Finite Group Databasebeta

← #29716315

Subgroups — #29716315

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455612C211yesmaximal, characteristic, complemented
2C4×#14519802C211yesmaximal, characteristic
3#21450993C211yesmaximal, characteristic, complemented
4#21447166C212yesmaximal, complemented
5#21446972C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446764C211yesmaximal, characteristic, complemented
8#21446195C212yesmaximal
9#21446052C212yesmaximal, complemented
10#21445839C211yesmaximal, characteristic, complemented
11C61×#3503C211yesmaximal, characteristic, complemented
12C2×#14519802C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840059—22no—
15#16840049—21no—
16#16840039—21no—
17#16839872—21no—
18#16839860—21no—
19C4×#11758201—21no—
20C4×#11758199—21no—
21#16839805C2211yescharacteristic
22#16839805C2212yes—
23#16839786C2211yescharacteristic
24C2×#14519704C2212yes—
25#16839766C2211yescharacteristic
26#16839766C2212yes—
27#16839763C2211yescharacteristic
28C4×#11758194C2211yescharacteristic
29#16839673—22no—
30#16839663C2212yes—
31#16839627C2212yes—
32#16839619—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839531C2212yes—
35#16839531C2212yes—
36#16839521C2212yes—
37#16839521C2212yes—
38#16839521C2212yes—
39Q82×C61—21no—
40C61×#557—21no—
41C61×#554—21no—
42C61×#554—22no—
43D4×Q8×C61—22no—
44D4×Q8×C61—21no—
45C61×#544—22no—
46C61×#544C2212yes—
47C61×#533—21no—
48C61×#533C2211yescharacteristic
49C2×C4⋊Q8×C61—21no—
50C2×C4⋊Q8×C61C2211yescharacteristic
51C61×#520C2212yes—
52C4×#41×C61C2211yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C4×D61—21no—
55#14519802—22no—
56#14519802Q814yescomplemented
57C2×#11758201—21no—
58C2×#11758199—21no—
59C2×#11758194C2311yescharacteristic
60C2×#11758194C2312yes—
61C2×#11758194C2311yescharacteristic
62C4×D244—21no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×#11758198—21no—
69C2×#11758198—24no—
70C2×C4×Dic61C2312yes—
71#14519710—22no—
72#14519710—21no—
73#14519708—22no—
74#14519708—21no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77#14519704—24no—
78#14519701—21no—
79#14519698—21no—
80#14519697—22no—
81#14519697—22no—
82#14519697—21no—
83#14519697—21no—
84#14519688—21no—
85C4×#92418—21no—
86#14519684—21no—
87#14519684—24no—
88C2×#41×C61C2311yescharacteristic
89C22×Q8×C61—22no—
90C22×Q8×C61D412yes—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—24no—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61D411yescharacteristic
101C4⋊Q8×C61D411yescharacteristic
102#125×C61—24no—
103#125×C61—22no—
104#125×C61—22no—
105#125×C61D412yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—24no—
109C22⋊Q8×C61—24no—
110C22⋊Q8×C61—21no—
111C22⋊Q8×C61—21no—
112C22⋊Q8×C61—24no—
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61—21no—
117C4×Q8×C61—21no—
118C4×Q8×C61D411yescharacteristic
119C4×Q8×C61D411yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—24no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61D411yescharacteristic
126#117×C61—22no—
127#117×C61—21no—
128#117×C61C2312yes—
129#117×C61C2311yescharacteristic
130C2×C4⋊C4×C61—22no—
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61D412yes—
133C2×C4⋊C4×C61C2312yes—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C23×D61—21no—
137#41×C61—21no—
138#41×C61—21no—
139#41×C61C2×Q812yes—
140#41×C61#4112yes—
141C2×Q8×C61—24no—
142C2×Q8×C61—21no—
143C2×Q8×C61—22no—
144C2×Q8×C61—21no—
145C2×Q8×C61—22no—
146C2×Q8×C61—82no—
147C2×Q8×C61—84no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—24no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D414yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—22no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C2×Q812yescomplemented
167C2×C4×D61—44no—
168C2×C4×D61—41no—
169C2×C4×D61—41no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C2×D412yes—
185C4⋊C4×C61C2×D412yes—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—24no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C42×C61—24no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—22no—
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202#11758201—24no—
