Finite Group Databasebeta

← #29734284

Subgroups — #29734284

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16840159C211yesmaximal, characteristic, complemented
3C2×#16840111C211yesmaximal, characteristic, complemented
4#21454170C211yesmaximal, characteristic
5#21447018C211yesmaximal, characteristic
6#21447017C211yesmaximal, characteristic
7#21447014C211yesmaximal, characteristic
8#21447013C211yesmaximal, characteristic
9#21446778C211yesmaximal, characteristic, complemented
10#21446619C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446087C211yesmaximal, characteristic, complemented
13#21446086C211yesmaximal, characteristic, complemented
14#21445576C211yesmaximal, characteristic
15C61×#1865C211yesmaximal, characteristic, complemented
16#16840316—24no—
17#16840316—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519776C2211yescharacteristic
20C2×#14519775C2211yescharacteristic
21C2×#14519767—21no—
22C2×#14519757C2211yescharacteristic
23C2×#14519755—21no—
24C2×#14519755—21no—
25C2×#14519753C2211yescharacteristic
26C2×C4×Dic122—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519734—21no—
29C2×C42×D61C2211yescharacteristic
30#16840159—24no—
31#16840111—24no—
32C2×#14519709C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519701C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519700C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42C2×#14519689C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45#16839666—21no—
46#16839660C2211yescharacteristic
47#16839657—21no—
48#16839631C2211yescharacteristic
49#16839621—21no—
50#16839619C2211yescharacteristic
51#16839619C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839517C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122C2311yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61D411yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519776—22no—
72#14519776—22no—
73#14519776—22no—
74#14519776D414yes—
75#14519775—22no—
76#14519775D414yes—
77#14519767—28no—
78#14519757—22no—
79#14519757—22no—
80#14519755—28no—
81#14519755—44no—
82#14519753—22no—
83#14519753—22no—
84C4×Dic122—28no—
85C4×Dic122—22no—
86C4×Dic122—22no—
87#14519734—28no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C42×D61—22no—
110C42×D61—22no—
111#14519709—22no—
112#14519708—22no—
113#14519708—22no—
114#14519702—22no—
115#14519702—22no—
116#14519701—22no—
117#14519701—22no—
118#14519700—22no—
119#14519700—22no—
120#14519689—22no—
121#14519689—22no—
122C8×Dic61—22no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684—21no—
126#14519684—21no—
127C22×Q8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C4×Q8×C61—22no—
131C4×Q8×C61—22no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139#106×C61—22no—
140#103×C61—22no—
141#103×C61—22no—
142#95×C61—21no—
143C23×D61C2×D411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61SD1614yes—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—24no—
182C2×Dic122—42no—
183C2×Dic122—42no—
184C2×Dic122—42no—
185C2×Dic122—42no—
186C2×Dic122—42no—
187C2×Dic122C2×D412yes—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61#4112yes—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—42no—
201C4⋊C4×C61C2×D412yes—
202C42×C61—24no—
203C42×C61—21no—
204C42×C61—21no—
205#11758198—24no—
206#11758198—24no—
207#11758198—42no—
208#11758198—42no—
209C4⋊Dic61—24no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61C2×D412yes—
216#11758196—24no—
217#11758196—24no—
218#11758196—24no—
219#11758196—24no—
220#11758196—24no—
221#11758196—42no—
222#11758196—42no—
223#11758196—42no—
224#11758196—42no—
225C4×Dic61—24no—
226C4×Dic61—22no—
227C4×Dic61—22no—
228C4×Dic61C2×D412yes—
