Finite Group Databasebeta

← #29681817

Subgroups — #29681817

Order 15616.

2030 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
1C2×#16840326C211yesmaximal, characteristic
2C2×#16840236C211yesmaximal, characteristic, complemented
3C2×#16839736C211yesmaximal, characteristic, complemented
4#21448399C211yesmaximal, characteristic, complemented
5#21446269C218yesmaximal, complemented
6#21445551C211yesmaximal, characteristic, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C2×C61×#410C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10#16840326—22no—
11#16840326—22no—
12#16840326—22no—
13#16840326—22no—
14C2×#14519773C412yescomplemented
15C2×#14519766C412yescomplemented
16C2×#14519751C414yescomplemented
17C2×#14519745C2214yes—
18#16840236—24no—
19#16840236—24no—
20C2×C4×D244—22no—
21C2×#14519739—22no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×C22⋊C4×D61C2214yes—
26#16839736—24no—
27C2×#14519692C2214yes—
28#16839664—22no—
29#16839663—22no—
30#16839617C2214yes—
31#16839528C2214yes—
32#16839528C2214yes—
33C22×#41×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#410—24no—
37C2×#99×C61C2214yes—
38C24×D61C2312yes—
39C22×D244—22no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244C2×C412yes—
44C22×D244C2×C412yes—
45C22×D244D412yes—
46C22×D244D412yes—
47C22×D244D412yes—
48C22×D244D412yes—
49C22×C22⋊D61C2×C418yescomplemented
50C22×C4×D61—24no—
51#14519773—24no—
52#14519773—24no—
53#14519766—24no—
54#14519766—24no—
55#14519751—28no—
56#14519751—28no—
57#14519745—28no—
58#14519745—28no—
59C4×D244—216no—
60#14519739—216no—
61#14519736—22no—
62#14519736—21no—
63#14519736—21no—
64#14519736—21no—
65#14519736—21no—
66#14519736—22no—
67#14519736—22no—
68#14519736D414yescomplemented
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C412yescomplemented
72C2×C22⋊Dic61C2312yes—
73C2×C22⋊Dic61C2312yes—
74C2×C22⋊Dic61C2312yes—
75C2×#11758198—24no—
76C2×#11758198—28no—
77C2×#11758198C2×C418yescomplemented
78C22⋊C4×D61—28no—
79C22⋊C4×D61—28no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2×C412yescomplemented
83#14519692—28no—
84C2×#41×C61—22no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—21no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C22×Q8×C61C2×C411yescharacteristic, complemented
93C22×D4×C61C2×C411yescharacteristic, complemented
94C22×D4×C61C2312yes—
95C23×C4×C61C2×C412yescomplemented
96C23×C4×C61C2311yescharacteristic
97#117×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100#117×C61D414yescomplemented
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C22⋊C4×C61C2312yes—
104C2×C22⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107#99×C61—28no—
108C24×C61C22×C412yes—
109C24×C61C2411yescharacteristic
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—84no—
115C23×D61C22×C414yes—
116C23×D61C2×D414yes—
117C23×D61C2×D414yes—
118#41×C61—42no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—42no—
124#41×C61—42no—
125#41×C61—42no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61#4112yes—
156C22×C4×C61#4112yes—
157C2×C22⋊D61—216no—
158C2×C22⋊D61—88no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C22×C414yes—
162C2×D244—24no—
163C2×D244—24no—
164C2×D244—24no—
165C2×D244—24no—
166C2×D244—24no—
167C2×D244—24no—
168C2×D244—24no—
169C2×D244—24no—
170C2×D244—24no—
171C2×D244—24no—
172C2×D244—24no—
173C2×D244—24no—
174C2×D244C22⋊C414yes—
175C2×D244C22⋊C414yes—
176C2×D244C22⋊C414yes—
177C2×D244C22⋊C414yes—
178C2×C4×D61—216no—
179C2×C4×D61—88no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61C2×D414yes—
186C42×C61—24no—
187C42×C61—42no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61C2×D414yes—
194#11758198—216no—
195#11758198—216no—
196#11758198—88no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—24no—
200C4⋊Dic61—42no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—24no—
204C4×Dic61—42no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—42no—
209C23×C61—44no—
210C23×C61C23×C411yescharacteristic
211C23×C61C22×D411yescharacteristic
212C23×C61C22×D411yescharacteristic
213C23×C61C22×D411yescharacteristic
214C23×C61C22×D411yescharacteristic
215C23×C61C2×#4111yescharacteristic
216C23×C61C2×#4111yescharacteristic
