Finite Group Databasebeta

← #29646153

Subgroups — #29646153

Order 15616.

1048 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840346C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21454847C211yesmaximal, characteristic, complemented
4#21444189C211yesmaximal, characteristic, complemented
5#21444186C211yesmaximal, characteristic, complemented
6#21444177C212yesmaximal, complemented
7#21444171C211yesmaximal, characteristic, complemented
8#21444164C212yesmaximal, complemented
9#21444154C212yesmaximal, complemented
10#21444144C212yesmaximal, complemented
11C61×#1469C211yesmaximal, characteristic, complemented
12#16840346—22no—
13#16840346—22no—
14#16840346—22no—
15#16840346—22no—
16C2×#14519771C2211yescharacteristic
17C2×#14519770C411yescharacteristic
18C2×#14519766C411yescharacteristic
19C2×#14519761C411yescharacteristic
20C2×#14519759C2211yescharacteristic
21C2×#14519753C411yescharacteristic
22C2×#14519750C411yescharacteristic
23C2×#14519750C2211yescharacteristic
24C2×#14519749C411yescharacteristic
25C2×#14519747C411yescharacteristic
26C2×#14519745C411yescharacteristic
27C2×C4×D244C2211yescharacteristic
28C2×#14519737C2211yescharacteristic
29C2×C22⋊C4×D61C2211yescharacteristic
30#16839841—22no—
31#16839839—22no—
32#16839835—22no—
33#16839832—22no—
34#16839819C2211yescharacteristic, complemented
35#16839815C2211yescharacteristic, complemented
36#16839724C2212yes—
37#16839724C2212yes—
38#16839724C2212yescomplemented
39C22⋊C8×D61C2212yescomplemented
40#16839487C2212yescomplemented
41#16839487C2212yescomplemented
42#16839486C2212yes—
43#16839486C2212yescomplemented
44#16839485C2212yes—
45#16839485C2212yes—
46C2×#124×C61C2211yescharacteristic
47C61×#432C2211yescharacteristic, complemented
48C61×#431C2211yescharacteristic, complemented
49C61×#323C2212yes—
50C61×#323C2212yescomplemented
51C24×D61C2×C411yescharacteristic
52C22×D244C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C411yescharacteristic
57C22×C22⋊D61C2×C411yescharacteristic
58C23×Dic61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—22no—
63C2×#11758189—22no—
64C2×#11758189—22no—
65C2×C8×D61—22no—
66#14519771—21no—
67#14519771—21no—
68#14519771—21no—
69#14519771—21no—
70#14519770—22no—
71#14519770—22no—
72#14519766—22no—
73#14519766—22no—
74#14519761—22no—
75#14519761—22no—
76#14519759—22no—
77#14519759—22no—
78#14519753—22no—
79#14519753—22no—
80#14519750—22no—
81#14519750—22no—
82#14519750—22no—
83#14519750—22no—
84#14519749—22no—
85#14519749—22no—
86#14519747—22no—
87#14519747—22no—
88#14519745—22no—
89#14519745—22no—
90C4×D244D412yes—
91C4×D244D412yes—
92C4×D244D412yes—
93C4×D244D412yes—
94#14519737—22no—
95#14519737—22no—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×#11758196C2311yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×C22⋊Dic61C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2×C411yescharacteristic
111C2×#11758198C2311yescharacteristic
112C22⋊C4×D61—22no—
113C22⋊C4×D61—22no—
114C2×C4×Dic61C2×C411yescharacteristic
115C4⋊#92418—21no—
116C4⋊#92418—21no—
117C22⋊#92418C2312yes—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519689—22no—
123#14519688—21no—
124C4×#92418—21no—
125C22×Q8×C61C2×C411yescharacteristic
126C22×D4×C61C2×C411yescharacteristic
127#124×C61—21no—
128#124×C61—21no—
129#124×C61—21no—
130#124×C61—21no—
131C2×C22⋊C4×C61C2×C411yescharacteristic
132C2×C22⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139C22⋊C8×C61C2312yes—
140#97×C61—21no—
141C4×C8×C61—21no—
142C24×C61C22×C411yescharacteristic
143C24×C61C22×C411yescharacteristic