229C4×Dic61C2×D412yes—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418#4112yes—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61C2×SD1612yes—
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61—41no—
247Q8×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C2×SD1612yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C2×C4×C612−1+411yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61#12012yes—
288C2×Dic61#12012yes—
289C2×Dic61C2×SD1612yes—
290C2×Dic61#13512yes—
291C2×Dic61#13712yes—
292C2×Dic61#13712yes—
293C4×D61—28no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298C4×D61—44no—
299Dic122—22no—
300Dic122—22no—
301Dic122—41no—
302Dic122—41no—
303Dic122—48no—
304Dic122—84no—
305C8×C61—22no—
306C8×C61—41no—
307#92418—22no—
308#92418—41no—
309#25672—611nomaximal, Sylow
310C22×C61C2×#12011yescharacteristic
311C22×C61#53211yescharacteristic
312C22×C61#53611yescharacteristic
313C22×C61C22×SD1611yescharacteristic
314C22×C61C2×#13511yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61C2×#13711yescharacteristic
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C4×C61—22no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61#54711yescharacteristic
331C4×C61#54711yescharacteristic
332C4×C61#54811yescharacteristic
333C4×C61#54811yescharacteristic
334Dic61—42no—
335Dic61—41no—
336Dic61—42no—
337Dic61—81no—
338Dic61—81no—
339Dic61#44814yes—
340Dic61#45214yes—
341C2×#533—611no—
342C2×#498—611no—
343C2×#487—611no—
344#2857—611no—
345#2035—611no—
346#2035—611no—
347#2035—611no—
348#2035—611no—
349#1990—611no—
350#1956—611no—
351#1865—611no—
352#1865—611no—
353#1865—611no—
354#1865D6111yescharacteristic, complemented
355C4×C22⋊C8—611no—
356C2×C61#243211yescharacteristic
357C2×C61C2×#44811yescharacteristic
358C2×C61C2×#45211yescharacteristic
359C2×C61#316711yescharacteristic
360C2×C61#318511yescharacteristic
361C2×C61#322911yescharacteristic
362C2×C61#323111yescharacteristic
363D61—42no—
364#533—1224no—
365#533—1224no—
366C2×C4⋊Q8—611no—
367C2×C4⋊Q8—611no—
368C2×C4⋊Q8—611no—
369C2×#125—1221no—
370C2×C22⋊Q8—611no—
371C2×C22⋊Q8—611no—
372C2×C22⋊Q8—1221no—
373C2×C22⋊Q8—1221no—
374C2×C4×Q8—611no—
375C2×C4×Q8—1221no—
376C2×C4×Q8C2×D6111yescharacteristic
377C2×#117—1221no—
378C22×C42—611no—
379#498—1224no—
380#487—1224no—
381C2×#106—611no—
382C2×#106—611no—
383C2×#106—611no—
384C2×#106C2×D6111yescharacteristic
385C2×#103—611no—
386C2×#103—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103—611no—
390C2×#103—611no—
391C2×#103C2×D6111yescharacteristic
392C2×#103C2×D6111yescharacteristic
393C2×C22⋊C8—611no—
394C2×C22⋊C8—611no—
395C2×C4×C8—611no—
396C2×C4×C8—611no—
397#391—1221no—
398#389—611no—
399#389—1221no—
400#389C2×D6111yescharacteristic
401#384—611no—
402#380—611no—
403#380—611no—
404#380—1221no—
405#340—611no—
406#340—611no—
407#340—611no—
408#340C2×D6111yescharacteristic
409#336—611no—
410#336C2×D6111yescharacteristic
411C61#2567211yescharacteristic, complemented, Sylow
412C22×Q8—611no—
413C22×Q8—1221no—
414C22×Q8—1221no—
415C22×Q8C22×D6111yescharacteristic
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—611no—
419C22×C8—611no—
420C22×C8—611no—
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423C4⋊Q8—614no—
424C4⋊Q8—614no—
425C4⋊Q8—1222no—
426C4⋊Q8—1222no—
427C4⋊Q8—1222no—
428C4⋊Q8—1222no—
429#125—1228no—
430C22⋊Q8—1222no—
431C22⋊Q8—1228no—
432C22⋊Q8—1222no—
433C22⋊Q8—1222no—
434C22⋊Q8—1222no—
435C22⋊Q8—2444no—
436C4×Q8—22no—
437C4×Q8—22no—
438C4×Q8—1222no—
439C4×Q8—1222no—
440C4×Q8—1228no—
441#117—1228no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C42—21no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—1222no—
469C2×C42—1222no—
470C2×C42—1221no—
