Finite Group Databasebeta

← #29686947

Subgroups — #29686947

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455540C211yesmaximal, characteristic
2#21454721C212yesmaximal, complemented
3C2×#16839878C211yesmaximal, characteristic, complemented
4#21454241C211yesmaximal, characteristic, complemented
5#21446717C211yesmaximal, characteristic, complemented
6#21446487C212yesmaximal, complemented
7#21446321C212yesmaximal, complemented
8#21445874C211yesmaximal, characteristic, complemented
9#21445701C211yesmaximal, characteristic, complemented
10#21445618C212yesmaximal, complemented
11C61×#1953C211yesmaximal, characteristic, complemented
12#16840562C2212yes—
13C2×#14519799C2211yescharacteristic
14C2×#14519785C2212yes—
15C2×#14519785C2211yescharacteristic
16C2×#14519783C2211yescharacteristic
17#16840230C2211yescharacteristic
18#16840038C412yescomplemented
19#16840033C412yescomplemented
20#16840026C414yescomplemented
21#16839883—24no—
22#16839878—24no—
23#16839878—21no—
24#16839878—21no—
25#16839878—21no—
26#16839878—21no—
27#16839800C2211yescharacteristic
28#16839778C2212yes—
29#16839778C2212yes—
30C2×#14519703C2212yes—
31#16839744C2212yes—
32#16839735C2212yes—
33#16839720C2212yes—
34#16839719—21no—
35#16839718C2211yescharacteristic
36#16839716—21no—
37#16839714—21no—
38M16×Dic61C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C4×C8×D61—21no—
41#16839533C2212yes—
42#16839523C2212yes—
43C61×#430C2212yes—
44C2×#108×C61C2211yescharacteristic
45C2×#108×C61C2211yescharacteristic
46C61×#405C2211yescharacteristic
47C61×#344C2212yes—
48C2×#11758222C2312yes—
49C2×#11758222C2311yescharacteristic
50#14519810—22no—
51#14519810C2×C414yes—
52#14519805—22no—
53#14519805C2×C414yes—
54#14519799—24no—
55#14519799—21no—
56#14519799—21no—
57#14519799—21no—
58#14519799—21no—
59C2×Dic244C2×C411yescharacteristic, complemented
60C2×#11758190C2×C412yescomplemented
61C2×D488C2×C411yescharacteristic, complemented
62#14519785—21no—
63#14519785—21no—
64#14519785—22no—
65#14519785—22no—
66#14519785—22no—
67#14519785—22no—
68#14519785D412yes—
69#14519785D412yes—
70#14519783—21no—
71#14519783—21no—
72#14519783C2×C412yes—
73#14519783C2×C412yes—
74C2×#11758194C2311yescharacteristic
75C2×#11758189—21no—
76C2×#11758189—21no—
77C2×#11758189—22no—
78C22×#92418C2311yescharacteristic
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81C2×C8×D61—22no—
82#14519742—21no—
83#14519739—21no—
84#14519737—22no—
85#14519736C2312yes—
86#14519734—21no—
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—21no—
89#14519711—21no—
90#14519709—21no—
91#14519703—24no—
92#14519703—24no—
93#14519703—22no—
94#14519703—22no—
95#14519695C2×C414yescomplemented
96#14519693C2×C414yescomplemented
97#14519691—24no—
98#14519688—21no—
99#14519687—21no—
100#14519687—22no—
101C4×#92418—21no—
102C8×Dic61—21no—
103C8×Dic61—22no—
104C8×Dic61D411yescharacteristic
105C8×Dic61D411yescharacteristic
106C8×Dic61D411yescharacteristic
107C8×Dic61D411yescharacteristic
108C2×M16×C61C2312yes—
109C2×M16×C61C2312yes—
