Finite Group Databasebeta

← #29606913

Subgroups — #29606913

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839680C211yesmaximal, characteristic
2C2×#16839664C211yesmaximal, characteristic
3C2×#16839654C212yesmaximal, complemented
4C2×C4×#11758196C211yesmaximal, characteristic, complemented
5#21444084C211yesmaximal, characteristic, complemented
6#21444073C211yesmaximal, characteristic, complemented
7#21444065C212yesmaximal, complemented
8#21444063C212yesmaximal, complemented
9#21444059C211yesmaximal, characteristic
10#21444055C212yesmaximal, complemented
11C61×#1434C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C22×C4⋊Dic61C2212yes—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×#11758196C2211yescharacteristic
16C22×#11758196C2211yescharacteristic
17C22×#11758198C2212yes—
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×#11758198C2212yes—
21C22×#11758198C2212yes—
22C22×#11758198C2212yes—
23C22×C4×Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839680C418yescomplemented
26#16839664—24no—
27#16839654—28no—
28C4×#11758196—24no—
29C2×#14519684C2212yes—
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C22×C42×C61C2211yescharacteristic
35C2×#95×C61C2212yes—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C24×D61C2312yes—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—42no—
43C22×D244—42no—
44C22×D244C2×C414yes—
45C23×Dic61C2312yes—
46C23×Dic61C2311yescharacteristic
47C23×Dic61C2311yescharacteristic
48C23×Dic61C2311yescharacteristic
49C23×Dic61C2311yescharacteristic
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—24no—
52C2×C4⋊Dic61—42no—
53C2×C4⋊Dic61—42no—
54C2×C4⋊Dic61—42no—
55C2×#11758196—22no—
56C2×#11758196—22no—
57C2×#11758196—22no—
58C2×#11758196—22no—
59C2×#11758196—22no—
60C2×#11758196C2×C414yes—
61C2×#11758198—24no—
62C2×#11758198—24no—
63C2×#11758198—24no—
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—24no—
67C2×#11758198—24no—
68C2×#11758198—24no—
69C2×#11758198—42no—
70C2×#11758198—42no—
71C2×#11758198—42no—
72C2×#11758198—42no—
73C2×#11758198—42no—
74C2×#11758198—42no—
75C2×#11758198—42no—
76C2×#11758198—42no—
77C2×#11758198C2×C418yes—
78C2×#11758198C2×C418yes—
79C2×C4×Dic61—22no—
80C2×C4×Dic61—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—42no—
83C2×C4×Dic61D414yescomplemented
84#14519684—22no—
85#14519684—22no—
86#14519684—24no—
87#14519684—42no—
88#14519684—42no—
89C23×C4×C61C2312yes—
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2311yescharacteristic
94C2×C42×C61—42no—
95C2×C42×C61D414yes—
96#95×C61—24no—
97#95×C61—22no—
98#95×C61—42no—
99#95×C61C2×C414yes—
100C24×C61C2411yescharacteristic
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61C22×C414yes—
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—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C22×C414yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61#4112yes—
144C22×C4×C61#4112yes—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
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×Dic61C22×C412yes—
169C22×Dic61C22×C412yes—
170C22×Dic61C2×D412yes—
171C22×Dic61C2×D412yes—
172C22×Dic61#4112yes—
173C22×Dic61#4112yes—
174C2×D244—28no—
175C2×D244—48no—
176C2×D244—44no—
177C2×D244—44no—
