Finite Group Databasebeta

← #29740571

Subgroups — #29740571

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455782C211yesmaximal, characteristic, complemented
2#21454703C211yesmaximal, characteristic
3#21454412C211yesmaximal, characteristic, complemented
4#21447360C211yesmaximal, characteristic, complemented
5#21447329C211yesmaximal, characteristic, complemented
6#21447267C211yesmaximal, characteristic, complemented
7#21447033C211yesmaximal, characteristic, complemented
8#21446471C211yesmaximal, characteristic, complemented
9#21446415C211yesmaximal, characteristic, complemented
10#21446336C211yesmaximal, characteristic, complemented
11#21446190C211yesmaximal, characteristic, complemented
12#21446148C211yesmaximal, characteristic
13#21446045C211yesmaximal, characteristic, complemented
14#21446043C211yesmaximal, characteristic, complemented
15C61×#3046C211yesmaximal, characteristic
16C2×#14519784C2211yescharacteristic
17#16840245C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840100—21no—
20#16840100—21no—
21#16840095—21no—
22#16840090—21no—
23#16839874—21no—
24#16839869—21no—
25SD16×Dic61—21no—
26SD16×Dic61—21no—
27#16839805—21no—
28#16839805C2211yescharacteristic
29#16839805C2211yescharacteristic
30#16839802—21no—
31#16839802C2211yescharacteristic
32#16839798—21no—
33#16839798C2211yescharacteristic
34C2×#14519709C2211yescharacteristic
35C2×#14519708—21no—
36C2×#14519708C2211yescharacteristic
37C2×#14519704C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39#16839771C2211yescharacteristic
40#16839763C2211yescharacteristic
41#16839763C2211yescharacteristic
42C2×#14519701C2211yescharacteristic
43C2×#14519696C2211yescharacteristic
44#16839729C2211yescharacteristic
45#16839717C2211yescharacteristic
46#16839669C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839530C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#137×C61C2211yescharacteristic
57C2×#135×C61C2211yescharacteristic
58C61×#536C2211yescharacteristic
59C61×#520C2211yescharacteristic
60C61×#464—21no—
61C61×#463—21no—
62C61×#462—21no—
63C61×#461—21no—
64C61×#424C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C2×#103×C61C2211yescharacteristic
67#14519784—22no—
68#14519784—22no—
69C2×#11758194—21no—
70C2×#11758194C2311yescharacteristic
71C2×#11758194C2311yescharacteristic
72C2×#11758194C2311yescharacteristic
73C22×#92418—21no—
74C22×#92418C2311yescharacteristic
75Q8×Dic61—21no—
76Q8×Dic61—21no—
77D4×Dic61—21no—
78D4×Dic61—21no—
79D4×Dic61—41no—
80D4×Dic61—41no—
81#14519736—21no—
82#14519736—21no—
83#14519736C2311yescharacteristic
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—41no—
89C2×C4×Dic61C2311yescharacteristic
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710D411yescharacteristic
94#14519710D411yescharacteristic
95#14519709—21no—
96#14519709—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708D411yescharacteristic
103#14519708D411yescharacteristic
104#14519704—22no—
105#14519704—22no—
106#14519701—21no—
107#14519701—21no—
108#14519701—21no—
109#14519701—21no—
110#14519701—21no—
111#14519701—21no—
112#14519698—21no—
113#14519698—21no—
114#14519696—22no—
115C22⋊#92418—21no—
116C22⋊#92418—21no—
117#14519687—21no—
118C8×Dic61—21no—
119#14519684—41no—
120C2×#41×C61C2311yescharacteristic
121C2×#41×C61C2311yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123#137×C61—22no—
124#137×C61—22no—
125#135×C61—22no—
