Finite Group Databasebeta

← #29681595

Subgroups — #29681595

Order 15616.

1974 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840326C211yesmaximal, characteristic
2C2×#16840236C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic, complemented
4C22×#14519693C211yesmaximal, characteristic, complemented
5#21446214C214yesmaximal, complemented
6#21446204C214yesmaximal, complemented
7#21445530C211yesmaximal, characteristic, complemented
8#21445528C211yesmaximal, characteristic, complemented
9C61×#2896C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840326—24no—
13#16840326C418yescomplemented
14C2×#14519773C2212yes—
15C2×#14519766C2212yes—
16C2×#14519751—22no—
17C2×#14519745—22no—
18#16840236—22no—
19#16840236—22no—
20#16840236—22no—
21#16840236C2214yes—
22C2×C4×D244—22no—
23C2×#14519739—22no—
24C2×#14519736C2211yescharacteristic
25C2×#14519736C2211yescharacteristic
26C2×C22⋊C4×D61—22no—
27#16839771C2212yescomplemented
28#16839767C2212yescomplemented
29#16839763—22no—
30#16839758—22no—
31C2×#14519693—24no—
32C2×#14519693C2214yes—
33#16839523C2214yescomplemented
34#16839521C2214yescomplemented
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#419C2212yescomplemented
39C61×#418C2212yescomplemented
40C61×#417—22no—
41C61×#416—22no—
42C24×D61D412yes—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244—21no—
47C22×D244—22no—
48C22×D244—22no—
49C22×D244C2×C414yes—
50C22×D244D412yes—
51C22×D244D414yes—
52C22×D244C2312yes—
53C22×C22⋊D61—24no—
54C22×C4×D61—24no—
55C2×#11758194—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194C2312yes—
59C22×#92418—21no—
60#14519773—24no—
61#14519773C2×C418yescomplemented
62#14519766—24no—
63#14519766C2×C418yescomplemented
64#14519751—216no—
65#14519745—216no—
66C4×D244—28no—
67C4×D244—28no—
68#14519739—28no—
69#14519739—28no—
70#14519736—21no—
71#14519736—21no—
72#14519736—22no—
73#14519736C2×C414yescomplemented
74#14519736D412yescomplemented
75#14519736C2312yes—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D412yescomplemented
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—24no—
81C2×#11758198—24no—
82C22⋊C4×D61—28no—
83C22⋊C4×D61—28no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2312yes—
86#14519701—24no—
87#14519698—24no—
88#14519693—28no—
89C2×#41×C61—22no—
90C2×#41×C61—21no—
91C2×#41×C61—21no—
92C2×#41×C61—21no—
93C2×#41×C61—21no—
94C2×#41×C61—22no—
95C2×#41×C61—22no—
96C2×#41×C61C2×C414yescomplemented
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C2×M16×C61—21no—
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2312yes—
106C22×C8×C61—21no—
107#117×C61—21no—
108#117×C61—21no—
109#117×C61D412yescomplemented
110#117×C61C2312yes—
111C2×C4⋊C4×C61D412yescomplemented
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—24no—
115#102×C61—24no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—24no—
121C23×D61—84no—
122C23×D61C22⋊C418yes—
123C23×D61C2×D414yes—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—24no—
127#41×C61—24no—
128#41×C61—42no—
129#41×C61—42no—
130#41×C61—42no—
131#41×C61—42no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61C22×C412yescomplemented
138C2×D4×C61—24no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—41no—
145C2×D4×C61—41no—
146C2×D4×C61—42no—
147C2×D4×C61C22×C412yescomplemented
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—24no—
