Finite Group Databasebeta

← #29710896

Subgroups — #29710896

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454753C211yesmaximal, characteristic, complemented
2#21454720C211yesmaximal, characteristic
3#21447401C211yesmaximal, characteristic, complemented
4#21447397C211yesmaximal, characteristic, complemented
5#21447275C211yesmaximal, characteristic, complemented
6#21447273C211yesmaximal, characteristic, complemented
7#21447123C211yesmaximal, characteristic
8#21447105C211yesmaximal, characteristic
9#21446992C211yesmaximal, characteristic, complemented
10#21446991C211yesmaximal, characteristic, complemented
11#21445656C211yesmaximal, characteristic, complemented
12#21445620C211yesmaximal, characteristic, complemented
13#21445619C211yesmaximal, characteristic, complemented
14#21445574C211yesmaximal, characteristic, complemented
15C61×#1764C211yesmaximal, characteristic, complemented
16#16840560C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840319C2211yescharacteristic
19#16839892—22no—
20C8×Dic122—22no—
21#16839882—22no—
22#16839877—22no—
23#16839810C2211yescharacteristic
24#16839788C2211yescharacteristic
25#16839788C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839779C2211yescharacteristic
28#16839778C2211yescharacteristic
29C2×#14519705C2211yescharacteristic
30#16839759C2211yescharacteristic
31#16839746C2211yescharacteristic
32#16839744C2211yescharacteristic
33#16839736C2211yescharacteristic
34#16839735C2211yescharacteristic
35#16839717C2211yescharacteristic
36#16839717C2211yescharacteristic
37#16839717C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839533C2211yescharacteristic
40#16839533C2211yescharacteristic
41#16839532C2211yescharacteristic
42#16839527C2211yescharacteristic
43#16839522C2211yescharacteristic
44#16839520C2211yescharacteristic
45#16839519C2211yescharacteristic
46#16839518C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#131×C61C2211yescharacteristic
49C2×#104×C61C2211yescharacteristic
50C61×#416C2211yescharacteristic
51C61×#405C2211yescharacteristic
52C2×C4×C8×C61C2211yescharacteristic
53C61×#341C2211yescharacteristic
54C61×#339C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56#14519808—21no—
57#14519807—21no—
58#14519803—21no—
59#14519802—21no—
60#14519797—22no—
61#14519783—21no—
62#14519783—21no—
63#14519783—21no—
64#14519783—21no—
65C2×#11758194C2311yescharacteristic
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758194C2311yescharacteristic
69C2×#11758189—21no—
70C2×C8×D61—21no—
71#14519777—21no—
72#14519769—21no—
73#14519756—22no—
74C4×Dic122—22no—
75#14519736C2311yescharacteristic
76#14519736C2311yescharacteristic
77#14519736C2311yescharacteristic
78#14519712—22no—
79C4⋊#92418—22no—
80#14519706—22no—
81#14519706—22no—
82#14519705—21no—
83#14519705—21no—
84#14519704—22no—
85#14519703—22no—
86#14519702—21no—
87#14519699—21no—
88#14519696—21no—
89#14519695—21no—
90#14519694—21no—
91#14519693—21no—
92#14519692—22no—
93#14519691—22no—
94#14519687—21no—
95#14519687—21no—
96#14519687D412yes—
97C8×Dic61—21no—
98C8×Dic61—21no—
99C8×Dic61D412yes—
100C2×#41×C61C2311yescharacteristic
101#131×C61—21no—
102#131×C61—21no—
103#131×C61—21no—
104#131×C61—21no—
105C2×M16×C61—21no—
106C2×M16×C61C2311yescharacteristic
107C2×M16×C61C2311yescharacteristic
108C22×C8×C61—21no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111#117×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#104×C61—21no—
114#104×C61—21no—
115#103×C61—21no—
116#102×C61—21no—
117#97×C61—21no—
118C4×C8×C61—21no—
119C4×C8×C61—21no—
120C4×C8×C61—21no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124#41×C61—21no—
125C2×Q8×C61#4111yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61#4111yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—21no—
132#11758222—21no—
133#11758222—21no—
134#11758222—21no—
135#11758222—21no—
136C2×D244#4111yescharacteristic
137C2×C4×D61—21no—
138C2×Dic122—22no—
139C2×Dic122#4111yescharacteristic
140M16×C61—21no—
141M16×C61—21no—
142M16×C61—21no—
143M16×C61—21no—
144M16×C61—41no—
