Finite Group Databasebeta

← #29680047

Subgroups — #29680047

Order 15616.

1402 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11758224C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21445912C212yesmaximal, complemented
5#21445825C211yesmaximal, characteristic, complemented
6#21445821C211yesmaximal, characteristic, complemented
7#21445819C211yesmaximal, characteristic, complemented
8#21445756C212yesmaximal
9#21445638C212yesmaximal, complemented
10#102×Dic61C212yesmaximal, complemented
11C61×#1802C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C4×#11758224—24no—
14C4×#11758224C418yescomplemented
15C2×C4×Dic122C2211yescharacteristic
16C2×#14519742C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×C4×C22⋊D61—21no—
19C2×#14519737—21no—
20C2×#14519735—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4×Dic61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839632C2211yescharacteristic
33#16839627—21no—
34#16839619—21no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×D4×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—22no—
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60C4×Dic122—22no—
61C4×Dic122C2×C414yes—
62#14519742—22no—
63#14519742C2×C414yes—
64D4×Dic61—22no—
65D4×Dic61C2×C414yes—
66C4×C22⋊D61—28no—
67#14519737—28no—
68#14519735—22no—
69#14519735—44no—
70#14519735C2×C414yes—
71#14519733—28no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—28no—
85C2×C4×Dic61—22no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C42×D61C2×C414yes—
90C42×D61D414yes—
91C4⋊#92418—22no—
92#14519702—24no—
93#14519700—24no—
94#14519698—24no—
95#14519689—24no—
96C8×Dic61—24no—
97#14519684—22no—
98#14519684—22no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C22×C8×C61C2312yes—
102C4×D4×C61—22no—
103C4×D4×C61C2×C414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111#102×C61—24no—
112C24×C61—21no—
113C23×D61C2×D411yescharacteristic
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61C22×C412yes—
118C2×D4×C61#4112yes—
119C2×D4×C61#4112yes—
120C22×C4×C61—21no—
121C22×C4×C61—24no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—24no—
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C22×Dic61—24no—
132C22×Dic61—24no—
133C22×Dic61—21no—
134C22×Dic61—24no—
135C22×Dic61—21no—
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143#11758224—28no—
144#11758224—28no—
145#11758224—28no—
146#11758224—28no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61C22⋊C414yes—
156C2×C4×D61C22⋊C414yes—
157C2×C4×D61C22×C412yes—
158C2×C4×D61C2×D412yes—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C22×C412yes—
163C2×Dic122#4112yes—
164C2×Dic122#4112yes—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61C22×C412yes—
174C22⋊C4×C61—24no—
175C42×C61C22×C412yes—
176C42×C61C2×D412yes—
177C22⋊Dic61—24no—
178#11758198—24no—
179#11758198—44no—
180C4⋊Dic61—21no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—42no—
184C4⋊Dic61—42no—
185C4⋊Dic61C22×C412yes—
186#11758196—24no—
187#11758196—24no—
188C4×Dic61—24no—
189C4×Dic61—24no—
190C4×Dic61—24no—
191C4×Dic61—44no—
192C4×Dic61C22⋊C414yes—
193C4×Dic61C22⋊C414yes—
194C4×Dic61C22×C412yes—
195C4×Dic61C2×D412yes—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61C22×D411yescharacteristic
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61C2×C22⋊C412yes—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61#11714yescomplemented
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—48no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61C2×C22⋊C412yes—
228C2×C4×C61C2×C22⋊C412yes—
229C2×C4×C61#13512yes—
230C2×C4×C61#13512yes—
231C2×C4×C61C23×C411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—216no—
238C2×Dic61—216no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—48no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61AGL(1,ℤ/8)12yes—
257C2×Dic61#13712yes—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—28no—
263C4×D61—28no—
264Dic122—22no—
265Dic122—22no—
266Dic122—22no—
267Dic122#11714yescomplemented
268C8×C61—42no—
269#92418—42no—
270#20737—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61#52511yescharacteristic
280C22×C61#52511yescharacteristic
281C22×C61C2×#13511yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61C2×#13711yescharacteristic
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—41no—
291C2×D61—41no—
292C4×C61—22no—
293C4×C61—42no—
294C4×C61—82no—
295C4×C61#41614yes—
296C4×C61C2×#11711yescharacteristic
297C4×C61C2×#11711yescharacteristic
298C4×C61#52511yescharacteristic
299C4×C61#52511yescharacteristic
300Dic61—22no—
301Dic61—24no—
302Dic61—22no—
303Dic61—42no—
304Dic61—82no—
305Dic61#41714yes—
306C2×C4×#41—611no—
307C2×#431—611no—
308#2285—611no—
309#1825—612no—
310#1803—611no—
311#1803—611no—
312#1802—611no—
313#1802D6111yescharacteristic, complemented
314#1788—612no—
315C4×#103—612no—
316C4×#102—612no—
317C2×C61—44no—
318C2×C61#228011yescharacteristic
319C2×C61C2×#41611yescharacteristic
320C2×C61C2×#41711yescharacteristic
321C2×C61#310611yescharacteristic
322C2×C61#310611yescharacteristic
