Finite Group Databasebeta

← #29737603

Subgroups — #29737603

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455666C211yesmaximal, characteristic, complemented
2#21454349C211yesmaximal, characteristic
3#21454348C211yesmaximal, characteristic
4#21447354C211yesmaximal, characteristic, complemented
5#21447185C211yesmaximal, characteristic, complemented
6#21447183C211yesmaximal, characteristic, complemented
7#21446977C211yesmaximal, characteristic, complemented
8#21446096C211yesmaximal, characteristic, complemented
9#21446093C211yesmaximal, characteristic, complemented
10#21446045C211yesmaximal, characteristic, complemented
11#21446042C211yesmaximal, characteristic, complemented
12#21445884C211yesmaximal, characteristic, complemented
13#21445881C211yesmaximal, characteristic, complemented
14#21445776C211yesmaximal, characteristic
15C61×#3083C211yesmaximal, characteristic, complemented
16C2×#14519824C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16840066—21no—
21#16840059—21no—
22#16840054—21no—
23#16840039—21no—
24C2×#14519713C2211yescharacteristic
25C2×#14519712C2211yescharacteristic
26#16839805C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28#16839789C2211yescharacteristic
29C2×#14519704C2211yescharacteristic
30C2×#14519703C2211yescharacteristic
31#16839770—21no—
32#16839770C2211yescharacteristic
33#16839770C2211yescharacteristic
34#16839769—21no—
35#16839763C2211yescharacteristic
36#16839761—21no—
37#16839761C2211yescharacteristic
38#16839761C2211yescharacteristic
39#16839760—21no—
40#16839756—21no—
41#16839756C2211yescharacteristic
42#16839755—21no—
43C2×#14519702C2211yescharacteristic
44C2×#14519700—21no—
45C2×#14519700C2211yescharacteristic
46#103×D61—21no—
47C2×#14519695C2211yescharacteristic
48#16839667C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839531C2211yescharacteristic
51#16839531C2211yescharacteristic
52#16839530C2211yescharacteristic
53#16839521C2211yescharacteristic
54#16839521C2211yescharacteristic
55#16839516C2211yescharacteristic
56C61×#544C2211yescharacteristic
57C61×#520C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#477—21no—
61C61×#474—21no—
62C61×#428C2211yescharacteristic
63C2×#107×C61C2211yescharacteristic
64C61×#424C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#416C2211yescharacteristic
67#14519824—22no—
68#14519824—22no—
69#14519824—41no—
70#14519824Q814yescomplemented
71C2×#11758222—21no—
72C2×#11758222C2311yescharacteristic
73C2×#11758224—21no—
74C2×#11758224—41no—
75C22×Dic122—21no—
76C22×Dic122C2311yescharacteristic
77C2×Q8×D61—21no—
78#14519784—22no—
79#14519784—41no—
80#14519783—22no—
81#14519783—41no—
82C2×#11758194—21no—
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85C2×#11758194C2311yescharacteristic
86C2×#11758189—21no—
87C22×#92418—21no—
88C22×#92418C2311yescharacteristic
89C2×C8×D61—21no—
90C4×Dic122—21no—
91C4×D244—21no—
92#14519737—41no—
93#14519735—21no—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—41no—
96C4⋊C4×D61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519713—41no—
99#14519712—41no—
100#14519710—21no—
101#14519710—21no—
102#14519710—21no—
103#14519708—21no—
104C4⋊#92418—21no—
105C4⋊#92418—21no—
106#14519704—22no—
107#14519703—22no—
108#14519702—21no—
109#14519702—21no—
110#14519701—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700—21no—
117#14519700—21no—
118#14519700—21no—
119#14519700D411yescharacteristic
120#14519700D411yescharacteristic
121#14519698—21no—
122#14519697—21no—
123#14519697—21no—
124#14519697D411yescharacteristic
125#14519697D411yescharacteristic
126#14519695—41no—
127#14519684—41no—
128C2×#41×C61C2311yescharacteristic
129C2×M16×C61C2311yescharacteristic
130C22×C8×C61C2311yescharacteristic
131C4⋊Q8×C61—21no—