110C2×M16×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112#117×C61C2311yescharacteristic
113#108×C61—24no—
114#108×C61—21no—
115#108×C61—21no—
116#108×C61C2×C412yescomplemented
117#108×C61C2×C412yescomplemented
118#97×C61—21no—
119C4×C8×C61—21no—
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—21no—
122C2×C22⋊D61—22no—
123C22×Dic61—21no—
124#11758222—21no—
125#11758222—21no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222C22×C412yes—
131#11758222C2×D412yes—
132C2×D244C22×C412yes—
133C2×D244#4111yescharacteristic
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×Dic122C22×C412yes—
138C2×Dic122#4111yescharacteristic
139M16×C61—22no—
140M16×C61—21no—
141M16×C61—22no—
142M16×C61—22no—
143M16×C61C22×C414yes—
144M16×C61C2×D412yes—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C22×C411yescharacteristic
150C2×C8×C61C2×D411yescharacteristic
151C2×C8×C61C2×D411yescharacteristic
152C2×C8×C61#4111yescharacteristic
153C2×C8×C61#4111yescharacteristic
154C4⋊C4×C61—21no—
155C22⋊C4×C61#4112yes—
156C42×C61—21no—
157C22⋊Dic61—22no—
158#11758198—22no—
159C4⋊Dic61—22no—
160#11758196—22no—
161C4×Dic61—22no—
162C4×Dic61—21no—
163C4×Dic61—22no—
164C4×Dic61C2×D411yescharacteristic
165C4×Dic61C2×D411yescharacteristic
166#11758194—21no—
167#11758194C22×C412yes—
168C2×#92418—21no—
169C2×#92418C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418#4111yescharacteristic
174C2×#92418#4111yescharacteristic
175Dic244—24no—
176Dic244—21no—
177Dic244—21no—
178D488—24no—
179D488—21no—
180D488—21no—
181#11758190—24no—
182#11758190—24no—
183#11758190—22no—
184#11758190—22no—
185#11758189—22no—
186#11758189—22no—
187#11758189—22no—
188#11758189—22no—
189#11758189—42no—
190#11758189—42no—
191C8×D61—22no—
192C8×D61—22no—
193C8×D61—22no—
194C8×D61—22no—
195C8×D61—42no—
196C8×D61—42no—
197C23×C61C2×#4111yescharacteristic
198C22×D61—21no—
199C22×D61—22no—
200C2×C4×C61—22no—
201C2×C4×C61C23×C411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C22⋊D61—22no—
208C22⋊D61—22no—
209C22⋊D61—42no—
210C22⋊D61—42no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—22no—
214C2×Dic61—21no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217D244—22no—
218D244—22no—
219D244—21no—
220D244C4×D412yes—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—44no—
224C4×D61—42no—
225C4×D61—42no—
226Dic122—21no—
227Dic122—22no—
228Dic122—22no—
229Dic122C4×D412yes—
230C8×C61—24no—
231C8×C61—22no—
232C8×C61—42no—
233C8×C61C4×D411yescharacteristic
234C8×C61C4×D411yescharacteristic
235C8×C61C4×D411yescharacteristic
236C8×C61C4×D411yescharacteristic
237#92418—22no—
238#92418#12111yescharacteristic
239#92418#12111yescharacteristic
240#92418#12111yescharacteristic
241#92418#12111yescharacteristic
242#21558—611nomaximal, Sylow
243C22×C61—41no—
244C22×C61C4×#4111yescharacteristic
245C22×C61#52511yescharacteristic
246C22×C61#53211yescharacteristic
247C2×D61—22no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—82no—
251C4×C61—42no—
252C4×C61C2×C4×D411yescharacteristic
253C4×C61C2×C4×D411yescharacteristic
254C4×C61C2×#12111yescharacteristic
255C4×C61#52911yescharacteristic
