Finite Group Databasebeta

← #29833988

Subgroups — #29833988

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C211yesmaximal, characteristic, complemented
2C2×#16840203C211yesmaximal, characteristic, complemented
3#21455539C211yesmaximal, characteristic, complemented
4#21447797C214yesmaximal, complemented
5#21447783C214yesmaximal, complemented
6C2×#16839578C211yesmaximal, characteristic, complemented
7#21447547C211yesmaximal, characteristic, complemented
8#21447447C211yesmaximal, characteristic, complemented
9C2×C61×#578C211yesmaximal, characteristic, complemented
10#16840560—21no—
11C2×#14519799C2211yescharacteristic
12#16840203—22no—
13#16840203—22no—
14#16840203—21no—
15#16840202C2212yescomplemented
16#16840198—21no—
17#16840197C2212yescomplemented
18#16840192—22no—
19C2×#14519723C2212yescomplemented
20C2×#14519720C2212yescomplemented
21C2×#14519717C2211yescharacteristic
22C2×#14519717C2211yescharacteristic
23#16840037C2212yescomplemented
24#16840032C2212yescomplemented
25#16840025—22no—
26C2×#14519713C2211yescharacteristic
27#16839744—21no—
28#16839578C2214yescomplemented
29#16839569C2214yescomplemented
30#16839546C2214yescomplemented
31C2×2−1+4×C61—21no—
32C61×#578—22no—
33C61×#578—22no—
34C61×#578—22no—
35C61×#578C2214yes—
36C2×#137×C61—22no—
37C2×#137×C61—21no—
38C2×#135×C61—21no—
39C2×#135×C61C2211yescharacteristic
40C22×Q16×C61—21no—
41C22×Q16×C61C2211yescharacteristic
42C2×#131×C61C2211yescharacteristic
43C2×#11758222—21no—
44#14519808—41no—
45#14519807—22no—
46#14519803—41no—
47#14519802—22no—
48#14519799—21no—
49#14519799—21no—
50#14519799—21no—
51#14519799—21no—
52#14519797—42no—
53C2×Dic244C2311yescharacteristic
54C2×#11758190—21no—
55C2×D488C2311yescharacteristic
56C2×#11758194—21no—
57#14519723—22no—
58#14519723—24no—
59#14519721—22no—
60#14519720—22no—
61#14519720—24no—
62#14519718—22no—
63#14519717—21no—
64#14519717—21no—
65#14519717—21no—
66#14519717C2312yes—
67C2×#11758185—21no—
68C2×#11758185C2312yes—
69#14519713—21no—
70#14519713C2312yes—
71#14519695—22no—
72#14519693—22no—
732−1+4×C61—24no—
742−1+4×C61—24no—
752−1+4×C61—42no—
762−1+4×C61—42no—
77C2×#41×C61—22no—
78C2×#41×C61—21no—
79C2×#41×C61—42no—
80C2×#41×C61—41no—
81C2×#41×C61D411yescharacteristic
82C22×Q8×C61—22no—
83C22×Q8×C61—21no—
84C22×Q8×C61—41no—
85#137×C61—24no—
86#137×C61—28no—
87#137×C61—28no—
88#137×C61—24no—
89#135×C61—24no—
90#135×C61—24no—
91#135×C61—21no—
92#135×C61—21no—
93#135×C61D412yes—
94#135×C61C2312yes—
95C2×Q16×C61—22no—
96C2×Q16×C61—24no—
97C2×Q16×C61—24no—
98C2×Q16×C61—21no—
99C2×Q16×C61—21no—
100C2×Q16×C61—21no—
101C2×Q16×C61D411yescharacteristic
102C2×Q16×C61D412yes—
103C2×Q16×C61D412yes—
104C2×Q16×C61C2312yes—
105C2×SD16×C61—22no—
106C2×SD16×C61—21no—
107C2×D8×C61D411yescharacteristic
108#131×C61—21no—
109#131×C61—21no—
110#131×C61D412yes—
111#131×C61C2312yes—
112C2×M16×C61—21no—
113C2×M16×C61D411yescharacteristic
114C22×C8×C61—21no—
115C22×C8×C61C2311yescharacteristic
