Finite Group Databasebeta

← #29712554

Subgroups — #29712554

Order 15616.

1250 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
1#21455210C211yesmaximal, characteristic, complemented
2#21455167C211yesmaximal, characteristic, complemented
3#21454367C211yesmaximal, characteristic, complemented
4#21446988C212yesmaximal, complemented
5#21446884C212yesmaximal, complemented
6#21446320C212yesmaximal, complemented
7#21446262C212yesmaximal, complemented
8#21446238C211yesmaximal, characteristic, complemented
9#21446203C211yesmaximal, characteristic, complemented
10#21445923C211yesmaximal, characteristic, complemented
11C61×#3444C211yesmaximal, characteristic, complemented
12#16840598C2211yescharacteristic
13#16840560C2212yes—
14#16840529C2211yescharacteristic
15#16840273C2212yes—
16#16839951—22no—
17#16839948—22no—
18#16839944—21no—
19#16839939—21no—
20#16839934—21no—
21#16839929—21no—
22#16839780—22no—
23#16839780C2212yes—
24#16839779C2212yescomplemented
25C2×#14519705—22no—
26C2×#14519705C2212yes—
27C2×#14519705C2212yes—
28#16839746C2211yescharacteristic
29#16839736C2211yescharacteristic
30#16839735C2212yes—
31#16839726C2212yescomplemented
32#16839532C2212yescomplemented
33#16839522C2212yescomplemented
34#16839522C2212yescomplemented
35#16839518C2212yescomplemented
36C61×#533—21no—
37C61×#533C2211yescharacteristic, complemented
38C61×#532—21no—
39C61×#532C2211yescharacteristic
40C61×#529—21no—
41C61×#529—21no—
42C61×#529—22no—
43C61×#529—22no—
44C61×#525—22no—
45C61×#525C2211yescharacteristic
46C61×#525C2211yescharacteristic, complemented
47C4×#41×C61—22no—
48C4×#41×C61C2211yescharacteristic
49C4×#41×C61C2211yescharacteristic, complemented
50C22×C42×C61C2211yescharacteristic, complemented
51#14519825—21no—
52#14519824—22no—
53#14519823—22no—
54#14519822—21no—
55#14519822—22no—
56C2×#11758222—22no—
57C2×#11758222C2312yes—
58#41×D61—42no—
59#14519808—22no—
60#14519807—22no—
61#14519803—22no—
62#14519802—22no—
63#14519797—42no—
64#14519784—42no—
65C2×#11758194—22no—
66C2×#11758194C2312yes—
67C2×#11758194C2312yes—
68#14519760—22no—
69#14519756—22no—
70#14519751—22no—
71#14519746—22no—
72C4×C22⋊D61—42no—
73#14519736C2312yes—
74#14519705—22no—
75#14519705—22no—
76#14519705—22no—
77#14519705—24no—
78#14519705—22no—
79#14519704—42no—
80#14519696—21no—
81#14519694—21no—
82#14519692—22no—
83#14519691—24no—
84C22⋊#92418—42no—
85C2×#41×C61—22no—
86C2×#41×C61C2311yescharacteristic
87C2×#41×C61C2311yescharacteristic
88C23×C4×C61—21no—
89C23×C4×C61C2311yescharacteristic
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61D411yescharacteristic, complemented
92#127×C61D411yescharacteristic, complemented
93#126×C61—22no—
94#126×C61—22no—
95#125×C61—22no—
96#125×C61—21no—
97#124×C61—22no—
98#124×C61D411yescharacteristic, complemented
99#124×C61D411yescharacteristic, complemented
100#123×C61—22no—
101#123×C61—22no—
102#123×C61—21no—
103#123×C61D412yes—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—21no—
107C22⋊Q8×C61—22no—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61—22no—
