Finite Group Databasebeta

← #29689846

Subgroups — #29689846

Order 15616.

1324 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2#21454450C211yesmaximal, characteristic, complemented
3#423×D61C211yesmaximal, characteristic, complemented
4#21446809C211yesmaximal, characteristic, complemented
5#21446659C211yesmaximal, characteristic, complemented
6#21446658C211yesmaximal, characteristic, complemented
7#21446281C214yesmaximal, complemented
8#21445810C214yesmaximal, complemented
9C61×#1997C211yesmaximal, characteristic, complemented
10C22×#11758226C412yes—
11C2×M16×D61C2212yes—
12#16840313—21no—
13#16840313—21no—
14#16840313—21no—
15#16840313—21no—
16#16840313—21no—
17#16840313—21no—
18#16840313—21no—
19#16840313—21no—
20C2×#14519774C2211yescharacteristic
21C2×#14519766C412yes—
22C2×#14519752C412yes—
23C2×#14519752C2212yes—
24C2×#14519739C412yes—
25C2×#14519739C2212yes—
26C2×C42×D61C2211yescharacteristic
27#16839817—21no—
28#16839817—21no—
29#16839813—21no—
30#16839813—21no—
31#16839786C2211yescharacteristic
32#16839785C2212yes—
33#16839785C2212yes—
34#16839784—21no—
35#16839784—21no—
36C4⋊C8×D61—21no—
37C4⋊C8×D61—21no—
38C2×#14519693C2214yes—
39C2×#14519693C2214yes—
40#16839725C2212yes—
41C4×#11758194C2211yescharacteristic
42#16839531C2214yes—
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#423C2211yescharacteristic
45C4×M16×C61C2211yescharacteristic
46C61×#343C2214yes—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226—22no—
51C2×#11758226—48no—
52C22×D244C2×C412yes—
53C22×D244C2×C414yes—
54C22×D244C2×C412yes—
55C22×D244C2312yes—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C414yes—
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C2×#11758194C2312yes—
61C2×#11758194C2312yes—
62C2×#11758189—22no—
63M16×D61—44no—
64C2×C8×D61—22no—
65#14519774—21no—
66#14519774—21no—
67#14519774—21no—
68#14519774—21no—
69#14519766—22no—
70#14519766—22no—
71#14519766—22no—
72#14519766—22no—
73#14519752—22no—
74#14519752—22no—
75#14519752—22no—
76#14519752—22no—
77#14519752—22no—
78#14519752—22no—
79#14519752—22no—
80#14519752—22no—
81#14519739—22no—
82#14519739—22no—
83#14519739—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—22no—
87#14519739—22no—
88#14519739—22no—
89C2×#11758198C2×C414yes—
90C2×#11758198C2×C414yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C4⋊#92418—21no—
103C4⋊#92418—21no—
104#14519706—22no—
105#14519706—22no—
106#14519706—22no—
107#14519706—22no—
108#14519693—44no—
109#14519693—44no—
110#14519689—22no—
111#14519689—22no—
112#14519688—21no—
113C4×#92418—21no—
114C22×Q8×C61C2×C412yes—
115C2×M16×C61C2312yes—
116C2×M16×C61C2312yes—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C2×C4⋊C4×C61C2×C412yes—
122C2×C4⋊C4×C61C2312yes—
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—21no—
125C4⋊C8×C61—21no—
126#97×C61—21no—
127C4×C8×C61—21no—
128C23×D61C22×C411yescharacteristic
129C23×D61C22×C414yes—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61—44no—
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C414yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140#11758226—48no—
141#11758226—48no—
142#11758226—48no—
143#11758226—48no—
144C2×D244—24no—
145C2×D244—24no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—22no—
149C2×D244—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—44no—
155C2×D244—44no—
156C2×D244—48no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—48no—
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176M16×C61—22no—
177M16×C61—24no—
178M16×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193C4×Dic61C22⋊C411yescharacteristic
194C4×Dic61C22⋊C411yescharacteristic
195C4×Dic61C22⋊C411yescharacteristic
196C4×Dic61C22⋊C411yescharacteristic
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205#11758194—22no—
206#11758194—24no—
207#11758194—22no—
208C2×#92418—22no—
209C2×#92418—22no—
210#11758189—44no—
211C8×D61—44no—
212C23×C61C23×C411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—24no—
216C22×D61—28no—
217C22×D61—41no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61C2×C22⋊C411yescharacteristic
221C22×D61C2×C22⋊C411yescharacteristic
222Q8×C61—44no—
223Q8×C61—44no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—44no—
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61H(5,𝔽2)12yes—
235C2×C4×C612−1+412yes—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245D244—44no—
246D244—44no—
247D244—48no—
248D244—48no—
249D244—42no—
250D244—42no—
251D244—44no—
252D244—44no—
253D244—44no—
254C4×D61—22no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—48no—
268C4×D61—48no—
269C4×D61#10014yescomplemented
270C8×C61—42no—
271C8×C61—42no—
272#92418—42no—
273#92418—42no—
274#21878—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61#52012yes—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—44no—
287C2×D61—44no—
288C2×D61—88no—
289C2×D61C2×#10012yes—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—44no—
293C4×C61C2×#10012yes—
294C4×C61#41312yes—
295C4×C61#53511yescharacteristic
296C4×C61#53611yescharacteristic
297C4×C61#53712yes—
298Dic61—21no—
299Dic61—21no—
