Finite Group Databasebeta

← #30547828

Subgroups — #30547828

Order 15616.

2346 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2#21455767C214yesmaximal, complemented
3#21455759C214yesmaximal, complemented
4#21455750C214yesmaximal, complemented
5#21455742C214yesmaximal, complemented
6#21455732C218yesmaximal, complemented
7#21454408C212yesmaximal, complemented
8#21454336C212yesmaximal, complemented
9C2×#16839726C211yesmaximal, characteristic, complemented
10C22×#117×C61C211yesmaximal, characteristic, complemented
11C22×#11758222C2211yescharacteristic
12C22×#11758194C2212yes—
13#16840273—21no—
14#16840273—21no—
15#16840273—22no—
16#16840273C2214yes—
17#16840273C2212yes—
18#16840273C2212yes—
19C2×#14519760—21no—
20C2×#14519756C2212yes—
21C2×#14519751C2212yes—
22C2×#14519746—21no—
23C2×C4×C22⋊D61—22no—
24C2×#14519736C2211yescharacteristic
25#16840103C22116yescomplemented
26#16840097C2218yescomplemented
27#16840092C2218yescomplemented
28#16840086C22116yescomplemented
29#16840080C2218yescomplemented
30#16840075C2218yescomplemented
31#16839805C2218yescomplemented
32#16839802C2218yescomplemented
33#16839798C2218yescomplemented
34#16839770C2214yescomplemented
35#16839766C2214yescomplemented
36#16839761C2218yescomplemented
37#16839756C2218yescomplemented
38#16839726C2214yes—
39#16839726C2212yescomplemented
40#16839726C2212yescomplemented
41C2×C22⋊#92418C2214yescomplemented
42C24×C4×C61C2211yescharacteristic
43C2×#117×C61—22no—
44C2×#117×C61—21no—
45C2×#117×C61—22no—
46C2×#117×C61—21no—
47C2×#117×C61C2212yes—
48C2×#117×C61C2214yes—
49C2×#117×C61C2214yescomplemented
50C2×#117×C61C2212yescomplemented
51C22×C4⋊C4×C61C2212yescomplemented
52C22×C22⋊C4×C61—21no—
53C22×C42×C61—21no—
54C2×#11758222—21no—
55C2×#11758222—21no—
56C2×#11758222—28no—
57C2×#11758222C2312yes—
58C2×#11758222C2312yes—
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—22no—
62C22×C22⋊D61—21no—
63C22×C4×D61—21no—
64C2×#11758194—28no—
65C2×#11758194C2314yes—
66C2×#11758194C2314yes—
67C22×#92418C2314yes—
68#14519760—22no—
69#14519760—22no—
70#14519760—24no—
71#14519756—22no—
72#14519756—22no—
73#14519756C2314yes—
74#14519756C2314yes—
75#14519751—22no—
76#14519751—22no—
77#14519751C2314yes—
78#14519751C2314yes—
79#14519746—22no—
80#14519746—24no—
81#14519746—22no—
82C4×C22⋊D61—24no—
83C4×C22⋊D61—24no—
84C4×C22⋊D61—24no—
85C4×C22⋊D61—24no—
86#14519736—21no—
87#14519736—21no—
88#14519736C2312yes—
89#14519736C2312yes—
90C2×C4⋊Dic61C2312yes—
91C2×#11758196—22no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—22no—
94C2×#11758198—22no—
95C2×C4×Dic61—21no—
96#14519710C23116yescomplemented
97#14519708C23116yescomplemented
98#14519700C23116yescomplemented
99#14519697C23116yescomplemented
100C22⋊#92418C2318yes—
101C22⋊#92418C2318yes—
102C25×C61D411yescharacteristic
103C23×C4×C61—28no—
104C23×C4×C61—22no—
105C23×C4×C61D412yes—
106C23×C4×C61D411yescharacteristic
107C23×C4×C61D411yescharacteristic
108C23×C4×C61D412yes—
109C23×C4×C61D411yescharacteristic
110C23×C4×C61C2312yes—
111C23×C4×C61C2312yes—
112C23×C4×C61C2311yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C23×C4×C61C2311yescharacteristic
115#117×C61—24no—
116#117×C61—24no—
117#117×C61—22no—
118#117×C61—22no—
119#117×C61—24no—
120#117×C61—24no—
121#117×C61—24no—
122#117×C61C2314yes—
123#117×C61C2314yes—
124#117×C61C2318yes—
125C2×C4⋊C4×C61—22no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61C2314yes—
129C2×C4⋊C4×C61C2314yes—
130C2×C4⋊C4×C61C2314yes—
131C2×C22⋊C4×C61—24no—
132C2×C22⋊C4×C61—22no—
133C2×C22⋊C4×C61—24no—
134C2×C22⋊C4×C61—22no—
135C2×C42×C61—24no—
136C2×C42×C61—22no—
137C2×C42×C61—24no—
138C2×C42×C61—22no—