112#120×C61—21no—
113#120×C61D412yes—
114C4×Q8×C61—22no—
115C4×Q8×C61—21no—
116C4×Q8×C61—21no—
117C4×Q8×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—22no—
124C4×D4×C61—22no—
125C4×D4×C61—22no—
126C4×D4×C61—22no—
127C4×D4×C61—22no—
128#117×C61—22no—
129#117×C61—22no—
130#117×C61—22no—
131#117×C61—21no—
132#117×C61D412yescomplemented
133#117×C61C2311yescharacteristic
134#117×C61C2311yescharacteristic
135C2×C42×C61—21no—
136C2×C42×C61—21no—
137C2×C42×C61—21no—
138C2×C42×C61—21no—
139C2×C42×C61—21no—
140C2×C42×C61—21no—
141C2×C42×C61D412yescomplemented
142C2×C42×C61C2311yescharacteristic
143C2×C42×C61C2311yescharacteristic
144C24×C61C2×D411yescharacteristic
145#41×C61—22no—
146#41×C61—22no—
147#41×C61—22no—
148#41×C61—22no—
149#41×C61—48no—
150C2×Q8×C61—22no—
151C2×Q8×C61C2×D411yescharacteristic
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61—22no—
154C2×D4×C61—22no—
155C2×D4×C61—22no—
156C2×D4×C61—21no—
157C2×D4×C61C2×D412yes—
158C2×D4×C61C2×D411yescharacteristic
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—22no—
165C22×C4×C61—22no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—44no—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D412yes—
176C22×C4×C61C2411yescharacteristic
177C2×C22⋊D61—22no—
178C2×C22⋊D61—42no—
179C2×C22⋊D61—42no—
180#11758226—42no—
181Q8×D61—42no—
182#11758224—42no—
183#11758224—44no—
184D4×D61—42no—
185D4×D61—44no—
186#11758222—24no—
187#11758222—22no—
188#11758222—22no—
189#11758222—22no—
190#11758222—22no—
191#11758222—42no—
192#11758222—42no—
193C2×D244—22no—
194C2×D244#4112yescomplemented
195C2×C4×D61—22no—
196C2×C4×D61—42no—
197C2×Dic122—22no—
198C2×Dic122#4112yescomplemented
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61—21no—
209C22⋊C4×C61—22no—
210C22⋊C4×C61—22no—
211C22⋊C4×C61—22no—
212C22⋊C4×C61—22no—
213C22⋊C4×C61—22no—
214C22⋊C4×C61—22no—
215C22⋊C4×C61—21no—
216C22⋊C4×C61—22no—
217C22⋊C4×C61—22no—
218C22⋊C4×C61C2×D412yes—
219C42×C61—21no—
220C42×C61—21no—
221C42×C61—21no—
222C42×C61—21no—
223C42×C61—22no—
224C42×C61—22no—
225C42×C61—21no—
226C42×C61—21no—
227C42×C61—21no—
228C42×C61C2×D411yescharacteristic
229C42×C61C2×D411yescharacteristic
230C42×C61C2×D411yescharacteristic
231C42×C61C2×D411yescharacteristic
232C42×C61C2×D412yes—
233C22⋊Dic61—22no—
234C22⋊Dic61—42no—
235#11758202—42no—
236#11758201—42no—
237#11758200—42no—
238#11758199—42no—
239#11758198—42no—
240C4⋊Dic61—22no—
241#11758196—42no—
242C4×Dic61—22no—
243#11758194—22no—
244#11758194—22no—
245#11758194—22no—
246#11758194—22no—
247#11758194—42no—
248C2×#92418—22no—
249C2×#92418—22no—
250C2×#92418—42no—
251C23×C61—21no—
252C23×C61—22no—
253C23×C61—21no—
254C23×C61—42no—
255C23×C61#12012yes—
256C23×C61C22×D411yescharacteristic
257C22×D61—22no—
258C22×D61—42no—
259Q8×C61—22no—
260Q8×C61—22no—
261Q8×C61—44no—
262D4×C61—24no—
263D4×C61—22no—
264D4×C61—22no—
265D4×C61—44no—
266D4×C61—44no—
267D4×C61—44no—
268D4×C61—42no—
269C2×C4×C61—24no—
270C2×C4×C61—21no—
271C2×C4×C61—21no—
272C2×C4×C61—22no—
273C2×C4×C61—22no—
