Finite Group Databasebeta

← #18004269

Subgroups — #18004269

Order 4736.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924563C211yesmaximal, characteristic, complemented
2C2×#11924542C211yesmaximal, characteristic, complemented
3#14558113C211yesmaximal, characteristic, complemented
4#14558099C211yesmaximal, characteristic, complemented
5#14557787C211yesmaximal, characteristic, complemented
6#14557784C211yesmaximal, characteristic, complemented
7#14557781C211yesmaximal, characteristic, complemented
8#14557776C211yesmaximal, characteristic, complemented
9#14557769C211yesmaximal, characteristic, complemented
10#14557768C211yesmaximal, characteristic, complemented
11#14557765C211yesmaximal, characteristic, complemented
12#14557572C211yesmaximal, characteristic, complemented
13#14557528C211yesmaximal, characteristic, complemented
14#102×D37C211yesmaximal, characteristic, complemented
15C37×#465C211yesmaximal, characteristic, complemented
16C2×#10596966C2211yescharacteristic
17C2×#10596962C2211yescharacteristic
18C2×D4×D37C2211yescharacteristic
19#11924563—24no—
20C2×#10596941C2211yescharacteristic
21C2×D296C2211yescharacteristic
22C2×#10596939C2211yescharacteristic
23#11924542—24no—
24C2×#10596934C2211yescharacteristic, complemented
25C2×#10596929C2211yescharacteristic
26C2×#10596929C2211yescharacteristic, complemented
27C2×C8×D37C2211yescharacteristic
28C2×C8×D37C2211yescharacteristic, complemented
29#11924532—21no—
30#11924522—21no—
31#11924514—21no—
32#11924512C2211yescharacteristic, complemented
33#11924511—21no—
34#11924511C2211yescharacteristic
35#11924511C2211yescharacteristic, complemented
36#11924510C2211yescharacteristic
37#11924510C2211yescharacteristic, complemented
38#11924509—21no—
39#11924509—21no—
40#11924507C2211yescharacteristic
41#11924506—21no—
42#11924504—21no—
43#11924494C2211yescharacteristic
44C4⋊C4×D37C2211yescharacteristic
45#11924470C2211yescharacteristic, complemented
46#11924469C2211yescharacteristic, complemented
47#11924467C2211yescharacteristic, complemented
48#11924460C2211yescharacteristic, complemented
49#11924458C2211yescharacteristic
50#11924458C2211yescharacteristic, complemented
51#11924457C2211yescharacteristic
52#11924456C2211yescharacteristic
53#11924456C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic
55C22×C8×C37C2211yescharacteristic, complemented
56C22⋊Q8×C37C2211yescharacteristic, complemented
57#121×C37C2211yescharacteristic
58#106×C37C2211yescharacteristic, complemented
59#103×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic
61C23×D37—21no—
62C2×Q8×C37C2311yescharacteristic
63C2×D4×C37—21no—
64C2×D4×C37C2311yescharacteristic
65C22×C4×C37C2311yescharacteristic
66C2×C22⋊D37—21no—
67C2×C22⋊D37—21no—
68C2×C22⋊D37—21no—
69C2×C22⋊D37—21no—
70C2×C22⋊D37D411yescharacteristic
71C2×C22⋊D37D411yescharacteristic
72#10596966—22no—
73#10596966—22no—
74D4×D37—22no—
75D4×D37—22no—
76#10596962—22no—
77#10596962—22no—
78C2×D148—21no—
79C2×D148—21no—
80C2×D148—21no—
81C2×D148D411yescharacteristic
82C2×D148C2311yescharacteristic
83C2×D148C2311yescharacteristic
84C2×C4×D37—21no—
85C2×C4×D37—21no—
86C2×C4×D37C2311yescharacteristic
87C2×C4×D37C2311yescharacteristic
88C2×Dic74D411yescharacteristic
89SD16×C37—22no—
90C2×C8×C37—22no—
91C2×C8×C37C2311yescharacteristic
92C2×C8×C37C2311yescharacteristic
93C4⋊C4×C37—21no—
94C4⋊C4×C37C2311yescharacteristic
95C4⋊C4×C37C2311yescharacteristic
96C22⋊C4×C37—21no—
97C22⋊C4×C37—21no—
98C22⋊Dic37—21no—
99#10596941—22no—
100#10596939—22no—
101#10596938—21no—
102#10596938—21no—