151C22×C4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61—42no—
154C22×C4×C61C22×C412yes—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D412yes—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61#4112yes—
162C22×C4×C61#4112yes—
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—216no—
165C2×C22⋊D61—88no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C2×D244—24no—
169C2×D244—24no—
170C2×D244—24no—
171C2×D244—24no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—24no—
175C2×D244—24no—
176C2×D244—24no—
177C2×D244—24no—
178C2×D244—24no—
179C2×D244—22no—
180C2×D244—22no—
181C2×D244—22no—
182C2×D244—22no—
183C2×D244C22⋊C418yes—
184C2×D244C22×C414yes—
185C2×D244C2×D414yes—
186C2×C4×D61—216no—
187C2×C4×D61—88no—
188M16×C61—22no—
189M16×C61—22no—
190M16×C61—42no—
191C2×C8×C61—22no—
192C2×C8×C61—42no—
193C4⋊C4×C61—22no—
194C4⋊C4×C61C2×D414yes—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—22no—
197C42×C61—22no—
198C42×C61—22no—
199C22⋊Dic61—24no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202#11758198—216no—
203#11758198—216no—
204C4⋊Dic61—24no—
205C4⋊Dic61—22no—
206C4⋊Dic61C2×D414yes—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22×C414yescomplemented
211#11758194—22no—
212#11758194—22no—
213#11758194—42no—
214C2×#92418—22no—
215C2×#92418—42no—
216C23×C61—22no—
217C23×C61—42no—
218C23×C61—44no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61C2×C22⋊C412yes—
222C23×C61#12011yescharacteristic
223C23×C61#12011yescharacteristic
224C23×C61#12312yes—
225C23×C61C22×D411yescharacteristic
226C22×D61—24no—
227C22×D61—28no—
228C22×D61—24no—
229C22×D61—28no—
230C22×D61—48no—
231C22×D61—48no—
232C22×D61—88no—
233C22×D61—84no—
234C22×D61—88no—
235C22×D61C2×C22⋊C418yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—44no—
243D4×C61—44no—
244D4×C61—41no—
245D4×C61—41no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—44no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—44no—
252C2×C4×C61—44no—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61C4×D412yes—
257C2×C4×C61C4×D412yes—
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12112yes—
264C2×C4×C61#12312yes—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C22⋊D61—816no—
272C2×Dic61—24no—
273C2×Dic61—24no—
274C2×Dic61—44no—
275C2×Dic61—44no—
276D244—24no—
277D244—24no—
278D244—24no—
279D244—24no—
280D244—28no—
281D244—28no—
282C4×D61—816no—
283C8×C61—42no—
284#92418—42no—
285#20953—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—44no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61#38612yes—
296C22×C61#38612yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C22×C61C2×#12011yescharacteristic
299C22×C61C2×#12311yescharacteristic
300C2×D61—28no—
301C2×D61—416no—
302C2×D61—48no—
303C2×D61—48no—
304C2×D61—84no—
305C2×D61—88no—
306C4×C61—44no—
307C4×C61—42no—
308C4×C61#38612yes—
309C4×C61#38612yes—
310C4×C61C2×C4×D411yescharacteristic
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C2×#12111yescharacteristic
313C4×C61C2×#12111yescharacteristic
314Dic61—44no—
315Dic61—44no—
316C2×#535—611no—
317C2×#518—611no—
318#2896—611no—
319#2896D6111yescharacteristic, complemented
320C22×#100—611no—
321#1887—614no—
322#1885—614no—
323#1733—611no—
324#1733—611no—
325C2×C61—21no—
326C2×C61—22no—
327C2×C61—21no—
328C2×C61—22no—
329C2×C61—82no—
330C2×C61C2×#38611yescharacteristic
