Finite Group Databasebeta

← #29729785

Subgroups — #29729785

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21447179C212yesmaximal, complemented
5#21447074C212yesmaximal, complemented
6#21446728C212yesmaximal, complemented
7#21446660C211yesmaximal, characteristic, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446484C212yesmaximal, complemented
10C4×#14519693C211yesmaximal, characteristic, complemented
11C61×#3138C211yesmaximal, characteristic, complemented
12C22×#11758199C2212yes—
13C2×C4×D244C2211yescharacteristic
14#16840059—22no—
15#16840049—21no—
16#16840039—21no—
17#16839954—21no—
18#16839954—21no—
19#16839954—22no—
20#16839893—21no—
21#16839893—22no—
22Q8×#92418—21no—
23C2×#14519712C2212yes—
24#16839805C2212yes—
25#16839789C2212yes—
26#16839786C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28#16839766C2212yes—
29C2×#14519698C2212yes—
30C2×#14519694C2212yes—
31C2×#14519694C2212yes—
32C2×#14519693C2211yescharacteristic
33C2×#14519693C2211yescharacteristic
34#16839718C2212yes—
35C4×#11758194C2211yescharacteristic
36#16839530C2211yescharacteristic
37#16839530C2211yescharacteristic
38#16839526C2212yes—
39C61×#533C2211yescharacteristic
40C2×#127×C61C2211yescharacteristic
41C61×#490—22no—
42C61×#486—21no—
43C61×#486—21no—
44C61×#418C2212yes—
45C61×#418C2212yes—
46C4×M16×C61C2211yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C4×D61—21no—
49C2×#11758199—24no—
50C2×#11758199—22no—
51C2×#11758199—22no—
52C2×#11758199—42no—
53C2×#11758194C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758194C2311yescharacteristic
56C22×#92418C2312yes—
57C4×D244—21no—
58C4×D244—21no—
59C4×D244—21no—
60C4×D244—21no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758198—21no—
63#14519712—42no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418—22no—
69C4⋊#92418—22no—
70C4⋊#92418D411yescharacteristic
71C4⋊#92418D412yes—
72C4⋊#92418D411yescharacteristic
73#14519698—22no—
74#14519698—22no—
75#14519697—22no—
76#14519697—22no—
77#14519694—24no—
78#14519694—42no—
79#14519693—22no—
80#14519693—22no—
81#14519688—21no—
82#14519688—22no—
83C4×#92418—21no—
84C4×#92418—22no—
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61C2312yes—
88C2×M16×C61C2311yescharacteristic
89C2×M16×C61C2311yescharacteristic
90C4⋊Q8×C61D411yescharacteristic
91C4⋊Q8×C61D411yescharacteristic
92#127×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#125×C61D412yes—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101#117×C61C2312yes—
102C2×C42×C61C2311yescharacteristic
103#102×C61—22no—
104#102×C61—22no—
105#102×C61—22no—
106#102×C61—22no—
107#97×C61—21no—
108C4×C8×C61—21no—
109C24×C61—22no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—82no—
124C2×D4×C61—82no—
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244#4112yes—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—82no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141M16×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C2×D412yes—
149C4⋊C4×C61C2×D412yes—
150C22⋊C4×C61—22no—
151C42×C61—22no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156#11758199—44no—
157#11758199—44no—
158#11758198—41no—
159#11758198—41no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—22no—
165#11758194—21no—
166#11758194—22no—
167#11758194#4112yes—
168C2×#92418—21no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—22no—
173C2×#92418C2×D412yes—
174C2×#92418C2×D412yes—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—82no—
179C23×C61—82no—
180C23×C61C2×#4111yescharacteristic
181C22×D61—22no—
182C22×D61—41no—
183C22×D61—41no—
184C22×D61—81no—
185C22×D61—81no—
186Q8×C61—42no—
187Q8×C61—81no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—82no—
193D4×C61—82no—
194D4×C61—82no—
195D4×C61—82no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61C2×#4111yescharacteristic
208C2×C4×C61H(5,𝔽2)11yescharacteristic
209C2×C4×C612−1+411yescharacteristic
210C2×Dic61—41no—
211C2×Dic61—81no—
212D244—22no—
213D244—22no—
214D244—41no—
215D244—41no—
216C4×D61—82no—
217C4×D61—82no—
218C8×C61—41no—
219C8×C61—41no—
220#92418—22no—
221#92418—22no—
222#92418—22no—
223#92418—41no—
224#92418—42no—
225#25312—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—82no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—162no—
232C22×C61—162no—
233C22×C61C2×#12111yescharacteristic
234C22×C61#52911yescharacteristic
235C22×C61#54211yescharacteristic
236C2×D61—42no—
237C2×D61—82no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—82no—
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61#47212yes—
247C4×C61#54611yescharacteristic
248C4×C61#54811yescharacteristic
249C4×C61#54811yescharacteristic
250C4×C61#55011yescharacteristic
251Dic61—81no—
252#3138D6111yescharacteristic, complemented
253#3077—611no—
254C2×#459—611no—
255#2962—611no—
256#2063—612no—
257#2045—612no—
258#1978—612no—
259#1965—611no—
260#1950—611no—
261#1931—612no—
262C4×#100—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—162no—
