Finite Group Databasebeta

← #29688615

Subgroups — #29688615

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839874C211yesmaximal, characteristic, complemented
2C2×#16839863C211yesmaximal, characteristic, complemented
3C2×SD16×Dic61C211yesmaximal, characteristic, complemented
4C2×D8×Dic61C211yesmaximal, characteristic, complemented
5#21448545C211yesmaximal, characteristic, complemented
6#21448532C211yesmaximal, characteristic, complemented
7#21446702C211yesmaximal, characteristic, complemented
8#21446696C211yesmaximal, characteristic, complemented
9#21446686C211yesmaximal, characteristic, complemented
10#21446675C211yesmaximal, characteristic, complemented
11#21446631C211yesmaximal, characteristic, complemented
12#21446626C211yesmaximal, characteristic, complemented
13#21445987C211yesmaximal, characteristic, complemented
14#21445507C211yesmaximal, characteristic, complemented
15C2×C61×#496C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16839874—24no—
21#16839863—24no—
22SD16×Dic61—24no—
23D8×Dic61—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×#14519687C2211yescharacteristic
38C2×C4×#92418C2211yescharacteristic
39C2×C8×Dic61C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839665C2211yescharacteristic
44#16839665C2211yescharacteristic
45#16839655—22no—
46#16839642—22no—
47#16839642—22no—
48#16839637—22no—
49#16839632C2211yescharacteristic
50#16839628C2211yescharacteristic
51#16839608C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C22×D8×C61C2211yescharacteristic
56C2×#127×C61C2211yescharacteristic
57C2×#124×C61C2211yescharacteristic
58C61×#496C418yescomplemented
59C2×#97×C61C2211yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65Q8×Dic61—22no—
66Q8×Dic61—22no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—22no—
83C2×C22⋊Dic61—22no—
84C2×C22⋊Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519711—22no—
89#14519711—22no—
90#14519709—22no—
91#14519709—22no—
92#14519701—22no—
93#14519701—22no—
94#14519698—22no—
95#14519698—22no—
96#14519698—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—22no—
100#14519687—22no—
101#14519687D414yescomplemented
102C4×#92418—22no—
103C8×Dic61—22no—
104C8×Dic61D414yescomplemented
105#14519684—22no—
106#14519684—22no—
107#14519684D412yes—
108#14519684D412yes—
109C22×Q8×C61C2311yescharacteristic
110C22×D4×C61—22no—
111C22×D4×C61C2311yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C2×SD16×C61—22no—
115C2×SD16×C61—22no—
116C2×SD16×C61—22no—
117C2×SD16×C61—22no—
118C2×SD16×C61C2×C414yescomplemented
119C2×SD16×C61C2×C414yescomplemented
120C2×D8×C61—22no—
121C2×D8×C61—22no—
122C2×D8×C61—22no—
123C2×D8×C61—22no—
124C2×D8×C61C2×C414yescomplemented
125C2×D8×C61C2×C414yescomplemented
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#127×C61—22no—
129#127×C61C2×C414yescomplemented
130#124×C61—22no—
131#124×C61C2×C414yescomplemented
132C2×C22⋊C4×C61—22no—
133C2×C42×C61C2311yescharacteristic
134#97×C61C2×C414yescomplemented
135C24×C61—21no—
136C24×C61—21no—
137C24×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—28no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—82no—
161C2×D4×C61—82no—
162C2×D4×C61C22×C412yes—
163C2×D4×C61C22×C412yes—
164C2×D4×C61C22×C412yes—
165C22×C4×C61—21no—
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2411yescharacteristic
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—42no—
173C22×Dic61—42no—
174C22×Dic61—42no—
175C22×Dic61—82no—
176C22×Dic61—82no—
177C22×Dic61—82no—
178C22×Dic61C2×D412yes—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181SD16×C61—28no—
182SD16×C61—28no—
183D8×C61—28no—
184D8×C61—28no—
185C2×C8×C61C22×C412yes—
186C2×C8×C61C22×C412yes—
187C2×C8×C61C2×D412yes—
188C2×C8×C61C2×D412yes—
189C2×C8×C61#4112yes—
190C2×C8×C61#4112yes—
191C22⋊C4×C61—28no—
192C42×C61C22×C412yes—