145M16×C61—41no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—41no—
152C2×C8×C61—41no—
153C2×C8×C61C2×D411yescharacteristic
154C2×C8×C61C2×D411yescharacteristic
155C2×C8×C61#4111yescharacteristic
156C2×C8×C61#4111yescharacteristic
157C2×C8×C61#4111yescharacteristic
158C2×C8×C61#4111yescharacteristic
159C4⋊C4×C61—21no—
160C22⋊C4×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61#4111yescharacteristic
164C42×C61#4111yescharacteristic
165C22⋊Dic61—21no—
166C22⋊Dic61—21no—
167C22⋊Dic61#4112yes—
168#11758202—22no—
169#11758201—22no—
170#11758200—22no—
171#11758199—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176#11758196—24no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61C2×D412yes—
180#11758194—21no—
181#11758194—21no—
182#11758194—21no—
183#11758194—21no—
184#11758194—21no—
185#11758194—21no—
186#11758194—21no—
187#11758194—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418C2×D412yes—
192#11758189—41no—
193#11758189—41no—
194C8×D61—41no—
195C8×D61—41no—
196C23×C61—21no—
197C23×C61C2×#4111yescharacteristic
198C22×D61—21no—
199Q8×C61—21no—
200Q8×C61—21no—
201D4×C61—21no—
202D4×C61—21no—
203D4×C61—41no—
204D4×C61—41no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C22⋊D61—41no—
219C22⋊D61—41no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—21no—
223C2×Dic61—41no—
224D244—21no—
225D244—21no—
226C4×D61—41no—
227C4×D61—41no—
228Dic122—21no—
229Dic122—21no—
230Dic122—44no—
231C8×C61—41no—
232C8×C61—41no—
233C8×C61—41no—
234C8×C61#12411yescharacteristic
235C8×C61#12411yescharacteristic
236C8×C61#12411yescharacteristic
237C8×C61#12411yescharacteristic
238#92418—22no—
239#92418—41no—
240#92418—41no—
241#92418—41no—
242#23833—611nomaximal, Sylow
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—81no—
246C22×C61#52511yescharacteristic
247C22×C61#52911yescharacteristic
248C22×C61#52911yescharacteristic
249C2×D61—41no—
250C2×D61—81no—
251C4×C61—22no—
252C4×C61—41no—
253C4×C61—41no—
254C4×C61—81no—
255C4×C61#52511yescharacteristic
256C4×C61C2×#12411yescharacteristic
257C4×C61#52911yescharacteristic
258C4×C61#52911yescharacteristic
259Dic61—42no—
260Dic61—42no—
261Dic61—41no—
262Dic61—81no—
263#2960—611no—
264#2955—611no—
265#2103—611no—
266#2102—611no—
267#2080—611no—
268#2080—611no—
269#2053—611no—
270#2050—611no—
271#2031—611no—
272#2031—611no—
273#1764—611no—
274#1764D6111yescharacteristic, complemented
275#1757—611no—
276#1757—611no—
277#1747—611no—
278C2×C61—21no—
279C2×C61—21no—
280C2×C61—21no—
281C2×C61—81no—
282C2×C61#239811yescharacteristic
283D61—81no—
284#575—611no—
285C2×#131—611no—
286C2×#131C2×D6111yescharacteristic
287#533—611no—
288#446—1222no—
289C8×Q8—1222no—
290#444—1222no—
291#443—1222no—
292#430—611no—
293#424—611no—
294#424—611no—
295#423—611no—
296#421—611no—
297#421—611no—
298C2×#104—611no—
299C2×#104C2×D6111yescharacteristic
300#416—611no—
301#416C2×D6111yescharacteristic
302#413—611no—
303#413—611no—
304#410—611no—
305#410—611no—
306#405—611no—
307#405—611no—
308#405—611no—
309#405C2×D6111yescharacteristic
310C2×C4×C8C2×D6111yescharacteristic
311#344—611no—
312#344—611no—
313#344—611no—
314#344—611no—
315#341—611no—
316#341C2×D6111yescharacteristic
317#339—611no—
318#339C2×D6111yescharacteristic
319#338—611no—
320H(3,ℤ/4)—611no—
321H(3,ℤ/4)—611no—
322H(3,ℤ/4)—611no—
323C61#2383311yescharacteristic, complemented, Sylow
324C2×#41—611no—
325C2×#41C22×D6111yescharacteristic
326#137—1221no—
327#137—1221no—
328AGL(1,ℤ/8)—1221no—
329AGL(1,ℤ/8)—1221no—
330#135—1222no—
331#131—21no—
332#131—21no—
333#131—21no—
334#131—21no—
335#131—1221no—
336#131—1221no—
337#131—1221no—
338#131—1221no—
339C2×M16—21no—
340C2×M16—611no—
341C2×M16—611no—
342C2×M16—611no—
343C2×M16—611no—
344C2×M16—1221no—
345C2×M16C22×D6111yescharacteristic
346C2×M16C22×D6111yescharacteristic
347C22×C8—21no—
348C22×C8—1221no—
349C22×C8C22×D6111yescharacteristic
350C22×C8C22×D6111yescharacteristic
351C4⋊Q8—1221no—
352#125—1221no—
353C22⋊Q8—1222no—
354C4×Q8—1222no—
355#117—611no—
356#117—611no—
357#117—611no—
358#117C22×D6111yescharacteristic
