Finite Group Databasebeta

← #18006090

Subgroups — #18006090

Order 4736.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924560C211yesmaximal, characteristic, complemented
2C2×SD16×D37C211yesmaximal, characteristic, complemented
3#14558191C211yesmaximal, characteristic, complemented
4#14558168C211yesmaximal, characteristic, complemented
5#14557776C211yesmaximal, characteristic, complemented
6#14557766C211yesmaximal, characteristic, complemented
7#14557745C211yesmaximal, characteristic, complemented
8#14557730C211yesmaximal, characteristic, complemented
9#14557682C211yesmaximal, characteristic, complemented
10#14557675C211yesmaximal, characteristic, complemented
11#14557670C211yesmaximal, characteristic, complemented
12#14557665C211yesmaximal, characteristic, complemented
13#14557635C211yesmaximal, characteristic, complemented
14#14557606C211yesmaximal, characteristic, complemented
15C37×#496C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic, complemented
17C2×Q8×D37C2211yescharacteristic, complemented
18C2×D4×D37—21no—
19C2×D4×D37C2211yescharacteristic
20C2×D4×D37C2211yescharacteristic
21#11924560—24no—
22C2×#10596930C2211yescharacteristic
23C2×#10596940C2211yescharacteristic, complemented
24C2×#10596940C2211yescharacteristic, complemented
25C2×#10596941C2211yescharacteristic
26SD16×D37—24no—
27C2×#10596939C2211yescharacteristic
28C2×#10596929C2211yescharacteristic, complemented
29C2×C8×D37C2211yescharacteristic, complemented
30#11924522—21no—
31#11924513—21no—
32#11924512C2211yescharacteristic, complemented
33#11924509—21no—
34#11924507—21no—
35#11924507C2211yescharacteristic
36#11924507C2211yescharacteristic
37#11924507C2211yescharacteristic, complemented
38#11924504—21no—
39#11924504—21no—
40C4×Dic74C2211yescharacteristic, complemented
41C4×D148C2211yescharacteristic
42C4×D148C2211yescharacteristic
43#11924496—21no—
44#11924470C2211yescharacteristic, complemented
45C4⋊#64436C2211yescharacteristic, complemented
46#11924461C2211yescharacteristic, complemented
47#11924460C2211yescharacteristic, complemented
48#11924458C2211yescharacteristic
49#11924457C2211yescharacteristic, complemented
50#11924457C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic, complemented
52#11924448C2211yescharacteristic, complemented
53#11924448C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic, complemented
55C2×SD16×C37C2211yescharacteristic, complemented
56C2×D8×C37C2211yescharacteristic, complemented
57C2×D8×C37C2211yescharacteristic, complemented
58#127×C37C2211yescharacteristic
59#124×C37C2211yescharacteristic, complemented
60#97×C37C2211yescharacteristic, complemented
61C23×D37—21no—
62C23×D37—21no—
63C2×Q8×C37C2311yescharacteristic
64C2×D4×C37—21no—
65C2×D4×C37—21no—
66C2×D4×C37C2311yescharacteristic
67C2×D4×C37C2311yescharacteristic
68C2×D4×C37C2311yescharacteristic
69C2×C22⋊D37—21no—
70C2×C22⋊D37—21no—
71C2×C22⋊D37—21no—
72C2×C22⋊D37—21no—
73C2×C22⋊D37—21no—
74C2×C22⋊D37—21no—
75C22×Dic37—21no—
76Q8×D37—22no—
77Q8×D37—41no—
78#10596964—22no—
79#10596964—41no—
80D4×D37—22no—
81D4×D37—22no—
82D4×D37—41no—
83D4×D37—41no—
84D4×D37—44no—
85C2×D148D411yescharacteristic
86C2×D148C2311yescharacteristic
87C2×C4×D37—21no—
88C2×C4×D37—21no—
89C2×C4×D37C2311yescharacteristic
90C2×C4×D37C2311yescharacteristic
91C2×Dic74—21no—
92C2×Dic74C2311yescharacteristic
93SD16×C37—22no—
94SD16×C37—41no—
95D8×C37—22no—
96D8×C37—41no—
97C2×C8×C37C2311yescharacteristic
98C2×C8×C37C2311yescharacteristic
99C22⋊C4×C37—21no—
100C22⋊C4×C37—21no—
101C42×C37C2311yescharacteristic
102C22⋊Dic37—21no—
103C22⋊Dic37—21no—
104C22⋊Dic37—21no—
105#10596941—22no—
106#10596940—22no—
107#10596940—22no—
108#10596939—22no—
109#10596938—21no—
110#10596938—21no—
111#10596938D411yescharacteristic
112#10596938D411yescharacteristic