193C42×C61#4112yes—
194C22⋊Dic61—42no—
195C22⋊Dic61—42no—
196C22⋊Dic61—42no—
197C22⋊Dic61—82no—
198C22⋊Dic61—82no—
199C22⋊Dic61—82no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61—82no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—42no—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C2×D412yes—
220C2×#92418C2×D412yes—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61C22×D411yescharacteristic
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—41no—
240Q8×C61—41no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61—22no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—41no—
252D4×C61—41no—
253D4×C61—88no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61C4×D414yes—
260C2×C4×C61C23×C411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—84no—
280C2×Dic61—84no—
281C2×Dic61—84no—
282C2×Dic61#12712yes—
283C2×Dic61#12712yes—
284C8×C61C4×D414yes—
285C8×C61C4×D414yes—
286#92418—22no—
287#92418—22no—
288#21704—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—44no—
293C22×C61—44no—
294C22×C61—44no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—82no—
302C22×C61—82no—
303C22×C61—82no—
304C22×C61C2×C4×D411yescharacteristic
305C22×C61C2×#12711yescharacteristic
306C22×C61#53211yescharacteristic
307C22×C61#57612yes—
308C22×C61#57712yes—
309C4×C61—24no—
310C4×C61—42no—
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C2×C4×D411yescharacteristic
313C4×C61#53211yescharacteristic
314C4×C61#53211yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—42no—
318Dic61—81no—
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
322C2×#496D6111yescharacteristic
323C2×#442—611no—
324C2×#440—611no—
325C2×C4×SD16—611no—
326C2×C4×D8—611no—
327#2328—611no—
328#2326—611no—
329#1973—611no—
330#1972—611no—
331#1970—611no—
332#1968—611no—
333#1958—611no—
334#1957—611no—
335#1847—611no—
336#1726—611no—
337C2×C61—82no—
338C2×C61—82no—
339C2×C61—82no—
340C2×C61C4×#12711yescharacteristic
341C2×C61#294612yes—
342C2×C61#315411yescharacteristic
343C2×C61#315511yescharacteristic
344C2×C61#317912yes—
345C22×SD16C2×D6111yescharacteristic
346C22×SD16C2×D6111yescharacteristic
347C22×D8C2×D6111yescharacteristic
348C22×D8C2×D6111yescharacteristic
349C2×#127C2×D6111yescharacteristic
350C2×#124C2×D6111yescharacteristic
351C2×C4×Q8—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355#496Dic6118yescomplemented
356#442—1224no—
357#440—1224no—
358C4×SD16—1224no—
359C4×D8—1224no—
360C2×#107—611no—
361C2×#107—611no—
362C2×#106—611no—
363C2×#106—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#97—611no—
373C2×#97—611no—
374C2×#97C2×D6111yescharacteristic
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8—611no—
378#391—611no—
379#390—611no—
380#390—611no—
381#390—611no—
382#388—1222no—
383#386—1222no—
384#386—1222no—
385#385—1222no—
386#384—611no—
387#382—611no—
388#376—611no—
389C61#2170411yescharacteristic, complemented, Sylow
390C22×Q8C22×D6111yescharacteristic
391C22×D4—22no—
392C22×D4C22×D6111yescharacteristic
393C22×D4C22×D6111yescharacteristic
394C22×D4C22×D6111yescharacteristic
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4—1221no—
398C2×SD16—22no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16C2×Dic6114yescomplemented
403C2×SD16C2×Dic6114yescomplemented
404C2×D8—22no—
405C2×D8—22no—
406C2×D8—22no—
407C2×D8—22no—
408C2×D8C2×Dic6114yescomplemented
409C2×D8C2×Dic6114yescomplemented
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#127—22no—
415#127C2×Dic6114yescomplemented
416#124—22no—
417#124C2×Dic6114yescomplemented
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443#107—1222no—
444#107—1222no—
445#106—1222no—
446#106—1222no—
447#103—1222no—
448#103—1222no—
449#102—1222no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#102—1222no—
455#97—614no—
456#97—1222no—