144C23×D61—22no—
145C23×D61—42no—
146C23×D61—81no—
147C23×D61—81no—
148C23×D61C22×C411yescharacteristic
149C23×D61C22×C411yescharacteristic
150C23×D61C22×C411yescharacteristic
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—81no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—81no—
157C22×C4×C61C22×C411yescharacteristic
158C22×C4×C61C22×C411yescharacteristic
159C22×C4×C61C22×C411yescharacteristic
160C22×C4×C61C22×C411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C2×C22⋊D61—22no—
166C2×C22⋊D61—42no—
167C2×C22⋊D61—81no—
168C2×C22⋊D61—81no—
169C2×C22⋊D61—81no—
170C22×Dic61—22no—
171C22×Dic61—81no—
172C22×Dic61C22×C411yescharacteristic
173C22×Dic61C22×C411yescharacteristic
174C22×Dic61C22×C411yescharacteristic
175C22×Dic61C22×C411yescharacteristic
176C22×Dic61C22×C411yescharacteristic
177C2×D244—22no—
178C2×D244—42no—
179C2×D244—81no—
180C2×D244C22⋊C412yes—
181C2×D244C22⋊C412yes—
182C2×C4×D61—22no—
183C2×C4×D61—81no—
184C2×C4×D61—81no—
185C2×C4×D61—81no—
186C2×C4×D61C2×D412yes—
187C2×C4×D61C2×D412yes—
188C2×C4×D61C2×D412yes—
189C2×C4×D61C2×D412yes—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C42×C61C2×D411yescharacteristic
203C42×C61C2×D411yescharacteristic
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—21no—
209#11758198—21no—
210#11758198—21no—
211#11758198—21no—
212#11758198—21no—
213#11758198—21no—
214#11758198—22no—
215#11758198C22⋊C412yes—
216#11758198C22⋊C412yes—
217#11758198C22⋊C412yes—
218#11758198C22⋊C412yes—
219C4⋊Dic61—22no—
220C4⋊Dic61C22⋊C412yes—
221C4⋊Dic61C22⋊C412yes—
222#11758196—21no—
223#11758196—21no—
224#11758196—21no—
225#11758196—21no—
226#11758196—21no—
227#11758196—21no—
228#11758196—22no—
229C4×Dic61—22no—
230C2×#92418—22no—
231C2×#92418—22no—
232#11758189—82no—
233#11758189—82no—
234#11758189—82no—
235C8×D61—82no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—22no—
239C23×C61—22no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61—41no—
243C23×C61—41no—
244C23×C61C23×C411yescharacteristic
245C22×D61—22no—
246C22×D61—22no—
247C22×D61—21no—
248C22×D61—22no—
249C22×D61—41no—
250C22×D61—41no—
251C22×D61—41no—
252C22×D61—41no—
253C22×D61—44no—
254C22×D61—41no—
255C22×D61—41no—
256C22×D61—82no—
257C22×D61—82no—
258C22×D61—88no—
259C22×D61C2×C22⋊C412yes—
260C22×D61C2×C22⋊C412yes—
261Q8×C61—81no—
262Q8×C61—81no—
263D4×C61—81no—
264D4×C61—81no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61C2×C22⋊C411yescharacteristic
273C2×C4×C61C2×C22⋊C411yescharacteristic
274C2×C4×C61C2×C22⋊C411yescharacteristic
275C2×C4×C61C2×C22⋊C411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C22⋊D61—44no—
279C22⋊D61—82no—
280C22⋊D61—82no—
281C22⋊D61—82no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—22no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—82no—
291C2×Dic61C2×C22⋊C412yes—
292C2×Dic61C2×C22⋊C412yes—
293D244—42no—
294D244—42no—
295D244—82no—
296C4×D61—42no—
297C4×D61—42no—
298C4×D61—42no—
299C4×D61—42no—
300C4×D61—82no—
301C8×C61—42no—
302C8×C61—42no—
303#92418—42no—
304#92418—42no—
305#17405—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—82no—
315C22×C61—82no—
316C22×C61C22×C22⋊C411yescharacteristic
317C22×C61#56812yes—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—22no—
324C2×D61—44no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—82no—
329C2×D61—88no—
330C2×D61—82no—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—21no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—21no—
337C4×C61—41no—
338C4×C61—41no—
339Dic61—22no—
340Dic61—22no—
341Dic61—41no—
342Dic61—41no—