217C22×D61—24no—
218C22×D61—24no—
219C22×D61—24no—
220C22×D61—24no—
221C22×D61—24no—
222C22×D61—24no—
223C22×D61—48no—
224C22×D61—48no—
225C22×D61—48no—
226C22×D61—84no—
227C22×D61—84no—
228C22×D61—84no—
229C22×D61—84no—
230C22×D61C2×C22⋊C418yes—
231Q8×C61—42no—
232Q8×C61—42no—
233D4×C61—42no—
234D4×C61—42no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—42no—
238D4×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—24no—
243C2×C4×C61—44no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61—44no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C4×D412yes—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C4×D412yes—
255C2×C4×C61C4×D412yes—
256C2×C4×C61#12012yes—
257C2×C4×C61#12012yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12112yes—
260C2×C4×C61#12112yes—
261C2×C4×C61#12112yes—
262C2×C4×C61#12312yes—
263C2×C4×C61#12312yes—
264C22⋊D61—816no—
265C2×Dic61—24no—
266C2×Dic61—24no—
267C2×Dic61—44no—
268C2×Dic61—44no—
269C2×Dic61—44no—
270C2×Dic61—42no—
271D244—28no—
272D244—28no—
273D244—28no—
274D244—28no—
275C4×D61—816no—
276#20985—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—82no—
287C22×C61—82no—
288C22×C61C22×C22⋊C411yescharacteristic
289C22×C61C2×C4×D411yescharacteristic
290C22×C61C2×C4×D411yescharacteristic
291C22×C61C2×#12011yescharacteristic
292C22×C61C2×#12111yescharacteristic
293C22×C61C2×#12111yescharacteristic
294C22×C61C2×#12311yescharacteristic
295C2×D61—216no—
296C2×D61—28no—
297C2×D61—48no—
298C2×D61—48no—
299C2×D61—84no—
300C2×D61—84no—
301C2×D61—84no—
302C2×D61—84no—
303C4×C61—44no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#38612yes—
307C4×C61#38612yes—
308C4×C61#38612yes—
309C4×C61#38612yes—
310Dic61—44no—
311Dic61—44no—
312Dic61—82no—
313C2×#535—611no—
314C2×#518—611no—
315C2×#410—611no—
316C2×#410D6111yescharacteristic, complemented
317#2309—611no—
318#1898—618no—
319#1740—611no—
320#1740—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—21no—
324C2×C61—21no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—82no—
328C2×C61C2×#38611yescharacteristic
329C2×C61#288312yes—
330D61—48no—
331C22×#41C2×D6111yescharacteristic
332C23×D4—611no—
333#535—1222no—
334#535—1222no—
335#535—1222no—
336#535—1222no—
337C2×#127—612no—
338C2×#124—612no—
339C2×#121—614no—
340C2×#120—614no—
341#518—1224no—
342#518—1224no—
343C2×C4×D4—1222no—
344C2×C4×D4—1222no—
345C2×#117—611no—
346C2×#117—611no—
347C2×#117—611no—
348C2×#117C2×D6111yescharacteristic
349C2×#117C2×D6111yescharacteristic
350C22×C22⋊C4—614no—
351#410—24no—
352#410—1224no—
353C2×#99—614no—
354C2×#99C2×D6114yes—
355#390—1222no—
356#390—1222no—
357#379—614no—
358#342—614no—
359#342—614no—
360C61#2098511yescharacteristic, complemented, Sylow
361C25—612no—
362C2×#41—22no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41—21no—
367C2×#41—22no—
368C2×#41—22no—
369C2×#41—22no—
370C22×Q8C4×D6111yescharacteristic, complemented
371C22×D4—618no—
372C22×D4—612no—
373C22×D4—612no—
374C22×D4—612no—
375C22×D4—612no—
376C22×D4—612no—
377C22×D4—612no—
378C22×D4—1222no—
379C22×D4—1222no—
380C22×D4—1222no—
381C22×D4—1222no—
382C22×D4C4×D6111yescharacteristic, complemented
383C22×D4C22×D6112yes—
384C23×C4—1224no—
385C23×C4C4×D6112yescomplemented
386C23×C4C22×D6111yescharacteristic
387#127—1224no—
388#127—1224no—
389#124—1224no—
390#124—1224no—
391#121—1228no—
392#121—1228no—
393#120—1228no—
394#120—1228no—
395C4×D4—12216no—
396C4×D4—12216no—
397#117—22no—
398#117—22no—
399#117—22no—
400#117—614no—
401#117—1221no—
402#117—1221no—
403#117—1221no—
404#117—1221no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117C22⋊D6114yescomplemented
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—612no—
416C2×C22⋊C4—618no—
417C2×C22⋊C4—612no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—1224no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4—1228no—
422C2×C22⋊C4C22×D6112yes—
423C2×C22⋊C4C22×D6112yes—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—612no—
427C2×C42—1221no—
428C2×C42—1221no—
429#99—28no—
430#99—1228no—
431C24—614no—
432C24—614no—
433C24—614no—
434C24—1222no—
435C24—1222no—
436C24—1222no—
437C24—1222no—
438C24—4884no—
439C24C2×C4×D6112yes—
440C24C23×D6111yescharacteristic
