Finite Group Databasebeta

← #29606449

Subgroups — #29606449

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839637C211yesmaximal, characteristic
2C22×#14519684C211yesmaximal, characteristic
3#21444087C212yesmaximal, complemented
4#21444081C212yesmaximal, complemented
5#21444076C212yesmaximal, complemented
6#21444072C212yesmaximal, complemented
7#21444042C211yesmaximal, characteristic
8#21444041C211yesmaximal, characteristic
9#21444041C211yesmaximal, characteristic
10#21444041C211yesmaximal, characteristic
11C2×C61×#385C211yesmaximal, characteristic
12C24×Dic61C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2211yescharacteristic
18#16839637—24no—
19C2×#14519684—24no—
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2211yescharacteristic
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C24×C4×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2212yes—
34C22×C22⋊C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2211yescharacteristic
36C61×#385C418yescomplemented
37C2×#95×C61C2212yes—
38C23×Dic61—22no—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—42no—
43C23×Dic61—42no—
44C23×Dic61—42no—
45C23×Dic61—42no—
46C23×Dic61D414yes—
47C23×Dic61Q814yes—
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2312yes—
51C23×Dic61C2311yescharacteristic
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2312yes—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—42no—
57C2×C4⋊Dic61—42no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64#14519684—24no—
65#14519684—24no—
66#14519684—22no—
67#14519684—22no—
68#14519684—42no—
69#14519684—42no—
70#14519684—42no—
71#14519684—42no—
72#14519684—42no—
73#14519684—42no—
74#14519684—42no—
75#14519684—42no—
76C25×C61C2311yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—22no—
79C23×C4×C61—42no—
80C23×C4×C61—42no—
81C23×C4×C61C2×C414yes—
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C23×C4×C61C2312yes—
85C23×C4×C61C2312yes—
86C2×C4⋊C4×C61—42no—
87C2×C4⋊C4×C61—42no—
88C2×C4⋊C4×C61C2×C418yescomplemented
89C2×C22⋊C4×C61—22no—
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61C2×C414yes—
94C2×C22⋊C4×C61C2×C414yes—
95#95×C61C2×C418yescomplemented
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61C22×C412yes—
101C24×C61C2×D412yes—
102C24×C61C2×Q812yes—
103C24×C61#4112yes—
104C24×C61#4112yes—
105C24×C61#4112yes—
106C24×C61#4112yes—
107C24×C61C2411yescharacteristic
108C22×C4×C61—28no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
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×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—48no—
130C22×C4×C61—44no—
131C22×C4×C61—44no—
132C22×C4×C61—48no—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C22×C414yes—
135C22×C4×C61C22×C414yes—
136C22×C4×C61C22×C412yes—
137C22×Dic61—28no—
138C22×Dic61—28no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
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—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—44no—
171C22×Dic61—44no—
172C22×Dic61—48no—
173C22×Dic61—44no—
174C22×Dic61—44no—
175C22×Dic61—48no—
176C22×Dic61—48no—
177C22×Dic61—48no—
178C22×Dic61C2×D412yes—
179C22×Dic61C2×D412yes—
180C22×Dic61C2×Q814yes—
181C4⋊C4×C61—48no—
182C22⋊C4×C61—28no—
183C22⋊C4×C61—28no—
184C22⋊Dic61—44no—
185C22⋊Dic61—44no—
186C4⋊Dic61—44no—
187C4⋊Dic61—44no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—24no—
195C23×C61—24no—
196C23×C61—24no—
197C23×C61—24no—
198C23×C61—24no—
199C23×C61—24no—
200C23×C61—24no—
201C23×C61—24no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61#11714yes—
205C23×C61C4×D414yes—
206C23×C61C4×Q814yes—
207C23×C61C22⋊Q814yes—
208C23×C61#12314yes—
209C23×C61#12514yes—
210C23×C61#12614yes—
211C23×C61C23×C411yescharacteristic