343Dic61—41no—
344C2×#538—611no—
345#2975—611no—
346#2973—611no—
347#1473—611no—
348#1473—611no—
349#1471—612no—
350#1469—611no—
351#1469D6111yescharacteristic, complemented
352#1468—612no—
353#1465—612no—
354#1462—612no—
355C2×C61—22no—
356C2×C61—81no—
357C2×C61—81no—
358C2×C61#287911yescharacteristic
359C2×C61#288311yescharacteristic
360C2×C61#288811yescharacteristic
361D61—42no—
362D61—42no—
363D61—81no—
364#538—1222no—
365#538—1222no—
366#538—1222no—
367#538—1222no—
368C2×#126—611no—
369C2×#126—611no—
370C2×#124—611no—
371C2×#124C2×D6111yescharacteristic
372C2×#123—611no—
373C2×#123—611no—
374C2×C22⋊Q8—611no—
375C2×#121—611no—
376C2×#121—611no—
377C2×#121—611no—
378C2×#120—611no—
379C2×#120—611no—
380C2×C4×D4—611no—
381C2×C4×D4—611no—
382C22×C22⋊C4—611no—
383#436—1222no—
384#436—1222no—
385#435—1222no—
386#435—1222no—
387#432—611no—
388#432C2×D6111yescharacteristic, complemented
389#431—611no—
390#431C2×D6111yescharacteristic, complemented
391#407—612no—
392#407—612no—
393#407—612no—
394C2×C22⋊C8—612no—
395#324—612no—
396#324—612no—
397#323—612no—
398#323—612no—
399#323—612no—
400#323—612no—
401#323C2×D6112yes—
402#323C2×D6112yescomplemented
403C61#1740511yescharacteristic, complemented, Sylow
404C25—611no—
405C22×Q8C4×D6111yescharacteristic
406C22×D4—611no—
407C22×D4—611no—
408C22×D4—611no—
409C22×D4—611no—
410C22×D4—611no—
411C22×D4—611no—
412C22×D4C4×D6111yescharacteristic
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—611no—
416C23×C4—611no—
417C2×M16—1222no—
418C2×M16—1222no—
419C2×M16—1222no—
420C22×C8—1222no—
421#126—1222no—
422#126—1222no—
423#126—1221no—
424#126—1221no—
425#126—1221no—
426#126—1221no—
427#124—21no—
428#124—21no—
429#124—21no—
430#124—21no—
431#124—1222no—
432#124—1222no—
433#123—1222no—
434#123—1222no—
435#123—1222no—
436#123—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439#121—1222no—
440#121—1222no—
441#121—1222no—
442#121—1222no—
443#121—1222no—
444#121—1222no—
445#120—1222no—
446#120—1222no—
447#120—1222no—
448#120—1222no—
449C4×D4—612no—
450C4×D4—612no—
451C4×D4—612no—
452C4×D4—612no—
453C4×D4—1222no—
454C4×D4—1222no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—611no—
460C2×C4⋊C4—611no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—611no—
464C2×C22⋊C4—611no—
465C2×C22⋊C4—611no—
466C2×C22⋊C4—611no—
467C2×C22⋊C4—611no—
468C2×C22⋊C4—611no—
469C2×C22⋊C4—611no—
470C2×C22⋊C4—611no—
471C2×C22⋊C4—1222no—
472C2×C22⋊C4—1222no—
473C2×C22⋊C4C4×D6111yescharacteristic
474C2×C22⋊C4C4×D6111yescharacteristic
475C2×C22⋊C4C22×D6111yescharacteristic
476C2×C22⋊C4C22×D6111yescharacteristic
477C2×C42—611no—
478C2×C42C22×D6111yescharacteristic
479C4⋊C8—21no—
480C4⋊C8—21no—
481C4⋊C8—1221no—
482C4⋊C8—1221no—
483C22⋊C8—612no—
484C22⋊C8—612no—
485C22⋊C8—1222no—
486C22⋊C8—1222no—
487C22⋊C8—1222no—
488C22⋊C8—1222no—
489C22⋊C8C22×D6112yes—
490C22⋊C8C22×D6112yes—
491#97—21no—
492#97—1221no—
493C4×C8—21no—
494C4×C8—1221no—
495C24—611no—
496C24—611no—
497C24—611no—
498C24—1222no—
499C24—4881no—
500C24—4881no—
501C24—2442no—
502C24C2×C4×D6111yescharacteristic
503C24C2×C4×D6111yescharacteristic
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—81no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—81no—
510C2×D4—612no—
511C2×D4—612no—
512C2×D4—1222no—
513C2×D4—1222no—
514C2×D4—1222no—
515C2×D4—1222no—
516C2×D4—1222no—
517C2×D4—4881no—
518C2×D4—4881no—
519C2×D4—4881no—
520C2×D4—4881no—
521C2×D4—2442no—
522C2×D4—2442no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—612no—
526C22×C4—612no—
527C22×C4—612no—
