Finite Group Databasebeta

← #18006201

Subgroups — #18006201

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×#11924560C211yesmaximal, characteristic, complemented
2#14558177C211yesmaximal, characteristic, complemented
3#14558171C211yesmaximal, characteristic, complemented
4#14557914C211yesmaximal, characteristic, complemented
5#14557779C211yesmaximal, characteristic, complemented
6#14557776C211yesmaximal, characteristic, complemented
7#14557750C211yesmaximal, characteristic, complemented
8#14557746C211yesmaximal, characteristic, complemented
9#14557695C211yesmaximal, characteristic, complemented
10#14557692C211yesmaximal, characteristic, complemented
11#14557667C211yesmaximal, characteristic, complemented
12#14557665C211yesmaximal, characteristic, complemented
13D8×Dic37C211yesmaximal, characteristic, complemented
14#14557609C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic
17C2×D4×D37C2211yescharacteristic, complemented
18#11924560—24no—
19C2×#10596942C2211yescharacteristic, complemented
20C2×#10596930C2211yescharacteristic, complemented
21C2×#10596940C2211yescharacteristic, complemented
22C2×#10596939C2211yescharacteristic
23C2×#10596939C2211yescharacteristic
24C2×#10596929C2211yescharacteristic, complemented
25#11924525C2211yescharacteristic, complemented
26#11924523C2211yescharacteristic, complemented
27#11924519—21no—
28#11924517—21no—
29#11924515C2211yescharacteristic, complemented
30#11924512C2211yescharacteristic, complemented
31#11924510C2211yescharacteristic
32#11924508—21no—
33#11924508—21no—
34#11924507C2211yescharacteristic
35#11924505—21no—
36#11924504—21no—
37D4×Dic37C2211yescharacteristic, complemented
38D4×Dic37C2211yescharacteristic, complemented
39C4×C22⋊D37C2211yescharacteristic
40C2×C22⋊Dic37—21no—
41C2×#10596938—21no—
42#11924470C2211yescharacteristic, complemented
43#11924465C2211yescharacteristic, complemented
44#11924461C2211yescharacteristic, complemented
45#11924460C2211yescharacteristic, complemented
46#11924458C2211yescharacteristic, complemented
47#11924457C2211yescharacteristic, complemented
48#11924457C2211yescharacteristic, complemented
49#11924457C2211yescharacteristic, complemented
50C22⋊#64436C2211yescharacteristic, complemented
51#11924448C2211yescharacteristic, complemented
52C8×Dic37C2211yescharacteristic, complemented
53C2×#41×C37C2211yescharacteristic
54C22×D4×C37C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C37—24no—
56C2×SD16×C37C2211yescharacteristic, complemented
57C2×SD16×C37C2211yescharacteristic, complemented
58C2×D8×C37C2211yescharacteristic
59C2×D8×C37C2211yescharacteristic
60C2×M16×C37C2211yescharacteristic, complemented
61C24×C37—21no—
62C23×D37—21no—
63#41×C37—22no—
64#41×C37—41no—
65C2×Q8×C37C2311yescharacteristic
66C2×D4×C37—22no—
67C2×D4×C37—21no—
68C2×D4×C37—21no—
69C2×D4×C37—41no—
70C2×D4×C37C2311yescharacteristic
71C2×D4×C37C2311yescharacteristic
72C2×D4×C37C2311yescharacteristic
73C22×C4×C37—21no—
74C22×C4×C37C2311yescharacteristic
75C2×C22⋊D37—21no—
76C2×C22⋊D37—21no—
77C2×C22⋊D37—21no—
78C2×C22⋊D37—21no—
79C2×C22⋊D37D411yescharacteristic
80C22×Dic37—21no—
81C22×Dic37—21no—
82C22×Dic37—21no—
83#10596964—22no—
84#10596964—41no—
85D4×D37—22no—
86D4×D37—41no—
87C2×D148C2311yescharacteristic
88C2×C4×D37C2311yescharacteristic
89C2×Dic74C2311yescharacteristic
90SD16×C37—22no—
91SD16×C37—22no—