212C23×C61C22×D411yescharacteristic
213C23×C61C22×Q811yescharacteristic
214C23×C61C2×#4111yescharacteristic
215C23×C61C2×#4111yescharacteristic
216C23×C61C2×#4111yescharacteristic
217C23×C61C2×#4111yescharacteristic
218C23×C61H(5,𝔽2)12yes—
219C23×C61H(5,𝔽2)12yes—
220C23×C61H(5,𝔽2)12yes—
221C23×C612−1+412yes—
222C2×C4×C61—24no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—416no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×Dic61—24no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—416no—
257C2×Dic61—416no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282C2×Dic61—42no—
283#14265—611nomaximal, Sylow
284C22×C61—28no—
285C22×C61—24no—
286C22×C61—24no—
287C22×C61—28no—
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—24no—
291C22×C61—24no—
292C22×C61—28no—
293C22×C61—28no—
294C22×C61#38718yes—
295C22×C61C2×#11711yescharacteristic
296C22×C61C2×C4×D411yescharacteristic
297C22×C61C2×C4×Q811yescharacteristic
298C22×C61#51812yes—
299C22×C61#52012yes—
300C22×C61C2×C22⋊Q811yescharacteristic
301C22×C61C2×#12311yescharacteristic
302C22×C61C2×#12511yescharacteristic
303C22×C61C2×#12611yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61#53411yescharacteristic
306C22×C61#53511yescharacteristic
307C22×C61#53611yescharacteristic
308C22×C61#53812yes—
309C22×C61#53812yes—
310C22×C61#53812yes—
311C22×C61#53812yes—
312C22×C61#53912yes—
313C22×C61#53912yes—
314C22×C61#53912yes—
315C22×C61#53912yes—
316C22×C61#54312yes—
317C22×C61#54412yes—
318C4×C61—44no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—44no—
325Dic61—81no—
326Dic61—81no—
327Dic61—82no—
328C2×#385—611no—
329C2×#385D6111yescharacteristic
330C22×#95—611no—
331#1445—612no—
332#1443—612no—
333#1442—612no—
334#1441—612no—
335C22⋊#95—611no—
336C22⋊#95—611no—
337C22⋊#95—611no—
338C22⋊#95—611no—
339C2×C61—28no—
340C2×C61C2×#38711yescharacteristic
341C2×C61#231812yes—
342C2×C61#232111yescharacteristic
343C2×C61#232211yescharacteristic
344C2×C61#232912yes—
345C2×C61#262211yescharacteristic
346C2×C61#262311yescharacteristic
347C2×C61#262611yescharacteristic
348C2×C61#262711yescharacteristic
349C2×C61#264312yes—
350C2×C61#264612yes—
351C24×C4—611no—
352C24×C4—611no—
353C24×C4C2×D6111yescharacteristic
354C22×C4⋊C4—611no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4C2×D6112yes—
357C22×C22⋊C4—611no—
358C22×C22⋊C4—611no—
359C22×C22⋊C4C2×D6111yescharacteristic
360C22×C22⋊C4C2×D6111yescharacteristic
361#385—1224no—
362#385Dic6118yescomplemented
363C2×#95—612no—
364C2×#95—612no—
365C2×#95—611no—
366C2×#95—611no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—612no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95—1224no—
376C2×#95C2×D6112yes—
377C61#1426511yescharacteristic, complemented, Sylow
378C25C22×D6111yescharacteristic
379C23×C4—22no—
380C23×C4—22no—
381C23×C4—42no—
382C23×C4—42no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—612no—
386C23×C4—614no—
387C23×C4—614no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—612no—
391C23×C4—1222no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4—1222no—
395C23×C4—2442no—
396C23×C4—2442no—
397C23×C4—2442no—
398C23×C4—2442no—
399C23×C4C2×Dic6114yes—
400C23×C4C22×D6111yescharacteristic
401C23×C4C22×D6111yescharacteristic
402C23×C4C22×D6112yes—
403C23×C4C22×D6112yes—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—42no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—2442no—
409C2×C4⋊C4—2442no—
410C2×C4⋊C4C2×Dic6118yescomplemented
411C2×C22⋊C4—22no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—22no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4—1222no—
417C2×C22⋊C4—1222no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4C2×Dic6114yes—
422C2×C22⋊C4C2×Dic6114yes—
