Finite Group Databasebeta

← #29714755

Subgroups — #29714755

Order 15616.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839964C211yesmaximal, characteristic, complemented
2C2×C4×#11758199C211yesmaximal, characteristic
3#21450737C211yesmaximal, characteristic, complemented
4#21447055C212yesmaximal, complemented
5#21446896C212yesmaximal
6#21446756C211yesmaximal, characteristic, complemented
7#21446622C211yesmaximal, characteristic, complemented
8#21446174C212yesmaximal, complemented
9#21446073C212yesmaximal, complemented
10#21445819C211yesmaximal, characteristic, complemented
11C2×C61×#556C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C22×#11758199C2211yescharacteristic
14C2×#14519753—21no—
15C2×C4×D244C2211yescharacteristic
16#16839964—24no—
17C4×#11758199—24no—
18C2×#14519710C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839684—22no—
28#16839664C2212yes—
29#16839660C2212yes—
30#16839632C2211yescharacteristic
31#16839619—21no—
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#556—24no—
36C61×#556—24no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×#126×C61—22no—
39C2×#124×C61C2212yes—
40C2×C22⋊Q8×C61—21no—
41C2×C4×Q8×C61C2212yes—
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61C2211yescharacteristic
44C2×#117×C61—21no—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758201—42no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C2×#11758199—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2312yes—
55C22×#92418C2311yescharacteristic
56#14519753—44no—
57C4×D244—22no—
58C4×D244—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×C4⋊Dic61C2312yes—
63C2×#11758196—22no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67#14519710—22no—
68C4⋊#92418—22no—
69#14519701—24no—
70#14519701—24no—
71#14519701—42no—
72#14519698—22no—
73#14519698—42no—
74#14519697—22no—
75C4×#92418—22no—
76#14519684—22no—
77C22×Q8×C61D411yescharacteristic
78C22×Q8×C61C2312yes—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C4⋊Q8×C61—22no—
82C4⋊Q8×C61D414yes—
83#126×C61—216no—
84#124×C61—24no—
85#124×C61—24no—
86C22⋊Q8×C61—28no—
87C4×Q8×C61—24no—
88C4×Q8×C61—24no—
89C4×Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61D414yes—
93#117×C61—28no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61D412yes—
98C2×C4⋊C4×C61D411yescharacteristic
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61—22no—
103C2×C42×C61—22no—
104C2×C42×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106C24×C61—21no—
107C23×D61—21no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D414yes—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—24no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—22no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244#4112yes—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C4⋊C4×C61—28no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C22⋊C4×C61—28no—
152C22⋊C4×C61—28no—
153C22⋊C4×C61—24no—
154C42×C61—28no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D412yes—
158C42×C61C2×D412yes—
159#11758201—48no—
160#11758199—28no—
161#11758198—44no—
162#11758198—42no—
163#11758198—82no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—42no—
169C4⋊Dic61—81no—
170#11758196—44no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418#4112yes—
180C23×C61—22no—
181C23×C61—41no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61C22×D411yescharacteristic
188C22×D61—22no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195Q8×C61—24no—
196Q8×C61—24no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—44no—
200D4×C61—22no—
201D4×C61—22no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—48no—
219C2×C4×C61#12012yes—
220C2×C4×C61#12012yes—
221C2×C4×C61#13512yes—
222C2×C4×C61AGL(1,ℤ/8)12yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—22no—
236D244—22no—
237D244—41no—
238D244—41no—
239C4×D61—84no—
240#92418—22no—
241#92418—41no—
242#92418—42no—
243#24185—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61C2×#12011yescharacteristic
252C22×C61#54811yescharacteristic
253C22×C61#55011yescharacteristic
254C22×C61C2×#13511yescharacteristic
255C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57811yescharacteristic
258C2×D61—42no—
259C2×D61—42no—
260C2×D61—42no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C4×C61—22no—
266C4×C61—44no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61#44914yes—
271C4×C61#52511yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61#54611yescharacteristic
274C4×C61#54711yescharacteristic
275Dic61—82no—
276Dic61—81no—
277C2×#556D6111yescharacteristic, complemented
278C2×#461—611no—
279C2×C4×D8—611no—
280#2538—611no—
281#2042—612no—
282#2012—612no—
283#1984—611no—
284#1956—611no—
285#1880—612no—
286#1863—612no—
287#1802—611no—
288C2×C61—82no—
289C2×C61#240711yescharacteristic
290C2×C61C2×#44911yescharacteristic
291C2×C61#301311yescharacteristic
292C2×C61#336011yescharacteristic
293C2×C61#336711yescharacteristic
294C2×C61#339311yescharacteristic
295C2×C61#339711yescharacteristic
296D61—82no—
297C22×SD16—612no—
298C22×D8—611no—