126#135×C61—22no—
127C2×Q16×C61—21no—
128C2×Q16×C61D411yescharacteristic
129C2×SD16×C61—21no—
130C2×SD16×C61D411yescharacteristic
131C2×SD16×C61D411yescharacteristic
132C2×D8×C61D411yescharacteristic
133C2×M16×C61C2311yescharacteristic
134C22×C8×C61C2311yescharacteristic
135C4⋊Q8×C61—21no—
136#124×C61—21no—
137#123×C61—41no—
138C22⋊Q8×C61—21no—
139C22⋊Q8×C61—21no—
140C4×Q8×C61—21no—
141C4×D4×C61—21no—
142C4×D4×C61—41no—
143#117×C61—21no—
144#117×C61C2311yescharacteristic
145C2×C4⋊C4×C61—21no—
146C2×C4⋊C4×C61C2311yescharacteristic
147C4⋊C8×C61—21no—
148C4⋊C8×C61—21no—
149#103×C61—21no—
150#103×C61—21no—
151#103×C61—21no—
152#102×C61—21no—
153#41×C61—21no—
154#41×C61—21no—
155#41×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61#4112yes—
160#41×C61#4112yes—
161C2×Q8×C61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61—21no—
165C2×Q8×C61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61C2×D411yescharacteristic
168C2×Q8×C61C2×D411yescharacteristic
169C2×D4×C61—21no—
170C2×D4×C61—21no—
171C2×D4×C61C2×D411yescharacteristic
172C2×D4×C61C2×D411yescharacteristic
173C22×C4×C61—21no—
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61—41no—
178C22×C4×C61C2411yescharacteristic
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61—41no—
183C22×Dic61—41no—
184Q16×C61—22no—
185Q16×C61—42no—
186SD16×C61—22no—
187SD16×C61—22no—
188SD16×C61—42no—
189D8×C61—22no—
190M16×C61—21no—
191M16×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61C2×D411yescharacteristic
196C2×C8×C61C2×D411yescharacteristic
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—41no—
200C4⋊C4×C61—41no—
201C4⋊C4×C61—41no—
202C4⋊C4×C61C2×D411yescharacteristic
203C4⋊C4×C61C2×D411yescharacteristic
204C22⋊C4×C61—21no—
205C22⋊C4×C61—41no—
206C22⋊C4×C61—41no—
207C22⋊C4×C61—41no—
208C42×C61—21no—
209C42×C61—41no—
210C22⋊Dic61—21no—
211C22⋊Dic61—41no—
212C22⋊Dic61—41no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218C4⋊Dic61—41no—
219C4⋊Dic61C2×D411yescharacteristic
220C4⋊Dic61C2×D411yescharacteristic
221#11758196—82no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—41no—
226C4×Dic61—41no—
227#11758194—24no—
228#11758194—21no—
229#11758194—21no—
230#11758194—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194#4112yes—
234C2×#92418—24no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418C2×D411yescharacteristic
241C2×#92418C2×D411yescharacteristic
242C23×C61—21no—
243C23×C61—21no—
244C23×C61H(5,𝔽2)11yescharacteristic
245Q8×C61—21no—
246Q8×C61—21no—
247Q8×C61—21no—
248Q8×C61—41no—
249Q8×C61—41no—
250Q8×C61—41no—
251Q8×C61—41no—
252Q8×C61#12312yes—
253D4×C61—22no—
254D4×C61—22no—
255D4×C61—21no—
256D4×C61—21no—
257D4×C61—21no—
258D4×C61—41no—
259D4×C61—41no—
260D4×C61#12312yes—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—81no—
274C2×C4×C61—161no—
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C612−1+411yescharacteristic
281C2×Dic61—22no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—81no—
289C2×Dic61—81no—
290C2×Dic61—161no—
291C2×Dic61—161no—
292C8×C61—21no—
293C8×C61—21no—
294C8×C61—41no—
295#92418—21no—
296#92418—22no—
297#92418—21no—
298#92418—41no—
299#92418—41no—
300#26197—611nomaximal, Sylow
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—22no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61#53911yescharacteristic
310C22×C61D4211yescharacteristic
311C22×C61#55311yescharacteristic