457#97C2×Dic6114yescomplemented
458C4×C8—614no—
459C4×C8—1222no—
460C4×C8—1222no—
461#95—612no—
462#95—612no—
463#95—1222no—
464#95—1222no—
465C24—21no—
466C24—21no—
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8C22×Dic6112yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—28no—
490C2×D4—82no—
491C2×D4—82no—
492C2×D4C22×Dic6112yes—
493C2×D4C22×Dic6112yes—
494C2×D4C22×Dic6112yes—
495C22×C4—21no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—612no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—4882no—
504C22×C4—4882no—
505C22×C4—4882no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4D4×D6112yes—
510C22×C4C23×D6111yescharacteristic
511SD16—28no—
512SD16—28no—
513D8—28no—
514D8—28no—
515C2×C8—612no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8D4×D6112yes—
522C2×C8D4×D6112yes—
523C2×C8#1175822412yes—
524C2×C8#1175822412yes—
525C2×C8C22×Dic6112yes—
526C2×C8C22×Dic6112yes—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—4882no—
536C4⋊C4—2442no—
537C22⋊C4—28no—
538C22⋊C4—4882no—
539C22⋊C4—4882no—
540C22⋊C4—4882no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C42—612no—
545C42—612no—
546C42—1221no—
547C42—1221no—
548C42—2442no—
549C42#1175822412yes—
550C42C22×Dic6112yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23C2×D4×D6111yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571D4—22no—
572D4—22no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—88no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—4884no—
597C2×C4—4884no—
598C2×C4—4884no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4D4×Dic6114yes—
608C2×C4C23×Dic6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×#1175822411yescharacteristic
612C2×C4C2×#1175822411yescharacteristic
613C2×C4C2×#1175822411yescharacteristic
614C8—1222no—
615C8—1222no—
616C8D4×Dic6114yes—
617C8D4×Dic6114yes—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×D4×Dic6111yescharacteristic
634C22#127×D6111yescharacteristic
635C22#1684031011yescharacteristic
636C22#1684056612yes—
637C22#1684057212yes—
638C4—24no—
639C4—42no—
640C4—1222no—
641C4—1222no—
642C4—4881no—
643C4—4881no—
644C4—4881no—
645C4—4881no—
646C4—2442no—
647C4C2×D4×Dic6111yescharacteristic
648C4C2×D4×Dic6111yescharacteristic
649C4#1684031011yescharacteristic
650C4#1684031011yescharacteristic
651C2—82no—
652C2—82no—
653C2—82no—
654C2#127×Dic6111yescharacteristic
655C2#2145465212yes—
656C2#2145629211yescharacteristic
657C2#2145630311yescharacteristic
658C2#2145649912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#496C2
socleC23×C61C22×D4
cosocleC22×C4×C61C24
hypercenterC2×#496D61
hypocenterC61#21704
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
1. C2×#16839874C2semidirect
2. C2×#16839863C2semidirect
3. C2×SD16×Dic61C2semidirect
4. C2×D8×Dic61C2semidirect
5. #21448545C2semidirect
6. #21448532C2semidirect
7. #21446702C2semidirect
8. #21446696C2semidirect
9. #21446686C2semidirect
10. #21446675C2semidirect
11. #21446631C2semidirect
12. #21446626C2semidirect
13. #21445987C2semidirect
14. #21445507C2semidirect
58. C61×#496C4semidirect
101. #14519687D4semidirect
104. C8×Dic61D4semidirect
118. C2×SD16×C61C2×C4semidirect
119. C2×SD16×C61C2×C4semidirect
124. C2×D8×C61C2×C4semidirect
125. C2×D8×C61C2×C4semidirect
129. #127×C61C2×C4semidirect
131. #124×C61C2×C4semidirect
134. #97×C61C2×C4semidirect
355. #496Dic61semidirect
389. C61#21704semidirect
402. C2×SD16C2×Dic61semidirect
403. C2×SD16C2×Dic61semidirect
408. C2×D8C2×Dic61semidirect
409. C2×D8C2×Dic61semidirect
415. #127C2×Dic61semidirect
417. #124C2×Dic61semidirect
457. #97C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398742
C2×#168398632
C2×SD16×Dic612
C2×D8×Dic612
#214485452
#214485322
#214467022
#214466962
#214466862
#214466752
#214466312
#214466262
#214459872
#214455072
C2×C61×#4962
#2170461