299#556—24no—
300#556—24no—
301C2×C4⋊Q8C2×D6111yescharacteristic
302C2×#126—22no—
303C2×#124C2×D6112yes—
304C2×C22⋊Q8—21no—
305C2×C22⋊Q8—1221no—
306C2×C4×Q8C2×D6112yes—
307C2×C4×Q8C2×D6111yescharacteristic
308C2×C4×D4—611no—
309C2×C4×D4C2×D6111yescharacteristic
310C2×#117—21no—
311#461—1224no—
312C4×D8—1224no—
313C2×#107—611no—
314C2×C4⋊C8—611no—
315C2×#103—612no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—612no—
321C2×C4×C8—611no—
322#396—1222no—
323#390—612no—
324#389—612no—
325#384—611no—
326#380—1221no—
327#340—612no—
328#336—612no—
329#336—612no—
330C61#2418511yescharacteristic, complemented, Sylow
331C22×Q8C22⋊D6111yescharacteristic
332C22×Q8C22×D6112yes—
333C22×D4—611no—
334C22×D4C22×D6111yescharacteristic
335C23×C4—21no—
336C23×C4—1221no—
337C2×SD16—1224no—
338C2×SD16—1224no—
339C2×SD16—2442no—
340C2×D8—1222no—
341C2×D8—1222no—
342C2×D8—1222no—
343C22×C8—611no—
344C22×C8—612no—
345C22×C8—611no—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D6114yes—
348#126—216no—
349#124—24no—
350#124—24no—
351C22⋊Q8—28no—
352C22⋊Q8—2444no—
353C4×Q8—24no—
354C4×Q8—24no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×D4—22no—
358C4×D4—1222no—
359C4×D4—1222no—
360C4×D4C22⋊D6114yes—
361#117—28no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—612no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4C22⋊D6111yescharacteristic
371C2×C4⋊C4C22⋊D6112yes—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1222no—
379C2×C42—22no—
380C2×C42C22×D6112yes—
381C2×C42C22×D6111yescharacteristic
382#107—1222no—
383C4⋊C8—1222no—
384#103—1224no—
385#103—1224no—
386#103—2442no—
387#102—1222no—
388#102—1222no—
389#102—2442no—
390C4×C8—1222no—
391#95—1222no—
392C24—21no—
393C24—1221no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8C2×C22⋊D6114yes—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—612no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4C2×C22⋊D6112yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—1221no—
421C22×C4—1222no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4C2×C22⋊D6112yes—
426C22×C4C2×C22⋊D6112yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C23×D6111yescharacteristic
430SD16—2448no—
431D8—1228no—
432C2×C8—612no—
433C2×C8—1221no—
434C2×C8—1221no—
435C2×C8—1221no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1221no—
440C2×C8—1221no—
441C4⋊C4—28no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—1221no—
448C4⋊C4—1221no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—4881no—
452C4⋊C4—2444no—
453C4⋊C4—2442no—
454C4⋊C4C2×C22⋊D6112yes—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—4882no—
459C22⋊C4—2444no—
460C22⋊C4—2442no—
461C42—28no—
462C42—22no—
463C42—22no—
464C42C2×C22⋊D6112yes—
465C42C2×C22⋊D6112yes—
466C23—22no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—1222no—
474C23—2441no—
475C23—2441no—
476C23—2441no—
477C23—2441no—
478C23—2441no—
479C23—2441no—
480C23C22×C22⋊D6111yescharacteristic
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490D4—1222no—
491D4—1222no—
492D4—2441no—
493D4—2441no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—48no—
509C2×C4—4884no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4#1451974612yes—
517C2×C4#1451974612yes—
518C2×C4#1451979712yes—
519C2×C4#1451980212yes—
520C2×C4C22×C22⋊D6111yescharacteristic
521C2×C4C22×C22⋊D6111yescharacteristic
522C2×C4#41×D6111yescharacteristic
523C2×C4#41×D6111yescharacteristic
524C2×C4#1451982311yescharacteristic
525C2×C4#1451982411yescharacteristic
526C8—1222no—
527C8—2441no—
528C8—2442no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22—4881no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22C2×#1451974611yescharacteristic
544C22#1684041911yescharacteristic
545C22#1684043111yescharacteristic
546C22C2×#1451979711yescharacteristic
547C22C2×#1451980211yescharacteristic
548C22#1684057011yescharacteristic
549C22#1684057711yescharacteristic
550C4—22no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—4882no—
556C4—4881no—
557C4#1683990714yes—
558C4#1684026911yescharacteristic
559C4#1684034011yescharacteristic
560C4#1684040011yescharacteristic
561C4#1684041111yescharacteristic
562C2—82no—
563C2—4882no—
564C2#2144915811yescharacteristic
565C2C2×#1683990711yescharacteristic
566C2#2145513411yescharacteristic
567C2#2145846411yescharacteristic
568C2#2145853311yescharacteristic
569C2#2145880311yescharacteristic
570C2#2145883911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#556C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC2×#556D61
hypocenterC61#24185
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. C2×#16839964C2semidirect
3. #21450737C2semidirect
4. #21447055C2semidirect
6. #21446756C2semidirect
7. #21446622C2semidirect
8. #21446174C2semidirect
9. #21446073C2semidirect
10. #21445819C2semidirect
11. C2×C61×#556C2semidirect
277. C2×#556D61semidirect
330. C61#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#168399642
C2×C4×#117581992
#214507372
#214470552
#214468962
#214467562
#214466222
#214461742
#214460732
#214458192
C2×C61×#5562
#2418561