132C4⋊Q8×C61—21no—
133#125×C61—21no—
134#125×C61—21no—
135C22⋊Q8×C61—41no—
136C4×Q8×C61—21no—
137C4×D4×C61—21no—
138C4×D4×C61—41no—
139#117×C61—21no—
140#117×C61C2311yescharacteristic
141#117×C61C2311yescharacteristic
142C2×C4⋊C4×C61—21no—
143C2×C4⋊C4×C61C2311yescharacteristic
144C2×C4⋊C4×C61C2311yescharacteristic
145#107×C61—21no—
146#107×C61—21no—
147#107×C61—21no—
148#106×C61—21no—
149C4⋊C8×C61—21no—
150C4⋊C8×C61—21no—
151#103×C61—21no—
152#103×C61D411yescharacteristic
153#103×C61D411yescharacteristic
154#102×C61—21no—
155#102×C61D411yescharacteristic
156#102×C61D411yescharacteristic
157#41×C61—21no—
158#41×C61—21no—
159#41×C61C2×Q812yes—
160#41×C61#4112yes—
161C2×Q8×C61—41no—
162C2×Q8×C61C2×D411yescharacteristic
163C2×D4×C61—21no—
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—41no—
169C22×C4×C61C2411yescharacteristic
170C2×C22⋊D61—21no—
171C2×C22⋊D61—41no—
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—41no—
176Q8×D61—24no—
177Q8×D61—42no—
178#11758224—24no—
179#11758224—42no—
180#11758224—44no—
181#11758224—44no—
182#11758222—21no—
183#11758222—21no—
184#11758222—24no—
185#11758222—42no—
186#11758222C2×Q812yes—
187#11758222#4112yes—
188C2×D244C2×D411yescharacteristic
189C2×C4×D61—21no—
190C2×C4×D61—41no—
191C2×C4×D61—41no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122—21no—
196C2×Dic122—24no—
197C2×Dic122—21no—
198C2×Dic122—42no—
199C2×Dic122—41no—
200C2×Dic122C2×D411yescharacteristic
201C2×Dic122C2×Q812yes—
202M16×C61—21no—
203M16×C61—21no—
204C2×C8×C61—21no—
205C2×C8×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61C2×D411yescharacteristic
208C2×C8×C61C2×D411yescharacteristic
209C4⋊C4×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—41no—
214C4⋊C4×C61—41no—
215C4⋊C4×C61—41no—
216C4⋊C4×C61—41no—
217C4⋊C4×C61C2×D411yescharacteristic
218C4⋊C4×C61C2×D411yescharacteristic
219C4⋊C4×C61C2×D411yescharacteristic
220C4⋊C4×C61C2×D411yescharacteristic
221C22⋊C4×C61—21no—
222C22⋊C4×C61—21no—
223C22⋊C4×C61—41no—
224C42×C61—21no—
225C42×C61—21no—
226C42×C61—41no—
227#11758198—41no—
228C4⋊Dic61—21no—
229C4⋊Dic61—21no—
230#11758196—41no—
231#11758196—82no—
232C4×Dic61—21no—
233C4×Dic61—21no—
234C4×Dic61—41no—
235#11758194—21no—
236#11758194—21no—
237#11758194—21no—
238#11758194—21no—
239#11758194—21no—
240#11758194—21no—
241#11758194—42no—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418—42no—
248C2×#92418C2×D411yescharacteristic
249C2×#92418C2×D411yescharacteristic
250#11758189—42no—
251C8×D61—42no—
252C23×C61—21no—
253C23×C61H(5,𝔽2)11yescharacteristic
254C22×D61—21no—
255Q8×C61—21no—
256Q8×C61—161no—
257Q8×C61C22⋊Q812yes—
258D4×C61—22no—
259D4×C61—22no—
260D4×C61—21no—
261D4×C61C22⋊Q812yes—
262C2×C4×C61—21no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—81no—
272C2×C4×C61—81no—
273C2×C4×C61—161no—
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×Q811yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61H(5,𝔽2)11yescharacteristic
280C22⋊D61—22no—
281C22⋊D61—22no—
282C22⋊D61—44no—
283C2×Dic61—21no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—44no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—81no—
293D244—21no—
294D244C22⋊Q812yes—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—44no—
298C4×D61—161no—
299Dic122—21no—
300Dic122—22no—
301Dic122—22no—
302Dic122—44no—
303Dic122—41no—
304Dic122—41no—
305Dic122C22⋊Q812yes—
306C8×C61—21no—
307C8×C61—21no—
308C8×C61—41no—
309#92418—21no—
310#92418—21no—
311#92418—41no—
312#92418—41no—
313#92418—41no—
314#25987—611nomaximal, Sylow
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—22no—
318C22×C61—41no—
319C22×C61—41no—
320C22×C61#54311yescharacteristic
321C22×C61D4211yescharacteristic
322C22×C61#55111yescharacteristic