274C2×C4×C61—21no—
275C2×C4×C61—21no—
276C2×C4×C61—21no—
277C2×C4×C61—21no—
278C2×C4×C61—22no—
279C2×C4×C61—22no—
280C2×C4×C61—22no—
281C2×C4×C61—42no—
282C2×C4×C61—42no—
283C2×C4×C61—42no—
284C2×C4×C61—42no—
285C2×C4×C61—44no—
286C2×C4×C61—44no—
287C2×C4×C61—42no—
288C2×C4×C61—41no—
289C2×C4×C61—41no—
290C2×C4×C61—41no—
291C2×C4×C61—41no—
292C2×C4×C61—42no—
293C2×C4×C61#12012yes—
294C2×C4×C61C22×D411yescharacteristic
295C2×C4×C61C22×D411yescharacteristic
296C2×C4×C61C2×#4112yes—
297C2×C4×C61H(5,𝔽2)12yes—
298C22⋊D61—44no—
299C22⋊D61—42no—
300C22⋊D61—42no—
301C22⋊D61—84no—
302C2×Dic61—22no—
303C2×Dic61—42no—
304C2×Dic61—42no—
305D244—22no—
306D244—22no—
307D244—42no—
308C4×D61—44no—
309C4×D61—42no—
310C4×D61—42no—
311Dic122—22no—
312Dic122—22no—
313Dic122—42no—
314#92418—42no—
315#92418—42no—
316#23994—611nomaximal, Sylow
317C22×C61—24no—
318C22×C61—22no—
319C22×C61—22no—
320C22×C61—41no—
321C22×C61—42no—
322C22×C61—41no—
323C22×C61—42no—
324C22×C61—42no—
325C22×C61—41no—
326C22×C61—41no—
327C22×C61C2×#12011yescharacteristic
328C22×C61#54612yes—
329C2×D61—42no—
330C2×D61—82no—
331C2×D61—162no—
332C4×C61—22no—
333C4×C61—22no—
334C4×C61—22no—
335C4×C61—41no—
336C4×C61—41no—
337C4×C61—42no—
338C4×C61—42no—
339C4×C61—41no—
340C4×C61—41no—
341C4×C61#52511yescharacteristic
342C4×C61#53511yescharacteristic
343C4×C61#54612yes—
344Dic61—42no—
345Dic61—82no—
346#3444D6111yescharacteristic, complemented
347#3025—611no—
348#3018—611no—
349#2901—611no—
350#2031—612no—
351#2010—612no—
352#1906—612no—
353#1896—612no—
354#1891—611no—
355#1884—611no—
356#1827—611no—
357C2×C61—22no—
358C2×C61—21no—
359C2×C61—42no—
360C2×C61—42no—
361C2×C61—81no—
362C2×C61#240411yescharacteristic
363D61—82no—
364#585—611no—
365#575—612no—
366#568—611no—
367#533—21no—
368#533C2×D6111yescharacteristic, complemented
369#532—21no—
370#532C2×D6111yescharacteristic
371#529—21no—
372#529—21no—
373#529—22no—
374#529—22no—
375#525—22no—
376#525—612no—
377#525C2×D6111yescharacteristic
378#525C2×D6111yescharacteristic, complemented
379C4×#41—22no—
380C4×#41C2×D6111yescharacteristic
381C4×#41C2×D6111yescharacteristic, complemented
382C22×C42C2×D6111yescharacteristic, complemented
383#458—1222no—
384B(4,𝔽2)—1222no—
385#456—1221no—
386#455—1221no—
387#454—1221no—
388#453—1221no—
389#421—612no—
390#421—612no—
391#421—1222no—
392C2×#104—612no—
393C2×#104—612no—
394C2×#104—1222no—
395#413—611no—
396#410—611no—
397#410—612no—
398#407—612no—
399#344—612no—
400#339—612no—
401#339—612no—
402H(3,ℤ/4)—612no—
403C61#2399411yescharacteristic, complemented, Sylow
4042−1+4—1222no—
4052−1+4—1221no—
406H(5,𝔽2)—1222no—
407H(5,𝔽2)—1222no—
408H(5,𝔽2)—1221no—
409C2×#41—22no—
410C2×#41—612no—
411C2×#41—1222no—
412C2×#41—2442no—
413C2×#41C22×D6111yescharacteristic
414C2×#41C22×D6111yescharacteristic
415C23×C4—21no—
416C23×C4C22×D6111yescharacteristic
417#137—1222no—
418#137—1222no—
419AGL(1,ℤ/8)—1222no—
420AGL(1,ℤ/8)—1222no—
421#135—2442no—
422#131—2442no—
423C2×M16—612no—
424C2×M16—612no—
425C2×M16—1222no—
426C4⋊Q8—21no—