178C2×D244—48no—
179C42×C61—44no—
180#11758198—216no—
181#11758198—48no—
182#11758198—48no—
183#11758198—48no—
184#11758198—48no—
185#11758198—48no—
186C4⋊Dic61—48no—
187C4⋊Dic61—44no—
188#11758196—28no—
189#11758196—28no—
190C4×Dic61—28no—
191C4×Dic61—44no—
192C23×C61C23×C411yescharacteristic
193C23×C61C22×D411yescharacteristic
194C23×C61C22×D411yescharacteristic
195C23×C61C2×#4111yescharacteristic
196C23×C61C2×#4111yescharacteristic
197C23×C61C2×#4111yescharacteristic
198C23×C61C2×#4111yescharacteristic
199C23×C61H(5,𝔽2)12yes—
200C23×C61H(5,𝔽2)12yes—
201C23×C61H(5,𝔽2)12yes—
202C23×C612−1+412yes—
203C22×D61—24no—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—42no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—42no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61—42no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—42no—
237C22×D61—42no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—24no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—24no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61C4×D414yes—
266C2×C4×C61#12114yes—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—24no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—24no—
278C2×Dic61—24no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—42no—
289C2×Dic61—42no—
290C2×Dic61—42no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—42no—
295C2×Dic61C4×D414yes—
296D244—416no—
297#14312—611nomaximal, Sylow
298C22×C61C2×C4×D411yescharacteristic
299C22×C61C2×C4×D411yescharacteristic
300C22×C61C4×#4111yescharacteristic
301C22×C61#51812yes—
302C22×C61#52012yes—
303C22×C61C2×#12111yescharacteristic
304C22×C61#52511yescharacteristic
305C22×C61#52911yescharacteristic
306C22×C61#53211yescharacteristic
307C22×C61#53411yescharacteristic
308C22×C61#53512yes—
309C22×C61#53511yescharacteristic
310C22×C61#53511yescharacteristic
311C22×C61#53611yescharacteristic
312C22×C61#53712yes—
313C22×C61#53812yes—
314C22×C61#53812yes—
315C22×C61#53912yes—
316C22×C61#54012yes—
317C22×C61#54012yes—
318C22×C61#54212yes—
319C22×C61#54212yes—
320C22×C61#54212yes—
321C2×D61—44no—
322C2×D61—44no—
323C2×D61—44no—
324C2×D61—44no—
325C2×D61—44no—
326C2×D61—44no—
327C2×D61—44no—
328C2×D61—82no—
329C2×D61—82no—
330C2×D61—82no—
331C2×D61—82no—
332C2×D61—82no—
333C2×D61—82no—
334C2×D61—82no—
335C2×D61—82no—
336C4×C61—24no—
337C4×C61—24no—
338C4×C61—44no—
339C4×C61—42no—
340C4×C61—82no—
341Dic61—24no—
342Dic61—42no—
343Dic61—42no—
344Dic61—42no—
345Dic61—82no—
346C2×#394—611no—
347C2×#390—611no—
348C2×#388—612no—
349C2×C4×C4⋊C4—611no—
350#1444—611no—
351#1441—611no—
352#1439—612no—
353#1439—612no—
354#1438—611no—
355#1437—612no—
356#1434D6111yescharacteristic, complemented
357C2×C61C4×#12111yescharacteristic
358C2×C61#232111yescharacteristic
359C2×C61#232412yes—
360C2×C61#232611yescharacteristic
361C2×C61#233512yes—
362C2×C61#261811yescharacteristic
363C2×C61#261911yescharacteristic
364C2×C61#263211yescharacteristic
365C2×C61#263511yescharacteristic
366C2×C61#265112yes—
367C2×C61#266212yes—
368D61—84no—
369C23×D4—611no—
370C22×C4⋊C4—611no—
371C22×C4⋊C4—611no—
372C22×C4⋊C4—612no—
373C22×C4⋊C4—611no—
374C22×C22⋊C4—612no—
375C22×C22⋊C4—612no—
376C22×C22⋊C4—612no—
377C22×C22⋊C4—612no—