359C2×C42C22×D6111yescharacteristic
360#108—1222no—
361C4⋊C8—1222no—
362C4⋊C8—1222no—
363C4⋊C8—1222no—
364#104—21no—
365#104—21no—
366#104—1222no—
367#104—1222no—
368#104—1221no—
369#104—1221no—
370#103—21no—
371#103—1221no—
372#102—21no—
373#102—1221no—
374#101—1221no—
375#101—1221no—
376#100—1221no—
377#100—1221no—
378#99—1222no—
379#99—1222no—
380#97—21no—
381#97—612no—
382#97—1221no—
383#97—1221no—
384C4×C8—21no—
385C4×C8—21no—
386C4×C8—21no—
387C4×C8—612no—
388C4×C8—1221no—
389C4×C8—1221no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1221no—
395#41—1221no—
396#41—1221no—
397#41—1221no—
398C2×Q8—611no—
399C2×Q8—1222no—
400C2×Q8#1175822211yescharacteristic
401C2×D4—21no—
402C2×D4—611no—
403C2×D4—1221no—
404C2×D4#1175822211yescharacteristic
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—1221no—
408C22×C4C23×D6111yescharacteristic
409Q16—1222no—
410SD16—1222no—
411SD16—1222no—
412D8—1222no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—41no—
418M16—41no—
419M16—1221no—
420M16—1221no—
421M16—1221no—
422M16—1221no—
423M16—1221no—
424M16—1221no—
425M16—1221no—
426M16—1221no—
427M16—2441no—
428M16—2441no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—612no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—2441no—
441C2×C8—2441no—
442C2×C8#1175822211yescharacteristic
443C2×C8#1175822211yescharacteristic
444C2×C8D4×D6111yescharacteristic
445C2×C8D4×D6111yescharacteristic
446C2×C8#1175822411yescharacteristic
447C2×C8#1175822411yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—1224no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C22⋊C4—21no—
455C22⋊C4—612no—
456C22⋊C4—1221no—
457C22⋊C4—1221no—
458C42—21no—
459C42—21no—
460C42—612no—
461C42—1221no—
462C42—1221no—
463C42#1175822211yescharacteristic
464C42#1175822211yescharacteristic
465C23—21no—
466C23—1221no—
467C23#41×D6111yescharacteristic
468Q8—21no—
469Q8—21no—
470Q8—1221no—
471Q8—1221no—
472Q8—2444no—
473D4—21no—
474D4—21no—
475D4—41no—
476D4—41no—
477D4—1221no—
478D4—1221no—
479D4—2441no—
480D4—2441no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1221no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4C2×D4×D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C2×C4C2×#1175822411yescharacteristic
497C2×C4C2×#1175822211yescharacteristic
498C2×C4C2×#1175822211yescharacteristic
499C2×C4C2×#1175822211yescharacteristic
500C8—41no—
501C8—41no—
502C8—41no—
503C8—1222no—
504C8—2441no—
505C8—2441no—
506C8—2441no—
507C8#1451976311yescharacteristic
508C8#1451976311yescharacteristic
509C8#1451976311yescharacteristic
510C8#1451976311yescharacteristic
511C22—41no—
512C22—41no—
513C22—81no—
514C22—4881no—
515C22—2441no—
516C22#1684027111yescharacteristic
517C22#1684029011yescharacteristic
518C22#1684029211yescharacteristic
519C4—22no—
520C4—41no—
521C4—41no—
522C4—81no—
523C4—4881no—
524C4—2442no—
525C4—2442no—
526C4—2441no—
527C4#1684027111yescharacteristic
528C4C2×#1451976311yescharacteristic
529C4#1684029411yescharacteristic
530C4#1684029611yescharacteristic
531C2—21no—
532C2—21no—
533C2—21no—
534C2—81no—
535C2—4881no—
536C2#2144910611yescharacteristic
subgroupG/H
centerC8#14519763
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1764C2
socleC2×C61#2398
prime radical#16840560C22
hypercenter#1764D61
hypocenterC61#23833
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1764

Complements

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

NH ≅ G/Ntype
1. #21454753C2semidirect
3. #21447401C2semidirect
4. #21447397C2semidirect
5. #21447275C2semidirect
6. #21447273C2semidirect
9. #21446992C2semidirect
10. #21446991C2semidirect
11. #21445656C2semidirect
12. #21445620C2semidirect
13. #21445619C2semidirect
14. #21445574C2semidirect
15. C61×#1764C2semidirect
274. #1764D61semidirect
323. C61#23833semidirect

Maximal subgroups

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

subgroupindex
#214547532
#214547202
#214474012
#214473972
#214472752
#214472732
#214471232
#214471052
#214469922
#214469912
#214456562
#214456202
#214456192
#214455742
C61×#17642
#2383361