139C24×C61—24no—
140C24×C61—44no—
141C24×C61C2×D411yescharacteristic
142C24×C61C2×D412yes—
143C24×C61C2×D412yes—
144C24×C61C2×D411yescharacteristic
145C24×C61C2×D411yescharacteristic
146C23×D61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—28no—
149C22×C4×C61—24no—
150C22×C4×C61—24no—
151C22×C4×C61—28no—
152C22×C4×C61—28no—
153C22×C4×C61—28no—
154C22×C4×C61—24no—
155C22×C4×C61—24no—
156C22×C4×C61—24no—
157C22×C4×C61—24no—
158C22×C4×C61—24no—
159C22×C4×C61—48no—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D412yes—
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61#4118yes—
174C22×C4×C61C2412yes—
175C22×C4×C61C2412yes—
176C22×C4×C61C2411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C22×C4×C61C2411yescharacteristic
179C2×C22⋊D61—22no—
180C2×C22⋊D61—22no—
181C2×C22⋊D61—24no—
182C2×C22⋊D61—24no—
183C2×C22⋊D61—48no—
184C2×C22⋊D61—44no—
185C22×Dic61—21no—
186C22×Dic61—22no—
187#11758222—28no—
188#11758222—28no—
189#11758222—44no—
190#11758222—44no—
191C2×D244—24no—
192C2×D244—21no—
193C2×D244C2412yes—
194C2×C4×D61—22no—
195C2×C4×D61—22no—
196C2×C4×D61—44no—
197C2×Dic122—24no—
198C2×Dic122—21no—
199C2×Dic122C2412yes—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—24no—
202C4⋊C4×C61—24no—
203C4⋊C4×C61C2418yes—
204C22⋊C4×C61—28no—
205C22⋊C4×C61—24no—
206C22⋊C4×C61—24no—
207C42×C61—28no—
208C42×C61—24no—
209C42×C61—24no—
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212C22⋊Dic61—24no—
213C22⋊Dic61—24no—
214#11758198—24no—
215#11758198—24no—
216C4⋊Dic61—22no—
217C4⋊Dic61C2414yes—
218#11758196—24no—
219#11758196—24no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222#11758194—28no—
223#11758194—28no—
224C2×#92418—28no—
225C2×#92418C2418yes—
226C23×C61—22no—
227C23×C61—24no—
228C23×C61—24no—
229C23×C61—24no—
230C23×C61—22no—
231C23×C61—24no—
232C23×C61—24no—
233C23×C61—42no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61—42no—
237C23×C61—416no—
238C23×C61#12314yes—
239C23×C61C22×D412yes—
240C23×C61C22×D411yescharacteristic
241C23×C61C22×D411yescharacteristic
242C23×C61C22×D411yescharacteristic
243C23×C61C22×D412yes—
244C22×D61—22no—
245C22×D61—44no—
246C22×D61—42no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—24no—
252C2×C4×C61—24no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—24no—
256C2×C4×C61—24no—
257C2×C4×C61—24no—
258C2×C4×C61—24no—
259C2×C4×C61—28no—
260C2×C4×C61—48no—
261C2×C4×C61—48no—
262C2×C4×C61—48no—
263C2×C4×C61#12314yes—
264C2×C4×C61#12314yes—
265C2×C4×C61#12314yes—
266C2×C4×C61C22×D412yes—
267C2×C4×C61C22×D412yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4114yes—
272C2×C4×C61C2×#4114yes—
273C2×C4×C61C2×#4114yes—
274C2×C4×C61C2511yescharacteristic
275C22⋊D61—48no—
276C22⋊D61—44no—
277C22⋊D61—44no—
278C22⋊D61—48no—
279C2×Dic61—24no—
280C2×Dic61—22no—
281C2×Dic61—44no—
282C2×Dic61—42no—
283D244—24no—
284D244—42no—
285C4×D61—44no—
286C4×D61—44no—
287Dic122—24no—
288Dic122—42no—
289#92418—28no—
290#60511—611nomaximal, Sylow
291C22×C61—28no—
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—44no—
301C22×C61—48no—
302C22×C61—48no—
303C22×C61—44no—
304C22×C61C2×#12312yes—
305C22×C61C2×#12312yes—
306C22×C61C2×#12312yes—
307C22×C61#57514yes—
308C22×C61C23×D411yescharacteristic
309C2×D61—42no—
310C2×D61—44no—
311C2×D61—84no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—48no—
315C4×C61C2×#12312yes—
316C4×C61C2×#12312yes—
317C4×C61C2×#12312yes—
318C4×C61C22×#4112yes—
319Dic61—42no—
320Dic61—44no—
321C2×#525—611no—
322C22×#117D6111yescharacteristic, complemented
323#3095—614no—
324#3094—614no—
325#3093—614no—
326#3092—614no—
327#3091—618no—
328#2911—612no—
329#2896—612no—