103#10596938—21no—
104#10596938—21no—
105#10596938—21no—
106#10596938—21no—
107C4⋊Dic37C2311yescharacteristic
108C4⋊Dic37C2311yescharacteristic
109#10596936—21no—
110#10596936—21no—
111C4×Dic37—21no—
112#10596934—22no—
113C2×#64436C2311yescharacteristic
114C2×#64436C2311yescharacteristic
115D296—22no—
116#10596929—22no—
117#10596929D414yescomplemented
118C8×D37—22no—
119C8×D37—22no—
120C23×C37—21no—
121C23×C37C2×D411yescharacteristic
122C22×D37—21no—
123C22×D37—21no—
124C22×D37—42no—
125C22×D37—81no—
126C22×D37—81no—
127C22×D37C2×D411yescharacteristic
128C22×D37C2×D411yescharacteristic
129Q8×C37—21no—
130Q8×C37—21no—
131D4×C37—21no—
132D4×C37—21no—
133D4×C37—81no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37C2×D411yescharacteristic
140C2×C4×C37C2411yescharacteristic
141C22⋊D37—22no—
142C22⋊D37—22no—
143C22⋊D37—42no—
144C22⋊D37—81no—
145C22⋊D37—81no—
146C22⋊D37—81no—
147C2×Dic37—21no—
148C2×Dic37—21no—
149C2×Dic37C2×D411yescharacteristic
150C2×Dic37C2×D411yescharacteristic
151D148—22no—
152D148—21no—
153D148—21no—
154D148—21no—
155D148—21no—
156D148—42no—
157D148—81no—
158D148—81no—
159C4×D37—21no—
160C4×D37—21no—
161C4×D37—42no—
162C4×D37—81no—
163C4×D37C2×D412yes—
164C4×D37#4112yes—
165Dic74—22no—
166C8×C37—21no—
167C8×C37C2×D412yes—
168#64436—21no—
169#64436C2×D412yes—
170C22×C37—21no—
171C22×C37—21no—
172C22×C37—21no—
173C22×C37—41no—
174C22×C37—41no—
175C22×C37—41no—
176C22×C37C22×D411yescharacteristic
177C2×D37—21no—
178C2×D37—21no—
179C2×D37—21no—
180C2×D37—21no—
181C2×D37—21no—
182C2×D37—41no—
183C2×D37—41no—
184C2×D37—41no—
185C2×D37—41no—
186C2×D37—41no—
187C2×D37—41no—
188C2×D37—84no—
189C2×D37#12112yes—
190C4×C37—21no—
191C4×C37—41no—
192C4×C37—41no—
193C4×C37—41no—
194C4×C37C22×D411yescharacteristic
195C4×C37C2×#4111yescharacteristic
196Dic37—21no—
197Dic37—41no—
198Dic37—41no—
199Dic37#12112yes—
200#3062—371nomaximal, Sylow
201C2×C37—41no—
202C2×C37—81no—
203C2×C37C2×#12111yescharacteristic
204C2×C37#57611yescharacteristic
205C2×C37#57711yescharacteristic
206D37—22no—
207D37—41no—
208D37—81no—
209D37—81no—
210C2×AGL(1,ℤ/8)—371no—
211C2×#131—371no—
212#537—371no—
213#535—371no—
214#469—371no—
215#469—371no—
216#468—371no—
217#468—371no—
218#466—371no—
219#466—371no—
220#465—371no—
221#465D3711yescharacteristic, complemented
222#428—371no—
223#416—371no—
224C2×#102—371no—
225C37#306211yescharacteristic, complemented, Sylow
226C2×#41—371no—
227C2×#41—371no—
228C22×D4—371no—
229AGL(1,ℤ/8)—744no—
230C2×SD16—371no—
231C2×SD16C2×D3711yescharacteristic
232C2×D8—371no—
233C2×D8—371no—
234#131—744no—
235C2×M16—371no—
236C2×M16—371no—
237C2×M16—371no—
238C22×C8—371no—
239C22×C8—371no—
240C22×C8C2×D3711yescharacteristic, complemented
241#127—741no—
242#124—741no—
243C22⋊Q8—371no—
244C22⋊Q8—741no—
245C22⋊Q8C2×D3711yescharacteristic, complemented
246#121—371no—
247#121—371no—
248#121—371no—
249#121—371no—
250#121—371no—
251#121—741no—
252#121—741no—
253#121—741no—
254#121C2×D3711yescharacteristic
255#120—741no—
256#120—741no—
257#117—371no—
258C2×C4⋊C4—371no—
259#107—371no—
260#107—371no—
261#106—371no—
262#106C2×D3711yescharacteristic, complemented
263#103—371no—
264#103C2×D3711yescharacteristic, complemented
265#102—371no—
266#102—371no—
267#102—371no—
268#102—371no—
269#102—371no—
270#102C2×D3711yescharacteristic
271C24—741no—
272#41—742no—
273#41—742no—
274#41—742no—
275#41—742no—
276C2×Q8—371no—
277C2×Q8C22×D3711yescharacteristic
278C2×D4—21no—
279C2×D4—371no—
280C2×D4—371no—
281C2×D4—371no—
282C2×D4—371no—
283C2×D4—371no—
284C2×D4—741no—
285C2×D4—741no—
