Finite Group Databasebeta

← #29610109

Subgroups — #29610109

Order 15616.

1706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839669C212yesmaximal, complemented
2C2×#16839637C212yesmaximal
3#21444084C212yesmaximal, complemented
4#21444076C212yesmaximal, complemented
5#21444064C211yesmaximal, characteristic
6#21444056C212yesmaximal
7#21444044C212yesmaximal, complemented
8#21444041C211yesmaximal, characteristic
9C2×C61×#393C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C22×C4⋊Dic61C2212yes—
12C22×C4⋊Dic61C2212yes—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2211yescharacteristic
16C22×C4×Dic61C2212yes—
17#16839669—28no—
18#16839637—28no—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2211yescharacteristic
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C23×Q8×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2212yes—
30C22×C22⋊C4×C61C2211yescharacteristic
31C61×#393C418yescomplemented
32C2×#95×C61C2212yes—
33C2×#95×C61C2211yescharacteristic
34C23×Dic61—22no—
35C23×Dic61—24no—
36C23×Dic61—42no—
37C23×Dic61—41no—
38C23×Dic61—41no—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2312yes—
43C2×C4⋊Dic61—24no—
44C2×C4⋊Dic61—24no—
45C2×C4⋊Dic61—24no—
46C2×C4⋊Dic61—42no—
47C2×C4⋊Dic61—42no—
48C2×C4⋊Dic61—42no—
49C2×C4⋊Dic61—42no—
50C2×C4⋊Dic61—41no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61—24no—
53C2×C22⋊Dic61—41no—
54C2×C22⋊Dic61—41no—
55C2×C22⋊Dic61—42no—
56C2×C22⋊Dic61Q814yescomplemented
57C2×C4×Dic61—24no—
58C2×C4×Dic61—24no—
59C2×C4×Dic61—42no—
60#14519684—24no—
61#14519684—24no—
62#14519684—24no—
63#14519684—24no—
64#14519684—42no—
65#14519684—42no—
66#14519684—41no—
67#14519684—42no—
68#14519684D418yes—
69C25×C61C2311yescharacteristic
70C22×Q8×C61—24no—
71C22×Q8×C61—24no—
72C22×Q8×C61—41no—
73C22×Q8×C61—41no—
74C22×Q8×C61C2×C414yescomplemented
75C23×C4×C61C2312yes—
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2312yes—
79C2×C22⋊C4×C61—24no—
80C2×C22⋊C4×C61—24no—
81C2×C22⋊C4×C61—41no—
82C2×C22⋊C4×C61—41no—
83C2×C22⋊C4×C61C2×C418yes—
84C2×C22⋊C4×C61C2×C414yes—
85#95×C61C2×C414yescomplemented
86#95×C61C2×C418yescomplemented
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—21no—
90C24×C61—21no—
91C24×C61—22no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61C22×C412yes—
96C24×C61C2×Q812yes—
97C24×C61#4112yes—
98C24×C61C2411yescharacteristic
99C2×Q8×C61—216no—
100C2×Q8×C61—44no—
101C2×Q8×C61—44no—
102C2×Q8×C61—48no—
103C2×Q8×C61—44no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C22×C414yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C2×D414yes—
127C22×C4×C61#4114yes—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—44no—
154C22×Dic61—48no—
155C22×Dic61—48no—
156C22×Dic61—44no—
157C22×Dic61—44no—
158C22×Dic61C2×D414yes—
159C22×Dic61C2×D414yes—
160C22×Dic61C2×Q814yes—
161C22⋊C4×C61—216no—
162C22⋊C4×C61—44no—
163C22⋊Dic61—44no—
164C22⋊Dic61—48no—
165C4⋊Dic61—48no—
166C4⋊Dic61—48no—
167C4⋊Dic61—44no—
168C4×Dic61—48no—
169C23×C61—24no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—24no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—42no—
190C23×C61—41no—
191C23×C61C4×Q814yes—
192C23×C61C23×C411yescharacteristic
193C23×C61C22×D412yes—
194C23×C61C22×Q811yescharacteristic
195C23×C61C2×#4112yes—
196C23×C61C2×#4111yescharacteristic
197C23×C61H(5,𝔽2)11yescharacteristic