256Dic61—22no—
257Dic61—22no—
258Dic61—41no—
259Dic61—42no—
260Dic61—82no—
261#3067—611no—
262#2955—612no—
263C2×#443—611no—
264C4×#131—611no—
265#1976—611no—
266#1953D6111yescharacteristic, complemented
267#1932—612no—
268#1906—612no—
269#1814—611no—
270#1772—611no—
271#1757—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61C4×#12111yescharacteristic
276D61—42no—
277D61—82no—
278#575—612no—
279C2×#135—611no—
280C2×#131—611no—
281C2×#131—612no—
282C2×#131—611no—
283C4×#41—611no—
284#473—612no—
285#472—612no—
286#471—614no—
287#444—1224no—
288#443—1221no—
289#443—1221no—
290#443—1221no—
291#443—1221no—
292#443—1224no—
293#430C2×D6112yes—
294C2×#108C2×D6111yescharacteristic
295C2×#108C2×D6111yescharacteristic
296#427—611no—
297#421—612no—
298#421—612no—
299C2×#104—612no—
300#413—612no—
301#410—612no—
302#405—612no—
303#405—611no—
304#405—1221no—
305#405—1221no—
306#405C2×D6111yescharacteristic
307C4×M16—611no—
308C4×M16—1221no—
309C2×C4×C8—611no—
310C2×C4×C8—1221no—
311#344—612no—
312#344C2×D6112yes—
313#339—612no—
314C61#2155811yescharacteristic, complemented, Sylow
315C2×#41—611no—
316C2×#41—612no—
317#137—614no—
318#137—1222no—
319AGL(1,ℤ/8)—614no—
320AGL(1,ℤ/8)—1222no—
321#135—1224no—
322#135—1221no—
323#135—1221no—
324#135—1221no—
325#135—1221no—
326C2×Q16—611no—
327C2×SD16—612no—
328C2×D8—611no—
329#131—612no—
330#131—612no—
331#131—612no—
332#131—612no—
333#131—1221no—
334#131—1221no—
335#131—1222no—
336#131—1222no—
337#131—1222no—
338#131—1222no—
339#131—1221no—
340#131—1221no—
341C2×M16—611no—
342C2×M16—1221no—
343C2×M16—1222no—
344C2×M16—1221no—
345C2×M16C22×D6112yes—
346C2×M16C22×D6112yes—
347C2×M16C22×D6111yescharacteristic
348C22×C8—611no—
349C22×C8—1221no—
350C22×C8—1222no—
351C22×C8—1221no—
352C22×C8C22×D6111yescharacteristic
353C4×Q8—1221no—
354C4×D4—1221no—
355C4×D4—1222no—
356#117—612no—
357#117—1221no—
358#117C22×D6111yescharacteristic
359C2×C42—611no—
360C2×C42—1221no—
361#108—24no—
362#108—21no—
363#108—21no—
364#108C4×D6112yescomplemented
365#108C4×D6112yescomplemented
366#107—1221no—
367#106—1221no—
368#104—1222no—
369#104—1222no—
370#104—1224no—
371#104—1224no—
372#101—614no—
373#100—614no—
374#99—1224no—
375#97—21no—
376#97—1221no—
377#97—1221no—
378#97—1222no—
379C4×C8—21no—
380C4×C8—611no—
381C4×C8—611no—
382C4×C8—611no—
383C4×C8—611no—
384C4×C8—1221no—
385C4×C8—1221no—
386C4×C8—1222no—
387#41—612no—
388#41—612no—
389#41—1221no—
390#41—1221no—
391#41—1222no—
392#41—1222no—
393#41—1222no—
394#41—1222no—
395C2×Q8—612no—
396C2×Q8—611no—
397C2×D4—612no—
398C2×D4—611no—
399C2×D4—1221no—
400C2×D4—1222no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1222no—
405C22×C4C23×D6111yescharacteristic
406Q16—1224no—
407Q16—1221no—
408Q16—1221no—
409SD16—1224no—
410SD16—1224no—
411SD16—1222no—
412SD16—1222no—
413D8—1224no—
414D8—1221no—
415D8—1221no—
416M16—21no—
417M16—22no—
418M16—22no—
419M16—22no—
420M16—612no—
421M16—1222no—
422M16—1222no—
423M16—1221no—
424M16—1222no—
425M16—1222no—
426M16—2442no—
427M16—2442no—
428M16C2×C4×D6114yes—
429M16D4×D6112yes—
