Finite Group Databasebeta

← #30602918

Subgroups — #30602918

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2#21462256C211yesmaximal, characteristic
3C2×#16840641C211yesmaximal, characteristic
4#21461508C211yesmaximal, characteristic
5C2×#16840633C211yesmaximal, characteristic
6#21460997C211yesmaximal, characteristic
7#339×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2211yescharacteristic
9C2×#14519856C2211yescharacteristic
10C22×C4⋊#6923C2211yescharacteristic
11C22×C4⋊#6923C2211yescharacteristic
12#16840773—22no—
13#16840773C414yescomplemented
14C2×#14519846—21no—
15C2×#14519845—21no—
16C4⋊C4×#6923—21no—
17C4⋊C4×#6923—21no—
18C4×C4⋊#6923—21no—
19C4×C4⋊#6923—21no—
20C22⋊C4×#6923—22no—
21C2×(C2×#6923)⋊C4—22no—
22#16840641C414yescomplemented
23#16840635—21no—
24#16840635—21no—
25#16840633—22no—
26C2×M16×D61C2211yescharacteristic
27#117×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839523C411yescharacteristic, complemented
30#16839522C411yescharacteristic, complemented
31#16839521C411yescharacteristic, complemented
32C61×#339C411yescharacteristic, complemented
33C23×#6923—21no—
34C23×#6923D412yes—
35C2×#11758213—21no—
36D61⋊M16—21no—
37D61⋊M16—21no—
38D61⋊M16—21no—
39D61⋊M16C2×C412yes—
40C2×#11758217—21no—
41C2×D61⋊C8—21no—
42#14519856—21no—
43#14519856—21no—
44#14519856C2×C412yes—
45#14519856D412yes—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—21no—
48C2×C4⋊#6923—21no—
49C2×C4⋊#6923—21no—
50C2×C4⋊#6923—21no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—21no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923C2×C412yes—
56C2×C4⋊#6923C2×C412yes—
57C2×C4⋊#6923D411yescharacteristic
58C2×C4⋊#6923D411yescharacteristic
59C2×C4⋊#6923Q812yes—
60C2×C22⋊#6923—21no—
61C2×C4×#6923—21no—
62C2×C4×#6923—22no—
63C2×C4×#6923—22no—
64#14519846—24no—
65#14519845—24no—
66(C2×#6923)⋊C4—22no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—44no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758194C2×C411yescharacteristic
72C2×#11758189—21no—
73M16×D61—21no—
74M16×D61—21no—
75M16×D61C2×C412yes—
76M16×D61C2×C412yes—
77C2×C8×D61—21no—
78#14519736C2×C411yescharacteristic
79#14519735Q811yescharacteristic
80#14519735Q811yescharacteristic
81#14519734—21no—
82#14519733—21no—
83C4⋊C4×D61D411yescharacteristic
84C4⋊C4×D61D411yescharacteristic
85C22⋊C4×D61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2×C411yescharacteristic
88C42×D61—21no—
89C42×D61—21no—
90C42×D61—21no—
91C42×D61—21no—
92C42×D61—21no—
93C2×M16×C61C2×C411yescharacteristic
94#117×C61C2×C411yescharacteristic
95C2×C42×C61C2×C411yescharacteristic
96C23×D61C2×D411yescharacteristic
97C22×#6923—21no—
98C22×#6923—21no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—42no—
103C22×#6923—42no—
104C22×#6923—42no—
105C22×#6923—42no—
106C22×#6923—41no—
107C22×#6923—41no—
108C22×#6923C4⋊C414yes—
109C22×C4×C61—21no—
110C22×C4×C61C22×C411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61#4111yescharacteristic
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C4×D61—41no—
121C2×C4×D61C22×C411yescharacteristic
122C2×C4×D61C22×C411yescharacteristic
123C2×C4×D61C2×D411yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×C4×D61C2×D411yescharacteristic
126C2×C4×D61C2×Q811yescharacteristic
127C2×C4×D61C2×Q811yescharacteristic
128C2×C4×D61#4111yescharacteristic
129C2×C4×D61#4111yescharacteristic
130C2×C4×D61#4111yescharacteristic
131C2×C4×D61#4111yescharacteristic
132C2×C4×D61#4111yescharacteristic
133C22⋊#6923—22no—
134C22⋊#6923—22no—
135#11758217—22no—
136#11758217—41no—
137C2×#92420—41no—
138C4⋊#6923—22no—
139C4⋊#6923—22no—
140C4⋊#6923—22no—
141C4⋊#6923—22no—
142C4⋊#6923—22no—
143C4⋊#6923—22no—
144C4⋊#6923—21no—
145C4⋊#6923—21no—
146C4⋊#6923—41no—
147C4⋊#6923—41no—
148C4⋊#6923C22⋊C412yes—
149C4⋊#6923C22⋊C412yes—
150C4×#6923—22no—
151C4×#6923—22no—
152C4×#6923—82no—
153C4×#6923—82no—
154#11758213—22no—
155#11758213—41no—
156D61⋊C8—22no—
157D61⋊C8—41no—
158M16×C61—21no—
159M16×C61C4212yescomplemented
160C2×C8×C61—21no—
161C4⋊C4×C61C4⋊C411yescharacteristic
162C4⋊C4×C61C4⋊C411yescharacteristic
