Finite Group Databasebeta

← #29737579

Subgroups — #29737579

Order 15616.

842 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455677C211yesmaximal, characteristic, complemented
2#21454349C211yesmaximal, characteristic
3#21454348C211yesmaximal, characteristic
4#21447364C211yesmaximal, characteristic, complemented
5#21447185C211yesmaximal, characteristic, complemented
6#21447183C211yesmaximal, characteristic, complemented
7#21446977C211yesmaximal, characteristic, complemented
8#21446096C211yesmaximal, characteristic, complemented
9#21446093C211yesmaximal, characteristic, complemented
10#21446052C211yesmaximal, characteristic, complemented
11#21446049C211yesmaximal, characteristic, complemented
12#21445884C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic, complemented
14#21445776C211yesmaximal, characteristic
15C61×#3083C211yesmaximal, characteristic, complemented
16C2×#14519824C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18C2×#14519783C2211yescharacteristic
19#16840242C2211yescharacteristic
20#16840066—21no—
21#16840059—21no—
22#16840049—21no—
23#16840044—21no—
24C2×#14519713C2211yescharacteristic
25C2×#14519712C2211yescharacteristic
26#16839805C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
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#16839627C2211yescharacteristic
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
95C4⋊C4×D61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519713—41no—
98#14519712—41no—
99#14519710—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103C4⋊#92418—21no—
104C4⋊#92418—21no—
105#14519704—22no—
106#14519703—22no—
107#14519702—21no—
108#14519702—21no—
109#14519701—21no—
110#14519700—21no—
111#14519700—21no—
112#14519700—21no—
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700—21no—
117#14519700—21no—
118#14519700D411yescharacteristic
119#14519700D411yescharacteristic
120#14519698—21no—
121#14519697—21no—
122#14519697—21no—
123#14519697D411yescharacteristic
124#14519697D411yescharacteristic
125#14519695—41no—
126#14519684—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—
231C4×Dic61—21no—
232C4×Dic61—21no—
233C4×Dic61—41no—
234#11758194—21no—
235#11758194—21no—
236#11758194—21no—
237#11758194—21no—
238#11758194—21no—
239#11758194—21no—
240#11758194—42no—
241C2×#92418—21no—
242C2×#92418—21no—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—42no—
247C2×#92418C2×D411yescharacteristic
248C2×#92418C2×D411yescharacteristic
249#11758189—42no—
250C8×D61—42no—
251C23×C61—21no—
252C23×C61H(5,𝔽2)11yescharacteristic
253C22×D61—21no—
254Q8×C61—21no—
255Q8×C61—161no—
256Q8×C61C22⋊Q812yes—
257D4×C61—22no—
258D4×C61—22no—
259D4×C61—21no—
260D4×C61C22⋊Q812yes—
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—21no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—81no—
271C2×C4×C61—81no—
272C2×C4×C61—161no—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×Q811yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61H(5,𝔽2)11yescharacteristic
279C22⋊D61—22no—
280C22⋊D61—22no—
281C22⋊D61—44no—
282C2×Dic61—21no—
283C2×Dic61—22no—
284C2×Dic61—44no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—81no—
292D244—21no—
293D244C22⋊Q812yes—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—44no—
297C4×D61—161no—
298Dic122—21no—
299Dic122—22no—
300Dic122—22no—
301Dic122—44no—
302Dic122—41no—
303Dic122—41no—
304Dic122C22⋊Q812yes—
305C8×C61—21no—
306C8×C61—21no—
307C8×C61—41no—
308#92418—21no—
309#92418—21no—
310#92418—41no—
311#92418—41no—
312#92418—41no—
313#25985—611nomaximal, Sylow
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—22no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61#54311yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#55111yescharacteristic
