Finite Group Databasebeta

← #29743440

Subgroups — #29743440

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455644C211yesmaximal, characteristic, complemented
2#21454654C211yesmaximal, characteristic
3#21454417C211yesmaximal, characteristic, complemented
4#21447344C211yesmaximal, characteristic, complemented
5#21447327C211yesmaximal, characteristic, complemented
6#21447178C211yesmaximal, characteristic, complemented
7#21447161C211yesmaximal, characteristic, complemented
8#21446602C211yesmaximal, characteristic, complemented
9#21446415C211yesmaximal, characteristic, complemented
10#21446325C211yesmaximal, characteristic
11#21446195C211yesmaximal, characteristic
12#21446096C211yesmaximal, characteristic, complemented
13#21446043C211yesmaximal, characteristic, complemented
14#21446027C211yesmaximal, characteristic, complemented
15C61×#3083C211yesmaximal, characteristic, complemented
16C2×#14519802C2211yescharacteristic
17C2×#14519784C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840236C2211yescharacteristic
20#16840059—21no—
21#16840059—21no—
22#16840049—21no—
23#16840039—21no—
24#16839871—21no—
25#16839865—21no—
26#16839847—21no—
27#16839842—21no—
28C2×#14519713C2211yescharacteristic
29#16839805—21no—
30#16839805C2211yescharacteristic
31#16839805C2211yescharacteristic
32#16839802—21no—
33#16839802C2211yescharacteristic
34#16839802C2211yescharacteristic
35#16839798—21no—
36#16839798C2211yescharacteristic
37C2×#14519710—21no—
38C2×#14519710C2211yescharacteristic
39#16839789C2211yescharacteristic
40C2×#14519704C2211yescharacteristic
41#16839766C2211yescharacteristic
42C2×#14519699C2211yescharacteristic
43C2×#14519697C2211yescharacteristic
44C2×#14519693C2211yescharacteristic
45#16839717C2211yescharacteristic
46#16839663C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839531C2211yescharacteristic
49#16839531C2211yescharacteristic
50#16839530C2211yescharacteristic
51#16839526C2211yescharacteristic
52#16839523C2211yescharacteristic
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
67C22×D244C2311yescharacteristic
68#14519802—22no—
69#14519802Q814yescomplemented
70C2×#11758201—21no—
71C2×#11758199—21no—
72#14519784—22no—
73#14519784—22no—
74C2×#11758194—21no—
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C22×#92418—21no—
79C22×#92418C2311yescharacteristic
80Q8×Dic61—21no—
81C4×D244—21no—
82C4×D244—21no—
83D4×Dic61—21no—
84D4×Dic61—41no—
85#14519739—21no—
86#14519739—21no—
87#14519736—21no—
88#14519736C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—41no—
92C2×#11758198—41no—
93C22⋊C4×D61—41no—
94C2×C4×Dic61C2311yescharacteristic
95#14519713—22no—
96#14519710—21no—
97#14519710—21no—
98#14519710—21no—
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519710D411yescharacteristic
103#14519710D411yescharacteristic
104#14519708—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708D411yescharacteristic
109#14519708D411yescharacteristic
110C4⋊#92418—21no—
111C4⋊#92418—21no—
112#14519704—22no—
113#14519699—21no—
114#14519699—21no—
115#14519697—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519693—22no—
120#14519687—21no—
121C8×Dic61—21no—
122C2×#41×C61C2311yescharacteristic
123C2×M16×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61—21no—
127#125×C61—21no—
128#125×C61—21no—
129C22⋊Q8×C61—41no—
130C4×Q8×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61—41no—
133#117×C61—21no—
134#117×C61C2311yescharacteristic
135#117×C61C2311yescharacteristic
136C2×C4⋊C4×C61—21no—
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C4⋊C4×C61C2311yescharacteristic
139#107×C61—21no—
140#107×C61—21no—
141#107×C61—21no—
142#106×C61—21no—
143C4⋊C8×C61—21no—
144C4⋊C8×C61—21no—
145#103×C61—21no—
146#103×C61D411yescharacteristic
147#103×C61D411yescharacteristic
148#102×C61—21no—
149#102×C61D411yescharacteristic
150#102×C61D411yescharacteristic
151C23×D61—21no—
152#41×C61—21no—
153#41×C61—21no—
154#41×C61C2×Q812yes—
155#41×C61#4112yes—
156C2×Q8×C61—41no—
157C2×Q8×C61C2×D411yescharacteristic
158C2×D4×C61—21no—
159C2×D4×C61C2×D411yescharacteristic
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—41no—
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—41no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244C2×Q812yes—
174C2×C4×D61—41no—
175C2×C4×D61—41no—
176M16×C61—21no—