323C2×D61—22no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—22no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61—41no—
331C4×C61—81no—
332C4×C61—81no—
333C4×C61C2×C22⋊Q811yescharacteristic
334C4×C61C2×C22⋊Q811yescharacteristic
335C4×C61#54611yescharacteristic
336C4×C61#54611yescharacteristic
337Dic61—22no—
338Dic61—42no—
339Dic61—42no—
340Dic61—81no—
341#3083—611no—
342#3083D6111yescharacteristic, complemented
343#2898—611no—
344#2898—611no—
345#2095—611no—
346#2064—611no—
347#2064—611no—
348#2028—611no—
349#1866—611no—
350#1866—611no—
351#1859—611no—
352#1859—611no—
353#1816—611no—
354#1816—611no—
355#1793—611no—
356C2×C61—21no—
357C2×C61—21no—
358C2×C61—42no—
359C2×C61#243711yescharacteristic
360C2×C61#328811yescharacteristic
361C2×C61#329311yescharacteristic
362D61—42no—
363C2×2−1+4—611no—
364C2×#131—611no—
365C2×#131—611no—
366#544C2×D6111yescharacteristic
367#520—611no—
368#520C2×D6111yescharacteristic
369#479—21no—
370#479—1221no—
371#478—21no—
372#478—1221no—
373#477—21no—
374#477—1221no—
375#474—21no—
376#474—1221no—
377C2×#108—611no—
378C2×#108—611no—
379#428—611no—
380#428C2×D6111yescharacteristic
381C2×#107—611no—
382C2×#107C2×D6111yescharacteristic
383#424—611no—
384#424C2×D6111yescharacteristic
385C2×#104—611no—
386C2×#104—611no—
387#419—611no—
388#419—611no—
389#419—1221no—
390#419—1221no—
391#417—611no—
392#417—611no—
393#417—611no—
394#417—1221no—
395#417—1221no—
396#417C2×D6111yescharacteristic
397#416—611no—
398#416—1221no—
399#416—1221no—
400#416C2×D6111yescharacteristic
401C2×#103—611no—
402C2×#103—611no—
403C2×#103—1221no—
404C2×#103—1221no—
405C2×#101—611no—
406#391—611no—
407#384—611no—
408#343—611no—
409#343—611no—
410#343—611no—
411#339—611no—
412#339—611no—
413#336—611no—
414C61#2598711yescharacteristic, complemented, Sylow
4152−1+4—614no—
4162−1+4—1222no—
4172−1+4—1222no—
4182−1+4—2441no—
419C2×#41—611no—
420C2×#41—1221no—
421C2×#41—1221no—
422C2×#41—2441no—
423C2×#41C22×D6111yescharacteristic
424C22×Q8—611no—
425C22×Q8—1221no—
426C22×Q8—1221no—
427#131—1222no—
428#131—1222no—
429#131—2441no—
430#131—2441no—
431C2×M16—611no—
432C2×M16—611no—
433C2×M16—611no—
434C2×M16—1221no—
435C2×M16—1221no—
436C2×M16C22×D6111yescharacteristic
437C22×C8—611no—
438C22×C8—1221no—
439C22×C8—1221no—
440C22×C8C22×D6111yescharacteristic
441C4⋊Q8—21no—
442C4⋊Q8—21no—
443#125—21no—
444#125—21no—
445C22⋊Q8—41no—
446C4×Q8—21no—
447C4×Q8—1221no—
448C4×D4—21no—
449C4×D4—41no—
450C4×D4—1221no—
451C4×D4—2441no—
452#117—21no—
453#117—1221no—
454#117C22×D6111yescharacteristic
455#117C22×D6111yescharacteristic
456C2×C4⋊C4—21no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—1221no—
459C2×C4⋊C4—2441no—
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C42—611no—
463#108—2441no—
464#108—2441no—
465#107—21no—
466#107—21no—
467#107—21no—
468#107—1221no—
469#107—1221no—
470#107—1221no—
471#106—21no—
472#106—1221no—
473C4⋊C8—21no—
474C4⋊C8—21no—
475C4⋊C8—1221no—
476C4⋊C8—1221no—
477#104—1222no—
478#104—1222no—
479#103—21no—
480#103—611no—
481#103—611no—
482#103—1221no—
483#103—1221no—
484#103—1221no—
485#103—1221no—
486#103—1221no—
487#103—1221no—
488#103—1221no—
489#103—1221no—
490#103—1221no—
491#103—1221no—
492#103—1221no—
493#103C22⋊D6111yescharacteristic
494#103C22⋊D6111yescharacteristic
495#102—21no—
496#102—611no—
497#102—611no—
498#102—1221no—
499#102—1221no—
500#102—1221no—
501#102C22⋊D6111yescharacteristic
502#102C22⋊D6111yescharacteristic
503#101—2441no—
504#95—2441no—
505#41—21no—
506#41—21no—
507#41—612no—
508#41—612no—
509#41—1224no—
510#41—1221no—
511#41—1221no—
512#41—1224no—
513#41—2442no—
514#41—2444no—
515#41—2442no—
516#41—2444no—
517#41Q8×D6112yes—
518#41#1175822612yes—
519C2×Q8—41no—
520C2×Q8—611no—
521C2×Q8—612no—
522C2×Q8—1224no—
523C2×Q8—1221no—