92D8×C37—22no—
93D8×C37D414yescomplemented
94M16×C37—22no—
95C2×C8×C37C2311yescharacteristic
96C2×C8×C37C2311yescharacteristic
97C22⋊Dic37—21no—
98C22⋊Dic37—21no—
99C22⋊Dic37—21no—
100C22⋊Dic37—21no—
101C22⋊Dic37—21no—
102C22⋊Dic37—21no—
103C22⋊Dic37D411yescharacteristic
104#10596942—41no—
105#10596940—22no—
106#10596939—22no—
107#10596939—41no—
108#10596938—21no—
109#10596938—21no—
110#10596938—21no—
111#10596938—21no—
112#10596938D411yescharacteristic
113C4⋊Dic37C2311yescharacteristic
114C4⋊Dic37C2311yescharacteristic
115#10596936—21no—
116#10596936—21no—
117#10596936D411yescharacteristic
118C4×Dic37C2311yescharacteristic
119C2×#64436C2311yescharacteristic
120C2×#64436C2311yescharacteristic
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
135Q8×C37—21no—
136Q8×C37—21no—
137D4×C37—21no—
138D4×C37—21no—
139D4×C37—21no—
140D4×C37—21no—
141D4×C37—42no—
142D4×C37—42no—
143D4×C37C2×D412yes—
144D4×C37C2×D412yes—
145C2×C4×C37—21no—
146C2×C4×C37—21no—
147C2×C4×C37—21no—
148C2×C4×C37—42no—
149C2×C4×C37C2×D411yescharacteristic
150C2×C4×C37C2411yescharacteristic
151C22⋊D37—41no—
152C22⋊D37—42no—
153C22⋊D37—42no—
154C22⋊D37—81no—
155C22⋊D37—81no—
156C2×Dic37—21no—
157C2×Dic37—21no—
158C2×Dic37—21no—
159C2×Dic37—42no—
160C2×Dic37—81no—
161C2×Dic37—81no—
162C2×Dic37C2×D411yescharacteristic
163C2×Dic37C2×D411yescharacteristic
164C2×Dic37C2×D411yescharacteristic
165D148—21no—
166D148—21no—
167C4×D37—21no—
168C4×D37—21no—
169Dic74—21no—
170Dic74—21no—
171C8×C37—21no—
172C8×C37C2×D412yes—
173#64436—21no—
174#64436—21no—
175C22×C37—22no—
176C22×C37—22no—
177C22×C37—21no—
178C22×C37—21no—
179C22×C37—21no—
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—21no—
191C2×D37—21no—
192C2×D37—41no—
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—42no—
208C2×C37—41no—
209C2×C37—42no—
210C2×C37—81no—
211C2×C37D4211yescharacteristic
212C2×C37#57511yescharacteristic
213C2×C37#57611yescharacteristic
214D37—41no—
215D37—81no—
216C2×AGL(1,ℤ/8)—371no—
217C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
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#447—371no—
229C4×D8—371no—
230#436—371no—
231C37#327311yescharacteristic, complemented, Sylow
232C2×#41—371no—
233C2×#41C2×D3711yescharacteristic
234C22×D4—371no—
235C22×D4C2×D3711yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—744no—
238C2×Q16—371no—
239C2×SD16—371no—
240C2×SD16—371no—
241C2×SD16C2×D3711yescharacteristic, complemented
242C2×SD16C2×D3711yescharacteristic, complemented
243C2×D8—371no—
244C2×D8—371no—
245C2×D8C2×D3711yescharacteristic
246C2×D8C2×D3711yescharacteristic
247C2×M16—371no—
248C2×M16C2×D3711yescharacteristic, complemented
249#124—371no—
250#124—371no—
251#123—741no—
252#123—741no—
253C22⋊Q8—371no—
254C22⋊Q8—371no—
255#121—371no—
256#121—371no—
257#121—741no—
258#121—741no—
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#102—371no—
273#102—371no—
274C22⋊C8—371no—
275C22⋊C8—371no—
276C4×C8—371no—
277C24—21no—
278C24—741no—
279#41—22no—
280#41—41no—
281#41—742no—
282#41—1481no—
283C2×Q8—371no—
284C2×Q8C22×D3711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—371no—
290C2×D4—371no—
291C2×D4—741no—
292C2×D4—741no—
293C2×D4—741no—
294C2×D4—742no—
295C2×D4—741no—
296C2×D4—1481no—
297C2×D4C22×D3711yescharacteristic
298C2×D4C22×D3711yescharacteristic
299C2×D4C22×D3711yescharacteristic
300C22×C4—21no—