322C2×D61—22no—
323C2×D61—41no—
324C2×D61—41no—
325C4×C61—22no—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—41no—
330C4×C61—81no—
331C4×C61—81no—
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61C2×C22⋊Q811yescharacteristic
334C4×C61#54611yescharacteristic
335C4×C61#54611yescharacteristic
336Dic61—22no—
337Dic61—42no—
338Dic61—81no—
339Dic61—81no—
340#3084—611no—
341#3083D6111yescharacteristic, complemented
342#2898—611no—
343#2898—611no—
344#2097—611no—
345#2064—611no—
346#2064—611no—
347#2028—611no—
348#1866—611no—
349#1866—611no—
350#1860—611no—
351#1860—611no—
352#1816—611no—
353#1815—611no—
354#1793—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61—42no—
358C2×C61#243711yescharacteristic
359C2×C61#328911yescharacteristic
360C2×C61#329011yescharacteristic
361D61—42no—
362C2×2−1+4—611no—
363C2×#131—611no—
364C2×#131—611no—
365#544C2×D6111yescharacteristic
366#520—611no—
367#520C2×D6111yescharacteristic
368#479—21no—
369#479—1221no—
370#478—21no—
371#478—1221no—
372#477—21no—
373#476—1221no—
374#475—1221no—
375#474—21no—
376C2×#108—611no—
377C2×#108—611no—
378#428—611no—
379#428C2×D6111yescharacteristic
380C2×#107C2×D6111yescharacteristic
381C2×#106—611no—
382#424—611no—
383#424C2×D6111yescharacteristic
384C2×#104—611no—
385C2×#104—611no—
386#419—611no—
387#419—611no—
388#419—1221no—
389#419—1221no—
390#417—611no—
391#417—611no—
392#417—611no—
393#417—1221no—
394#417—1221no—
395#417C2×D6111yescharacteristic
396#416—611no—
397#416—1221no—
398#416—1221no—
399#416C2×D6111yescharacteristic
400C2×#103—611no—
401C2×#103—611no—
402C2×#103—1221no—
403C2×#103—1221no—
404C2×#101—611no—
405#391—611no—
406#382—611no—
407#343—611no—
408#343—611no—
409#343—611no—
410#339—611no—
411#339—611no—
412#336—611no—
413C61#2598511yescharacteristic, complemented, Sylow
4142−1+4—614no—
4152−1+4—1222no—
4162−1+4—1222no—
4172−1+4—2441no—
418C2×#41—611no—
419C2×#41—1221no—
420C2×#41—1221no—
421C2×#41—2441no—
422C2×#41C22×D6111yescharacteristic
423C22×Q8—611no—
424C22×Q8—1221no—
425C22×Q8—1221no—
426#131—1222no—
427#131—1222no—
428#131—2441no—
429#131—2441no—
430C2×M16—611no—
431C2×M16—611no—
432C2×M16—611no—
433C2×M16—1221no—
434C2×M16—1221no—
435C2×M16C22×D6111yescharacteristic
436C22×C8—611no—
437C22×C8—1221no—
438C22×C8—1221no—
439C22×C8C22×D6111yescharacteristic
440C4⋊Q8—21no—
441C4⋊Q8—21no—
442#125—21no—
443#125—21no—
444C22⋊Q8—41no—
445C4×Q8—21no—
446C4×Q8—1221no—
447C4×D4—21no—
448C4×D4—41no—
449C4×D4—1221no—
450C4×D4—2441no—
451#117—21no—
452#117—1221no—
453#117C22×D6111yescharacteristic
454#117C22×D6111yescharacteristic
455C2×C4⋊C4—21no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C42—611no—
461#108—2441no—
462#108—2441no—
463#107—21no—
464#107—21no—
465#107—21no—
466#107—1221no—
467#106—21no—
468#106—1221no—
469#106—1221no—
470#106—1221no—
471C4⋊C8—21no—
472C4⋊C8—21no—
473C4⋊C8—1221no—
474C4⋊C8—1221no—
475#104—1222no—
476#104—1222no—
477#103—21no—
478#103—611no—
479#103—611no—
480#103—1221no—
481#103—1221no—
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#103C22⋊D6111yescharacteristic
492#103C22⋊D6111yescharacteristic
493#102—21no—
494#102—611no—
495#102—611no—
496#102—1221no—
497#102—1221no—
498#102—1221no—
499#102C22⋊D6111yescharacteristic
500#102C22⋊D6111yescharacteristic
501#101—2441no—
502#95—2441no—
503#95—2441no—
504#41—21no—
505#41—21no—
506#41—612no—
507#41—612no—
508#41—1224no—
509#41—1221no—
510#41—1221no—
511#41—1224no—
512#41—2442no—
513#41—2444no—
514#41—2442no—
515#41—2444no—
516#41Q8×D6112yes—
517#41#1175822612yes—
518C2×Q8—41no—
519C2×Q8—611no—
520C2×Q8—612no—
521C2×Q8—1224no—
522C2×Q8—1221no—