300Dic61—21no—
301Dic61—21no—
302Dic61—21no—
303Dic61C2×#10012yes—
304C2×#532—611no—
305#2920—611no—
306C2×#423—611no—
307#1997—611no—
308#1997D6111yescharacteristic, complemented
309#1965—611no—
310#1965—611no—
311#1900—614no—
312#1800—614no—
313C2×C61—22no—
314C2×C61#231111yescharacteristic
315C2×C61C22×#10011yescharacteristic
316C2×C61C2×#41311yescharacteristic
317D61—22no—
318D61—41no—
319D61—84no—
320C22×#41—612no—
321C22×M16—612no—
322#532—1221no—
323#532—1221no—
324#532—1221no—
325#532—1221no—
326#532—1221no—
327#532—1221no—
328#532—1221no—
329#532—1221no—
330C2×C4⋊Q8C2×D6111yescharacteristic
331C2×#127—611no—
332C2×#124—612no—
333C2×#121—612no—
334C2×#121—612no—
335C2×C4×D4—612no—
336C2×C4×D4—612no—
337C22×C42—611no—
338#432—1221no—
339#432—1221no—
340#431—1221no—
341#431—1221no—
342#423—612no—
343#423—612no—
344#423—611no—
345#423—1221no—
346#423—1221no—
347#423C2×D6111yescharacteristic
348C2×C4⋊C8—1221no—
349C2×C4⋊C8—1221no—
350C2×#100—614no—
351C2×#100—614no—
352#407—612no—
353C4×M16—611no—
354C4×M16C2×D6111yescharacteristic
355#343—614no—
356#343C2×D6114yes—
357C61#2187811yescharacteristic, complemented, Sylow
358C2×#41—1222no—
359C2×#41—1222no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—2448no—
363C22×Q8C4×D6112yes—
364C22×D4—612no—
365C22×D4—614no—
366C22×D4—612no—
367C22×D4—612no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C2×M16—612no—
373C2×M16—612no—
374C2×M16—1222no—
375C2×M16—2444no—
376C2×M16C22×D6112yes—
377C2×M16C22×D6112yes—
378C22×C8—1222no—
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—21no—
383#127—1221no—
384#127—1221no—
385#127—1221no—
386#127—1221no—
387#124—1222no—
388#124—1222no—
389#124—1222no—
390#124—1222no—
391#121—1222no—
392#121—1222no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#121—1222no—
397#121—1222no—
398#121—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—1222no—
407C2×C4⋊C4C4×D6112yes—
408C2×C4⋊C4C22×D6112yes—
409C2×C22⋊C4—614no—
410C2×C22⋊C4—614no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8—1222no—
426C4⋊C8—1222no—
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431#100—2444no—
432#100—2444no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435#97—21no—
436#97—1221no—
437C4×C8—21no—
438C4×C8—1221no—
439C24—611no—
440C24—614no—
441#41—2448no—
442#41—2448no—
443#41—2448no—
444#41—2448no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×D4—1224no—
449C2×D4—1224no—
450C2×D4—1222no—
451C2×D4—1222no—
452C2×D4—1224no—
453C2×D4—1224no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—2444no—
459C2×D4—2444no—
460C2×D4—2448no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—2442no—
482C22×C4—2448no—
483C22×C4C2×C4×D6111yescharacteristic
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4C2×C4×D6114yes—
486C22×C4C23×D6111yescharacteristic
487M16—22no—
488M16—24no—
489M16—22no—
490M16—1222no—
491M16—1224no—
492M16—1222no—
493M16—2444no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—2444no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C22⋊C4—1224no—
504C22⋊C4—1224no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C42D4×D6111yescharacteristic
522C42D4×D6111yescharacteristic
523C23—611no—
524C23—611no—
525C23—1222no—
526C23—1222no—
527C23—1228no—
528C23—1224no—
529C23—2441no—
530C23—2444no—
531C23—2444no—
532C23C22×C4×D6111yescharacteristic
533Q8—44no—
534Q8—44no—
535D4—2444no—
536D4—2444no—
537D4—2448no—
538D4—2448no—
539D4—2444no—
540D4—2444no—
541D4—2442no—
542D4—2442no—
543D4—2444no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—44no—
548C2×C4—611no—
549C2×C4—614no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2448no—
571C2×C4—2448no—
572C2×C4—2441no—
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22⋊C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4#1451982312yes—
580C2×C4#1451982512yes—
581C8—42no—
582C8—42no—
583C8—2442no—
584C8—2442no—
585C22—21no—
586C22—21no—
587C22—612no—
588C22—1221no—
589C22—1221no—
590C22—1221no—
591C22—1221no—
592C22—1221no—
593C22—4888no—
594C22—2444no—
595C22—2444no—
596C22—2444no—
597C22—2444no—
598C22C2×C22⋊C4×D6111yescharacteristic
599C22#1684024412yes—
600C4—21no—
601C4—21no—
602C4—44no—
603C4—612no—
604C4—1221no—
605C4—1221no—
606C4—1221no—
607C4—1221no—
608C4—1221no—
609C4#100×D6112yes—
610C4#1683974512yes—
611C4#1684033011yescharacteristic
612C4#1684033711yescharacteristic
613C4#1684034212yes—
614C2—22no—
615C2—1222no—
616C2—4884no—
617C2—2441no—
618C2#2144841011yescharacteristic
619C2C2×#100×D6111yescharacteristic
620C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840313C2
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#21878
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
2. #21454450C2semidirect
3. #423×D61C2semidirect
4. #21446809C2semidirect
5. #21446659C2semidirect
6. #21446658C2semidirect
7. #21446281C2semidirect
8. #21445810C2semidirect
9. C61×#1997C2semidirect
269. C4×D61#100semidirect
308. #1997D61semidirect
357. C61#21878semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
#214544502
#423×D612
#214468092
#214466592
#214466582
#214462812
#214458102
C61×#19972
#2187861