Finite Group Databasebeta

← #29679804

Subgroups — #29679804

Order 15616.

1502 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D61C211yesmaximal, characteristic
2C22×#14519689C211yesmaximal, characteristic
3#21448198C211yesmaximal, characteristic
4#21445913C212yesmaximal, complemented
5#21445912C212yesmaximal, complemented
6#21445908C212yesmaximal, complemented
7#21445904C212yesmaximal, complemented
8#21445536C211yesmaximal, characteristic, complemented
9#21445535C211yesmaximal, characteristic, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C61×#1735C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16C2×#14519735C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519734—21no—
19C2×#14519733—21no—
20#117×D61—24no—
21#117×D61C418yescomplemented
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C22⋊C4×D61—21no—
25C2×C42×D61—21no—
26C2×#14519689—24no—
27C2×#14519689C2212yes—
28C2×#14519689C2212yes—
29#16839623—21no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839617—21no—
33C4×#11758198—21no—
34C4×#11758198—21no—
35#16839525C2212yescomplemented
36#16839525C2212yescomplemented
37#16839524C2212yescomplemented
38#16839524C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C23×C8×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#117×C61C2211yescharacteristic
44C61×#340C2212yescomplemented
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C23×Dic61D411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61—21no—
53C22×C4×D61—21no—
54C22×C4×D61C2×C414yes—
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D414yes—
58C22×C4×D61C2311yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418—22no—
63C22×#92418C2312yes—
64#14519736—22no—
65#14519736—22no—
66#14519736—22no—
67#14519736C2×C414yes—
68#14519735—22no—
69#14519735—22no—
70#14519735C2×C414yes—
71#14519735C2×C414yes—
72#14519734—28no—
73#14519733—28no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61—22no—
81C4⋊C4×D61C2×C414yes—
82C4⋊C4×D61C2×C414yes—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—22no—
88C2×#11758198—22no—
89C22⋊C4×D61—28no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C42×D61—28no—
95#14519689—24no—
96#14519689—24no—
97#14519684—22no—
98#14519684—22no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61—22no—
103C22×C8×C61C2312yes—
104#117×C61—22no—
105#117×C61—22no—
106#117×C61—22no—
107#117×C61C2×C414yes—
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
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×D61C22⋊C414yes—
121C23×D61C2×D411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—28no—
143C2×C4×D61—28no—
144C2×C4×D61—28no—
145C2×C4×D61—28no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
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—21no—
157C2×C4×D61—21no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—44no—
161C2×C4×D61—44no—
162C2×C4×D61C22⋊C414yes—
163C2×C4×D61C22⋊C414yes—
164C2×C4×D61C22×C412yes—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61C2×D412yes—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—44no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—42no—
180C42×C61—24no—
181C42×C61—42no—
182C22⋊Dic61—24no—
183C22⋊Dic61—42no—
184#11758198—24no—
185#11758198—24no—
186#11758198—44no—
187#11758198—44no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22×C412yes—
193C4⋊Dic61C22×C412yes—
194#11758196—24no—
195#11758196—24no—
196C4×Dic61—24no—
197C4×Dic61—24no—
198C4×Dic61—24no—
199C4×Dic61—42no—
200C2×#92418—22no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—44no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61C2×C22⋊C412yes—
207C23×C61#12312yes—
208C23×C61#12312yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—22no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—28no—
218C22×D61—28no—
219C22×D61—44no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C412yes—
224C22×D61#13514yes—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61#11714yes—
234C2×C4×C61#11714yes—
235C2×C4×C61#12314yes—
236C2×C4×C61#12314yes—
237C2×C4×C61#12312yes—
238C2×C4×C61#12312yes—
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—28no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—28no—
268C4×D61—28no—
269C4×D61—48no—
270C4×D61—48no—
271C8×C61—42no—
272#92418—42no—
273#20699—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—24no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61#38114yes—
279C22×C61C22×C22⋊C411yescharacteristic
280C22×C61C2×#12311yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C22×C61C2×#13512yes—
283C22×C61#57512yes—
284C2×D61—24no—
285C2×D61—24no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—416no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61#41614yes—
295C4×C61—42no—
296C4×C61—42no—
297C4×C61—82no—
298C4×C61#38114yes—
299C4×C61C2×#11711yescharacteristic
300C4×C61C2×#11711yescharacteristic
301C4×C61C2×#12311yescharacteristic
302C4×C61C2×#12311yescharacteristic
303Dic61—22no—
304Dic61—22no—
305Dic61—42no—
306Dic61—42no—
307Dic61—82no—
308C22×#117—611no—
309C22×C22⋊C8—611no—
310#2280—611no—
311#1825—612no—
312#1825—612no—
313#1824—612no—
314#1823—612no—
315#1735—611no—
316#1735—611no—
317#1735—611no—
318#1735D6111yescharacteristic, complemented
319C2×C61—24no—
320C2×C61C2×#38111yescharacteristic
321C2×C61C2×#41611yescharacteristic
322C2×C61#289611yescharacteristic
323C2×C61#309112yes—
324C2×C61#309112yes—
325D61—24no—
326D61—42no—
327C24×C4—611no—
328C23×C8—611no—
329C23×C8C2×D6111yescharacteristic
330C2×#117—611no—
331C2×#117—618no—