203#11758199—24no—
204#11758198—41no—
205#11758198—41no—
206#11758198—84no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—42no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212#11758194—21no—
213#11758194—22no—
214#11758194—21no—
215#11758194—21no—
216#11758194—22no—
217#11758194C2×Q812yescomplemented
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—22no—
221C23×C61—21no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—41no—
227C22×D61—41no—
228Q8×C61—21no—
229Q8×C61—44no—
230Q8×C61—42no—
231Q8×C61—42no—
232Q8×C61—82no—
233Q8×C61—84no—
234Q8×C61C22⋊Q812yes—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—21no—
238D4×C61C22⋊Q812yes—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—24no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—44no—
252C2×C4×C61—84no—
253C2×C4×C61#12011yescharacteristic
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12011yescharacteristic
256C2×C4×C61#12011yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×Q811yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—41no—
267C2×Dic61—42no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—82no—
273C4×D61—82no—
274#92418—22no—
275#92418—41no—
276#92418—42no—
277#24321—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—22no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C2×#12011yescharacteristic
284C22×C61#54811yescharacteristic
285C22×C61D4×Q811yescharacteristic
286C2×D61—44no—
287C2×D61—42no—
288C2×D61—82no—
289C2×D61—81no—
290C2×D61—81no—
291C4×C61—22no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61—41no—
298C4×C61—44no—
299C4×C61—42no—
300C4×C61#45412yes—
301C4×C61C2×C22⋊Q811yescharacteristic
302C4×C61#53611yescharacteristic
303C4×C61#54811yescharacteristic
304C4×C61D4×Q811yescharacteristic
305Dic61—82no—
306Dic61—81no—
307#3503D6111yescharacteristic, complemented
308#3077—611no—
309C4×AGL(1,ℤ/8)—611no—
310#2563—611no—
311#2061—612no—
312#2027—612no—
313#1997—611no—
314#1986—611no—
315#1883—612no—
316#1860—612no—
317#1806—611no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—42no—
321C2×C61#240911yescharacteristic
322C2×C61C2×#45411yescharacteristic
323C2×C61#301911yescharacteristic
324D61—82no—
325C2×AGL(1,ℤ/8)—611no—
326Q82—21no—
327#557—21no—
328#554—21no—
329#554—22no—
330D4×Q8—22no—
331D4×Q8—21no—
332#544—22no—
333#544C2×D6112yes—
334#533—21no—
335#533C2×D6111yescharacteristic
336C2×C4⋊Q8—21no—
337C2×C4⋊Q8C2×D6111yescharacteristic
338#520C2×D6112yes—
339C4×#41C2×D6111yescharacteristic
340C2×C4×D4—611no—
341#478—1222no—
342#476—1221no—
343#474—1221no—
344#442—1221no—
345#440—1221no—
346C4×SD16—1221no—
347C4×D8—1221no—
348#428—612no—
349#428—611no—
350#423—611no—
351C2×#104—612no—
352#418—612no—
353#418—611no—
354#417—611no—
355C4×M16—611no—
356#393—1222no—
357#390—612no—
358#382—612no—
359#380—1222no—
360C4×C4⋊C4—611no—
361#343—612no—
362#343—612no—
363#339—612no—
364#339—612no—
365#339—612no—
366C61#2432111yescharacteristic, complemented, Sylow
367C2×#41C22×D6111yescharacteristic
368C22×Q8—22no—
369C22×Q8C22⋊D6112yes—
370C22×D4—611no—
371C23×C4—1221no—
372AGL(1,ℤ/8)—614no—
373AGL(1,ℤ/8)—1222no—
374C2×SD16—1221no—
375C2×D8—1221no—
376C2×M16—611no—
377C2×M16—611no—
378C2×M16—612no—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381C4⋊Q8—24no—
382C4⋊Q8—22no—
383C4⋊Q8—22no—
384C4⋊Q8—21no—
385C4⋊Q8—21no—
386C4⋊Q8—21no—
387C4⋊Q8—21no—
388C4⋊Q8C22⋊D6111yescharacteristic
389C4⋊Q8C22⋊D6111yescharacteristic
390#125—24no—
391#125—22no—
392#125—22no—
393#125C22⋊D6112yes—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—24no—
397C22⋊Q8—24no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—24no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8C22⋊D6111yescharacteristic
407C4×Q8C22⋊D6111yescharacteristic
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—24no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4C22⋊D6111yescharacteristic
417C4×D4C22⋊D6111yescharacteristic
418#117—22no—
419#117—21no—
420#117C22×D6111yescharacteristic
421#117C22×D6112yes—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4C22⋊D6112yes—
427C2×C4⋊C4C22×D6112yes—