116#41×C61—21no—
117#41×C61—21no—
118#41×C61—28no—
119#41×C61—22no—
120#41×C61—22no—
121#41×C61—44no—
122#41×C61—48no—
123#41×C61—48no—
124#41×C61—44no—
125#41×C61—44no—
126#41×C61—41no—
127#41×C61—41no—
128#41×C61D812yes—
129#41×C61C2×D412yes—
130C2×Q8×C61—28no—
131C2×Q8×C61—22no—
132C2×Q8×C61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—22no—
135C2×Q8×C61—42no—
136C2×Q8×C61—44no—
137C2×Q8×C61—44no—
138C2×Q8×C61—44no—
139C2×Q8×C61—41no—
140C2×Q8×C61—41no—
141C2×Q8×C61—41no—
142C2×Q8×C61D811yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—42no—
146C2×D4×C61—41no—
147C2×D4×C61D811yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C2×C22⋊D61—41no—
155#11758222—41no—
156#11758222—41no—
157#11758222—41no—
158#11758222—41no—
159C2×D244—21no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162Q16×C61—28no—
163Q16×C61—21no—
164Q16×C61—22no—
165Q16×C61—22no—
166Q16×C61—22no—
167Q16×C61C2×D412yes—
168Q16×C61C2×D414yes—
169SD16×C61—28no—
170SD16×C61—22no—
171SD16×C61—22no—
172D8×C61—21no—
173D8×C61C2×D412yes—
174M16×C61—21no—
175M16×C61—22no—
176M16×C61—22no—
177M16×C61C2×D412yes—
178C2×C8×C61—21no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C2×C8×C61C2×D411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2411yescharacteristic
187#11758202—42no—
188#11758201—42no—
189#11758200—42no—
190#11758199—42no—
191#11758194—22no—
192#11758194—41no—
193C2×#92418—41no—
194Dic244—21no—
195Dic244—21no—
196D488—21no—
197D488—21no—
198#11758190—41no—
199#11758190—41no—
200#11758185—22no—
201#11758185—22no—
202C23×C61—21no—
203C23×C61—41no—
204C23×C61AGL(1,ℤ/8)11yescharacteristic
205C22×D61—41no—
206Q8×C61—21no—
207Q8×C61—24no—
208Q8×C61—22no—
209Q8×C61—48no—
210Q8×C61—42no—
211Q8×C61—41no—
212Q8×C61—42no—
213Q8×C61—42no—
214Q8×C61C2×D812yes—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—22no—
218D4×C61—21no—
219D4×C61—48no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—41no—
223D4×C61C2×D812yes—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—48no—
228C2×C4×C61—48no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C2×D811yescharacteristic
238C2×C4×C61C2×D811yescharacteristic
239C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C22⋊D61—81no—
242C22⋊D61—81no—
243C2×Dic61—41no—
244D244—41no—
245D244—41no—
246C4×D61—81no—
247C4×D61—81no—
248Dic122—41no—
249Dic122—41no—
250C8×C61—22no—
251C8×C61#12011yescharacteristic
252C8×C61#12011yescharacteristic
253C8×C61C22×D411yescharacteristic
254C8×C61C22×D411yescharacteristic
255#92418—42no—
256#33540—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—82no—
262C22×C61#44711yescharacteristic
263C22×C61#44711yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C2×D61—81no—
266C2×D61—161no—
267C4×C61—22no—
268C4×C61—44no—
269C4×C61—42no—
270C4×C61#44711yescharacteristic
271C4×C61#44711yescharacteristic
272C4×C61C2×#12011yescharacteristic
273C4×C61C22×D811yescharacteristic
274Dic61—81no—
275C2×#578D6111yescharacteristic, complemented