332C2×#117—611no—
333C2×#117—611no—
334C2×#117—611no—
335C2×#117—1221no—
336C2×#117—1221no—
337C2×#117—1224no—
338C2×#117C2×D6111yescharacteristic
339C2×#117C2×D6111yescharacteristic
340C22×C4⋊C4—611no—
341C22×C4⋊C4—611no—
342C22×C22⋊C4—1221no—
343C22×C42—1221no—
344C2×C22⋊C8—612no—
345C2×C22⋊C8—612no—
346C2×C22⋊C8—1224no—
347#381—1221no—
348#380—612no—
349#380—612no—
350#379—1221no—
351C4×C22⋊C4—1221no—
352C4×C22⋊C4—1221no—
353#340—612no—
354#340—612no—
355#340—612no—
356#340—612no—
357#340—612no—
358#340—612no—
359#340C2×D6112yescomplemented
360#340C2×D6112yescomplemented
361C61#2069911yescharacteristic, complemented, Sylow
362C25—611no—
363C23×C4—611no—
364C23×C4—614no—
365C23×C4—614no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—1221no—
371C23×C4—1221no—
372C23×C4—1222no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—1222no—
376C23×C4C22×D6111yescharacteristic
377C22×C8—22no—
378C22×C8—22no—
379C22×C8—22no—
380C22×C8—612no—
381C22×C8—1222no—
382C22×C8—1222no—
383C22×C8—1222no—
384C22×C8C22×D6112yes—
385#117—22no—
386#117—22no—
387#117—22no—
388#117—614no—
389#117—614no—
390#117—614no—
391#117—1228no—
392#117—1228no—
393#117—1222no—
394#117—1222no—
395#117—1222no—
396#117—1222no—
397#117—1222no—
398#117C4×D6114yes—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—614no—
402C2×C4⋊C4—614no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4C22×D6112yes—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1228no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1221no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42—1228no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42—1221no—
427C22⋊C8—1224no—
428C22⋊C8—1224no—
429#95—1222no—
430#95—1222no—
431C24—611no—
432C24—611no—
433C24—614no—
434C24—1224no—
435C24—1222no—
436C24—1222no—
437C24—2442no—
438C24D4×D6111yescharacteristic
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—614no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—614no—
450C22×C4—614no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1228no—
464C22×C4—1228no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1228no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1228no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1222no—
474C22×C4—2444no—
475C22×C4—2444no—
476C22×C4C2×C4×D6112yes—
477C22×C4#1175822212yes—
478C22×C4#1175822212yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4#1175822412yes—
482C22×C4#1175822612yes—
483C22×C4C23×D6111yescharacteristic
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—2444no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4C2×C4×D6112yes—
505C4⋊C4C2×C4×D6112yes—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C22⋊C4—2444no—
512C22⋊C4—2444no—
513C22⋊C4—2442no—
514C42—24no—
515C42—42no—
516C42—1224no—
517C42—1224no—
518C42—1224no—
519C42—2442no—
520C23—22no—
521C23—22no—
522C23—612no—
523C23—614no—
524C23—612no—
525C23—1228no—
526C23—1221no—
527C23—1221no—
528C23—1222no—
529C23—1221no—
530C23—1221no—
531C23—1222no—
532C23—1222no—
533C23—1228no—
534C23—2444no—
535C23—2448no—
536C23—2448no—
537C23C22⋊C4×D6112yes—
538C23#1451975912yes—
539C23#1451976212yes—
540C23C2×D4×D6111yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1228no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1228no—
564C2×C4—1228no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—2448no—
568C2×C4—2448no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4C22⋊C4×D6112yes—
575C2×C4#1451973414yes—
576C2×C4#1451973514yes—
577C2×C4#1451975914yes—
578C2×C4#1451975912yes—
579C2×C4#1451976214yes—
580C2×C4#1451976212yes—
581C2×C4C22×C4×D6111yescharacteristic
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×#1175822411yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4C2×#1175822611yescharacteristic
587C8—42no—
588C8—2442no—
589C22—24no—
590C22—24no—
591C22—22no—
592C22—22no—
593C22—614no—
594C22—1222no—
595C22—1224no—
596C22—1224no—
597C22—1222no—
598C22—1222no—
599C22—1222no—
600C22—1222no—
601C22—24416no—
602C22—2442no—
603C22—2442no—
604C22#1683962414yes—
605C22C2×C22⋊C4×D6111yescharacteristic
606C22C2×#1451975911yescharacteristic
607C22C2×#1451976211yescharacteristic
608C22#135×D6112yes—
609C22#1684055912yes—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—1222no—
614C4—1222no—
615C4—4882no—
616C4—2442no—
617C4—2442no—
618C4#1683962414yes—
619C4C2×#1451973411yescharacteristic
620C4C2×#1451973511yescharacteristic
621C4C2×#1451975911yescharacteristic
622C4C2×#1451976211yescharacteristic
623C2—24no—
624C2—1224no—
625C2—2442no—
626C2C2×#1683962411yescharacteristic
627C2#416×D6111yescharacteristic
628C2#2145433411yescharacteristic
629C2#2145573012yes—
630C2#2145573612yes—
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#20699
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
4. #21445913C2semidirect
5. #21445912C2semidirect
6. #21445908C2semidirect
7. #21445904C2semidirect
8. #21445536C2semidirect
9. #21445535C2semidirect
10. #21445535C2semidirect
11. C61×#1735C2semidirect
21. #117×D61C4semidirect
35. #16839525C22semidirect
36. #16839525C22semidirect
37. #16839524C22semidirect
38. #16839524C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
44. C61×#340C22semidirect
45. C61×#340C22semidirect
318. #1735D61semidirect
359. #340C2×D61semidirect
360. #340C2×D61semidirect
361. C61#20699semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D612
C22×#145196892
#214481982
#214459132
#214459122
#214459082
#214459042
#214455362
#214455352
C61×#17352
#2069961