430C2×C8—21no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—611no—
434C2×C8—611no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8—1221no—
442C2×C8—1222no—
443C2×C8—1222no—
444C2×C8—2442no—
445C2×C8—2442no—
446C2×C8C2×C4×D6111yescharacteristic
447C2×C8C2×C4×D6111yescharacteristic
448C2×C8D4×D6111yescharacteristic
449C2×C8D4×D6111yescharacteristic
450C2×C8#1175822611yescharacteristic
451C2×C8#1175822611yescharacteristic
452C4⋊C4—21no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C22⋊C4—1222no—
456C22⋊C4—1222no—
457C22⋊C4#1175822612yes—
458C42—21no—
459C42—611no—
460C42—611no—
461C42—1222no—
462C42—1221no—
463C42—1222no—
464C23—1221no—
465C23—1222no—
466C23C2×#1175822611yescharacteristic
467Q8—612no—
468Q8—1221no—
469Q8—1222no—
470Q8—1222no—
471D4—612no—
472D4—1222no—
473D4—1222no—
474D4—1222no—
475D4—1222no—
476D4—1221no—
477D4—2442no—
478D4—2442no—
479C2×C4—22no—
480C2×C4—1221no—
481C2×C4—1221no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1221no—
486C2×C4—2444no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2442no—
491C2×C4C22×C4×D6111yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4#41×D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C2×C4C2×#1175822611yescharacteristic
497C8—24no—
498C8—22no—
499C8—42no—
500C8—611no—
501C8—611no—
502C8—611no—
503C8—611no—
504C8—1222no—
505C8#1451973911yescharacteristic
506C8#1451973911yescharacteristic
507C8#1451973911yescharacteristic
508C8#1451973911yescharacteristic
509C22—41no—
510C22—1222no—
511C22—4882no—
512C22—2442no—
513C22—2442no—
514C22C4×#1175822611yescharacteristic
515C22#1684027511yescharacteristic
516C22#1684031311yescharacteristic
517C4—42no—
518C4—1222no—
519C4—1222no—
520C4—4882no—
521C4—2441no—
522C4—2442no—
523C4C4×D4×D6111yescharacteristic
524C4C2×#1451973911yescharacteristic
525C4#121×D6111yescharacteristic
526C4#1684029711yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—4882no—
531C2—2442no—
532C2#2144832711yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1953C2
socleC2×C61C4×#121
prime radical#21455540C2
cosocleC22×C4×C61C24
hypercenter#1953D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1953

Complements

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

NH ≅ G/Ntype
2. #21454721C2semidirect
3. C2×#16839878C2semidirect
4. #21454241C2semidirect
5. #21446717C2semidirect
6. #21446487C2semidirect
7. #21446321C2semidirect
8. #21445874C2semidirect
9. #21445701C2semidirect
10. #21445618C2semidirect
11. C61×#1953C2semidirect
18. #16840038C4semidirect
19. #16840033C4semidirect
20. #16840026C4semidirect
59. C2×Dic244C2×C4semidirect
60. C2×#11758190C2×C4semidirect
61. C2×D488C2×C4semidirect
95. #14519695C2×C4semidirect
96. #14519693C2×C4semidirect
116. #108×C61C2×C4semidirect
117. #108×C61C2×C4semidirect
266. #1953D61semidirect
314. C61#21558semidirect
364. #108C4×D61semidirect
365. #108C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214555402
#214547212
C2×#168398782
#214542412
#214467172
#214464872
#214463212
#214458742
#214457012
#214456182
C61×#19532
#2155861