Finite Group Databasebeta

← #18004221

Subgroups — #18004221

Order 4736.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924565C211yesmaximal, characteristic, complemented
2C2×M16×D37C211yesmaximal, characteristic, complemented
3#14558043C211yesmaximal, characteristic
4C22⋊Q8×D37C211yesmaximal, characteristic, complemented
5#14557791C211yesmaximal, characteristic, complemented
6#14557789C211yesmaximal, characteristic, complemented
7#14557777C211yesmaximal, characteristic, complemented
8#14557775C211yesmaximal, characteristic, complemented
9#14557773C211yesmaximal, characteristic, complemented
10#14557761C211yesmaximal, characteristic, complemented
11#14557759C211yesmaximal, characteristic, complemented
12#14557574C211yesmaximal, characteristic, complemented
13#14557540C211yesmaximal, characteristic, complemented
14#14557531C211yesmaximal, characteristic, complemented
15C37×#468C211yesmaximal, characteristic, complemented
16C2×#10596964C2211yescharacteristic
17C2×Q8×D37C2211yescharacteristic
18C22×C4×D37C2211yescharacteristic
19#11924565—24no—
20C2×Dic148C2211yescharacteristic, complemented
21C2×#10596942C2211yescharacteristic, complemented
22C2×#10596940C2211yescharacteristic, complemented
23C2×#10596934C2211yescharacteristic
24C2×#10596929C2211yescharacteristic
25C2×#10596929C2211yescharacteristic
26M16×D37—24no—
27C2×C8×D37C2211yescharacteristic
28C2×C8×D37C2211yescharacteristic
29#11924518—21no—
30#11924516C2211yescharacteristic
31#11924516C2211yescharacteristic
32#11924515C2211yescharacteristic
33#11924515C2211yescharacteristic
34#11924514—21no—
35#11924513—21no—
36#11924512C2211yescharacteristic
37#11924507C2211yescharacteristic
38#11924504—21no—
39D4×Dic37—21no—
40#11924496—21no—
41#11924494C2211yescharacteristic
42C4⋊C4×D37—21no—
43C4⋊C4×D37C2211yescharacteristic
44C22⋊C4×D37—21no—
45#11924469C2211yescharacteristic, complemented
46#11924468C2211yescharacteristic, complemented
47#11924467C2211yescharacteristic, complemented
48#11924461C2211yescharacteristic, complemented
49#11924461C2211yescharacteristic, complemented
50#11924460C2211yescharacteristic, complemented
51#11924459C2211yescharacteristic, complemented
52#11924459C2211yescharacteristic, complemented
53#11924457C2211yescharacteristic, complemented
54C2×SD16×C37C2211yescharacteristic, complemented
55C2×M16×C37C2211yescharacteristic
56C22⋊Q8×C37C2211yescharacteristic
57#121×C37C2211yescharacteristic
58#107×C37C2211yescharacteristic, complemented
59#103×C37C2211yescharacteristic, complemented
60#102×C37C2211yescharacteristic, complemented
61C23×D37D411yescharacteristic
62C2×Q8×C37C2311yescharacteristic
63C2×D4×C37—21no—
64C2×D4×C37C2311yescharacteristic
65C22×C4×C37C2311yescharacteristic
66C2×C22⋊D37—21no—
67C2×C22⋊D37—21no—
68C22×Dic37—21no—
69C22×Dic37D411yescharacteristic
70Q8×D37—22no—
71Q8×D37—22no—
72#10596964—22no—
73#10596964—22no—
74C2×C4×D37—22no—
75C2×C4×D37—22no—
76C2×C4×D37—21no—
77C2×C4×D37—21no—
78C2×C4×D37D411yescharacteristic
79C2×C4×D37D411yescharacteristic
80C2×C4×D37C2311yescharacteristic
81C2×C4×D37C2311yescharacteristic
82C2×Dic74—21no—
83C2×Dic74C2311yescharacteristic
84C2×Dic74C2311yescharacteristic
85SD16×C37—22no—
86M16×C37—22no—
87C2×C8×C37C2311yescharacteristic
88C2×C8×C37C2311yescharacteristic
89C4⋊C4×C37—21no—
90C4⋊C4×C37C2311yescharacteristic
91C4⋊C4×C37C2311yescharacteristic
92C22⋊C4×C37—21no—
93C22⋊C4×C37—21no—
94C22⋊Dic37—21no—
95C22⋊Dic37—21no—
96C22⋊Dic37—21no—
97#10596942—22no—
98#10596940—22no—
