Finite Group Databasebeta

← #29680741

Subgroups — #29680741

Order 15616.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D61C211yesmaximal, characteristic
2C2×#16839800C211yesmaximal, characteristic, complemented
3C2×#16839798C211yesmaximal, characteristic, complemented
4C22×#14519689C211yesmaximal, characteristic
5#21448198C211yesmaximal, characteristic
6#21446021C212yesmaximal, complemented
7#21446019C212yesmaximal, complemented
8#21446013C212yesmaximal, complemented
9#21446008C212yesmaximal, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C61×#1735C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16C2×#14519735C2212yes—
17C2×#14519734—21no—
18C2×#14519733—21no—
19#117×D61—24no—
20#117×D61—24no—
21C2×C4⋊C4×D61C2212yes—
22C2×C22⋊C4×D61—21no—
23C2×C42×D61—21no—
24#16839800—24no—
25#16839798—24no—
26C2×#14519711C2212yescomplemented
27C2×#14519710C2212yescomplemented
28C2×#14519709C2212yescomplemented
29C2×#14519708C2212yescomplemented
30C2×#14519689C418yescomplemented
31C2×#14519689C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839623—21no—
36#16839619C2212yes—
37#16839619C2212yes—
38#16839617—21no—
39C4×#11758198—21no—
40C4×#11758198—21no—
41#16839524C2212yescomplemented
42#16839524C2212yescomplemented
43C23×C8×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C2×#117×C61C2211yescharacteristic
46C61×#340C2212yescomplemented
47C61×#340C2212yescomplemented
48C24×D61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—22no—
55C22×C4×D61C2×C414yes—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61Q814yes—
59C22×C4×D61C2311yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418C2×C414yescomplemented
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519736—22no—
67#14519736—22no—
68#14519736—22no—
69#14519736D414yes—
70#14519735—24no—
71#14519735—24no—
72#14519734—28no—
73#14519733—28no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×#11758198—22no—
82C2×#11758198—22no—
83C2×#11758198—22no—
84C22⋊C4×D61—28no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61—21no—
88C42×D61—28no—
89#14519711—24no—
90#14519710—24no—
91#14519709—24no—
92#14519708—24no—
93#14519689C2×C414yes—
94#14519689C2×C414yes—
95#14519689C2×C414yescomplemented
96#14519689C2×C414yescomplemented
97#14519684—22no—
98#14519684—22no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61C2×C414yescomplemented
103C22×C8×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105#117×C61—22no—
106#117×C61—22no—
107#117×C61—22no—
108#117×C61D414yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2312yes—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C24×C61C2×D411yescharacteristic
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—42no—
120C23×D61C4⋊C414yes—
121C23×D61C2×D411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61C4⋊C414yes—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×C4×D61—28no—
141C2×C4×D61—28no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—28no—
145C2×C4×D61—28no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—28no—
157C2×C4×D61—44no—
158C2×C4×D61—44no—
159C2×C4×D61C4⋊C414yes—
160C2×C4×D61C4⋊C414yes—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C22×C412yes—
163C2×C4×D61C2×Q812yes—
164C2×C4×D61C2×Q812yes—
165C2×C8×C61—28no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C2×C8×C61C22×C412yes—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2×D414yes—
176C22⋊C4×C61—24no—
177C22⋊C4×C61—42no—
178C42×C61—24no—
179C42×C61—42no—
180C22⋊Dic61—24no—
181C22⋊Dic61—42no—
182#11758198—28no—
183#11758198—44no—
184#11758198—44no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C2×D414yes—
189#11758196—28no—
190C4×Dic61—28no—
191C4×Dic61—24no—
192C4×Dic61—42no—
193C2×#92418—28no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C22×C412yes—
199C2×#92418C22×C412yes—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61C2×C4⋊C412yes—
203C23×C61#12012yes—