428C2×C22⋊C4—1224no—
429C2×C22⋊C4—1221no—
430C2×C42—21no—
431C2×C42—612no—
432C2×C42C22×D6111yescharacteristic
433#107—1221no—
434#107—1222no—
435#106—1221no—
436#106—1222no—
437C4⋊C8—1221no—
438C4⋊C8—1221no—
439#104—1224no—
440#103—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1222no—
445#102—1222no—
446#97—1221no—
447C4×C8—1221no—
448#95—1221no—
449#95—1224no—
450C24—1221no—
451#41—21no—
452#41—21no—
453#41Q8×D6112yes—
454#41#1175822612yes—
455C2×Q8—24no—
456C2×Q8—21no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—82no—
461C2×Q8—84no—
462C2×Q8C2×C22⋊D6111yescharacteristic
463C2×D4—21no—
464C2×D4—612no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4C2×C22⋊D6111yescharacteristic
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—2444no—
478C22×C4—2441no—
479C22×C4—2441no—
480C22×C4C2×C22⋊D6111yescharacteristic
481C22×C4C2×C22⋊D6114yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C23×D6111yescharacteristic
484SD16—1224no—
485D8—1224no—
486M16—612no—
487M16—1221no—
488M16—1222no—
489M16—1221no—
490M16—1221no—
491M16—1222no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1222no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2442no—
512C4⋊C4C2×C22⋊D6112yes—
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C4⋊C4C2×C22⋊D6112yes—
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C22⋊C4—24no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—4884no—
522C22⋊C4—2441no—
523C22⋊C4—2441no—
524C42—24no—
525C42—21no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—1222no—
530C42—1222no—
531C42C2×C22⋊D6111yescharacteristic
532C42C2×C22⋊D6111yescharacteristic
533C42C2×C22⋊D6111yescharacteristic
534C42C2×C22⋊D6111yescharacteristic
535C23—21no—
536C23—1222no—
537C23—2442no—
538C23—2442no—
539C23—2441no—
540C23—2441no—
541C23C22×C22⋊D6111yescharacteristic
542Q8—21no—
543Q8—44no—
544Q8—42no—
545Q8—42no—
546Q8—82no—
547Q8—84no—
548Q8#1451975312yes—
549D4—22no—
550D4—22no—
551D4—21no—
552D4—1222no—
553D4—1222no—
554D4—2441no—
555D4—2441no—
556D4#1451975312yes—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—84no—
571C2×C4—4882no—
572C2×C4—4882no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2441no—
577C2×C4—2442no—
578C2×C4#1451974611yescharacteristic
579C2×C4#1451974611yescharacteristic
580C2×C4#1451974611yescharacteristic
581C2×C4#1451974611yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4C2×Q8×D6111yescharacteristic
585C2×C4C2×#1175822611yescharacteristic
586C2×C4#1451982511yescharacteristic
587C2×C4#1451982511yescharacteristic
588C8—1222no—
589C8—2441no—
590C8—2442no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—4882no—
597C22—4881no—
598C22—4881no—
599C22—2444no—
600C22—2442no—
601C22C2×#1451974611yescharacteristic
602C22#1684042311yescharacteristic
603C22Q8×C22⋊D6111yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—22no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—44no—
612C4—42no—
613C4—4882no—
614C4—4881no—
615C4#1683993512yes—
616C4C2×#1451975311yescharacteristic
617C4#1684033611yescharacteristic
618C4#1684042311yescharacteristic
619C4Q8×C22⋊D6111yescharacteristic
620C2—21no—
621C2—21no—
622C2—42no—
623C2—4882no—
624C2#2144917811yescharacteristic
625C2C2×#1683993511yescharacteristic
626C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3503C2
socleC22×C61C2×#120
prime radicalC2×#14519802C22
hypercenter#3503D61
hypocenterC61#24321
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3503

Complements

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

NH ≅ G/Ntype
1. #21455612C2semidirect
3. #21450993C2semidirect
4. #21447166C2semidirect
5. #21446972C2semidirect
6. #21446809C2semidirect
7. #21446764C2semidirect
9. #21446052C2semidirect
10. #21445839C2semidirect
11. C61×#3503C2semidirect
56. #14519802Q8semidirect
166. C2×D244C2×Q8semidirect
217. #11758194C2×Q8semidirect
307. #3503D61semidirect
366. C61#24321semidirect

Maximal subgroups

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

subgroupindex
#214556122
C4×#145198022
#214509932
#214471662
#214469722
#214468092
#214467642
#214461952
#214460522
#214458392
C61×#35032
#2432161