C22semidirect
47. #11924465C22semidirect
48. #11924461C22semidirect
49. #11924459C22semidirect
50. #11924458C22semidirect
51. #11924457C22semidirect
52. #11924448C22semidirect
53. C8×Dic37C22semidirect
54. C22×D4×C37C22semidirect
58. #102×C37C22semidirect
59. #102×C37C22semidirect
60. C22⋊C8×C37C22semidirect
108. #10596939D4semidirect
228. #447D37semidirect
231. C37#3273semidirect
235. C22×D4C2×D37semidirect
272. #102C2×D37semidirect
273. #102C2×D37semidirect
275. C22⋊C8C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245622
C2×#119245602
#145581802
#145581772
#145579162
#145577782
#145577772
#145577472
#145577462
#145576972
#145576962
#145576652
#145576132
#145576082
C37×#4472
#327337