286C2×D4—741no—
287C2×D4—742no—
288C2×D4—742no—
289C2×D4—741no—
290C2×D4—741no—
291C2×D4—741no—
292C2×D4—741no—
293C2×D4C22×D3711yescharacteristic
294C22×C4—371no—
295C22×C4—371no—
296C22×C4—741no—
297C22×C4—741no—
298C22×C4C22×D3711yescharacteristic
299SD16—22no—
300SD16—742no—
301D8—742no—
302D8—742no—
303M16—374no—
304M16—742no—
305M16—742no—
306C2×C8—22no—
307C2×C8—371no—
308C2×C8—371no—
309C2×C8—742no—
310C2×C8—742no—
311C2×C8C22×D3711yescharacteristic
312C2×C8C22×D3711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—371no—
315C4⋊C4—371no—
316C4⋊C4—741no—
317C4⋊C4—741no—
318C4⋊C4C22×D3711yescharacteristic
319C4⋊C4C22×D3711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—741no—
323C22⋊C4—741no—
324C22⋊C4—741no—
325C22⋊C4—741no—
326C22⋊C4—741no—
327C22⋊C4—741no—
328C22⋊C4—741no—
329C42—741no—
330C23—21no—
331C23—371no—
332C23—371no—
333C23—2961no—
334C23—2961no—
335C23—741no—
336C23—741no—
337C23—1482no—
338C23D4×D3711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—742no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—2961no—
346D4—2961no—
347D4—2961no—
348D4—2961no—
349D4—2961no—
350D4—742no—
351D4—742no—
352D4—742no—
353D4—741no—
354D4—741no—
355D4—741no—
356D4—741no—
357D4—1482no—
358D4—1482no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—371no—
365C2×C4—372no—
366C2×C4—372no—
367C2×C4—371no—
368C2×C4—2961no—
369C2×C4—741no—
370C2×C4—741no—
371C2×C4—741no—
372C2×C4—741no—
373C2×C4—1482no—
374C2×C4D4×D3711yescharacteristic
375C2×C4C23×D3711yescharacteristic
376C8—21no—
377C8—372no—
378C8—741no—
379C8D4×D3712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—372no—
387C22—2964no—
388C22—741no—
389C22—741no—
390C22—741no—
391C22—741no—
392C22—741no—
393C22—1481no—
394C22—1481no—
395C22—1481no—
396C22—1481no—
397C22—1481no—
398C22—1481no—
399C22C2×D4×D3711yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—372no—
405C4—741no—
406C4—1481no—
407C4—1481no—
408C4C2×D4×D3711yescharacteristic
409C4#41×D3711yescharacteristic
410C2—41no—
411C2—81no—
412C2—2961no—
413C2—2961no—
414C2—742no—
415C2—1481no—
416C2#121×D3711yescharacteristic
417C2#1455833511yescharacteristic
418C2#1455833911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#465C2
socleC22×C37C22×D4
hypercenter#465D37
hypocenterC37#3062
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#11924563C2semidirect
2. C2×#11924542C2semidirect
3. #14558113C2semidirect
4. #14558099C2semidirect
5. #14557787C2semidirect
6. #14557784C2semidirect
7. #14557781C2semidirect
8. #14557776C2semidirect
9. #14557769C2semidirect
10. #14557768C2semidirect
11. #14557765C2semidirect
12. #14557572C2semidirect
13. #14557528C2semidirect
14. #102×D37C2semidirect
15. C37×#465C2semidirect
24. C2×#10596934C22semidirect
26. C2×#10596929C22semidirect
28. C2×C8×D37C22semidirect
32. #11924512C22semidirect
35. #11924511C22semidirect
37. #11924510C22semidirect
45. #11924470C22semidirect
46. #11924469C22semidirect
47. #11924467C22semidirect
48. #11924460C22semidirect
50. #11924458C22semidirect
53. #11924456C22semidirect
55. C22×C8×C37C22semidirect
56. C22⋊Q8×C37C22semidirect
58. #106×C37C22semidirect
59. #103×C37C22semidirect
117. #10596929D4semidirect
221. #465D37semidirect
225. C37#3062semidirect
240. C22×C8C2×D37semidirect
245. C22⋊Q8C2×D37semidirect
262. #106C2×D37semidirect
264. #103C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245632
C2×#119245422
#145581132
#145580992
#145577872
#145577842
#145577812
#145577762
#145577692
#145577682
#145577652
#145575722
#145575282
#102×D372
C37×#4652
#306237