323C2×C61#310711yescharacteristic
324C2×C61#310711yescharacteristic
325D61—42no—
326C22×#41—611no—
327C4×#41—618no—
328C4×#41—1224no—
329C2×C4×Q8—611no—
330C2×C4×Q8—611no—
331C2×C4×D4—611no—
332C2×C4×D4—1221no—
333C2×C4×D4—1221no—
334C2×C4×D4C2×D6111yescharacteristic
335C2×#117—611no—
336C2×#117—1221no—
337C2×#117—1221no—
338C22×C42—611no—
339C22×C42—1221no—
340#431—1224no—
341C2×C4⋊C8—611no—
342C2×C4⋊C8C2×D6111yescharacteristic
343C2×#103—612no—
344C2×#103—612no—
345C2×#102—612no—
346C2×#102C2×D6112yes—
347C2×C22⋊C8—612no—
348C2×C4×C8—612no—
349#384—611no—
350#384C2×D6111yescharacteristic
351#382—1221no—
352#380—612no—
353#380—1221no—
354C4×C4⋊C4—612no—
355C4×C4⋊C4—1221no—
356#340—612no—
357#340—612no—
358#340—612no—
359#340C2×D6112yes—
360C61#2073711yescharacteristic, complemented, Sylow
361C2×#41—614no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1222no—
365C2×#41—1222no—
366C2×#41—1222no—
367C22×Q8—611no—
368C22×D4—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—21no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—1221no—
375C23×C4—1221no—
376C22×C8—612no—
377C22×C8C22×D6112yes—
378C4×Q8—614no—
379C4×Q8—614no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382C4×D4—22no—
383C4×D4—614no—
384C4×D4—1222no—
385C4×D4—1228no—
386C4×D4—1228no—
387C4×D4C4×D6114yes—
388#117—614no—
389#117—1228no—
390#117—1222no—
391#117—2444no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22×D6112yes—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1222no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—614no—
410C2×C42—614no—
411C2×C42—1228no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1222no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—22no—
418C4⋊C8—1222no—
419#103—1224no—
420#103—1224no—
421#102—24no—
422#102—1224no—
423C22⋊C8—1224no—
424C4×C8—1224no—
425#95—1222no—
426#95—1222no—
427C24—21no—
428C24—611no—
429#41—1228no—
430#41—1228no—
431#41—1228no—
432#41—1228no—
433C2×Q8—612no—
434C2×Q8—612no—
435C2×Q8—612no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4C2×C4×D6112yes—
446C2×D4#1175822412yes—
447C2×D4#1175822612yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—614no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1224no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—1222no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—612no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1224no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4C2×C4×D6112yes—
500C22⋊C4—24no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C22⋊C4—2444no—
504C42—612no—
505C42—612no—
506C42—614no—
507C42—614no—
508C42—1224no—
509C42—1224no—
510C42—1224no—
511C42—2444no—
512C42C2×C4×D6112yes—
513C42D4×D6112yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—612no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23C2×D4×D6111yescharacteristic
529Q8—614no—
530Q8—1222no—
531Q8—1222no—
532Q8—1222no—
533D4—22no—
534D4—22no—
535D4—22no—
536D4—12216no—
537D4#1451973514yescomplemented
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1228no—
553C2×C4—12216no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1228no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—2448no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C22⋊C4×D6112yes—
575C2×C4#1451979612yes—
576C2×C4#1451979812yes—
577C2×C4C22×C4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4#41×D6111yescharacteristic
580C2×C4#41×D6111yescharacteristic
581C2×C4C2×#1175822411yescharacteristic
582C2×C4C2×#1175822611yescharacteristic
583C8—42no—
584C8—2442no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—2441no—
596C22—2441no—
597C22—2441no—
598C22—2441no—
599C22C2×C22⋊C4×D6111yescharacteristic
600C22#1684027211yescharacteristic
601C22#1684027511yescharacteristic
602C22C2×#1451979611yescharacteristic
603C22C2×#1451979811yescharacteristic
604C22AGL(1,ℤ/8)×D6111yescharacteristic
605C22#137×D6111yescharacteristic
606C4—22no—
607C4—42no—
608C4—82no—
609C4—614no—
610C4—1222no—
611C4—1224no—
612C4—1222no—
613C4—4882no—
614C4—2442no—
615C4#1683975514yes—
616C4#117×D6111yescharacteristic
617C4C2×#1451973511yescharacteristic
618C4#1684027211yescharacteristic
619C4#1684027511yescharacteristic
620C2—44no—
621C2—2442no—
622C2#2144820111yescharacteristic
623C2C2×#1683975511yescharacteristic
624C2#417×D6111yescharacteristic
625C2#2145587611yescharacteristic
626C2#2145588111yescharacteristic
627C2#2145588711yescharacteristic
628C2#2145589211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#20737
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. #21448224C2semidirect
4. #21445912C2semidirect
5. #21445825C2semidirect
6. #21445821C2semidirect
7. #21445819C2semidirect
9. #21445638C2semidirect
10. #102×Dic61C2semidirect
11. C61×#1802C2semidirect
14. C4×#11758224C4semidirect
217. D4×C61#117semidirect
267. Dic122#117semidirect
313. #1802D61semidirect
360. C61#20737semidirect
537. D4#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117582242
C2×#168398142
#214482242
#214459122
#214458252
#214458212
#214458192
#214457562
#214456382
#102×Dic612
C61×#18022
#2073761