523C2×Q8—1224no—
524C2×Q8—1221no—
525C2×Q8—1221no—
526C2×Q8—1221no—
527C2×Q8—1221no—
528C2×Q8—2441no—
529C2×Q8—2442no—
530C2×Q8—2442no—
531C2×Q8C2×C22⋊D6111yescharacteristic
532C2×D4—21no—
533C2×D4—611no—
534C2×D4—1221no—
535C2×D4—2441no—
536C2×D4C2×C22⋊D6111yescharacteristic
537C22×C4—21no—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—41no—
541C22×C4—1221no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—1221no—
545C22×C4—2441no—
546C22×C4—2441no—
547C22×C4—2441no—
548C22×C4C23×D6111yescharacteristic
549M16—21no—
550M16—21no—
551M16—1221no—
552M16—1221no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—2442no—
558M16—2442no—
559C2×C8—21no—
560C2×C8—21no—
561C2×C8—21no—
562C2×C8—611no—
563C2×C8—611no—
564C2×C8—1221no—
565C2×C8—1221no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—2442no—
570C2×C8—2442no—
571C2×C8C2×C22⋊D6111yescharacteristic
572C2×C8C2×C22⋊D6111yescharacteristic
573C4⋊C4—21no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—41no—
578C4⋊C4—41no—
579C4⋊C4—41no—
580C4⋊C4—41no—
581C4⋊C4—1221no—
582C4⋊C4—1221no—
583C4⋊C4—2441no—
584C4⋊C4D4×D6111yescharacteristic
585C4⋊C4D4×D6111yescharacteristic
586C4⋊C4C2×C22⋊D6111yescharacteristic
587C4⋊C4C2×C22⋊D6111yescharacteristic
588C22⋊C4—21no—
589C22⋊C4—21no—
590C22⋊C4—41no—
591C22⋊C4—2441no—
592C42—21no—
593C42—21no—
594C42—41no—
595C42—1221no—
596C42—1221no—
597C42—2441no—
598C23—21no—
599C23—1221no—
600C23#1451982211yescharacteristic
601Q8—21no—
602Q8—161no—
603Q8—612no—
604Q8—1221no—
605Q8—1222no—
606Q8—1222no—
607Q8—2444no—
608Q8—2441no—
609Q8—2441no—
610Q8#1451975312yes—
611D4—22no—
612D4—22no—
613D4—21no—
614D4—612no—
615D4—1222no—
616D4—1222no—
617D4—1221no—
618D4—2444no—
619D4#1451975312yes—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—161no—
632C2×C4—1221no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—9761no—
637C2×C4—4881no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C22×C22⋊D6111yescharacteristic
648C2×C4C2×Q8×D6111yescharacteristic
649C2×C4#41×D6111yescharacteristic
650C2×C4C2×#1175822611yescharacteristic
651C2×C4#1451982311yescharacteristic
652C8—21no—
653C8—21no—
654C8—41no—
655C8—1221no—
656C8—1221no—
657C8—2441no—
658C8—2441no—
659C8—2441no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—41no—
664C22—41no—
665C22—1222no—
666C22—2441no—
667C22—2441no—
668C22#1684038711yescharacteristic
669C22D4×C22⋊D6111yescharacteristic
670C22#1684043911yescharacteristic
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—81no—
677C4—81no—
678C4—1222no—
679C4—4881no—
680C4—4881no—
681C4—2442no—
682C4C22⋊Q8×D6111yescharacteristic
683C4C2×#1451975311yescharacteristic
684C4#1684040011yescharacteristic
685C4#1684040311yescharacteristic
686C2—21no—
687C2—21no—
688C2—42no—
689C2—2442no—
690C2#2144948211yescharacteristic
691C2#2145759011yescharacteristic
692C2#2145760111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3083C2
socleC22×C61D42
prime radicalC2×#14519824C22
hypercenter#3083D61
hypocenterC61#25985
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. #21455677C2semidirect
4. #21447364C2semidirect
5. #21447185C2semidirect
6. #21447183C2semidirect
7. #21446977C2semidirect
8. #21446096C2semidirect
9. #21446093C2semidirect
10. #21446052C2semidirect
11. #21446049C2semidirect
12. #21445884C2semidirect
13. #21445876C2semidirect
15. C61×#3083C2semidirect
70. #14519824Q8semidirect
341. #3083D61semidirect
413. C61#25985semidirect

Maximal subgroups

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

subgroupindex
#214556772
#214543492
#214543482
#214473642
#214471852
#214471832
#214469772
#214460962
#214460932
#214460522
#214460492
#214458842
#214458762
#214457762
C61×#30832
#2598561