312C4×C61—22no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61C2×#12311yescharacteristic
319C4×C61#52511yescharacteristic
320C4×C61#54711yescharacteristic
321C4×C61#54811yescharacteristic
322Dic61—42no—
323Dic61—41no—
324Dic61—41no—
325Dic61—81no—
326Dic61—81no—
327#3097—611no—
328#3046D6111yescharacteristic
329#2952—611no—
330#2912—611no—
331#2096—611no—
332#2090—611no—
333#2079—611no—
334#2038—611no—
335#1930—611no—
336#1922—611no—
337#1908—611no—
338#1882—611no—
339#1876—611no—
340#1859—611no—
341#1859—611no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61—42no—
345C2×C61—81no—
346C2×C61#244311yescharacteristic
347C2×C61#328911yescharacteristic
348C2×C61#329111yescharacteristic
349C2×#137C2×D6111yescharacteristic
350C2×#135C2×D6111yescharacteristic
351C2×#131—611no—
352#536C2×D6111yescharacteristic
353#520—611no—
354#520—611no—
355#520C2×D6111yescharacteristic
356#485—1221no—
357#485—1221no—
358#484—1221no—
359#483—1221no—
360#464—21no—
361#463—21no—
362#462—21no—
363#461—21no—
364#442—1221no—
365#441—1221no—
366C4×SD16—1221no—
367C4×SD16—1221no—
368#428—611no—
369#428—611no—
370#428—611no—
371#428—1221no—
372#428—1221no—
373#427—611no—
374#427—1221no—
375C2×#106—611no—
376C2×#106—611no—
377C2×#106—1221no—
378#424C2×D6111yescharacteristic
379C2×#104—611no—
380C2×#104—611no—
381#419—611no—
382#417—611no—
383#417—611no—
384#417C2×D6111yescharacteristic
385C2×#103—611no—
386C2×#103C2×D6111yescharacteristic
387C2×#101—611no—
388#408—611no—
389#405—611no—
390#391—611no—
391#384—611no—
392#343—611no—
393#343—611no—
394#343—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#336—611no—
400C61#2619711yescharacteristic, complemented, Sylow
401C2×#41C22×D6111yescharacteristic
402C2×#41C22×D6111yescharacteristic
403C22×Q8C22×D6111yescharacteristic
404#137—22no—
405#137—22no—
406#135—22no—
407#135—22no—
408C2×Q16—21no—
409C2×Q16C22⋊D6111yescharacteristic
410C2×SD16—21no—
411C2×SD16C22⋊D6111yescharacteristic
412C2×SD16C22⋊D6111yescharacteristic
413C2×D8C22⋊D6111yescharacteristic
414#131—1222no—
415#131—1222no—
416C2×M16—611no—
417C2×M16—611no—
418C2×M16—611no—
419C2×M16—1221no—
420C2×M16C22×D6111yescharacteristic
421C22×C8—611no—
422C22×C8—1221no—
423C22×C8C22×D6111yescharacteristic
424C4⋊Q8—21no—
425#124—21no—
426#123—41no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C4×Q8—21no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×D4—21no—
433C4×D4—41no—
434C4×D4—1221no—
435C4×D4—1221no—
436C4×D4—2441no—
437C4×D4—2441no—
438#117—21no—
439#117—611no—
440#117—1221no—
441#117—1221no—
442#117C22×D6111yescharacteristic
443C2×C4⋊C4—21no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—2441no—
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C42—611no—
451#107—611no—
452#107—611no—
453#107—1221no—
454#107—1221no—
455#107—1221no—
456#106—611no—
457#106—611no—
458#106—1221no—
459#106—1221no—
460#106—1221no—
461#106—1221no—
462#106—1221no—
463#106—1221no—
464#106—1221no—
465C4⋊C8—21no—
466C4⋊C8—21no—
467#104—1222no—
468#104—1222no—
469#103—21no—
470#103—21no—
471#103—21no—
472#103—1221no—
473#103—1221no—
474#103—1221no—
475#103—1221no—
476#103—1221no—
477#103—1221no—
478#102—21no—
479#102—1221no—
480#102—1221no—
481#101—1222no—
482C22⋊C8—1221no—
483C22⋊C8—1221no—
484#97—1221no—
485C4×C8—1221no—
486#95—2441no—
487#41—21no—
488#41—21no—
489#41—21no—
490#41—21no—
491#41—21no—
492#41—21no—