330C2×#407—611no—
331C2×C61—22no—
332C2×C61—22no—
333C2×C61—24no—
334C2×C61—22no—
335C2×C61—42no—
336C2×C61C22×#12311yescharacteristic
337C2×C61C2×#57512yes—
338D61—82no—
339C22×#41—611no—
340C24×C4C2×D6111yescharacteristic
341C22×M16—612no—
342#525—614no—
343#525—612no—
344#525—612no—
345#525—1222no—
346#525—1221no—
347#525—1221no—
348C2×#123—1221no—
349C2×C22⋊Q8—612no—
350C2×#121—612no—
351C2×#120—1221no—
352C2×C4×D4—1222no—
353C2×#117—22no—
354C2×#117—21no—
355C2×#117—22no—
356C2×#117—21no—
357C2×#117—611no—
358C2×#117C2×D6114yes—
359C2×#117C2×D6112yes—
360C2×#117C2×D6114yescomplemented
361C2×#117C2×D6112yescomplemented
362C22×C4⋊C4C2×D6112yescomplemented
363C22×C22⋊C4—21no—
364C22×C42—21no—
365#485—6116no—
366#484—618no—
367#483—618no—
368#482—6116no—
369#481—618no—
370#480—618no—
371#428—618no—
372#428—618no—
373#427—618no—
374#419—614no—
375#418—614no—
376#417—618no—
377#416—618no—
378#407—612no—
379#407—612no—
380#407—614no—
381C2×C22⋊C8—614no—
382C61#6051111yescharacteristic, complemented, Sylow
383C25C22⋊D6111yescharacteristic
384C2×#41—612no—
385C2×#41—612no—
386C2×#41—1221no—
387C2×#41—1221no—
388C2×#41—1228no—
389C22×Q8—611no—
390C22×D4—611no—
391C22×D4—1222no—
392C22×D4—1221no—
393C23×C4—28no—
394C23×C4—22no—
395C23×C4—1221no—
396C23×C4C22⋊D6112yes—
397C23×C4C22⋊D6111yescharacteristic
398C23×C4C22⋊D6112yes—
399C23×C4C22⋊D6111yescharacteristic
400C23×C4C22⋊D6111yescharacteristic
401C23×C4C22×D6111yescharacteristic
402C23×C4C22×D6112yes—
403C23×C4C22×D6112yes—
404C23×C4C22×D6111yescharacteristic
405C23×C4C22×D6111yescharacteristic
406C2×M16—614no—
407C2×M16—614no—
408C2×M16—1228no—
409C22×C8—614no—
410#123—1222no—
411#123—1222no—
412#123—1224no—
413C22⋊Q8—614no—
414C22⋊Q8—614no—
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417#121—614no—
418#121—614no—
419#121—1222no—
420#121—1222no—
421#120—1222no—
422#120—1224no—
423#120—1222no—
424C4×D4—1224no—
425C4×D4—1224no—
426C4×D4—1224no—
427C4×D4—1224no—
428#117—24no—
429#117—24no—
430#117—22no—
431#117—22no—
432#117—24no—
433#117—24no—
434#117—24no—
435#117—612no—
436#117—612no—
437#117—1221no—
438#117—1221no—
439#117C22×D6114yes—
440#117C22×D6114yes—
441#117C22×D6118yes—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—22no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4C22×D6114yes—
448C2×C4⋊C4C22×D6114yes—
449C2×C4⋊C4C22×D6114yes—
450C2×C22⋊C4—24no—
451C2×C22⋊C4—22no—
452C2×C22⋊C4—24no—
453C2×C22⋊C4—22no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1222no—
456C2×C22⋊C4—1222no—
457C2×C42—24no—
458C2×C42—22no—
459C2×C42—22no—
460C2×C42—24no—
461C2×C42—1221no—
462#107—6116no—
463#106—6116no—
464#103—6116no—
465#102—6116no—
466C22⋊C8—618no—
467C22⋊C8—618no—
468C24—24no—
469C24—44no—
470C24—1221no—
471C24C2×C22⋊D6111yescharacteristic
472C24C2×C22⋊D6112yes—
473C24C2×C22⋊D6112yes—
474C24C2×C22⋊D6111yescharacteristic
475C24C2×C22⋊D6111yescharacteristic
476#41—1228no—
477#41—1228no—
478#41—2444no—
479#41—2444no—
480C2×Q8—612no—
481C2×Q8—1221no—
482C2×Q8—1224no—
483C2×D4—612no—
484C2×D4—1224no—
485C2×D4—1224no—
486C2×D4—1222no—
487C2×D4—1222no—
488C2×D4—1221no—
489C2×D4—1224no—
490C2×D4—2448no—
491C2×D4—2444no—
492C22×C4—24no—
493C22×C4—28no—
494C22×C4—28no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—28no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—28no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—48no—
505C22×C4—1222no—
506C22×C4—1221no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—2444no—
510C22×C4#1175822218yes—
511C22×C4C2×C22⋊D6112yes—
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6112yes—
514C22×C4C2×C22⋊D6112yes—