163C22⋊C4×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C22⋊Dic61—21no—
168#11758198—21no—
169#11758198—21no—
170C4⋊Dic61C4⋊C411yescharacteristic
171C4⋊Dic61C4⋊C411yescharacteristic
172#11758196—21no—
173#11758196—21no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—41no—
180C4×Dic61—41no—
181#11758194—21no—
182#11758194C4212yescomplemented
183C2×#92418—21no—
184#11758189—22no—
185#11758189—22no—
186C8×D61—22no—
187C8×D61—22no—
188C23×C61C4×D411yescharacteristic
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61C2×C4⋊C411yescharacteristic
194C22×D61C4×D411yescharacteristic
195C22×D61#12011yescharacteristic
196C22×D61#12111yescharacteristic
197C22×D61C22⋊Q811yescharacteristic
198C22×D61#12311yescharacteristic
199C2×#6923—24no—
200C2×#6923—22no—
201C2×#6923—22no—
202C2×#6923—41no—
203C2×#6923—42no—
204C2×#6923—41no—
205C2×#6923—41no—
206C2×#6923—42no—
207C2×#6923—41no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—42no—
211C2×#6923—82no—
212C2×#6923—82no—
213C2×#6923—81no—
214C2×#6923—81no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C2×C4211yescharacteristic
221C2×C4×C61C2×C4⋊C411yescharacteristic
222C2×C4×C61#11711yescharacteristic
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C4×Q811yescharacteristic
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61#11711yescharacteristic
232C2×Dic61C4×Q811yescharacteristic
233C2×Dic61C22⋊Q811yescharacteristic
234C2×Dic61#12311yescharacteristic
235C2×Dic61#12411yescharacteristic
236C2×Dic61#12611yescharacteristic
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61C2×C22⋊C411yescharacteristic
246C4×D61#11711yescharacteristic
247C4×D61C4×D411yescharacteristic
248C4×D61C4×D411yescharacteristic
249C4×D61C4×D411yescharacteristic
250C4×D61C4×D411yescharacteristic
251C4×D61C4×Q811yescharacteristic
252C4×D61C4×Q811yescharacteristic
253C4×D61#12111yescharacteristic
254C4×D61C22⋊Q811yescharacteristic
255C4×D61C22⋊Q811yescharacteristic
256C4×D61#12311yescharacteristic
257C4×D61#12511yescharacteristic
258C4×D61#12511yescharacteristic
259C4×D61#12611yescharacteristic
260C4×D61C4⋊Q811yescharacteristic
261#92420—42no—
262C8×C61—22no—
263#92418—22no—
264#10899—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61C4×C4⋊C411yescharacteristic
268C22×C61#38511yescharacteristic
269C22×C61#38811yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—21no—
280C2×D61#38511yescharacteristic
281C2×D61#38611yescharacteristic
282C2×D61#38711yescharacteristic
283C2×D61#38911yescharacteristic
284C2×D61#45311yescharacteristic
285C2×D61#45411yescharacteristic
286C2×D61#45511yescharacteristic
287C2×D61#45611yescharacteristic
288#6923—44no—
289#6923—42no—
290#6923—42no—
291#6923—82no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61C4×C22⋊C411yescharacteristic
296C4×C61C4×C4⋊C411yescharacteristic
297C4×C61#38711yescharacteristic
298C4×C61#38911yescharacteristic
299Dic61—42no—
300Dic61—41no—
301Dic61—41no—
302Dic61#38711yescharacteristic
303Dic61#38711yescharacteristic
304Dic61#38811yescharacteristic
305Dic61#39111yescharacteristic
306C2×#428—611no—
307#2274—611no—
308C2×#343—611no—
309#1736—611no—
310C2×#339—611no—
311C2×#339—611no—
312C4⋊#95—611no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61#143611yescharacteristic
316C2×C61#185911yescharacteristic
317C2×C61#186011yescharacteristic
318D61—21no—
319D61—21no—
320D61#185611yescharacteristic
321D61#185711yescharacteristic
322D61#188211yescharacteristic
323D61#188311yescharacteristic
324C22×M16—611no—
325C22×M16—611no—
326C2×#117—611no—
327C2×#117—611no—
328C22×C4⋊C4—611no—
329C22×C4⋊C4—611no—
330C22×C42—611no—
331#428—614no—
332#428—1222no—
333C2×#107—1221no—
334C2×#106—1221no—
335C4×C4⋊C4—1221no—
336C4×C4⋊C4—1221no—
337C4×C4⋊C4—1221no—
338C4×C4⋊C4—1221no—
339C4×C22⋊C4—1222no—
340C2×#95—1222no—
341#343—614no—
342#340—1221no—
343#340—1221no—
344#339—611no—
345#339—611no—
346#339—611no—
347#339—1222no—
348#339#692311yescharacteristic, complemented
349C61#1089911yescharacteristic, complemented, Sylow
350C23×C4—611no—
351C23×C4—612no—
352C23×C4—1221no—
353C23×C4—1221no—
354C2×M16—611no—