423#95—1224no—
424#95—1224no—
425#95—1222no—
426#95—1222no—
427#95—2442no—
428#95—2442no—
429#95—2442no—
430#95—2442no—
431#95—2442no—
432#95—2442no—
433#95—2442no—
434#95—2442no—
435#95C2×Dic6118yescomplemented
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24#1175822212yes—
441C24#1175822212yes—
442C24D4×D6112yes—
443C24#1175822412yes—
444C24Q8×D6112yes—
445C24#1175822612yes—
446C24C22×Dic6112yes—
447C24C23×D6111yescharacteristic
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—614no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1228no—
491C22×C4—1228no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—2444no—
510C22×C4—2444no—
511C22×C4—2448no—
512C22×C4—2444no—
513C22×C4—2444no—
514C22×C4—2448no—
515C22×C4—2448no—
516C22×C4—2448no—
517C22×C4C22×Dic6112yes—
518C22×C4C22×Dic6114yes—
519C22×C4C22×Dic6114yes—
520C22×C4C22×Dic6112yes—
521C4⋊C4—48no—
522C4⋊C4—2444no—
523C4⋊C4—2444no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—2444no—
527C22⋊C4—2444no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23#1451973614yes—
545C23D4×Dic6114yes—
546C23Q8×Dic6114yes—
547C23#1451975514yes—
548C23#1451976214yes—
549C23#1451976714yes—
550C23#1451977214yes—
551C23C23×Dic6111yescharacteristic
552C23C2×D4×D6111yescharacteristic
553C23C2×Q8×D6111yescharacteristic
554C23C2×#1175822411yescharacteristic
555C23C2×#1175822211yescharacteristic
556C23C2×#1175822211yescharacteristic
557C23C2×#1175822611yescharacteristic
558C23#1451982212yes—
559C23#1451982212yes—
560C23#1451982312yes—
561C23#1451982412yes—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—416no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2442no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—24416no—
614C2×C4—24416no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C22—28no—
624C22—28no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—24no—
631C22—24no—
632C22—28no—
633C22#1683964818yes—
634C22C2×#1451973611yescharacteristic
635C22C2×D4×Dic6111yescharacteristic
636C22C2×Q8×Dic6111yescharacteristic
637C22#1684023712yes—
638C22#1684024512yes—
639C22C2×#1451975511yescharacteristic
640C22C2×#1451976211yescharacteristic
641C22C2×#1451976711yescharacteristic
642C22C2×#1451977211yescharacteristic
643C22#1684032211yescharacteristic
644C22#1684032311yescharacteristic
645C22#1684032711yescharacteristic
646C22#1684033411yescharacteristic
647C22#1684034512yes—
648C22#1684034512yes—
649C22#1684034712yes—
650C22#1684035212yes—
651C22#1684035412yes—
652C22#1684035412yes—
653C22#1684035612yes—
654C22#1684036112yes—
655C22#1684038712yes—
656C22#1684039112yes—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—42no—
661C4—4881no—
662C4—4881no—
663C4—4882no—
664C4—2442no—
665C4—2442no—
666C4—2444no—
667C2—28no—
668C2C2×#1683964811yescharacteristic
669C2#2144845212yes—
670C2#2144848011yescharacteristic
671C2#2144849111yescharacteristic
672C2#2144855412yes—
673C2#2145159711yescharacteristic
674C2#2145160511yescharacteristic
675C2#2145163011yescharacteristic
676C2#2145163811yescharacteristic
677C2#2145177512yes—
678C2#2145179112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#385C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×#385D61
hypocenterC61#14265
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. #21444087C2semidirect
4. #21444081C2semidirect
5. #21444076C2semidirect
6. #21444072C2semidirect
36. C61×#385C4semidirect
88. C2×C4⋊C4×C61C2×C4semidirect
95. #95×C61C2×C4semidirect
362. #385Dic61semidirect
377. C61#14265semidirect
410. C2×C4⋊C4C2×Dic61semidirect
435. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396372
C22×#145196842
#214440872
#214440812
#214440762
#214440722
#214440422
#214440412
C2×C61×#3852
#1426561