524C2×Q8—1224no—
525C2×Q8—1221no—
526C2×Q8—1221no—
527C2×Q8—1221no—
528C2×Q8—1221no—
529C2×Q8—2441no—
530C2×Q8—2442no—
531C2×Q8—2442no—
532C2×Q8C2×C22⋊D6111yescharacteristic
533C2×D4—21no—
534C2×D4—611no—
535C2×D4—1221no—
536C2×D4—2441no—
537C2×D4C2×C22⋊D6111yescharacteristic
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—21no—
541C22×C4—41no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—1221no—
546C22×C4—2441no—
547C22×C4—2441no—
548C22×C4—2441no—
549C22×C4C23×D6111yescharacteristic
550M16—21no—
551M16—21no—
552M16—1221no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16—2442no—
559M16—2442no—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—611no—
564C2×C8—611no—
565C2×C8—1221no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—2442no—
571C2×C8—2442no—
572C2×C8C2×C22⋊D6111yescharacteristic
573C2×C8C2×C22⋊D6111yescharacteristic
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—41no—
579C4⋊C4—41no—
580C4⋊C4—41no—
581C4⋊C4—41no—
582C4⋊C4—1221no—
583C4⋊C4—1221no—
584C4⋊C4—4882no—
585C4⋊C4—2441no—
586C4⋊C4D4×D6111yescharacteristic
587C4⋊C4D4×D6111yescharacteristic
588C4⋊C4C2×C22⋊D6111yescharacteristic
589C4⋊C4C2×C22⋊D6111yescharacteristic
590C22⋊C4—21no—
591C22⋊C4—21no—
592C22⋊C4—41no—
593C22⋊C4—2441no—
594C42—21no—
595C42—21no—
596C42—41no—
597C42—1221no—
598C42—1221no—
599C42—2441no—
600C23—21no—
601C23—1221no—
602C23#1451982211yescharacteristic
603Q8—21no—
604Q8—161no—
605Q8—612no—
606Q8—1221no—
607Q8—1222no—
608Q8—1222no—
609Q8—2444no—
610Q8—2441no—
611Q8—2441no—
612Q8#1451975312yes—
613D4—22no—
614D4—22no—
615D4—21no—
616D4—612no—
617D4—1222no—
618D4—1222no—
619D4—1221no—
620D4—2444no—
621D4#1451975312yes—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—161no—
634C2×C4—1221no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—9761no—
640C2×C4—4881no—
641C2×C4—2444no—
642C2×C4—2444no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C22×C22⋊D6111yescharacteristic
650C2×C4C2×Q8×D6111yescharacteristic
651C2×C4#41×D6111yescharacteristic
652C2×C4C2×#1175822611yescharacteristic
653C2×C4#1451982311yescharacteristic
654C8—21no—
655C8—21no—
656C8—41no—
657C8—1221no—
658C8—1221no—
659C8—2441no—
660C8—2441no—
661C8—2441no—
662C22—21no—
663C22—21no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—1222no—
668C22—2441no—
669C22—2441no—
670C22#1684038711yescharacteristic
671C22D4×C22⋊D6111yescharacteristic
672C22#1684043911yescharacteristic
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—81no—
679C4—81no—
680C4—1222no—
681C4—4881no—
682C4—2442no—
683C4—2442no—
684C4C22⋊Q8×D6111yescharacteristic
685C4C2×#1451975311yescharacteristic
686C4#1684040011yescharacteristic
687C4#1684040311yescharacteristic
688C2—21no—
689C2—21no—
690C2—42no—
691C2—2442no—
692C2#2144948211yescharacteristic
693C2#2145758311yescharacteristic
694C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3083C2
socleC22×C61D42
prime radicalC2×#14519824C22
hypercenter#3083D61
hypocenterC61#25987
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3083

Complements

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

NH ≅ G/Ntype
1. #21455666C2semidirect
4. #21447354C2semidirect
5. #21447185C2semidirect
6. #21447183C2semidirect
7. #21446977C2semidirect
8. #21446096C2semidirect
9. #21446093C2semidirect
10. #21446045C2semidirect
11. #21446042C2semidirect
12. #21445884C2semidirect
13. #21445881C2semidirect
15. C61×#3083C2semidirect
70. #14519824Q8semidirect
342. #3083D61semidirect
414. C61#25987semidirect

Maximal subgroups

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

subgroupindex
#214556662
#214543492
#214543482
#214473542
#214471852
#214471832
#214469772
#214460962
#214460932
#214460452
#214460422
#214458842
#214458812
#214457762
C61×#30832
#2598761