Finite Group Databasebeta

← #29740554

Subgroups — #29740554

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#21455788C211yesmaximal, characteristic, complemented
2#21455367C211yesmaximal, characteristic
3#21454417C211yesmaximal, characteristic, complemented
4#21447354C211yesmaximal, characteristic, complemented
5#21447325C211yesmaximal, characteristic, complemented
6#21447262C211yesmaximal, characteristic, complemented
7#21447034C211yesmaximal, characteristic, complemented
8#21446474C211yesmaximal, characteristic, complemented
9#21446418C211yesmaximal, characteristic, complemented
10#21446332C211yesmaximal, characteristic, complemented
11#21446188C211yesmaximal, characteristic, complemented
12#21446148C211yesmaximal, characteristic
13#21446045C211yesmaximal, characteristic, complemented
14#21446043C211yesmaximal, characteristic, complemented
15C61×#2951C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840333C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840103—21no—
21#16840103—21no—
22#16840097—21no—
23#16840092—21no—
24#16839982—21no—
25#16839975—21no—
26#16839970—21no—
27#16839965—21no—
28#16839805—21no—
29#16839805C2211yescharacteristic
30#16839802—21no—
31#16839802C2211yescharacteristic
32#16839802C2211yescharacteristic
33#16839798—21no—
34#16839798C2211yescharacteristic
35C2×#14519710—21no—
36C2×#14519710C2211yescharacteristic
37#16839788C2211yescharacteristic
38C2×#14519704C2211yescharacteristic
39C2×#14519704C2211yescharacteristic
40#16839770C2211yescharacteristic
41#16839763C2211yescharacteristic
42C2×#14519702C2211yescharacteristic
43C2×#14519700C2211yescharacteristic
44C2×#14519695C2211yescharacteristic
45#16839729C2211yescharacteristic
46#16839667C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839527C2211yescharacteristic
52#16839526C2211yescharacteristic
53#16839523C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#135×C61C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#520C2211yescharacteristic
59C61×#440—21no—
60C61×#440—21no—
61C4×Q16×C61—21no—
62C4×D8×C61—21no—
63C2×#107×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#405C2211yescharacteristic
67C22×Dic122C2311yescharacteristic
68#14519807—22no—
69#14519807—22no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72#14519784—22no—
73#14519784—22no—
74C2×#11758194—21no—
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C22×#92418—21no—
79C22×#92418C2311yescharacteristic
80#14519775—21no—
81#14519764—21no—
82#14519758—41no—
83#14519756—21no—
84#14519756—21no—
85Q8×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—41no—
88#14519736—21no—
89#14519736C2311yescharacteristic
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519710—21no—
95#14519710—21no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710D411yescharacteristic
100#14519710D411yescharacteristic
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104#14519708D411yescharacteristic
105#14519708D411yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519704—22no—
109#14519704—22no—
110#14519702—21no—
111#14519702—21no—
112#14519701—21no—
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700—21no—
117#14519698—21no—
118#14519695—22no—
119C22⋊#92418—21no—
120C22⋊#92418—21no—
121#14519684—41no—
122C2×#41×C61C2311yescharacteristic
123C2×#41×C61C2311yescharacteristic
124#135×C61—22no—
125#135×C61—22no—
126C2×Q16×C61D411yescharacteristic
127C2×SD16×C61D411yescharacteristic
128C2×SD16×C61D411yescharacteristic
129C2×D8×C61D411yescharacteristic
130C2×M16×C61C2311yescharacteristic
131C22×C8×C61C2311yescharacteristic
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—41no—
137C4×D4×C61—41no—
138#117×C61—21no—
139#117×C61—21no—
140#117×C61C2311yescharacteristic