355C2×M16—612no—
356C2×M16—612no—
357C2×M16—612no—
358C2×M16—1221no—
359C2×M16—1221no—
360C2×M16—1221no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16—1221no—
366C2×M16C2×#692311yescharacteristic
367C22×C8—1221no—
368C22×C8—1221no—
369#117—612no—
370#117—612no—
371#117—611no—
372#117—611no—
373#117—611no—
374#117—1221no—
375#117—1221no—
376#117—1221no—
377#117—1221no—
378#117C2×#692311yescharacteristic
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—612no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—611no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42—1221no—
407C2×C42C2×#692311yescharacteristic
408#107—1224no—
409#106—1224no—
410#95—1222no—
411#95—1222no—
412#95—2444no—
413C24—611no—
414C22×C4—21no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—614no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1221no—
438C22×C4—1222no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2441no—
444C22×C4—2441no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4C22×#692311yescharacteristic
450M16—21no—
451M16—612no—
452M16—1222no—
453M16—1222no—
454M16—1221no—
455M16—1222no—
456M16—1222no—
457M16—2441no—
458M16—2441no—
459M16C4×#692312yescomplemented
460C2×C8—21no—
461C2×C8—1222no—
462C2×C8—1222no—
463C2×C8—1221no—
464C2×C8—1222no—
465C2×C8—2441no—
466C2×C8—2441no—
467C4⋊C4—612no—
468C4⋊C4—612no—
469C4⋊C4—611no—
470C4⋊C4—611no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4C4⋊#692311yescharacteristic
484C4⋊C4C4⋊#692311yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—1221no—
487C22⋊C4—1221no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C22⋊C4—1221no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—1221no—
495C42—1221no—
496C42—1222no—
497C42—1222no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—4882no—
502C42—4882no—
503C42—2441no—
504C42—2441no—
505C23—611no—
506C23—611no—
507C23—611no—
508C23—611no—
509C23—611no—
510C23—611no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23D4×#692311yescharacteristic
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1224no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—4881no—
550C2×C4—4881no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2444no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2444no—
572C2×C4C2×C4×#692311yescharacteristic
573C2×C4C2×C4⋊#692311yescharacteristic
574C2×C4#1451985611yescharacteristic
575C2×C4D4×#692311yescharacteristic
576C2×C4D4×#692311yescharacteristic
577C2×C4Q8×#692311yescharacteristic
578C8—22no—
579C8—1222no—
580C8—2442no—
581C22—21no—
582C22—21no—
583C22—611no—
584C22—611no—
585C22—611no—
586C22—611no—
587C22—611no—
588C22—611no—
589C22—611no—
590C22—611no—
591C22—1221no—
592C22—1221no—
593C22—1221no—
594C22—1221no—
595C22—1221no—
596C22—1221no—
597C22—1221no—
598C22—1221no—
599C22—1221no—
600C22—1221no—
601C22C4⋊C4×#692311yescharacteristic
602C22#1684070911yescharacteristic
603C22#1684071211yescharacteristic
604C4—42no—
605C4—41no—
606C4—41no—
607C4—611no—
608C4—611no—
609C4—611no—
610C4—611no—
611C4—4882no—
612C4—2442no—
613C4—2444no—
614C4—2442no—
615C4—2442no—
616C4—2441no—
617C4—2441no—
618C4C22⋊C4×#692311yescharacteristic
619C4C4×C4⋊#692311yescharacteristic
620C4#1684071111yescharacteristic
621C4#1684071311yescharacteristic
622C2—21no—
623C2—21no—
624C2—611no—
625C2—611no—
626C2—611no—
627C2—611no—
628C2—1221no—
629C2—1221no—
630C2#2146099911yescharacteristic
631C2#2146177811yescharacteristic
632C2#2146177911yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#339C4
socleC22×C61#385
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#339#6923
hypocenterC61#10899
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
13. #16840773C4semidirect
22. #16840641C4semidirect
29. #16839523C4semidirect
30. #16839522C4semidirect
31. #16839521C4semidirect
32. C61×#339C4semidirect
159. M16×C61C42semidirect
182. #11758194C42semidirect
348. #339#6923semidirect
349. C61#10899semidirect
459. M16C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
#214622562
C2×#168406412
#214615082
C2×#168406332
#214609972
#339×D612
#1089961