113C4⋊Dic37D411yescharacteristic
114C4⋊Dic37C2311yescharacteristic
115C4⋊Dic37C2311yescharacteristic
116#10596936—21no—
117#10596936—21no—
118C4×Dic37—21no—
119C2×#64436C2311yescharacteristic
120C2×#64436C2311yescharacteristic
121#10596930D414yescomplemented
122#10596929—22no—
123C8×D37—22no—
124C23×C37—21no—
125C23×C37—21no—
126C23×C37—21no—
127C22×D37—21no—
128C22×D37—42no—
129C22×D37—42no—
130C22×D37—42no—
131C22×D37—42no—
132C22×D37—81no—
133C22×D37—81no—
134C22×D37C2×D411yescharacteristic
135C22×D37C2×D411yescharacteristic
136Q8×C37—21no—
137Q8×C37—21no—
138D4×C37—21no—
139D4×C37—21no—
140D4×C37—21no—
141D4×C37—21no—
142D4×C37—21no—
143D4×C37—21no—
144D4×C37—42no—
145D4×C37—81no—
146C2×C4×C37—21no—
147C2×C4×C37C2×D411yescharacteristic
148C2×C4×C37C2×D411yescharacteristic
149C2×C4×C37C2411yescharacteristic
150C22⋊D37—42no—
151C22⋊D37—42no—
152C22⋊D37—42no—
153C22⋊D37—42no—
154C22⋊D37—42no—
155C22⋊D37—42no—
156C2×Dic37—21no—
157C2×Dic37—21no—
158C2×Dic37—21no—
159C2×Dic37—42no—
160C2×Dic37C2×D411yescharacteristic
161C2×Dic37C2×D411yescharacteristic
162D148—22no—
163D148—21no—
164D148C2×D412yes—
165C4×D37—21no—
166C4×D37—21no—
167C4×D37—21no—
168C4×D37—21no—
169C4×D37—42no—
170C4×D37—42no—
171Dic74—21no—
172Dic74—42no—
173Dic74C2×D412yescomplemented
174C8×C37—21no—
175C8×C37C2×D412yescomplemented
176#64436—21no—
177#64436—21no—
178C22×C37—41no—
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×C37C22×D411yescharacteristic
188C2×D37—21no—
189C2×D37—21no—
190C2×D37—21no—
191C2×D37—22no—
192C2×D37—21no—
193C2×D37—21no—
194C2×D37—41no—
195C2×D37—41no—
196C2×D37—84no—
197C2×D37—84no—
198C2×D37AGL(1,ℤ/8)12yescomplemented
199C4×C37—21no—
200C4×C37—21no—
201C4×C37—41no—
202C4×C37C22×D411yescharacteristic
203C4×C37H(5,𝔽2)11yescharacteristic
204Dic37—22no—
205Dic37—21no—
206Dic37—21no—
207Dic37—41no—
208Dic37—41no—
209#3265—371nomaximal, Sylow
210C2×C37—81no—
211C2×C37—81no—
212C2×C37—81no—
213C2×C37D4211yescharacteristic
214C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C37#57711yescharacteristic
216D37—22no—
217D37—41no—
218D37—42no—
219C2×AGL(1,ℤ/8)—371no—
220C22×SD16—371no—
221#548—371no—
222D42—371no—
223#496D3711yescharacteristic, complemented
224#468—371no—
225#466—371no—
226#463—371no—
227#460—371no—
228#450—371no—
229#449—371no—
230#448—371no—
231#447—371no—
232#440—371no—
233#435—371no—
234C37#326511yescharacteristic, complemented, Sylow
235C2×#41—371no—
236C22×Q8—371no—
237C22×D4—371no—
238C22×D4—371no—
239C22×D4—741no—
240AGL(1,ℤ/8)—744no—
241C2×SD16—371no—
242C2×SD16—371no—
243C2×SD16—371no—
244C2×SD16—371no—
245C2×SD16—744no—
246C2×SD16C2×D3711yescharacteristic, complemented
247C2×SD16C2×D3711yescharacteristic, complemented
248C2×D8—371no—
249C2×D8C2×D3711yescharacteristic, complemented
250C2×D8C2×D3711yescharacteristic, complemented
251C2×M16—371no—
252C22×C8—371no—
253#127C2×D3711yescharacteristic
254#124—741no—
255#124C2×D3711yescharacteristic, complemented
256C22⋊Q8—371no—
257C22⋊Q8—741no—
258#121—371no—
259#121—371no—
260#121—371no—
261#121—741no—
262#121—741no—
263#120—741no—
264#120—741no—
265C4×Q8—371no—
266C4×D4—371no—
267C4×D4—371no—
268C4×D4—741no—
269#107—371no—
270C4⋊C8—371no—
271#103—371no—
272#103—371no—
273#102—371no—
274#102—371no—
275#102—371no—
276#102—371no—
277C22⋊C8—371no—
278C22⋊C8—371no—
279#97C2×D3711yescharacteristic, complemented
280C24—741no—
281C24—741no—
282#41—742no—
283#41—1481no—
284C2×Q8—371no—
285C2×Q8—741no—
286C2×Q8—742no—
287C2×Q8—1481no—
288C2×Q8C22×D3711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—371no—
292C2×D4—371no—
293C2×D4—741no—
294C2×D4—741no—
295C2×D4—742no—
296C2×D4—742no—
297C2×D4—741no—
298C2×D4—741no—
299C2×D4—741no—
300C2×D4—741no—
301C2×D4—1481no—
302C2×D4—1481no—
303C2×D4—1484no—
304C2×D4C22×D3711yescharacteristic