177M16×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61—41no—
190C4⋊C4×C61—41no—
191C4⋊C4×C61C2×D411yescharacteristic
192C4⋊C4×C61C2×D411yescharacteristic
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C22⋊C4×C61—21no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—41no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—41no—
201C22⋊Dic61—21no—
202C22⋊Dic61—41no—
203#11758201—24no—
204#11758199—24no—
205#11758198—41no—
206#11758198—41no—
207#11758198—82no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—41no—
211C4⋊Dic61C2×D411yescharacteristic
212C4⋊Dic61C2×D411yescharacteristic
213#11758196—82no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—41no—
218#11758194—24no—
219#11758194—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194C2×Q812yes—
224#11758194#4112yes—
225C2×#92418—24no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418C2×D411yescharacteristic
232C2×#92418C2×D411yescharacteristic
233C23×C61—21no—
234C23×C61H(5,𝔽2)11yescharacteristic
235C22×D61—22no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—81no—
239C22×D61—81no—
240Q8×C61—21no—
241Q8×C61—161no—
242Q8×C61C22⋊Q812yes—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—21no—
246D4×C61C22⋊Q812yes—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—81no—
257C2×C4×C61—81no—
258C2×C4×C61—161no—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×Q811yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C612−1+411yescharacteristic
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—81no—
272C2×Dic61—161no—
273D244—22no—
274D244—22no—
275D244—41no—
276D244—41no—
277C4×D61—161no—
278C4×D61—161no—
279C8×C61—21no—
280C8×C61—21no—
281C8×C61—41no—
282#92418—22no—
283#92418—21no—
284#92418—22no—
285#92418—21no—
286#92418—41no—
287#26400—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61#53911yescharacteristic
294C22×C61D4211yescharacteristic
295C22×C61#55411yescharacteristic
296C2×D61—42no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—82no—
300C2×D61—161no—
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C22⋊Q811yescharacteristic
309C4×C61#52511yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—42no—
313Dic61—41no—
314Dic61—41no—
315Dic61—81no—
316#3083D6111yescharacteristic, complemented
317#3081—611no—
318#2946—611no—
319#2913—611no—
320#2093—611no—
321#2089—611no—
322#2063—611no—
323#2060—611no—
324#1952—611no—
325#1922—611no—
326#1907—611no—
327#1883—611no—
328#1866—611no—
329#1859—611no—
330#1856—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—42no—
334C2×C61#244911yescharacteristic
335C2×C61#328911yescharacteristic
336C2×C61#329211yescharacteristic
337D61—82no—
338C2×AGL(1,ℤ/8)—611no—
339C2×#131—611no—
340#544C2×D6111yescharacteristic
341#520—611no—
342#520C2×D6111yescharacteristic
343#518—611no—
344#479—21no—
345#478—21no—
346#478—1221no—
347#478—1221no—
348#477—21no—
349#476—1221no—
350#474—21no—
351#474—1221no—
352#442—1221no—
353#440—1221no—
354C4×SD16—1221no—
355C4×D8—1221no—
356C2×#108—611no—
357#428—611no—
358#428—611no—
359#428—611no—
360#428—611no—
361#428—1221no—
362#428—1221no—
363#428C2×D6111yescharacteristic
364#427—611no—
365#427—1221no—
366C2×#107—611no—
367C2×#107—1221no—
368C2×#107C2×D6111yescharacteristic
369#424—611no—
370#424C2×D6111yescharacteristic
371C2×#104—611no—
372#418—611no—
373#417C2×D6111yescharacteristic
374#416C2×D6111yescharacteristic
375C2×#102—611no—
376C2×#102—611no—
377C2×#100—611no—
378#405—611no—
379#390—611no—
380#384—611no—
381#343—611no—
382#343—611no—
383#343—611no—
384#341—611no—
385#339—611no—
386#339—611no—
387#339—611no—
388#336—611no—
389C61#2640011yescharacteristic, complemented, Sylow
390C2×#41C22×D6111yescharacteristic
391C22×D4—611no—
392AGL(1,ℤ/8)—614no—
393AGL(1,ℤ/8)—1222no—
394C2×SD16—1221no—
395C2×D8—1221no—
396#131—1222no—
397#131—1222no—
398C2×M16—611no—
399C2×M16—611no—
400C2×M16—611no—
401C2×M16—1221no—
402C2×M16C22×D6111yescharacteristic
403C22×C8—611no—
404C22×C8—1221no—
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—21no—
407C4⋊Q8—21no—
408#125—21no—
409#125—21no—
410C22⋊Q8—41no—
411C4×Q8—21no—
412C4×Q8—1221no—
413C4×D4—21no—
414C4×D4—41no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—2441no—
421#117—21no—
422#117—611no—
423#117—1221no—
424#117C22×D6111yescharacteristic