198C23×C61H(5,𝔽2)11yescharacteristic
199C23×C61H(5,𝔽2)11yescharacteristic
200C23×C61H(5,𝔽2)12yes—
201C23×C612−1+411yescharacteristic
202C23×C612−1+412yes—
203Q8×C61—416no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—24no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61C4×D418yes—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—416no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257#14575—611nomaximal, Sylow
258C22×C61—24no—
259C22×C61—28no—
260C22×C61—44no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—44no—
266C22×C61—44no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61C2×C4×D412yes—
270C22×C61C2×C4×Q811yescharacteristic
271C22×C61#51811yescharacteristic
272C22×C61#52011yescharacteristic
273C22×C61#52012yes—
274C22×C61#53512yes—
275C22×C61#53612yes—
276C22×C61#53811yescharacteristic
277C22×C61#53911yescharacteristic
278C22×C61#54212yes—
279C22×C61#54212yes—
280C22×C61#54311yescharacteristic
281C22×C61#54411yescharacteristic
282C22×C61D4211yescharacteristic
283C22×C61#54612yes—
284C22×C61#54612yes—
285C22×C61D4×Q812yes—
286C22×C61#55111yescharacteristic
287C22×C61#55212yes—
288C22×C61#55412yes—
289C22×C61H(3,𝔽4)11yescharacteristic
290C22×C61#56212yes—
291C22×C61#56311yescharacteristic
292C4×C61—28no—
293C4×C61—44no—
294C4×C61—42no—
295C4×C61—42no—
296Dic61—44no—
297Dic61—44no—
298Dic61—44no—
299Dic61—82no—
300C2×#393D6111yescharacteristic
301C2×#391—612no—
302C2×#385—612no—
303#1444—612no—
304#1442—612no—
305#1439—611no—
306#1437—612no—
307#1434—612no—
308C22⋊#95—611no—
309C2×C61—44no—
310C2×C61#232612yes—
311C2×C61#232911yescharacteristic
312C2×C61#235811yescharacteristic
313C2×C61#236012yes—
314C2×C61#237911yescharacteristic
315C2×C61#267011yescharacteristic
316C2×C61#267711yescharacteristic
317C2×C61#268712yes—
318C2×C61#269212yes—
319C2×C61#276311yescharacteristic
320C2×C61#276411yescharacteristic
321C23×Q8C2×D6111yescharacteristic
322C24×C4—611no—
323C22×C4⋊C4—612no—
324C22×C4⋊C4—612no—
325C22×C4⋊C4—611no—
326C22×C22⋊C4—611no—
327C22×C22⋊C4—612no—
328C22×C22⋊C4C2×D6112yes—
329C22×C22⋊C4C2×D6111yescharacteristic
330C22×C42—612no—
331#393Dic6118yescomplemented
332#391—1228no—
333#385—1228no—
334C2×#95—612no—
335C2×#95—612no—
336C2×#95—612no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—611no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95C2×D6112yes—
344C2×#95C2×D6111yescharacteristic
345C61#1457511yescharacteristic, complemented, Sylow
346C25C22×D6111yescharacteristic
347C22×Q8—24no—
348C22×Q8—24no—
349C22×Q8—41no—
350C22×Q8—41no—
351C22×Q8C2×Dic6114yescomplemented
352C23×C4—612no—
353C23×C4—612no—
354C23×C4—612no—
355C23×C4—612no—
356C23×C4—1224no—
357C23×C4—1222no—
358C23×C4—2441no—
359C23×C4—2442no—
360C23×C4—2441no—
361C23×C4C22×D6112yes—
362C23×C4C22×D6111yescharacteristic
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6112yes—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1224no—
368C2×C4⋊C4—2442no—
369C2×C4⋊C4—2442no—
370C2×C4⋊C4—2442no—
371C2×C4⋊C4—2441no—
372C2×C4⋊C4—2442no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—614no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—2441no—
381C2×C22⋊C4—2441no—
382C2×C22⋊C4—2442no—
383C2×C22⋊C4C2×Dic6118yes—
384C2×C22⋊C4C2×Dic6114yes—
385C2×C42—1224no—
386C2×C42—1224no—
387C2×C42—2442no—
388#95—618no—
389#95—1224no—
390#95—1224no—
391#95—1224no—
392#95—1224no—