266C2×C61#242611yescharacteristic
267C2×C61C2×#47211yescharacteristic
268C2×C61#306811yescharacteristic
269D61—82no—
270C22×D8—612no—
271#533C2×D6111yescharacteristic
272C2×#127C2×D6111yescharacteristic
273C2×C4×D4—611no—
274#490—22no—
275#486—21no—
276#486—21no—
277#478—1222no—
278#476—1221no—
279#474—1221no—
280#459—1221no—
281#459—1222no—
282#459—1221no—
283#446—1222no—
284#446—1221no—
285C8×Q8—1221no—
286C2×#108—612no—
287#428—612no—
288#424—612no—
289#423—611no—
290C2×C4⋊C8—611no—
291#418—612no—
292#418C2×D6112yes—
293#418C2×D6112yes—
294C2×#102—612no—
295C2×#100—612no—
296C2×#100—611no—
297C2×#100—611no—
298C2×#100—612no—
299#405—612no—
300C4×M16—611no—
301C4×M16C2×D6111yescharacteristic
302#343—611no—
303#343—611no—
304#341—612no—
305C61#2531211yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—611no—
309C22×D4C22×D6112yes—
310C23×C4—1221no—
311C2×D8—1222no—
312C2×D8—1222no—
313C2×D8—1224no—
314C2×D8—2442no—
315C2×M16—611no—
316C2×M16—612no—
317C2×M16—611no—
318C2×M16C22×D6111yescharacteristic
319C2×M16C22×D6111yescharacteristic
320C22×C8—612no—
321C4⋊Q8C22⋊D6111yescharacteristic
322C4⋊Q8C22⋊D6111yescharacteristic
323#127—21no—
324#127—21no—
325#127—21no—
326#127—21no—
327#125C22⋊D6112yes—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—1221no—
333C4×D4—1221no—
334C4×D4—1221no—
335C4×D4—1221no—
336#117C22×D6112yes—
337C2×C4⋊C4—611no—
338C2×C22⋊C4—1221no—
339C2×C42C22×D6111yescharacteristic
340#108—2442no—
341#107—1222no—
342#106—1222no—
343C4⋊C8—611no—
344C4⋊C8—611no—
345C4⋊C8—612no—
346C4⋊C8—1222no—
347C4⋊C8—1222no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350#102—22no—
351#102—22no—
352#102—22no—
353#102—22no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1222no—
360#100—1224no—
361#100—2442no—
362#97—21no—
363#97—1222no—
364#97—1221no—
365C4×C8—21no—
366C4×C8—1222no—
367C4×C8—1221no—
368C24—22no—
369C24—1221no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—612no—
385C2×D4—1221no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—1221no—
389C2×D4—1221no—
390C22×C4—1221no—
391C22×C4—4882no—
392C22×C4—2441no—
393C22×C4—2441no—
394C22×C4#1175822411yescharacteristic
395C22×C4#1175822411yescharacteristic
396C22×C4C23×D6111yescharacteristic
397D8—2444no—
398D8—2444no—
399M16—21no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—612no—
404M16—1221no—
405M16—1221no—
406M16—1222no—
407M16—1221no—
408M16—1222no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—612no—
412C2×C8—612no—
413C2×C8—1221no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1221no—
417C2×C8—1222no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4C2×C22⋊D6112yes—
425C4⋊C4C2×C22⋊D6112yes—
426C22⋊C4—22no—
427C22⋊C4—2441no—
428C22⋊C4—2441no—
429C42—22no—
430C42D4×D6111yescharacteristic
431C42D4×D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C23—42no—
435C23—42no—
436C23—42no—
437C23—82no—
438C23—82no—
439C23—1222no—
440C23—4881no—
441C23—4881no—
442C23—2441no—
443C23—2441no—
444C23C2×#1175822411yescharacteristic
445Q8—42no—
446Q8—81no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—82no—
453D4—82no—
454D4—82no—
455D4—1222no—
456D4—1222no—
457D4—2441no—
458D4—2441no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—4881no—
466C2×C4—2441no—
467C2×C4#1451974911yescharacteristic
468C2×C4#1451974911yescharacteristic
469C2×C4#1451974911yescharacteristic
470C2×C4#1451974911yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4#41×D6111yescharacteristic
475C2×C4#1451982311yescharacteristic
476C2×C4#1451982411yescharacteristic
477C8—41no—
478C8—41no—
479C8—1222no—
480C8—1222no—
481C8—1222no—
482C8—2441no—
483C8—2442no—
484C22—21no—
485C22—21no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—162no—
490C22—162no—
491C22—4882no—
492C22—4882no—
493C22—4881no—
494C22—4881no—
495C22—2442no—
496C22C2×#1451974911yescharacteristic
497C22#1684029111yescharacteristic
498C22#1684038111yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—42no—
503C4—42no—
504C4—4881no—
505C4#1684003012yes—
506C4#1684040511yescharacteristic
507C4#1684041911yescharacteristic
508C4#1684042011yescharacteristic
509C4#1684043111yescharacteristic
510C2—21no—
511C2—21no—
512C2—162no—
513C2—4882no—
514C2#2144938811yescharacteristic
515C2C2×#1684003011yescharacteristic
516C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3138C2
socleC22×C61C2×#121
prime radicalC2×#16839954C2
hypercenter#3138D61
hypocenterC61#25312
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21454768C2semidirect
4. #21447179C2semidirect
5. #21447074C2semidirect
6. #21446728C2semidirect
7. #21446660C2semidirect
8. #21446594C2semidirect
9. #21446484C2semidirect
10. C4×#14519693C2semidirect
11. C61×#3138C2semidirect
252. #3138D61semidirect
305. C61#25312semidirect

Maximal subgroups

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

subgroupindex
#214556122
C2×#168399542
#214547682
#214471792
#214470742
#214467282
#214466602
#214465942
#214464842
C4×#145196932
C61×#31382
#2531261