425#117C22×D6111yescharacteristic
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—2441no—
433C2×C22⋊C4—2441no—
434C2×C42—611no—
435#108—1222no—
436#107—21no—
437#107—21no—
438#107—21no—
439#107—611no—
440#107—611no—
441#107—1221no—
442#107—1221no—
443#107—1221no—
444#107—1221no—
445#107—1221no—
446#107—1221no—
447#106—21no—
448#106—611no—
449#106—611no—
450#106—1221no—
451#106—1221no—
452#106—1221no—
453#106—1221no—
454C4⋊C8—21no—
455C4⋊C8—21no—
456C4⋊C8—1221no—
457C4⋊C8—1221no—
458#104—1222no—
459#103—21no—
460#103C22⋊D6111yescharacteristic
461#103C22⋊D6111yescharacteristic
462#102—21no—
463#102—1221no—
464#102—1221no—
465#102—1221no—
466#102—1221no—
467#102—1221no—
468#102—1221no—
469#102C22⋊D6111yescharacteristic
470#102C22⋊D6111yescharacteristic
471#100—1222no—
472#97—1221no—
473C4×C8—1221no—
474C24—1221no—
475#41—21no—
476#41—21no—
477#41Q8×D6112yes—
478#41#1175822612yes—
479C2×Q8—41no—
480C2×Q8C2×C22⋊D6111yescharacteristic
481C2×D4—21no—
482C2×D4—612no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4C2×C22⋊D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—41no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—2441no—
496C22×C4—2441no—
497C22×C4—2441no—
498C22×C4C23×D6111yescharacteristic
499SD16—1224no—
500D8—1224no—
501M16—21no—
502M16—21no—
503M16—612no—
504M16—612no—
505M16—1224no—
506M16—1221no—
507M16—1221no—
508M16—1221no—
509M16—1221no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—611no—
514C2×C8—611no—
515C2×C8—1224no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8C2×C22⋊D6111yescharacteristic
522C2×C8C2×C22⋊D6111yescharacteristic
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—611no—
532C4⋊C4—611no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—4882no—
536C4⋊C4—2441no—
537C4⋊C4D4×D6111yescharacteristic
538C4⋊C4D4×D6111yescharacteristic
539C4⋊C4C2×C22⋊D6111yescharacteristic
540C4⋊C4C2×C22⋊D6111yescharacteristic
541C22⋊C4—21no—
542C22⋊C4—21no—
543C22⋊C4—41no—
544C22⋊C4—1221no—
545C22⋊C4—4882no—
546C22⋊C4—2441no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—21no—
550C42—21no—
551C42—41no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—2441no—
556C23—21no—
557C23—1222no—
558C23—4881no—
559C23—4881no—
560C23—2441no—
561C23—2441no—
562C23#1451982211yescharacteristic
563Q8—21no—
564Q8—161no—
565Q8#1451975312yes—
566D4—22no—
567D4—22no—
568D4—21no—
569D4—1222no—
570D4—1222no—
571D4—2441no—
572D4—2441no—
573D4#1451975312yes—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—161no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—9761no—
590C2×C4—9761no—
591C2×C4—9761no—
592C2×C4—4881no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822611yescharacteristic
601C2×C4#1451982411yescharacteristic
602C8—21no—
603C8—21no—
604C8—41no—
605C8—1222no—
606C8—1221no—
607C8—1221no—
608C8—1222no—
609C8—2441no—
610C22—21no—
611C22—21no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—9761no—
616C22—4882no—
617C22—4881no—
618C22—4881no—
619C22—2442no—
620C22#1684036111yescharacteristic
621C22D4×C22⋊D6111yescharacteristic
622C22#1684046611yescharacteristic
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—4881no—
631C4—2442no—
632C4—2441no—
633C4—2441no—
634C4C2×#1451975311yescharacteristic
635C4#1684027511yescharacteristic
636C4#1684041111yescharacteristic
637C4#1684042511yescharacteristic
638C2—21no—
639C2—21no—
640C2—42no—
641C2—4882no—
642C2#2144962411yescharacteristic
643C2#2145759011yescharacteristic
644C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3083C2
socleC22×C61D42
prime radicalC2×#14519802C22
hypercenter#3083D61
hypocenterC61#26400
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. #21455644C2semidirect
3. #21454417C2semidirect
4. #21447344C2semidirect
5. #21447327C2semidirect
6. #21447178C2semidirect
7. #21447161C2semidirect
8. #21446602C2semidirect
9. #21446415C2semidirect
12. #21446096C2semidirect
13. #21446043C2semidirect
14. #21446027C2semidirect
15. C61×#3083C2semidirect
69. #14519802Q8semidirect
316. #3083D61semidirect
389. C61#26400semidirect

Maximal subgroups

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

subgroupindex
#214556442
#214546542
#214544172
#214473442
#214473272
#214471782
#214471612
#214466022
#214464152
#214463252
#214461952
#214460962
#214460432
#214460272
C61×#30832
#2640061