528C22×C4—612no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—4881no—
535C22×C4—4881no—
536C22×C4—4881no—
537C22×C4—4881no—
538C22×C4C2×C4×D6111yescharacteristic
539C22×C4C2×C4×D6111yescharacteristic
540C22×C4C2×C4×D6111yescharacteristic
541C22×C4C2×C4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543M16—4882no—
544M16—4882no—
545M16—4882no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—1222no—
549C2×C8—1222no—
550C2×C8—4882no—
551C4⋊C4—612no—
552C4⋊C4—612no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—612no—
570C22⋊C4—612no—
571C22⋊C4—612no—
572C22⋊C4—612no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1221no—
579C22⋊C4—1221no—
580C22⋊C4—1221no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1221no—
584C42—1222no—
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42D4×D6111yescharacteristic
588C42D4×D6111yescharacteristic
589C23—21no—
590C23—21no—
591C23—22no—
592C23—22no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—612no—
598C23—612no—
599C23—1222no—
600C23—1222no—
601C23—1222no—
602C23—1221no—
603C23—4882no—
604C23—4882no—
605C23—4888no—
606C23—2441no—
607C23—2441no—
608C23—2441no—
609C23—2441no—
610C23—2441no—
611C23—2441no—
612C23—2444no—
613C23C22×C4×D6111yescharacteristic
614Q8—81no—
615Q8—81no—
616D4—81no—
617D4—81no—
618D4—4882no—
619D4—4882no—
620D4—4882no—
621D4—4882no—
622D4—2442no—
623D4—2442no—
624D4—2444no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—612no—
633C2×C4—612no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1222no—
638C2×C4—4882no—
639C2×C4—4882no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4C22⋊C4×D6111yescharacteristic
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22⋊C4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4C2×D4×D6111yescharacteristic
654C8—42no—
655C8—42no—
656C8—2442no—
657C8—2442no—
658C22—21no—
659C22—21no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—82no—
667C22—82no—
668C22—1222no—
669C22—1222no—
670C22—1222no—
671C22—1222no—
672C22—1222no—
673C22—1222no—
674C22—4882no—
675C22—4882no—
676C22—4888no—
677C22—2444no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22C2×C22⋊C4×D6111yescharacteristic
682C22#1684052712yes—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—41no—
690C4—41no—
691C4—1222no—
692C4—1222no—
693C4—2441no—
694C4—2441no—
695C4—2441no—
696C2—22no—
697C2—81no—
698C2—81no—
699C2—4881no—
700C2—2442no—
701C2—2442no—
702C2#2145426111yescharacteristic
703C2#2145427811yescharacteristic
704C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1469C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840346C2
cosocleC22×C4×C61C24
hypercenter#1469D61
hypocenterC61#17405
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21454847C2semidirect
4. #21444189C2semidirect
5. #21444186C2semidirect
6. #21444177C2semidirect
7. #21444171C2semidirect
8. #21444164C2semidirect
9. #21444154C2semidirect
10. #21444144C2semidirect
11. C61×#1469C2semidirect
34. #16839819C22semidirect
35. #16839815C22semidirect
38. #16839724C22semidirect
39. C22⋊C8×D61C22semidirect
40. #16839487C22semidirect
41. #16839487C22semidirect
43. #16839486C22semidirect
47. C61×#432C22semidirect
48. C61×#431C22semidirect
50. C61×#323C22semidirect
351. #1469D61semidirect
388. #432C2×D61semidirect
390. #431C2×D61semidirect
402. #323C2×D61semidirect
403. C61#17405semidirect

Maximal subgroups

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

subgroupindex
C2×#168403462
#214548592
#214548472
#214441892
#214441862
#214441772
#214441712
#214441642
#214441542
#214441442
C61×#14692
#1740561