331C2×C61#301912yes—
332D61—48no—
333C22×#41C2×D6111yescharacteristic
334C23×D4—611no—
335C22×M16—611no—
336C22×M16C2×D6111yescharacteristic
337#535—618no—
338#535—1224no—
339C2×#127—612no—
340C2×#124—612no—
341C2×#121—1222no—
342C2×#120—1222no—
343#518—614no—
344#518—1222no—
345#518—1222no—
346#518—1222no—
347C2×C4×D4—1222no—
348C2×C4×D4—1222no—
349C2×#117—611no—
350C2×#117—611no—
351C2×#117C2×D6111yescharacteristic
352C22×C22⋊C4—1222no—
353#419—612no—
354#419C2×D6112yescomplemented
355#418—612no—
356#418C2×D6112yescomplemented
357#417—22no—
358#417—1222no—
359#416—22no—
360#416—1222no—
361C2×#100—614no—
362C2×#100—1224no—
363#339—614no—
364#339—614no—
365C61#2095311yescharacteristic, complemented, Sylow
366C25—612no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41—22no—
372C2×#41—21no—
373C2×#41—21no—
374C2×#41C4×D6114yescomplemented
375C22×Q8C22×D6111yescharacteristic
376C22×D4—21no—
377C22×D4—614no—
378C22×D4—612no—
379C22×D4—614no—
380C22×D4—612no—
381C22×D4—1224no—
382C22×D4—1222no—
383C22×D4—1221no—
384C22×D4—1221no—
385C22×D4—1222no—
386C22×D4—1221no—
387C22×D4—1221no—
388C22×D4C22×D6111yescharacteristic
389C23×C4—21no—
390C23×C4—1224no—
391C23×C4C22×D6111yescharacteristic
392C2×M16—21no—
393C2×M16—22no—
394C2×M16—22no—
395C2×M16—612no—
396C2×M16—1221no—
397C2×M16—1222no—
398C2×M16—1222no—
399C2×M16C22×D6112yes—
400C22×C8—21no—
401C22×C8—1221no—
402#127—618no—
403#127—1224no—
404#124—618no—
405#124—1224no—
406#121—12216no—
407#120—12216no—
408C4×D4—1228no—
409C4×D4—1228no—
410C4×D4—1228no—
411C4×D4—1228no—
412#117—21no—
413#117—21no—
414#117—614no—
415#117—612no—
416#117—612no—
417#117—1221no—
418#117—1222no—
419#117—1221no—
420#117C22⋊D6112yescomplemented
421#117C22×D6112yes—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22⋊D6112yescomplemented
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1228no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4—1224no—
429C2×C22⋊C4—1224no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—21no—
433C2×C42—612no—
434C2×C42—1221no—
435#103—24no—
436#103—1224no—
437#102—24no—
438#102—1224no—
439#100—1228no—
440C24—21no—
441C24—618no—
442C24—614no—
443C24—1222no—
444C24—1224no—
445C24—1222no—
446C24—4884no—
447C24D4×D6111yescharacteristic
448#41—24no—
449#41—24no—
450#41—24no—
451#41—24no—
452#41—42no—
453#41—42no—
454#41—42no—
455#41—42no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8C2×C4×D6112yescomplemented
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—614no—
472C2×D4—614no—
473C2×D4—618no—
474C2×D4—12216no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4—1222no—
479C2×D4—1224no—
480C2×D4—1224no—
481C2×D4—1224no—
482C2×D4—1224no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1224no—
486C2×D4—1224no—
487C2×D4—1224no—
488C2×D4—1224no—
489C2×D4—1224no—
490C2×D4—4888no—
491C2×D4C2×C4×D6112yescomplemented
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—42no—
498C22×C4—12216no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—4888no—
502C22×C4C2×C4×D6112yes—
503C22×C4#1175822212yes—
504C22×C4D4×D6112yes—
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4#1175822412yes—
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—22no—
513M16—22no—
514M16—42no—
515M16—1222no—
516M16—1222no—
517M16—2442no—
518C2×C8—22no—
519C2×C8—42no—
520C2×C8—1222no—
521C2×C8—2442no—
522C4⋊C4—22no—