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C4⋊C4×C61C2311yescharacteristic
145#107×C61—21no—
146#107×C61—21no—
147#103×C61—21no—
148#103×C61—21no—
149#102×C61—21no—
150#102×C61—21no—
151#97×C61—21no—
152C4×C8×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×C61C2×D411yescharacteristic
162C2×Q8×C61C2×D411yescharacteristic
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61C2×D411yescharacteristic
166C2×D4×C61C2×D411yescharacteristic
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61—41no—
172C22×C4×C61—41no—
173C22×C4×C61C2411yescharacteristic
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—41no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184Q16×C61—22no—
185SD16×C61—22no—
186SD16×C61—22no—
187D8×C61—22no—
188M16×C61—21no—
189M16×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D411yescharacteristic
194C2×C8×C61C2×D411yescharacteristic
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—41no—
200C4⋊C4×C61—41no—
201C4⋊C4×C61C2×D411yescharacteristic
202C4⋊C4×C61C2×D411yescharacteristic
203C22⋊C4×C61—21no—
204C22⋊C4×C61—41no—
205C22⋊C4×C61—41no—
206C42×C61—21no—
207C42×C61—41no—
208C42×C61—41no—
209C22⋊Dic61—21no—
210C22⋊Dic61—41no—
211C22⋊Dic61—41no—
212C22⋊Dic61—41no—
213#11758202—42no—
214#11758200—42no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—41no—
218C4⋊Dic61C2×D411yescharacteristic
219C4⋊Dic61C2×D411yescharacteristic
220#11758196—41no—
221#11758196—41no—
222#11758196—82no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
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#12312yes—
249D4×C61—22no—
250D4×C61—22no—
251D4×C61—21no—
252D4×C61—21no—
253D4×C61—21no—
254D4×C61—41no—
255D4×C61—41no—
256D4×C61#12312yes—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—81no—
268C2×C4×C61—81no—
269C2×C4×C61—161no—
270C2×C4×C61—161no—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C612−1+411yescharacteristic
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—81no—
287C2×Dic61—161no—
288Dic122—41no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
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#26196—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#3046—611no—
329#2951D6111yescharacteristic
330#2913—611no—
331#2095—611no—
332#2089—611no—
333#2078—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#328811yescharacteristic
348C2×C61#329311yescharacteristic
349C2×#137—611no—
350C2×#135C2×D6111yescharacteristic
351C2×#131—611no—
352#536—611no—
353#520—611no—
354#520C2×D6111yescharacteristic
355#520C2×D6111yescharacteristic
356#485—1221no—
357#485—1221no—
358#484—1221no—
359#483—1221no—
360#464—1221no—
361#463—1221no—
362#462—1221no—
363#461—1221no—
364#440—21no—
365#440—21no—
366C4×Q16—21no—
367C4×D8—21no—
368#428—611no—
369#428—611no—
370#428—611no—
371#428—1221no—
372#428—1221no—
373#427—611no—
374#427—1221no—
375C2×#107—611no—
376C2×#107—1221no—
377C2×#107C2×D6111yescharacteristic
378#424—611no—
379C2×#104—611no—
380C2×#104—611no—
381#419—611no—
382#417—611no—
383#417C2×D6111yescharacteristic
384#417C2×D6111yescharacteristic
385C2×#103—611no—
386C2×#103—611no—
387C2×#101—611no—
388#408—611no—
389#405C2×D6111yescharacteristic
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#2619611yescharacteristic, complemented, Sylow
401C2×#41C22×D6111yescharacteristic
402C2×#41C22×D6111yescharacteristic
403C22×Q8—611no—
404#137—1222no—
405#137—1222no—
406#135—22no—
407#135—22no—
408C2×Q16—1221no—
409C2×Q16C22⋊D6111yescharacteristic
410C2×SD16—1221no—
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—1221no—
425#124—1221no—
426#123—2441no—
427C22⋊Q8—1221no—
428C22⋊Q8—1221no—
429C4×Q8—21no—
430C4×Q8—21no—
431C4×Q8—1221no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—41no—
