Finite Group Databasebeta

← #18003909

Subgroups — #18003909

Order 4736.

712 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924583C211yesmaximal, characteristic, complemented
2C2×#11924565C211yesmaximal, characteristic, complemented
3C2×#11924555C211yesmaximal, characteristic, complemented
4#14558112C211yesmaximal, characteristic, complemented
5#14557696C211yesmaximal, characteristic, complemented
6#14557695C211yesmaximal, characteristic, complemented
7#14557691C211yesmaximal, characteristic, complemented
8#14557690C211yesmaximal, characteristic, complemented
9#14557675C211yesmaximal, characteristic, complemented
10#14557671C211yesmaximal, characteristic, complemented
11#14557670C211yesmaximal, characteristic, complemented
12#14557531C211yesmaximal, characteristic, complemented
13#103×D37C211yesmaximal, characteristic, complemented
14#14557498C211yesmaximal, characteristic, complemented
15C37×#449C211yesmaximal, characteristic, complemented
16#11924583—24no—
17#11924583—24no—
18C2×#10596962—21no—
19C2×#10596962C2211yescharacteristic
20C2×#10596964—21no—
21C2×#10596964—21no—
22C2×#10596964C2211yescharacteristic
23C2×#10596964C2211yescharacteristic
24C2×#10596964C2211yescharacteristic
25C22×Dic74—21no—
26C22×Dic74C2211yescharacteristic
27C2×Q8×D37C2211yescharacteristic
28C2×Q8×D37C2211yescharacteristic
29#11924565—24no—
30#11924555—24no—
31C2×Dic148C2211yescharacteristic
32C2×Dic148C2211yescharacteristic
33C2×#10596942C2211yescharacteristic
34C2×#10596940C2211yescharacteristic
35C2×#10596940C2211yescharacteristic
36C2×#10596940C2211yescharacteristic
37C2×#10596929C2211yescharacteristic
38C2×C8×D37C2211yescharacteristic
39#11924516C2211yescharacteristic
40#11924515C2211yescharacteristic
41#11924513—21no—
42#11924509—21no—
43#11924508—21no—
44#11924507—21no—
45#11924507C2211yescharacteristic
46C4⋊C4×D37C2211yescharacteristic
47#11924461C2211yescharacteristic
48#11924461C2211yescharacteristic
49#11924460C2211yescharacteristic
50#11924459C2211yescharacteristic
51#11924459C2211yescharacteristic
52#11924457C2211yescharacteristic
53C22⋊#64436C2211yescharacteristic
54#11924448C2211yescharacteristic
55#11924448C2211yescharacteristic
56C2×#41×C37C2211yescharacteristic
57C2×SD16×C37C2211yescharacteristic
58C2×D8×C37C2211yescharacteristic
59#121×C37C2211yescharacteristic
60#103×C37C2211yescharacteristic
61#102×C37C2211yescharacteristic
62C22⋊C8×C37C2211yescharacteristic
63#41×C37—22no—
64#41×C37—22no—
65C2×Q8×C37C2311yescharacteristic
66C2×D4×C37—21no—
67C2×D4×C37—21no—
68C2×D4×C37C2311yescharacteristic
69C2×D4×C37C2311yescharacteristic
70C22×C4×C37—21no—
71C22×C4×C37C2311yescharacteristic
72C2×C22⋊D37—21no—
73C2×C22⋊D37—21no—
74C2×C22⋊D37—21no—
75C2×C22⋊D37—21no—
76C2×C22⋊D37D411yescharacteristic
77C2×C22⋊D37D411yescharacteristic
78C22×Dic37—21no—
79C22×Dic37—21no—
80C22×Dic37—21no—
81C22×Dic37—21no—
82Q8×D37—22no—
83Q8×D37—22no—
84Q8×D37—22no—
85Q8×D37D414yescomplemented
86#10596964—28no—
87#10596964—28no—
88#10596964—22no—
89#10596964—22no—
90#10596964—22no—
91#10596964—22no—
92#10596964—22no—
93#10596964D414yescomplemented
94#10596962—28no—
95#10596962—22no—
96#10596962—22no—
97C2×D148D411yescharacteristic
98C2×C4×D37—21no—
99C2×C4×D37—21no—
100C2×C4×D37—21no—
101C2×C4×D37C2311yescharacteristic
102C2×C4×D37C2311yescharacteristic
103C2×Dic74—28no—
104C2×Dic74—21no—
105C2×Dic74—21no—
106C2×Dic74—21no—
107C2×Dic74—22no—
108C2×Dic74—22no—
109C2×Dic74D411yescharacteristic
110C2×Dic74C2311yescharacteristic
111C2×Dic74C2311yescharacteristic
112C2×Dic74C2311yescharacteristic
113SD16×C37—22no—
114D8×C37—22no—