99#10596938—21no—
100#10596938—21no—
101#10596938—21no—
102#10596938—21no—
103C4⋊Dic37—21no—
104C4⋊Dic37—21no—
105C4⋊Dic37C2311yescharacteristic
106C4⋊Dic37C2311yescharacteristic
107#10596936—21no—
108#10596936—21no—
109#10596936—21no—
110#10596936—21no—
111C4×Dic37—21no—
112#10596934—22no—
113C2×#64436C2311yescharacteristic
114C2×#64436C2311yescharacteristic
115Dic148—22no—
116#10596929—22no—
117#10596929D414yescomplemented
118C8×D37—22no—
119C8×D37—22no—
120C23×C37—21no—
121C23×C37C2×D411yescharacteristic
122C22×D37—22no—
123C22×D37—22no—
124C22×D37—41no—
125C22×D37C2×D411yescharacteristic
126C22×D37C2×D411yescharacteristic
127Q8×C37—21no—
128Q8×C37—21no—
129D4×C37—21no—
130D4×C37—21no—
131D4×C37—42no—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37C2×D411yescharacteristic
138C2×C4×C37C2411yescharacteristic
139C22⋊D37—42no—
140C22⋊D37—42no—
141C2×Dic37—22no—
142C2×Dic37—21no—
143C2×Dic37—21no—
144C2×Dic37—21no—
145C2×Dic37—21no—
146C2×Dic37—42no—
147C2×Dic37C2×D411yescharacteristic
148C2×Dic37C2×D411yescharacteristic
149C4×D37—22no—
150C4×D37—21no—
151C4×D37—21no—
152C4×D37—22no—
153C4×D37—42no—
154C4×D37—42no—
155C4×D37C2×D412yes—
156C4×D37#4112yes—
157Dic74—21no—
158Dic74—21no—
159Dic74—21no—
160Dic74—21no—
161Dic74—42no—
162C8×C37—21no—
163C8×C37C2×D412yes—
164#64436—21no—
165#64436C2×D412yes—
166C22×C37—21no—
167C22×C37—21no—
168C22×C37—21no—
169C22×C37—41no—
170C22×C37—41no—
171C22×C37—41no—
172C22×C37C22×D411yescharacteristic
173C2×D37—21no—
174C2×D37—21no—
175C2×D37—21no—
176C2×D37—21no—
177C2×D37—44no—
178C2×D37#12112yes—
179C2×D37#13712yescomplemented
180C4×C37—21no—
181C4×C37—41no—
182C4×C37—41no—
183C4×C37—41no—
184C4×C37C22×D411yescharacteristic
185C4×C37C2×#4111yescharacteristic
186Dic37—21no—
187Dic37—41no—
188Dic37—41no—
189Dic37—41no—
190Dic37—41no—
191Dic37#12112yes—
192#3056—371nomaximal, Sylow
193C2×C37—41no—
194C2×C37—81no—
195C2×C37C2×#12111yescharacteristic
196C2×C37C2×#13711yescharacteristic
197C2×C37#57511yescharacteristic
198D37—22no—
199D37—22no—
200C2×#137—371no—
201C22×M16—371no—
202#525—371no—
203C2×C22⋊Q8—371no—
204#470—371no—
205#470—371no—
206#468—371no—
207#468D3711yescharacteristic, complemented
208#467—371no—
209#467—371no—
210#465—371no—
211#465—371no—
212#428—371no—
213#419—371no—
214#417—371no—
215C37#305611yescharacteristic, complemented, Sylow
216C2×#41—371no—
217C22×Q8—371no—
218C23×C4—371no—
219#137—744no—
220C2×Q16—371no—
221C2×Q16—371no—
222C2×SD16—371no—
223C2×SD16C2×D3711yescharacteristic, complemented
224C2×M16—371no—
225C2×M16—371no—
226C2×M16—371no—
227C2×M16—744no—
228C2×M16C2×D3711yescharacteristic
229C22×C8—371no—
230C22×C8—371no—
231#123—741no—
232C22⋊Q8—371no—
233C22⋊Q8—371no—
234C22⋊Q8—371no—
235C22⋊Q8—371no—
236C22⋊Q8—371no—
237C22⋊Q8—741no—
238C22⋊Q8—741no—
239C22⋊Q8C2×D3711yescharacteristic
240#121—371no—
241#121C2×D3711yescharacteristic
242#120—741no—
243C4×D4—741no—
244C4×D4—741no—
245#117—371no—
246C2×C4⋊C4—371no—
247C2×C4⋊C4—741no—
248C2×C22⋊C4—741no—
249#107—371no—
250#107C2×D3711yescharacteristic, complemented
251#106—371no—
252#106—371no—
253#103—371no—
254#103—371no—
255#103—371no—
256#103—371no—
257#103—371no—
258#103C2×D3711yescharacteristic, complemented
259#102—371no—
260#102C2×D3711yescharacteristic, complemented
261C24—371no—
262#41—742no—
263#41—742no—
264C2×Q8—371no—
265C2×Q8—371no—
266C2×Q8—741no—
267C2×Q8—742no—
268C2×Q8—742no—
269C2×Q8C22×D3711yescharacteristic
270C2×D4—21no—
271C2×D4—741no—
272C2×D4—741no—