493#41#1175822412yes—
494#41#1175822412yes—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8C2×C22⋊D6111yescharacteristic
502C2×Q8C2×C22⋊D6111yescharacteristic
503C2×D4—21no—
504C2×D4—21no—
505C2×D4C2×C22⋊D6111yescharacteristic
506C2×D4C2×C22⋊D6111yescharacteristic
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—41no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—2441no—
516C22×C4—2441no—
517C22×C4C23×D6111yescharacteristic
518Q16—22no—
519Q16—42no—
520SD16—22no—
521SD16—22no—
522SD16—42no—
523D8—22no—
524M16—21no—
525M16—21no—
526M16—612no—
527M16—1224no—
528M16—1221no—
529M16—1221no—
530M16—1221no—
531M16—1221no—
532M16—1221no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—611no—
537C2×C8—611no—
538C2×C8—1224no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8C2×C22⋊D6111yescharacteristic
545C2×C8C2×C22⋊D6111yescharacteristic
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—611no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—4882no—
558C4⋊C4—2441no—
559C4⋊C4—2441no—
560C4⋊C4D4×D6111yescharacteristic
561C4⋊C4D4×D6111yescharacteristic
562C22⋊C4—21no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—1221no—
567C22⋊C4—2441no—
568C22⋊C4—2441no—
569C42—21no—
570C42—41no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42—2441no—
575C42—2441no—
576C23—21no—
577C23—21no—
578C23#1451982211yescharacteristic
579Q8—21no—
580Q8—21no—
581Q8—21no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8#1451975812yes—
587D4—22no—
588D4—22no—
589D4—21no—
590D4—21no—
591D4—21no—
592D4—41no—
593D4—41no—
594D4#1451975812yes—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—81no—
608C2×C4—161no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—9761no—
613C2×C4—9761no—
614C2×C4—4881no—
615C2×C4—4881no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822411yescharacteristic
625C2×C4#1451982411yescharacteristic
626C8—21no—
627C8—21no—
628C8—41no—
629C8—1221no—
630C8—1221no—
631C8—1222no—
632C8—2441no—
633C8—2441no—
634C22—21no—
635C22—21no—
636C22—22no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22#1684035711yescharacteristic
643C22D4×C22⋊D6111yescharacteristic
644C22#1684045511yescharacteristic
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4—2441no—
655C4—2441no—
656C4C2×#1451975811yescharacteristic
657C4#1684027211yescharacteristic
658C4#1684041111yescharacteristic
659C4#1684041811yescharacteristic
660C2—21no—
661C2—21no—
662C2—42no—
663C2—81no—
664C2#2144955011yescharacteristic
665C2#2145759011yescharacteristic
666C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3046C2
socleC22×C61D42
prime radicalC2×#135×C61C22
hypercenter#3046D61
hypocenterC61#26197
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3046

Complements

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

NH ≅ G/Ntype
1. #21455782C2semidirect
3. #21454412C2semidirect
4. #21447360C2semidirect
5. #21447329C2semidirect
6. #21447267C2semidirect
7. #21447033C2semidirect
8. #21446471C2semidirect
9. #21446415C2semidirect
10. #21446336C2semidirect
11. #21446190C2semidirect
13. #21446045C2semidirect
14. #21446043C2semidirect
400. C61#26197semidirect

Maximal subgroups

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

subgroupindex
#214557822
#214547032
#214544122
#214473602
#214473292
#214472672
#214470332
#214464712
#214464152
#214463362
#214461902
#214461482
#214460452
#214460432
C61×#30462
#2619761