393#95—2441no—
394#95—2442no—
395#95—2442no—
396#95—2442no—
397#95C2×Dic6114yescomplemented
398#95C2×Dic6118yescomplemented
399C24—22no—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24Q8×D6112yes—
408C24#1175822612yes—
409C24C22×Dic6112yes—
410C24C23×D6111yescharacteristic
411C2×Q8—216no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—44no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—614no—
435C22×C4—614no—
436C22×C4—614no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—2444no—
463C22×C4—2448no—
464C22×C4—2448no—
465C22×C4—2444no—
466C22×C4—2444no—
467C22×C4D4×D6114yes—
468C22×C4#1175822414yes—
469C22×C4C22×Dic6114yes—
470C22×C4C22×Dic6112yes—
471C22×C4C22×Dic6112yes—
472C22×C4C22×Dic6114yes—
473C4⋊C4—2448no—
474C4⋊C4—2448no—
475C4⋊C4—2444no—
476C22⋊C4—216no—
477C22⋊C4—44no—
478C22⋊C4—2444no—
479C22⋊C4—2448no—
480C42—2448no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—41no—
503C23Q8×Dic6114yes—
504C23C23×Dic6111yescharacteristic
505C23C2×D4×D6112yes—
506C23C2×Q8×D6111yescharacteristic
507C23C2×#1175822412yes—
508C23C2×#1175822611yescharacteristic
509C23#1451982211yescharacteristic
510C23#1451982211yescharacteristic
511C23#1451982312yes—
512C23#1451982311yescharacteristic
513C23#1451982412yes—
514C23#1451982411yescharacteristic
515Q8—416no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—24416no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4D4×Dic6118yes—
569C22—24no—
570C22—28no—
571C22—42no—
572C22—44no—
573C22—42no—
574C22—44no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22C2×D4×Dic6112yes—
581C22C2×Q8×Dic6111yescharacteristic
582C22#1684023711yescharacteristic
583C22#1684024512yes—
584C22#1684024511yescharacteristic
585C22#1684033012yes—
586C22#1684033212yes—
587C22#1684035211yescharacteristic
588C22#1684036111yescharacteristic
589C22#1684038112yes—
590C22#1684038112yes—
591C22#1684038711yescharacteristic
592C22#1684039111yescharacteristic
593C22#1684039411yescharacteristic
594C22#1684039912yes—
595C22#1684040312yes—
596C22#1684042512yes—
597C22#1684044311yescharacteristic
598C22#1684045412yes—
599C22#1684047012yes—
600C22#1684050411yescharacteristic
601C22#1684050612yes—
602C22#1684051211yescharacteristic
603C4—28no—
604C4—42no—
605C4—44no—
606C4—42no—
607C4—4882no—
608C4—2444no—
609C4—2444no—
610C4—2444no—
611C2—44no—
612C2#2144853212yes—
613C2#2144855411yescharacteristic
614C2#2144880311yescharacteristic
615C2#2144882812yes—
616C2#2144899311yescharacteristic
617C2#2145203911yescharacteristic
618C2#2145212811yescharacteristic
619C2#2145223912yes—
620C2#2145231412yes—
621C2#2145323611yescharacteristic
622C2#2145324511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#393C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#393D61
hypocenterC61#14575
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×#16839669C2semidirect
3. #21444084C2semidirect
4. #21444076C2semidirect
7. #21444044C2semidirect
31. C61×#393C4semidirect
56. C2×C22⋊Dic61Q8semidirect
74. C22×Q8×C61C2×C4semidirect
85. #95×C61C2×C4semidirect
86. #95×C61C2×C4semidirect
331. #393Dic61semidirect
345. C61#14575semidirect
351. C22×Q8C2×Dic61semidirect
397. #95C2×Dic61semidirect
398. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396692
C2×#168396372
#214440842
#214440762
#214440642
#214440562
#214440442
#214440412
C2×C61×#3932
#1457561