273C2×D4C22×D3711yescharacteristic
274C22×C4—371no—
275C22×C4—371no—
276C22×C4—371no—
277C22×C4—371no—
278C22×C4—371no—
279C22×C4—742no—
280C22×C4—742no—
281C22×C4—741no—
282C22×C4—741no—
283C22×C4—741no—
284C22×C4C22×D3711yescharacteristic
285Q16—742no—
286Q16—742no—
287SD16—22no—
288SD16—742no—
289M16—22no—
290M16—374no—
291M16—742no—
292M16—742no—
293C2×C8—371no—
294C2×C8—371no—
295C2×C8—742no—
296C2×C8—742no—
297C2×C8C22×D3711yescharacteristic
298C2×C8C22×D3711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—371no—
301C4⋊C4—371no—
302C4⋊C4—741no—
303C4⋊C4—741no—
304C4⋊C4—741no—
305C4⋊C4—741no—
306C4⋊C4—741no—
307C4⋊C4—741no—
308C4⋊C4C22×D3711yescharacteristic
309C4⋊C4C22×D3711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—741no—
313C22⋊C4—741no—
314C22⋊C4—741no—
315C22⋊C4—741no—
316C22⋊C4—741no—
317C22⋊C4—741no—
318C22⋊C4—741no—
319C42—741no—
320C23—21no—
321C23—371no—
322C23—371no—
323C23—742no—
324C23—742no—
325C23—1481no—
326C23D4×D3711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—741no—
330Q8—741no—
331Q8—741no—
332Q8—741no—
333Q8—1482no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—1482no—
338D4—1482no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—371no—
345C2×C4—372no—
346C2×C4—372no—
347C2×C4—371no—
348C2×C4—742no—
349C2×C4—742no—
350C2×C4—742no—
351C2×C4—741no—
352C2×C4—741no—
353C2×C4—741no—
354C2×C4—741no—
355C2×C4—741no—
356C2×C4—741no—
357C2×C4—1482no—
358C2×C4—1482no—
359C2×C4—1482no—
360C2×C4D4×D3711yescharacteristic
361C2×C4C23×D3711yescharacteristic
362C8—21no—
363C8—372no—
364C8—741no—
365C8D4×D3712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—372no—
373C22—372no—
374C22—741no—
375C22—741no—
376C22—741no—
377C22—741no—
378C22—1484no—
379C22C2×D4×D3711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—372no—
385C4—741no—
386C4—1481no—
387C4—1481no—
388C4—1481no—
389C4—1481no—
390C4C2×D4×D3711yescharacteristic
391C4#41×D3711yescharacteristic
392C2—41no—
393C2—81no—
394C2—742no—
395C2—742no—
396C2#121×D3711yescharacteristic
397C2#137×D3711yescharacteristic
398C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#468C2
socleC22×C37C22×D4
prime radical#14558043C2
hypercenter#468D37
hypocenterC37#3056
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#11924565C2semidirect
2. C2×M16×D37C2semidirect
4. C22⋊Q8×D37C2semidirect
5. #14557791C2semidirect
6. #14557789C2semidirect
7. #14557777C2semidirect
8. #14557775C2semidirect
9. #14557773C2semidirect
10. #14557761C2semidirect
11. #14557759C2semidirect
12. #14557574C2semidirect
13. #14557540C2semidirect
14. #14557531C2semidirect
15. C37×#468C2semidirect
20. C2×Dic148C22semidirect
21. C2×#10596942C22semidirect
22. C2×#10596940C22semidirect
45. #11924469C22semidirect
46. #11924468C22semidirect
47. #11924467C22semidirect
48. #11924461C22semidirect
49. #11924461C22semidirect
50. #11924460C22semidirect
51. #11924459C22semidirect
52. #11924459C22semidirect
53. #11924457C22semidirect
54. C2×SD16×C37C22semidirect
58. #107×C37C22semidirect
59. #103×C37C22semidirect
60. #102×C37C22semidirect
117. #10596929D4semidirect
179. C2×D37#137semidirect
207. #468D37semidirect
215. C37#3056semidirect
223. C2×SD16C2×D37semidirect
250. #107C2×D37semidirect
258. #103C2×D37semidirect
260. #102C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245652
C2×M16×D372
#145580432
C22⋊Q8×D372
#145577912
#145577892
#145577772
#145577752
#145577732
#145577612
#145577592
#145575742
#145575402
#145575312
C37×#4682
#305637