276#3414—611no—
277C2×#510—611no—
278#3067—611no—
279#2195—614no—
280#2193—614no—
281C2×#362—611no—
282#2143—611no—
283#2117—611no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—42no—
288C2×C61—82no—
289C2×C61C2×#44711yescharacteristic
290D61—161no—
291C2×2−1+4—21no—
292#578—22no—
293#578—22no—
294#578—22no—
295#578C2×D6114yes—
296#575—1221no—
297C2×#137—22no—
298C2×#137—21no—
299C2×#135—21no—
300C2×#135—611no—
301C2×#135C2×D6111yescharacteristic
302C22×Q16—21no—
303C22×Q16C2×D6111yescharacteristic
304C2×#131C2×D6111yescharacteristic
305#510—612no—
306#510—1222no—
307#510—1222no—
308#510—1221no—
309#509—612no—
310#509—1221no—
311#508—1222no—
312C2×Q32—612no—
313C2×SD32—612no—
314C2×M32—611no—
315C2×M32—611no—
316#473—612no—
317#472—612no—
318#471—1222no—
319C2×#108—611no—
320#413—1221no—
321#362—614no—
322#359—614no—
323#350—614no—
324C61#3354011yescharacteristic, complemented, Sylow
3252−1+4—24no—
3262−1+4—24no—
3272−1+4—42no—
3282−1+4—42no—
329C2×#41—22no—
330C2×#41—21no—
331C2×#41—42no—
332C2×#41—41no—
333C2×#41—1221no—
334C2×#41C22⋊D6111yescharacteristic
335C22×Q8—22no—
336C22×Q8—21no—
337C22×Q8—41no—
338#137—24no—
339#137—28no—
340#137—28no—
341#137—24no—
342#137—1222no—
343#137—2441no—
344AGL(1,ℤ/8)—1222no—
345AGL(1,ℤ/8)—2441no—
346#135—24no—
347#135—24no—
348#135—21no—
349#135—21no—
350#135—1221no—
351#135—1221no—
352#135—1221no—
353#135—1221no—
354#135—2442no—
355#135C22⋊D6112yes—
356#135C22×D6112yes—
357C2×Q16—22no—
358C2×Q16—24no—
359C2×Q16—24no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×Q16—21no—
363C2×Q16—611no—
364C2×Q16C22⋊D6111yescharacteristic
365C2×Q16C22⋊D6112yes—
366C2×Q16C22⋊D6112yes—
367C2×Q16C22×D6112yes—
368C2×SD16—22no—
369C2×SD16—21no—
370C2×SD16—1221no—
371C2×D8—611no—
372C2×D8C22⋊D6111yescharacteristic
373#131—21no—
374#131—21no—
375#131C22⋊D6112yes—
376#131C22×D6112yes—
377C2×M16—21no—
378C2×M16—1221no—
379C2×M16C22⋊D6111yescharacteristic
380C22×C8—21no—
381C22×C8C22×D6111yescharacteristic
382Q32—1224no—
383Q32—1222no—
384SD32—1224no—
385SD32—1222no—
386SD32—1222no—
387D16—1222no—
388M32—612no—
389M32—1221no—
390M32—1221no—
391M32—1221no—
392C2×C16—612no—
393C2×C16—1221no—
394#108—612no—
395#108—1221no—
396#101—1222no—
397#100—1222no—
398#41—21no—
399#41—21no—
400#41—28no—
401#41—22no—
402#41—22no—
403#41—44no—
404#41—48no—
405#41—48no—
406#41—44no—
407#41—44no—
408#41—41no—
409#41—41no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41#1175819912yes—
415#41C2×C22⋊D6112yes—
416C2×Q8—28no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—42no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—41no—
426C2×Q8—41no—
427C2×Q8—41no—
428C2×Q8—1221no—
429C2×Q8#1175819911yescharacteristic
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—1221no—
435C2×D4—2441no—
436C2×D4#1175819911yescharacteristic
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—2441no—
443C22×C4C2×C22⋊D6111yescharacteristic
444Q16—28no—
445Q16—22no—