204C23×C61#12312yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—28no—
214C22×D61—28no—
215C22×D61—44no—
216C22×D61—48no—
217C22×D61—48no—
218C22×D61C2×C4⋊C412yes—
219C22×D61C2×C4⋊C412yes—
220C22×D61#13514yes—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C4⋊C412yes—
228C2×C4×C61C4×D414yes—
229C2×C4×C61C4×D414yes—
230C2×C4×C61#12012yes—
231C2×C4×C61C22⋊Q814yes—
232C2×C4×C61C22⋊Q814yes—
233C2×C4×C61#12312yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×Q811yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—28no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61C2×C4⋊C412yes—
252C2×Dic61C2×C4⋊C412yes—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—28no—
262C4×D61—28no—
263C4×D61—416no—
264C8×C61—22no—
265C8×C61—22no—
266#92418—22no—
267#92418—22no—
268#20843—611nomaximal, Sylow
269C22×C61—24no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—24no—
273C22×C61#38514yes—
274C22×C61C22×C4⋊C411yescharacteristic
275C22×C61C2×#12011yescharacteristic
276C22×C61C2×#12311yescharacteristic
277C22×C61C2×#13512yes—
278C22×C61#57512yes—
279C2×D61—24no—
280C2×D61—22no—
281C2×D61—24no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—416no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61#42714yescomplemented
290C4×C61—44no—
291C4×C61—82no—
292C4×C61#38514yes—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×C22⋊Q811yescharacteristic
297Dic61—22no—
298Dic61—22no—
299Dic61—44no—
300Dic61—82no—
301C22×#117—611no—
302C2×#427—611no—
303C2×#427—611no—
304C22×C22⋊C8—611no—
305#2280—611no—
306#1854—612no—
307#1854—612no—
308#1853—612no—
309#1852—612no—
310#1735—611no—
311#1735D6111yescharacteristic, complemented
312C2×C61—24no—
313C2×C61C2×#38511yescharacteristic
314C2×C61C2×#42711yescharacteristic
315C2×C61#291111yescharacteristic
316C2×C61#300212yes—
317C2×C61#309112yes—
318D61—24no—
319D61—42no—
320C24×C4—611no—
321C23×C8—611no—
322C23×C8C2×D6111yescharacteristic
323C2×#117—611no—
324C2×#117—611no—
325C2×#117—612no—
326C2×#117—1221no—
327C2×#117—1221no—
328C2×#117—1224no—
329C2×#117—1224no—
330C2×#117C2×D6111yescharacteristic
331C2×#117C2×D6111yescharacteristic
332C22×C4⋊C4—612no—
333C22×C22⋊C4—1221no—
334C22×C42—1221no—
335#427—1224no—
336#427—1224no—
337C2×#107—612no—
338C2×#107—612no—
339C2×#106—612no—
340C2×#106—612no—
341C2×C22⋊C8—618no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×C22⋊C8—611no—
345C2×C22⋊C8—611no—
346#381—1221no—
347#380—612no—
348#380—612no—
349#379—1221no—
350C4×C22⋊C4—1221no—
351C4×C22⋊C4—1221no—
352#340—612no—
353#340—612no—
354#340C2×D6112yescomplemented
355#340C2×D6112yescomplemented
356C61#2084311yescharacteristic, complemented, Sylow
357C25—611no—
358C23×C4—611no—
359C23×C4—614no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—614no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—1222no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—1222no—
369C23×C4—1222no—
370C23×C4C22×D6111yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—614no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8—1222no—
377C22×C8—1222no—
378C22×C8C4×D6114yescomplemented
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—614no—
385#117—1228no—
386#117—1228no—
387#117—1224no—
388#117—1222no—
389#117—1224no—
390#117—1222no—
391#117—1222no—
392#117D24414yes—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4C22×D6112yes—
399C2×C4⋊C4C22×D6112yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1228no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1221no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1228no—
411C2×C42—1222no—
412C2×C42—1221no—
413C2×C42—1221no—
414#107—1224no—
415#107—1224no—
416#106—1224no—
417#106—1224no—
418C22⋊C8—614no—
419C22⋊C8—614no—
420C22⋊C8—614no—
421C22⋊C8—614no—
422#95—1222no—
423#95—1222no—
424C24—611no—
425C24—611no—
426C24—614no—
427C24—1224no—
428C24—1222no—
429C24—1222no—
430C24—2442no—