523C4⋊C4—614no—
524C4⋊C4—1224no—
525C4⋊C4—1222no—
526C4⋊C4C2×C22⋊D6114yes—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—12216no—
530C22⋊C4—12216no—
531C22⋊C4—1222no—
532C22⋊C4—1224no—
533C22⋊C4—1222no—
534C42—22no—
535C42—22no—
536C42—614no—
537C42—1222no—
538C42—1222no—
539C42—1222no—
540C23—22no—
541C23—42no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—618no—
546C23—1224no—
547C23—1228no—
548C23—1224no—
549C23—1228no—
550C23—4884no—
551C23—4888no—
552C23—4888no—
553C23—2448no—
554C23—2448no—
555C23C22⋊C4×D6112yes—
556C23#1451974511yescharacteristic
557C23#1451974511yescharacteristic
558C23#1451975912yes—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—44no—
567D4—44no—
568D4—41no—
569D4—41no—
570D4—1224no—
571D4—1224no—
572D4—1224no—
573D4—1224no—
574D4—1228no—
575D4—1228no—
576D4—48816no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—48816no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4C22⋊C4×D6112yes—
590C2×C4#1451973712yes—
591C2×C4#1451973712yes—
592C2×C4C4×C22⋊D6112yes—
593C2×C4C4×C22⋊D6112yes—
594C2×C4#1451974511yescharacteristic
595C2×C4#1451974511yescharacteristic
596C2×C4#1451974812yes—
597C2×C4#1451974812yes—
598C2×C4#1451975012yes—
599C2×C4#1451975012yes—
600C2×C4#1451975912yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4C22×C22⋊D6111yescharacteristic
605C2×C4C2×#1175822411yescharacteristic
606C2×C4C2×#1175822211yescharacteristic
607C8—42no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—82no—
617C22—82no—
618C22—1228no—
619C22—4888no—
620C22—4884no—
621C22—2448no—
622C22—24416no—
623C22—2448no—
624C22#1683963812yes—
625C22#1683963812yes—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22C2×#1451974511yescharacteristic
628C22C2×#1451975911yescharacteristic
629C4—42no—
630C4—44no—
631C4—2444no—
632C4—2444no—
633C4#1683963812yes—
634C4#1683963812yes—
635C4C2×#1451973711yescharacteristic
636C4C2×C4×C22⋊D6111yescharacteristic
637C4C2×#1451974811yescharacteristic
638C4C2×#1451975011yescharacteristic
639C2—21no—
640C2—21no—
641C2—22no—
642C2—22no—
643C2—82no—
644C2—2448no—
645C2C2×#1683963811yescharacteristic
646C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenter#2896D61
hypocenterC61#20953
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#16840236C2semidirect
3. #21454338C2semidirect
4. C22×#14519693C2semidirect
5. #21446214C2semidirect
6. #21446204C2semidirect
7. #21445530C2semidirect
8. #21445528C2semidirect
9. C61×#2896C2semidirect
13. #16840326C4semidirect
27. #16839771C22semidirect
28. #16839767C22semidirect
33. #16839523C22semidirect
34. #16839521C22semidirect
38. C61×#419C22semidirect
39. C61×#418C22semidirect
61. #14519773C2×C4semidirect
63. #14519766C2×C4semidirect
73. #14519736C2×C4semidirect
74. #14519736D4semidirect
77. C2×C4⋊Dic61D4semidirect
96. C2×#41×C61C2×C4semidirect
109. #117×C61D4semidirect
111. C2×C4⋊C4×C61D4semidirect
137. C2×Q8×C61C22×C4semidirect
147. C2×D4×C61C22×C4semidirect
210. C4×Dic61C22×C4semidirect
319. #2896D61semidirect
354. #419C2×D61semidirect
356. #418C2×D61semidirect
365. C61#20953semidirect
374. C2×#41C4×D61semidirect
420. #117C22⋊D61semidirect
424. C2×C4⋊C4C22⋊D61semidirect
461. C2×Q8C2×C4×D61semidirect
491. C2×D4C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
C2×#168402362
#214543382
C22×#145196932
#214462142
#214462042
#214455302
#214455282
C61×#28962
#2095361