435C4×D4—41no—
436C4×D4—1221no—
437C4×D4—2441no—
438#117—21no—
439#117—21no—
440#117—611no—
441#117—1221no—
442#117C22×D6111yescharacteristic
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—2441no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C42—611no—
451#107—21no—
452#107—21no—
453#107—611no—
454#107—611no—
455#107—1221no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#107—1221no—
460#106—611no—
461#106—611no—
462#106—1221no—
463#106—1221no—
464#106—1221no—
465C4⋊C8—1221no—
466C4⋊C8—1221no—
467#104—1222no—
468#104—1222no—
469#103—21no—
470#103—21no—
471#103—1221no—
472#103—1221no—
473#103—1221no—
474#103—1221no—
475#103—1221no—
476#103—1221no—
477#103—1221no—
478#102—21no—
479#102—21no—
480#102—1221no—
481#101—1222no—
482C22⋊C8—1221no—
483C22⋊C8—1221no—
484#97—21no—
485C4×C8—21no—
486#95—2441no—
487#41—21no—
488#41—21no—
489#41—21no—
490#41—21no—
491#41—21no—
492#41—21no—
493#41#1175822212yes—
494#41#1175822212yes—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
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—41no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—2441no—
517C22×C4C23×D6111yescharacteristic
518Q16—22no—
519Q16—2442no—
520SD16—22no—
521SD16—22no—
522SD16—2442no—
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—21no—
549C4⋊C4—21no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—611no—
553C4⋊C4—611no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—4882no—
557C4⋊C4—2441no—
558C4⋊C4—2441no—
559C4⋊C4—2441no—
560C4⋊C4D4×D6111yescharacteristic
561C4⋊C4D4×D6111yescharacteristic
562C22⋊C4—21no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—1221no—
566C22⋊C4—2441no—
567C22⋊C4—2441no—
568C22⋊C4—2441no—
569C42—21no—
570C42—41no—
571C42—41no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42—2441no—
576C23—21no—
577C23—21no—
578C23#1451982211yescharacteristic
579Q8—21no—
580Q8—21no—
581Q8—21no—
582Q8—2441no—
583Q8—2441no—
584Q8—2441no—
585Q8—2441no—
586Q8#1451976012yes—
587D4—22no—
588D4—22no—
589D4—21no—
590D4—21no—
591D4—21no—
592D4—41no—
593D4—41no—
594D4#1451976012yes—
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—81no—
606C2×C4—81no—
607C2×C4—161no—
608C2×C4—161no—
609C2×C4—1222no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—9761no—
613C2×C4—4881no—
614C2×C4—2441no—
615C2×C4—2441no—
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×#1175822211yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
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#1684035411yescharacteristic
643C22D4×C22⋊D6111yescharacteristic
644C22#1684046111yescharacteristic
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4—2441no—
655C4—2441no—
656C4C2×#1451976011yescharacteristic
657C4#1684027111yescharacteristic
658C4#1684041111yescharacteristic
659C4#1684042111yescharacteristic
660C2—21no—
661C2—21no—
662C2—42no—
663C2—81no—
664C2#2144954811yescharacteristic
665C2#2145758311yescharacteristic
666C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2951C2
socleC22×C61D42
prime radicalC2×#135×C61C22
hypercenter#2951D61
hypocenterC61#26196
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #21455788C2semidirect
3. #21454417C2semidirect
4. #21447354C2semidirect
5. #21447325C2semidirect
6. #21447262C2semidirect
7. #21447034C2semidirect
8. #21446474C2semidirect
9. #21446418C2semidirect
10. #21446332C2semidirect
11. #21446188C2semidirect
13. #21446045C2semidirect
14. #21446043C2semidirect
400. C61#26196semidirect

Maximal subgroups

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

subgroupindex
#214557882
#214553672
#214544172
#214473542
#214473252
#214472622
#214470342
#214464742
#214464182
#214463322
#214461882
#214461482
#214460452
#214460432
C61×#29512
#2619661