301C22×C4—371no—
302C22×C4—741no—
303C22×C4—741no—
304C22×C4—741no—
305C22×C4C22×D3711yescharacteristic
306Q16—1481no—
307SD16—22no—
308SD16—22no—
309SD16—742no—
310SD16—742no—
311D8—22no—
312D8—742no—
313D8—1481no—
314D8C22⋊D3714yescomplemented
315M16—22no—
316M16—742no—
317C2×C8—371no—
318C2×C8—371no—
319C2×C8C22×D3711yescharacteristic
320C2×C8C22×D3711yescharacteristic
321C4⋊C4—371no—
322C4⋊C4—371no—
323C4⋊C4—371no—
324C4⋊C4—741no—
325C4⋊C4—741no—
326C22⋊C4—371no—
327C22⋊C4—371no—
328C22⋊C4—741no—
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—21no—
352Q8—21no—
353Q8—741no—
354Q8—741no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—2961no—
362D4—2961no—
363D4—741no—
364D4—741no—
365D4—1481no—
366D4—1482no—
367D4—1482no—
368D4C2×C22⋊D3712yes—
369D4C2×C22⋊D3712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—371no—
375C2×C4—371no—
376C2×C4—371no—
377C2×C4—2961no—
378C2×C4—2961no—
379C2×C4—741no—
380C2×C4—741no—
381C2×C4—741no—
382C2×C4—741no—
383C2×C4—741no—
384C2×C4—1482no—
385C2×C4D4×D3711yescharacteristic
386C2×C4C23×D3711yescharacteristic
387C8—21no—
388C8—741no—
389C8—741no—
390C8C2×C22⋊D3712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
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—1481no—
409C22—1481no—
410C22—1481no—
411C22C2×D4×D3711yescharacteristic
412C4—21no—
413C4—41no—
414C4—741no—
415C4—741no—
416C4—741no—
417C4—1481no—
418C4—1481no—
419C4—1481no—
420C4C22×C22⋊D3711yescharacteristic
421C4#1192458111yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—2961no—
427C2—1481no—
428C2D4×C22⋊D3711yescharacteristic
429C2#1455832911yescharacteristic
430C2#1455833711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37C22×D4
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3273
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11924560C2semidirect
2. #14558177C2semidirect
3. #14558171C2semidirect
4. #14557914C2semidirect
5. #14557779C2semidirect
6. #14557776C2semidirect
7. #14557750C2semidirect
8. #14557746C2semidirect
9. #14557695C2semidirect
10. #14557692C2semidirect
11. #14557667C2semidirect
12. #14557665C2semidirect
13. D8×Dic37C2semidirect
14. #14557609C2semidirect
15. C2×AGL(1,ℤ/8)×C37C2semidirect
17. C2×D4×D37C22semidirect
19. C2×#10596942C22semidirect
20. C2×#10596930C22semidirect
21. C2×#10596940C22semidirect
24. C2×#10596929C22semidirect
25. #11924525C22semidirect
26. #11924523C22semidirect
29. #11924515C22semidirect
30. #11924512C22semidirect
37. D4×Dic37C22semidirect
38. D4×Dic37C22semidirect
42. #11924470C22semidirect
43. #11924465C22semidirect
44. #11924461C22semidirect
45. #11924460C22semidirect
46. #11924458C22semidirect
47. #11924457C22semidirect
48. #11924457C22semidirect
49. #11924457C22semidirect
50. C22⋊#64436C22semidirect
51. #11924448C22semidirect
52. C8×Dic37C22semidirect
54. C22×D4×C37C22semidirect
56. C2×SD16×C37C22semidirect
57. C2×SD16×C37C22semidirect
60. C2×M16×C37C22semidirect
93. D8×C37D4semidirect
217. C2×AGL(1,ℤ/8)D37semidirect
231. C37#3273semidirect
235. C22×D4C2×D37semidirect
241. C2×SD16C2×D37semidirect
242. C2×SD16C2×D37semidirect
248. C2×M16C2×D37semidirect
314. D8C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245602
#145581772
#145581712
#145579142
#145577792
#145577762
#145577502
#145577462
#145576952
#145576922
#145576672
#145576652
D8×Dic372
#145576092
C2×AGL(1,ℤ/8)×C372
#327337