446Q16—21no—
447Q16—22no—
448Q16—22no—
449Q16—1221no—
450Q16—1221no—
451Q16—2442no—
452Q16C2×C22⋊D6112yes—
453Q16C2×C22⋊D6114yes—
454SD16—28no—
455SD16—22no—
456SD16—22no—
457SD16—2442no—
458SD16—2442no—
459SD16—2441no—
460SD16—2441no—
461D8—21no—
462D8—1221no—
463D8—1221no—
464D8—2442no—
465D8C2×C22⋊D6112yes—
466M16—21no—
467M16—22no—
468M16—22no—
469M16—1222no—
470M16—2441no—
471M16C2×C22⋊D6112yes—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—2441no—
476C2×C8C2×C22⋊D6111yescharacteristic
477C2×C8C2×C22⋊D6111yescharacteristic
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C2×C8C2×C22⋊D6111yescharacteristic
481C2×C8C23×D6111yescharacteristic
482C16—1222no—
483C16—1222no—
484C23—21no—
485C23—41no—
486C23—2441no—
487C23#1451980311yescharacteristic
488Q8—21no—
489Q8—24no—
490Q8—22no—
491Q8—48no—
492Q8—42no—
493Q8—41no—
494Q8—42no—
495Q8—42no—
496Q8—2441no—
497Q8—2441no—
498Q8C2×#1175819912yes—
499D4—22no—
500D4—22no—
501D4—22no—
502D4—21no—
503D4—48no—
504D4—42no—
505D4—42no—
506D4—41no—
507D4—4881no—
508D4—4881no—
509D4—2441no—
510D4—2441no—
511D4C2×#1175819912yes—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—4881no—
524C2×C4—4881no—
525C2×C4—2441no—
526C2×C4#1451974611yescharacteristic
527C2×C4#1451974611yescharacteristic
528C2×C4C2×#1175819911yescharacteristic
529C2×C4C2×#1175819911yescharacteristic
530C2×C4#1451980311yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C8—22no—
533C8—2442no—
534C8#1451974611yescharacteristic
535C8#1451974611yescharacteristic
536C8C22×C22⋊D6111yescharacteristic
537C8C22×C22⋊D6111yescharacteristic
538C22—22no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—9761no—
544C22—4881no—
545C22#1683989611yescharacteristic
546C22#1683989611yescharacteristic
547C22C2×#1451980311yescharacteristic
548C4—22no—
549C4—44no—
550C4—42no—
551C4—4881no—
552C4#1683989611yescharacteristic
553C4#1683989611yescharacteristic
554C4C2×#1451974611yescharacteristic
555C4C22×#1175819911yescharacteristic
556C2—21no—
557C2—21no—
558C2—21no—
559C2—42no—
560C2—82no—
561C2—9761no—
562C2C2×#1683989611yescharacteristic
subgroupG/H
centerC2C2×#16839896
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#578C2
socleC2×C61C2×#447
hypercenterC2×#578D61
hypocenterC61#33540
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. C2×#16840203C2semidirect
3. #21455539C2semidirect
4. #21447797C2semidirect
5. #21447783C2semidirect
6. C2×#16839578C2semidirect
7. #21447547C2semidirect
8. #21447447C2semidirect
9. C2×C61×#578C2semidirect
15. #16840202C22semidirect
17. #16840197C22semidirect
19. C2×#14519723C22semidirect
20. C2×#14519720C22semidirect
23. #16840037C22semidirect
24. #16840032C22semidirect
28. #16839578C22semidirect
29. #16839569C22semidirect
30. #16839546C22semidirect
275. C2×#578D61semidirect
324. C61#33540semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168402032
#214555392
#214477972
#214477832
C2×#168395782
#214475472
#214474472
C2×C61×#5782
#3354061