115C2×C8×C37C2311yescharacteristic
116C2×C8×C37C2311yescharacteristic
117C4⋊C4×C37C2311yescharacteristic
118C22⋊C4×C37—21no—
119C22⋊Dic37—21no—
120C22⋊Dic37—21no—
121#10596942—22no—
122#10596940—22no—
123#10596940—22no—
124#10596940—22no—
125#10596938—21no—
126C4⋊Dic37—21no—
127C4⋊Dic37C2311yescharacteristic
128#10596936—21no—
129#10596936—21no—
130C2×#64436C2311yescharacteristic
131C2×#64436C2311yescharacteristic
132Dic148—22no—
133Dic148—22no—
134#10596929—22no—
135C8×D37—22no—
136C23×C37—21no—
137C23×C37—21no—
138C23×C37C2×D411yescharacteristic
139C22×D37C2×D411yescharacteristic
140C22×D37C2×D411yescharacteristic
141Q8×C37—21no—
142Q8×C37C2×D412yes—
143D4×C37—24no—
144D4×C37—21no—
145D4×C37—21no—
146D4×C37—21no—
147D4×C37—81no—
148D4×C37C2×D412yes—
149C2×C4×C37—24no—
150C2×C4×C37—21no—
151C2×C4×C37—21no—
152C2×C4×C37—21no—
153C2×C4×C37—21no—
154C2×C4×C37C2×D411yescharacteristic
155C2×C4×C37C2411yescharacteristic
156C22⋊D37—24no—
157C22⋊D37—22no—
158C22⋊D37—24no—
159C22⋊D37—22no—
160C22⋊D37—42no—
161C22⋊D37—81no—
162C2×Dic37—24no—
163C2×Dic37—24no—
164C2×Dic37—21no—
165C2×Dic37—21no—
166C2×Dic37—24no—
167C2×Dic37—21no—
168C2×Dic37—21no—
169C2×Dic37—42no—
170C2×Dic37C2×D411yescharacteristic
171C2×Dic37C2×D411yescharacteristic
172D148—22no—
173C4×D37—24no—
174C4×D37—24no—
175C4×D37—21no—
176C4×D37—21no—
177C4×D37—81no—
178C4×D37C2×D412yes—
179C4×D37C2×D412yes—
180Dic74—24no—
181Dic74—24no—
182Dic74—21no—
183Dic74—21no—
184Dic74—22no—
185Dic74—24no—
186Dic74—21no—
187Dic74—21no—
188Dic74C2×D412yes—
189Dic74C2×D412yes—
190C8×C37—21no—
191C8×C37—21no—
192#64436—21no—
193#64436—21no—
194C22×C37—21no—
195C22×C37—22no—
196C22×C37—21no—
197C22×C37—21no—
198C22×C37—41no—
199C22×C37—41no—
200C22×C37—41no—
201C22×C37—41no—
202C22×C37—41no—
203C22×C37C22×D411yescharacteristic
204C2×D37—21no—
205C2×D37—21no—
206C2×D37—21no—
207C2×D37—21no—
208C2×D37—21no—
209C2×D37#12012yes—
210C4×C37—21no—
211C4×C37—22no—
212C4×C37—41no—
213C4×C37C22×D411yescharacteristic
214C4×C37C22×D411yescharacteristic
215Dic37—21no—
216Dic37—22no—
217Dic37—22no—
218Dic37—41no—
219Dic37—41no—
220Dic37#12012yes—
221#3013—371nomaximal, Sylow
222C2×C37—41no—
223C2×C37—42no—
224C2×C37—81no—
225C2×C37C2×#12011yescharacteristic
226C2×C37#57711yescharacteristic
227C2×C37#57811yescharacteristic
228D37—22no—
229D37—41no—
230C2×2−1+4—371no—
231C2×#137—371no—
232C2×#135—371no—
233#537—371no—
234#452—371no—
235#452—371no—
236#451—371no—
237#451—371no—
238#449—371no—
239#449D3711yescharacteristic, complemented
240#448—371no—
241#448—371no—
242#417—371no—
243C2×#103—371no—
244#408—371no—
245C37#301311yescharacteristic, complemented, Sylow
2462−1+4—744no—
2472−1+4—744no—
248C2×#41—371no—
249C2×#41—371no—
250C2×#41—371no—
251C2×#41—371no—
252C2×#41—741no—
253C2×#41—741no—
254C2×#41—741no—
255C2×#41C2×D3711yescharacteristic
256C22×Q8—371no—
257C22×Q8—371no—
258C22×Q8—371no—
259C22×Q8—741no—
260#137—744no—
261#135—744no—
262C2×Q16—371no—
263C2×Q16—371no—
264C2×Q16—371no—
265C2×SD16—371no—
266C2×SD16—371no—
267C2×SD16—371no—
268C2×SD16C2×D3711yescharacteristic
269C2×D8C2×D3711yescharacteristic
270C2×M16—371no—
271C22×C8—371no—
272C22⋊Q8—371no—
273C22⋊Q8—371no—
274C22⋊Q8—741no—
275#121—371no—
276#121—741no—
277#121—741no—
278#121—741no—
279#121C2×D3711yescharacteristic
280C2×C4⋊C4—371no—
281#103—371no—
282#103—371no—
283#103—371no—
284#103—371no—
285#103—371no—
286#103C2×D3711yescharacteristic
287#102—371no—
288#102C2×D3711yescharacteristic
289C22⋊C8—371no—
290C22⋊C8—371no—
291C22⋊C8—371no—
292C22⋊C8C2×D3711yescharacteristic
293#41—22no—
294#41—22no—