471C2×C42C22×D6111yescharacteristic
472#106—22no—
473#106—1222no—
474#106—1222no—
475#106—1222no—
476#103—22no—
477#103—22no—
478#103—1222no—
479#103—1222no—
480#103—1222no—
481#103—1222no—
482#103—1222no—
483#103—1222no—
484C22⋊C8—1222no—
485C22⋊C8—1222no—
486C4×C8—1222no—
487C4×C8—1222no—
488#95—21no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492C24—611no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—612no—
499C2×Q8—1224no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—2442no—
506C2×Q8—2442no—
507C2×Q8—2442no—
508C2×Q8—2442no—
509C2×Q8—2442no—
510C2×Q8D4×D6112yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—614no—
518C22×C4—611no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4D4×D6111yescharacteristic
535C22×C4D4×D6111yescharacteristic
536C22×C4C23×D6111yescharacteristic
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—612no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8#1175822212yes—
549C4⋊C4—24no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—42no—
556C4⋊C4—612no—
557C4⋊C4—1224no—
558C4⋊C4—1224no—
559C4⋊C4—1224no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1224no—
563C4⋊C4—1221no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C4⋊C4D4×D6112yes—
573C22⋊C4—1224no—
574C22⋊C4—1224no—
575C22⋊C4—2442no—
576C22⋊C4—2442no—
577C42—24no—
578C42—21no—
579C42—21no—
580C42—612no—
581C42—612no—
582C42—1222no—
583C42—1222no—
584C42—1224no—
585C23—612no—
586C23—1221no—
587C23—1221no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23C2×D4×D6111yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597Q8—1222no—
598Q8—1222no—
599Q8—4884no—
600Q8—2448no—
601Q8—2441no—
602Q8—2441no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—612no—
621C2×C4—612no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—612no—
626C2×C4—1228no—
627C2×C4—1221no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—2444no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4SD16×D6112yes—
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4#41×D6111yescharacteristic
651C2×C4C2×#1175822211yescharacteristic
652C2×C4#1451982411yescharacteristic
653C2×C4#1451982511yescharacteristic
654C8—22no—
655C8—41no—
656C8—1222no—
657C8—2441no—
658C22—1222no—
659C22—1222no—
660C22—1222no—
661C22—2441no—
662C22—2441no—
663C22—2441no—
664C22—2441no—
665C22#120×D6111yescharacteristic
666C22#1684030911yescharacteristic
667C22#1684033511yescharacteristic
668C22C2×SD16×D6111yescharacteristic
669C22#135×D6111yescharacteristic
670C22#137×D6111yescharacteristic
671C22#137×D6111yescharacteristic
672C4—22no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—614no—
679C4—614no—
680C4—4881no—
681C4—4881no—
682C4—2442no—
683C4—2441no—
684C4—2442no—
685C4#1684041011yescharacteristic
686C4#1684041311yescharacteristic
687C4#1684042111yescharacteristic
688C4#1684042211yescharacteristic
689C2—2442no—
690C2#2144944511yescharacteristic
691C2#448×D6111yescharacteristic
692C2#452×D6111yescharacteristic
693C2#2145637511yescharacteristic
694C2#2145656111yescharacteristic
695C2#2145696211yescharacteristic
696C2#2145697711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1865D61
hypocenterC61#25672
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×#16840159C2semidirect
3. C2×#16840111C2semidirect
9. #21446778C2semidirect
10. #21446619C2semidirect
11. #21446090C2semidirect
12. #21446087C2semidirect
13. #21446086C2semidirect
15. C61×#1865C2semidirect
354. #1865D61semidirect
411. C61#25672semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168401592
C2×#168401112
#214541702
#214470182
#214470172
#214470142
#214470132
#214467782
#214466192
#214460902
#214460872
#214460862
#214455762
C61×#18652
#2567261