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6112yes—
517C22×C4C2×C22⋊D6112yes—
518C22×C4C2×C22⋊D6112yes—
519C22×C4C2×C22⋊D6112yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C23×D6112yes—
525C22×C4C23×D6112yes—
526C22×C4C23×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528C22×C4C23×D6111yescharacteristic
529M16—1228no—
530M16—1228no—
531C2×C8—618no—
532C2×C8—1228no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—614no—
537C4⋊C4—1222no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4C23×D6118yes—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1224no—
547C22⋊C4—1224no—
548C22⋊C4—1224no—
549C22⋊C4—1224no—
550C42—28no—
551C42—24no—
552C42—24no—
553C42—1222no—
554C42—1222no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—416no—
567C23—1222no—
568C23—2444no—
569C23—2442no—
570C23#1451976014yes—
571C23C22×C22⋊D6112yes—
572C23C22×C22⋊D6111yescharacteristic
573C23C22×C22⋊D6111yescharacteristic
574C23C22×C22⋊D6111yescharacteristic
575C23C22×C22⋊D6112yes—
576Q8—1224no—
577Q8—2442no—
578D4—1224no—
579D4—2448no—
580D4—2444no—
581D4—2444no—
582D4—2448no—
583D4—2442no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—1224no—
601C2×C4—1222no—
602C2×C4—2444no—
603C2×C4—2442no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4#1451976014yes—
607C2×C4#1451976014yes—
608C2×C4#1451976014yes—
609C2×C4C22×C22⋊D6112yes—
610C2×C4C22×C22⋊D6112yes—
611C2×C4C22×C22⋊D6111yescharacteristic
612C2×C4C22×C22⋊D6111yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C2×#1175822214yes—
615C2×C4C2×#1175822214yes—
616C2×C4C2×#1175822214yes—
617C2×C4C24×D6111yescharacteristic
618C8—1228no—
619C22—28no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—44no—
629C22—48no—
630C22—48no—
631C22—44no—
632C22—4884no—
633C22—2444no—
634C22—2442no—
635C22C2×#1451976012yes—
636C22C2×#1451976012yes—
637C22C2×#1451976012yes—
638C22#1684056014yes—
639C22C23×C22⋊D6111yescharacteristic
640C4—22no—
641C4—22no—
642C4—48no—
643C4—2444no—
644C4—2442no—
645C4C2×#1451976012yes—
646C4C2×#1451976012yes—
647C4C2×#1451976012yes—
648C4C22×#1175822212yes—
649C2—22no—
650C2—24no—
651C2—22no—
652C2—22no—
653C2—42no—
654C2—4882no—
655C2C22×#1451976011yescharacteristic
656C2C2×#1684056012yes—
subgroupG/H
centerC2×C4C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC22×#117×C61C2
socleC22×C61C23×D4
prime radicalC2×#16840273C2
hypercenterC22×#117D61
hypocenterC61#60511
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. #21455767C2semidirect
3. #21455759C2semidirect
4. #21455750C2semidirect
5. #21455742C2semidirect
6. #21455732C2semidirect
7. #21454408C2semidirect
8. #21454336C2semidirect
9. C2×#16839726C2semidirect
10. C22×#117×C61C2semidirect
25. #16840103C22semidirect
26. #16840097C22semidirect
27. #16840092C22semidirect
28. #16840086C22semidirect
29. #16840080C22semidirect
30. #16840075C22semidirect
31. #16839805C22semidirect
32. #16839802C22semidirect
33. #16839798C22semidirect
34. #16839770C22semidirect
35. #16839766C22semidirect
36. #16839761C22semidirect
37. #16839756C22semidirect
39. #16839726C22semidirect
40. #16839726C22semidirect
41. C2×C22⋊#92418C22semidirect
49. C2×#117×C61C22semidirect
50. C2×#117×C61C22semidirect
51. C22×C4⋊C4×C61C22semidirect
96. #14519710C23semidirect
97. #14519708C23semidirect
98. #14519700C23semidirect
99. #14519697C23semidirect
322. C22×#117D61semidirect
360. C2×#117C2×D61semidirect
361. C2×#117C2×D61semidirect
362. C22×C4⋊C4C2×D61semidirect
382. C61#60511semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
#214557672
#214557592
#214557502
#214557422
#214557322
#214544082
#214543362
C2×#168397262
C22×#117×C612
#6051161