295#41—374no—
296#41—742no—
297#41—742no—
298#41—748no—
299#41—742no—
300#41—742no—
301#41—742no—
302#41—742no—
303#41—748no—
304#41—748no—
305#41—742no—
306C2×Q8—371no—
307C2×Q8—374no—
308C2×Q8—371no—
309C2×Q8—371no—
310C2×Q8—371no—
311C2×Q8—741no—
312C2×Q8—741no—
313C2×Q8—742no—
314C2×Q8—742no—
315C2×Q8—748no—
316C2×Q8—742no—
317C2×Q8—742no—
318C2×Q8—742no—
319C2×Q8—741no—
320C2×Q8C22×D3711yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—371no—
324C2×D4—371no—
325C2×D4—371no—
326C2×D4—741no—
327C2×D4—741no—
328C2×D4—741no—
329C2×D4—741no—
330C2×D4C22×D3711yescharacteristic
331C2×D4C22×D3711yescharacteristic
332C22×C4—21no—
333C22×C4—371no—
334C22×C4—371no—
335C22×C4—741no—
336C22×C4—741no—
337C22×C4—741no—
338C22×C4—741no—
339C22×C4—741no—
340C22×C4—741no—
341C22×C4—741no—
342C22×C4C22×D3711yescharacteristic
343Q16—742no—
344Q16—742no—
345Q16—742no—
346SD16—22no—
347SD16—742no—
348SD16—742no—
349SD16—742no—
350D8—22no—
351M16—742no—
352C2×C8—371no—
353C2×C8—371no—
354C2×C8—742no—
355C2×C8C22×D3711yescharacteristic
356C2×C8C22×D3711yescharacteristic
357C4⋊C4—371no—
358C4⋊C4—741no—
359C4⋊C4—741no—
360C4⋊C4—741no—
361C4⋊C4C22×D3711yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—741no—
364C22⋊C4—741no—
365C22⋊C4—741no—
366C23—21no—
367C23—21no—
368C23—371no—
369C23—371no—
370C23D4×D3711yescharacteristic
371Q8—21no—
372Q8—372no—
373Q8—372no—
374Q8—742no—
375Q8—744no—
376Q8—744no—
377Q8—744no—
378Q8—741no—
379Q8—741no—
380Q8—741no—
381Q8—741no—
382Q8D4×D3712yes—
383D4—24no—
384D4—21no—
385D4—21no—
386D4—21no—
387D4—81no—
388D4—2961no—
389D4—742no—
390D4—742no—
391D4—742no—
392D4—744no—
393D4—744no—
394D4—1482no—
395D4D4×D3712yes—
396C2×C4—24no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—371no—
402C2×C4—372no—
403C2×C4—372no—
404C2×C4—371no—
405C2×C4—2961no—
406C2×C4—744no—
407C2×C4—744no—
408C2×C4—744no—
409C2×C4—744no—
410C2×C4—744no—
411C2×C4—741no—
412C2×C4—741no—
413C2×C4—741no—
414C2×C4—741no—
415C2×C4—741no—
416C2×C4—741no—
417C2×C4—1482no—
418C2×C4D4×D3711yescharacteristic
419C2×C4C23×D3711yescharacteristic
420C8—21no—
421C8—21no—
422C8—741no—
423C8—741no—
424C22—21no—
425C22—22no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—372no—
434C22—741no—
435C22—741no—
436C22—741no—
437C22—741no—
438C22—741no—
439C22C2×D4×D3711yescharacteristic
440C4—21no—
441C4—22no—
442C4—41no—
443C4—372no—
444C4—741no—
445C4—742no—
446C4—742no—
447C4—1481no—
448C4—1481no—
449C4C2×D4×D3711yescharacteristic
450C4C2×D4×D3711yescharacteristic
451C2—41no—
452C2—42no—
453C2—81no—
454C2—742no—
455C2—1481no—
456C2#120×D3711yescharacteristic
457C2#1455834311yescharacteristic
458C2#1455834711yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#449C2
socleC22×C37C22×D4
hypercenter#449D37
hypocenterC37#3013
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#11924583C2semidirect
2. C2×#11924565C2semidirect
3. C2×#11924555C2semidirect
4. #14558112C2semidirect
5. #14557696C2semidirect
6. #14557695C2semidirect
7. #14557691C2semidirect
8. #14557690C2semidirect
9. #14557675C2semidirect
10. #14557671C2semidirect
11. #14557670C2semidirect
12. #14557531C2semidirect
13. #103×D37C2semidirect
14. #14557498C2semidirect
15. C37×#449C2semidirect
85. Q8×D37D4semidirect
93. #10596964D4semidirect
239. #449D37semidirect
245. C37#3013semidirect

Maximal subgroups

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

subgroupindex
C2×#119245832
C2×#119245652
C2×#119245552
#145581122
#145576962
#145576952
#145576912
#145576902
#145576752
#145576712
#145576702
#145575312
#103×D372
#145574982
C37×#4492
#301337