427C4⋊Q8C22⋊D6111yescharacteristic, complemented
428#127C22⋊D6111yescharacteristic, complemented
429#126—22no—
430#126—22no—
431#125—22no—
432#125—21no—
433#124—22no—
434#124C22⋊D6111yescharacteristic, complemented
435#124C22⋊D6111yescharacteristic, complemented
436#123—22no—
437#123—22no—
438#123—21no—
439#123—1222no—
440#123C22⋊D6112yes—
441C22⋊Q8—22no—
442C22⋊Q8—21no—
443C22⋊Q8—21no—
444C22⋊Q8—22no—
445C22⋊Q8—1222no—
446#121—22no—
447#121—22no—
448#121—21no—
449#121—21no—
450#121—1222no—
451#120—21no—
452#120—1222no—
453#120C22⋊D6112yes—
454C4×Q8—22no—
455C4×Q8—21no—
456C4×Q8—21no—
457C4×Q8—22no—
458C4×D4—22no—
459C4×D4—22no—
460C4×D4—22no—
461C4×D4—22no—
462C4×D4—21no—
463C4×D4—21no—
464C4×D4—22no—
465C4×D4—22no—
466C4×D4—22no—
467C4×D4—22no—
468C4×D4—2442no—
469#117—22no—
470#117—22no—
471#117—22no—
472#117—21no—
473#117—612no—
474#117C22⋊D6112yescomplemented
475#117C22×D6111yescharacteristic
476#117C22×D6111yescharacteristic
477C2×C42—21no—
478C2×C42—21no—
479C2×C42—21no—
480C2×C42—21no—
481C2×C42—21no—
482C2×C42—21no—
483C2×C42C22⋊D6112yescomplemented
484C2×C42C22×D6111yescharacteristic
485C2×C42C22×D6111yescharacteristic
486#104—1222no—
487#104—1222no—
488#104—1222no—
489#104—1222no—
490#104—1224no—
491#104—2442no—
492#101—1221no—
493#100—1221no—
494#99—1222no—
495#99—1224no—
496C22⋊C8—2442no—
497C24C2×C22⋊D6111yescharacteristic
498#41—22no—
499#41—22no—
500#41—22no—
501#41—22no—
502#41—48no—
503#41—1224no—
504#41—1222no—
505#41—1222no—
506#41—1222no—
507#41—1222no—
508#41—2442no—
509#41—2444no—
510#41—2442no—
511#41—2442no—
512#41—2442no—
513C2×Q8—22no—
514C2×Q8—612no—
515C2×Q8—1222no—
516C2×Q8—2442no—
517C2×Q8C2×C22⋊D6111yescharacteristic
518C2×Q8C2×C22⋊D6111yescharacteristic
519C2×D4—22no—
520C2×D4—21no—
521C2×D4—22no—
522C2×D4—22no—
523C2×D4—612no—
524C2×D4—1222no—
525C2×D4—1222no—
526C2×D4—2442no—
527C2×D4—2444no—
528C2×D4—2442no—
529C2×D4—2442no—
530C2×D4C2×C22⋊D6111yescharacteristic
531C2×D4C2×C22⋊D6111yescharacteristic
532C2×D4C2×C22⋊D6112yes—
533C22×C4—21no—
534C22×C4—21no—
535C22×C4—21no—
536C22×C4—21no—
537C22×C4—22no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—22no—
541C22×C4—21no—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—22no—
545C22×C4—22no—
546C22×C4—44no—
547C22×C4—1222no—
548C22×C4—2442no—
549C22×C4C2×C22⋊D6111yescharacteristic
550C22×C4C2×C22⋊D6112yes—
551C22×C4C23×D6111yescharacteristic
552Q16—2442no—
553SD16—2442no—
554SD16—2442no—
555D8—2442no—
556M16—1222no—
557M16—1222no—
558M16—1222no—
559M16—1222no—
560M16—2442no—
561C2×C8—1222no—
562C2×C8—1222no—
563C2×C8—2442no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—1222no—
575C4⋊C4—2442no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—21no—
580C22⋊C4—22no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—22no—
584C22⋊C4—22no—
585C22⋊C4—1222no—
586C22⋊C4—2442no—
587C22⋊C4—2442no—
588C22⋊C4C2×C22⋊D6112yes—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—22no—
594C42—21no—
595C42—21no—
596C42—21no—
597C42—22no—
598C42—1222no—