441#41—42no—
442#41—42no—
443#41—42no—
444#41—42no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—42no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—614no—
464C2×D4—614no—
465C2×D4—614no—
466C2×D4—614no—
467C2×D4—12216no—
468C2×D4—1224no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—1224no—
473C2×D4—1224no—
474C2×D4—1224no—
475C2×D4—1224no—
476C2×D4—1224no—
477C2×D4—1224no—
478C2×D4—1224no—
479C2×D4—1224no—
480C2×D4—4888no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—614no—
488C22×C4—12216no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—4888no—
492C22×C4C2×C4×D6112yes—
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4#1175822212yes—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6112yes—
501C22×C4#1175822412yes—
502C22×C4C2×C22⋊D6112yes—
503C4⋊C4—24no—
504C4⋊C4—42no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1224no—
508C4⋊C4—2442no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—614no—
513C22⋊C4—12216no—
514C22⋊C4—12216no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—4888no—
521C22⋊C4C2×C22⋊D6114yes—
522C42—24no—
523C42—42no—
524C42—1222no—
525C42—1222no—
526C42—1224no—
527C42—2442no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—42no—
532C23—44no—
533C23—618no—
534C23—1224no—
535C23—1224no—
536C23—1224no—
537C23—1224no—
538C23—1224no—
539C23—1224no—
540C23—4884no—
541C23—4884no—
542C23—4884no—
543C23—4884no—
544C23—2448no—
545C23—2448no—
546C23—2448no—
547C23C22×C4×D6111yescharacteristic
548C23C2×D4×D6111yescharacteristic
549C23C2×D4×D6111yescharacteristic
550C23C2×D4×D6111yescharacteristic
551C23C22×C22⋊D6111yescharacteristic
552C23C2×#1175822411yescharacteristic
553C23C2×#1175822211yescharacteristic
554Q8—42no—
555Q8—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—1228no—
563D4—1228no—
564D4—1228no—
565D4—1228no—
566D4—48816no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—48816no—
579C2×C4—2444no—
580C2×C4—2442no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4C22⋊C4×D6111yescharacteristic
584C2×C4C22⋊C4×D6111yescharacteristic
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4#1451973712yes—
588C2×C4#1451973712yes—
589C2×C4C4×C22⋊D6112yes—
590C2×C4C4×C22⋊D6112yes—
591C2×C4#1451974512yes—
592C2×C4#1451974512yes—
593C2×C4#1451974812yes—
594C2×C4#1451974812yes—
595C2×C4#1451975012yes—
596C2×C4#1451975012yes—
597C2×C4#1451975912yes—
598C2×C4#1451975912yes—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—1228no—
611C22—12216no—
612C22—4884no—
613C22—4884no—
614C22—4884no—
615C22—4884no—
616C22—2448no—
617C22—2448no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C22C2×#1451973711yescharacteristic
620C22C2×C4×C22⋊D6111yescharacteristic
621C22C2×#1451974511yescharacteristic
622C22C2×#1451974811yescharacteristic
623C22C2×#1451975011yescharacteristic
624C22C2×#1451975911yescharacteristic
625C4—44no—
626C4—42no—
627C4—82no—
628C4—4882no—
629C4—2444no—
630C4—2444no—
631C4#1683963812yes—
632C4#1683963812yes—
633C4#1683963812yes—
634C4#1683963812yes—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—82no—
642C2—2448no—
643C2C2×#1683963811yescharacteristic
644C2#2145427812yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC2×C61×#410C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840326C2
cosocleC24×C61C24
hypercenterC2×#410D61
hypocenterC61#20985
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16840236C2semidirect
3. C2×#16839736C2semidirect
4. #21448399C2semidirect
5. #21446269C2semidirect
6. #21445551C2semidirect
7. #21445551C2semidirect
8. C2×C61×#410C2semidirect
14. C2×#14519773C4semidirect
15. C2×#14519766C4semidirect
16. C2×#14519751C4semidirect
49. C22×C22⋊D61C2×C4semidirect
68. #14519736D4semidirect
71. C2×C4⋊Dic61C2×C4semidirect
77. C2×#11758198C2×C4semidirect
82. C2×C4×Dic61C2×C4semidirect
92. C22×Q8×C61C2×C4semidirect
93. C22×D4×C61C2×C4semidirect
95. C23×C4×C61C2×C4semidirect
100. #117×C61D4semidirect
316. C2×#410D61semidirect
360. C61#20985semidirect
370. C22×Q8C4×D61semidirect
382. C22×D4C4×D61semidirect
385. C23×C4C4×D61semidirect
408. #117C22⋊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
C2×#168397362
#214483992
#214462692
#214455512
C2×C61×#4102
#2098561