431C24D4×D6111yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—614no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—1228no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1228no—
455C22×C4—1228no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1228no—
459C22×C4—1228no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—2444no—
465C22×C4—2444no—
466C22×C4C2×C4×D6112yes—
467C22×C4C2×D24412yes—
468C22×C4#1175822212yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4#1175822412yes—
472C22×C4Q8×D6112yes—
473C22×C4C23×D6111yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—28no—
479C2×C8—612no—
480C2×C8—612no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1228no—
486C2×C8C2×C4×D6112yes—
487C2×C8C2×C4×D6112yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—614no—
492C4⋊C4—1228no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4C2×D24414yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—1228no—
500C22⋊C4—1224no—
501C22⋊C4—2444no—
502C22⋊C4—2444no—
503C22⋊C4—2442no—
504C42—24no—
505C42—42no—
506C42—1228no—
507C42—1224no—
508C42—2442no—
509C23—22no—
510C23—22no—
511C23—612no—
512C23—614no—
513C23—612no—
514C23—1228no—
515C23—1221no—
516C23—1221no—
517C23—1222no—
518C23—1221no—
519C23—1221no—
520C23—1222no—
521C23—1222no—
522C23—1228no—
523C23—2444no—
524C23—2448no—
525C23—2448no—
526C23C4⋊C4×D6112yes—
527C23#1451974712yes—
528C23#1451975912yes—
529C23C2×D4×D6111yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1228no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1228no—
552C2×C4—1228no—
553C2×C4—1224no—
554C2×C4—24416no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4C4⋊C4×D6112yes—
561C2×C4#1451973714yes—
562C2×C4C4×D24414yes—
563C2×C4#1451974712yes—
564C2×C4#1451975514yes—
565C2×C4#1451975714yes—
566C2×C4#1451975912yes—
567C2×C4C22×C4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C22×D24411yescharacteristic
570C2×C4C2×Q8×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C2×C4C2×#1175822211yescharacteristic
573C8—22no—
574C8—22no—
575C8—1222no—
576C8—1222no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—24no—
581C22—614no—
582C22—1222no—
583C22—1222no—
584C22—1224no—
585C22—1224no—
586C22—1222no—
587C22—1222no—
588C22—1222no—
589C22—24416no—
590C22—2442no—
591C22—2442no—
592C22#1683963614yes—
593C22C2×C4⋊C4×D6111yescharacteristic
594C22C2×#1451974711yescharacteristic
595C22C2×#1451975911yescharacteristic
596C22#135×D6112yes—
597C22#1684055912yes—
598C4—44no—
599C4—82no—
600C4—1222no—
601C4—1222no—
602C4—4882no—
603C4—2444no—
604C4#1683963614yes—
605C4C2×#1451973711yescharacteristic
606C4C2×C4×D24411yescharacteristic
607C4C2×#1451975511yescharacteristic
608C4C2×#1451975711yescharacteristic
609C2—24no—
610C2—1224no—
611C2—2442no—
612C2C2×#1683963611yescharacteristic
613C2#427×D6111yescharacteristic
614C2#2145440711yescharacteristic
615C2#2145503912yes—
616C2#2145573012yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1735C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1735D61
hypocenterC61#20843
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C2×#16839800C2semidirect
3. C2×#16839798C2semidirect
6. #21446021C2semidirect
7. #21446019C2semidirect
8. #21446013C2semidirect
9. #21446008C2semidirect
10. #21445535C2semidirect
11. C61×#1735C2semidirect
26. C2×#14519711C22semidirect
27. C2×#14519710C22semidirect
28. C2×#14519709C22semidirect
29. C2×#14519708C22semidirect
30. C2×#14519689C4semidirect
41. #16839524C22semidirect
42. #16839524C22semidirect
46. C61×#340C22semidirect
47. C61×#340C22semidirect
63. C22×#92418C2×C4semidirect
95. #14519689C2×C4semidirect
96. #14519689C2×C4semidirect
102. C22×C8×C61C2×C4semidirect
289. C2×D61#427semidirect
311. #1735D61semidirect
354. #340C2×D61semidirect
355. #340C2×D61semidirect
356. C61#20843semidirect
378. C22×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D612
C2×#168398002
C2×#168397982
C22×#145196892
#214481982
#214460212
#214460192
#214460132
#214460082
#214455352
C61×#17352
#2084361