599C42C2×C22⋊D6111yescharacteristic
600C42C2×C22⋊D6111yescharacteristic
601C42C2×C22⋊D6111yescharacteristic
602C42C2×C22⋊D6111yescharacteristic
603C42C2×C22⋊D6112yes—
604C23—21no—
605C23—22no—
606C23—21no—
607C23—42no—
608C23—1222no—
609C23—2442no—
610C23#1451974612yes—
611C23C22×C22⋊D6111yescharacteristic
612Q8—22no—
613Q8—22no—
614Q8—44no—
615Q8—1222no—
616Q8—1222no—
617Q8—2442no—
618D4—24no—
619D4—22no—
620D4—22no—
621D4—44no—
622D4—44no—
623D4—44no—
624D4—42no—
625D4—1222no—
626D4—1222no—
627D4—4884no—
628D4—2442no—
629D4—2444no—
630D4—2442no—
631D4—2442no—
632C2×C4—24no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—1222no—
657C2×C4—2444no—
658C2×C4—2442no—
659C2×C4—2442no—
660C2×C4—2442no—
661C2×C4—2442no—
662C2×C4#1451974612yes—
663C2×C4C22×C22⋊D6111yescharacteristic
664C2×C4C22×C22⋊D6111yescharacteristic
665C2×C4#41×D6112yes—
666C2×C4#1451982312yes—
667C8—2442no—
668C8—2442no—
669C22—22no—
670C22—24no—
671C22—22no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—42no—
676C22—42no—
677C22—41no—
678C22—41no—
679C22—9762no—
680C22—4882no—
681C22—2442no—
682C22C2×#1451974611yescharacteristic
683C22#1684040012yes—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—41no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—41no—
693C4—4882no—
694C4—2442no—
695C4#1684026911yescharacteristic
696C4#1684032611yescharacteristic
697C4#1684040012yes—
698C2—21no—
699C2—22no—
700C2—42no—
701C2—42no—
702C2—81no—
703C2—4882no—
704C2#2144913411yescharacteristic
subgroupG/H
centerC4#16840326
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3444C2
socleC2×C61#2404
hypercenter#3444D61
hypocenterC61#23994
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3444

Complements

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

NH ≅ G/Ntype
1. #21455210C2semidirect
2. #21455167C2semidirect
3. #21454367C2semidirect
4. #21446988C2semidirect
5. #21446884C2semidirect
6. #21446320C2semidirect
7. #21446262C2semidirect
8. #21446238C2semidirect
9. #21446203C2semidirect
10. #21445923C2semidirect
11. C61×#3444C2semidirect
24. #16839779C22semidirect
31. #16839726C22semidirect
32. #16839532C22semidirect
33. #16839522C22semidirect
34. #16839522C22semidirect
35. #16839518C22semidirect
37. C61×#533C22semidirect
46. C61×#525C22semidirect
49. C4×#41×C61C22semidirect
50. C22×C42×C61C22semidirect
91. C4⋊Q8×C61D4semidirect
92. #127×C61D4semidirect
98. #124×C61D4semidirect
99. #124×C61D4semidirect
132. #117×C61D4semidirect
141. C2×C42×C61D4semidirect
194. C2×D244#41semidirect
198. C2×Dic122#41semidirect
346. #3444D61semidirect
368. #533C2×D61semidirect
378. #525C2×D61semidirect
381. C4×#41C2×D61semidirect
382. C22×C42C2×D61semidirect
403. C61#23994semidirect
427. C4⋊Q8C22⋊D61semidirect
428. #127C22⋊D61semidirect
434. #124C22⋊D61semidirect
435. #124C22⋊D61semidirect
474. #117C22⋊D61semidirect
483. C2×C42C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214552102
#214551672
#214543672
#214469882
#214468842
#214463202
#214462622
#214462382
#214462032
#214459232
C61×#34442
#2399461