305C2×D4C22×D3711yescharacteristic
306C2×D4C22×D3711yescharacteristic
307C22×C4—371no—
308C22×C4—371no—
309C22×C4—741no—
310C22×C4—741no—
311C22×C4—741no—
312SD16—22no—
313SD16—41no—
314SD16—374no—
315SD16—742no—
316SD16—742no—
317SD16—742no—
318D8—22no—
319D8—41no—
320D8—742no—
321M16—742no—
322C2×C8—371no—
323C2×C8—371no—
324C2×C8—742no—
325C2×C8C22×D3711yescharacteristic
326C2×C8C22×D3711yescharacteristic
327C4⋊C4—371no—
328C4⋊C4—371no—
329C4⋊C4—371no—
330C4⋊C4—741no—
331C4⋊C4—741no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—371no—
335C22⋊C4—371no—
336C22⋊C4—741no—
337C22⋊C4—741no—
338C22⋊C4—741no—
339C22⋊C4—741no—
340C22⋊C4—741no—
341C42—741no—
342C42C22×D3711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—371no—
347C23—371no—
348C23—2961no—
349C23—2961no—
350C23—741no—
351C23—1482no—
352C23—1482no—
353C23—1482no—
354C23—1482no—
355Q8—21no—
356Q8—21no—
357Q8—372no—
358Q8—741no—
359Q8—1482no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—372no—
369D4—742no—
370D4—741no—
371D4—1482no—
372D4—1482no—
373D4—1482no—
374D4—1482no—
375D4—1482no—
376D4—1482no—
377C2×C4—21no—
378C2×C4—371no—
379C2×C4—371no—
380C2×C4—741no—
381C2×C4—741no—
382C2×C4—741no—
383C2×C4—741no—
384C2×C4—741no—
385C2×C4—741no—
386C2×C4—741no—
387C2×C4—1482no—
388C2×C4—1482no—
389C2×C4—1482no—
390C2×C4D4×D3711yescharacteristic
391C2×C4D4×D3711yescharacteristic
392C2×C4C23×D3711yescharacteristic
393C8—21no—
394C8—741no—
395C8—741no—
396C8D4×D3712yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—372no—
407C22—2964no—
408C22—2964no—
409C22—741no—
410C22—741no—
411C22—741no—
412C22—741no—
413C22—741no—
414C22—742no—
415C22—1481no—
416C22—1481no—
417C22C2×D4×D3711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—741no—
422C4—741no—
423C4—742no—
424C4—1481no—
425C4—1481no—
426C4C2×D4×D3711yescharacteristic
427C4#1192458111yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—742no—
432C2—1481no—
433C2—1482no—
434C2#1455816811yescharacteristic
435C2AGL(1,ℤ/8)×D3711yescharacteristic
436C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#496C2
socleC22×C37C22×D4
hypercenter#496D37
hypocenterC37#3265
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#11924560C2semidirect
2. C2×SD16×D37C2semidirect
3. #14558191C2semidirect
4. #14558168C2semidirect
5. #14557776C2semidirect
6. #14557766C2semidirect
7. #14557745C2semidirect
8. #14557730C2semidirect
9. #14557682C2semidirect
10. #14557675C2semidirect
11. #14557670C2semidirect
12. #14557665C2semidirect
13. #14557635C2semidirect
14. #14557606C2semidirect
15. C37×#496C2semidirect
16. C2×#10596964C22semidirect
17. C2×Q8×D37C22semidirect
23. C2×#10596940C22semidirect
24. C2×#10596940C22semidirect
28. C2×#10596929C22semidirect
29. C2×C8×D37C22semidirect
32. #11924512C22semidirect
37. #11924507C22semidirect
40. C4×Dic74C22semidirect
44. #11924470C22semidirect
45. C4⋊#64436C22semidirect
46. #11924461C22semidirect
47. #11924460C22semidirect
49. #11924457C22semidirect
50. #11924457C22semidirect
51. #11924457C22semidirect
52. #11924448C22semidirect
53. #11924448C22semidirect
54. C2×SD16×C37C22semidirect
55. C2×SD16×C37C22semidirect
56. C2×D8×C37C22semidirect
57. C2×D8×C37C22semidirect
59. #124×C37C22semidirect
60. #97×C37C22semidirect
121. #10596930D4semidirect
173. Dic74C2×D4semidirect
175. C8×C37C2×D4semidirect
198. C2×D37AGL(1,ℤ/8)semidirect
223. #496D37semidirect
234. C37#3265semidirect
246. C2×SD16C2×D37semidirect
247. C2×SD16C2×D37semidirect
249. C2×D8C2×D37semidirect
250. C2×D8C2×D37semidirect
255. #124C2×D37semidirect
279. #97C2×D37semidirect
396. C8D4×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245602
C2×SD16×D372
#145581912
#145581682
#145577762
#145577662
#145577452
#145577302
#145576822
#145576752
#145576702
#145576652
#145576352
#145576062
C37×#4962
#326537