378C22×C22⋊C4—612no—
379C22×C22⋊C4—612no—
380C22×C42—611no—
381C22×C42—611no—
382C22×C42C2×D6111yescharacteristic
383#394—618no—
384#390—1224no—
385#388—1228no—
386C4×C4⋊C4—1224no—
387C2×#95—611no—
388C2×#95—611no—
389C2×#95—612no—
390C2×#95—612no—
391C2×#95—612no—
392C2×#95C2×D6112yes—
393C2×#95C2×D6111yescharacteristic
394C2×#95C2×D6112yes—
395C2×#95C2×D6111yescharacteristic
396C61#1431211yescharacteristic, complemented, Sylow
397C25—612no—
398C22×D4—614no—
399C22×D4—1222no—
400C22×D4—1222no—
401C22×D4—2442no—
402C22×D4—2442no—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—612no—
408C23×C4C22×D6112yes—
409C23×C4C22×D6112yes—
410C23×C4C22×D6111yescharacteristic
411C23×C4C22×D6111yescharacteristic
412C23×C4C22×D6111yescharacteristic
413C2×C4⋊C4—614no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1224no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—2442no—
422C2×C4⋊C4—2442no—
423C2×C4⋊C4—2442no—
424C2×C22⋊C4—618no—
425C2×C22⋊C4—618no—
426C2×C22⋊C4—1224no—
427C2×C22⋊C4—1224no—
428C2×C22⋊C4—1224no—
429C2×C22⋊C4—1224no—
430C2×C22⋊C4—1224no—
431C2×C22⋊C4—1224no—
432C2×C22⋊C4—1224no—
433C2×C22⋊C4—1224no—
434C2×C22⋊C4—2442no—
435C2×C22⋊C4—2442no—
436C2×C22⋊C4—2442no—
437C2×C22⋊C4—2442no—
438C2×C22⋊C4—2442no—
439C2×C22⋊C4—2442no—
440C2×C22⋊C4—2442no—
441C2×C22⋊C4—2442no—
442C2×C42—42no—
443C2×C42—614no—
444C2×C42—1222no—
445C2×C42—1222no—
446C2×C42—1222no—
447C2×C42—2442no—
448C2×C42D24414yes—
449#95—22no—
450#95—24no—
451#95—42no—
452#95—1224no—
453#95—1222no—
454#95—1222no—
455#95—2442no—
456#95—2442no—
457#95C4×D6114yes—
458C24—614no—
459C24—1222no—
460C24—1222no—
461C24—1222no—
462C24—1222no—
463C24—1222no—
464C24—1222no—
465C24—1222no—
466C24—1222no—
467C24—1222no—
468C24—1222no—
469C24—1222no—
470C24—1222no—
471C24—1222no—
472C24—1222no—
473C24C23×D6111yescharacteristic
474C2×D4—1228no—
475C2×D4—2448no—
476C2×D4—2444no—
477C2×D4—2444no—
478C2×D4—2448no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—612no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4C2×C4×D6114yes—
530C22×C4C2×C4×D6112yes—
531C22×C4C2×D24412yes—
532C22×C4C2×D24412yes—
533C22×C4C2×D24412yes—
534C22×C4#1175822212yes—
535C22×C4D4×D6112yes—
536C22×C4#1175822612yes—
537C4⋊C4—1228no—
538C4⋊C4—1228no—
539C4⋊C4—2444no—
540C4⋊C4—2448no—
541C22⋊C4—12216no—
542C22⋊C4—2448no—
543C22⋊C4—2448no—
544C22⋊C4—2448no—
545C22⋊C4—2448no—
546C22⋊C4—2448no—
547C42—44no—
548C42—1228no—
549C42—2444no—
550C23—1224no—
551C23—1224no—
552C23—1224no—
553C23—1224no—
554C23—1224no—
555C23—1224no—
556C23—1224no—
557C23—2442no—
558C23—2442no—
559C23—2442no—
560C23—2442no—
561C23—2442no—
562C23—2442no—
563C23—2442no—
564C23—2442no—
565C23—2442no—
566C23—2442no—
567C23—2442no—
568C23—2442no—
569C23—2442no—
570C23—2442no—
571C23—2442no—
572C23—2442no—
573C23—2442no—
574C23—2442no—
575C23—2442no—
576C23—2442no—
577C23—2442no—
578C23—2442no—
579C23—2442no—
580C23—2442no—
581C23—2442no—
582C23—2442no—
583C23—2442no—
584C23—2442no—
585C23C22×C4×D6111yescharacteristic
586C23C2×D4×D6111yescharacteristic
587C23C22×D24411yescharacteristic
588C23#41×D6111yescharacteristic
589C23#41×D6111yescharacteristic
590C23C2×#1175822211yescharacteristic
591C23C2×#1175822611yescharacteristic
592C23#1451982212yes—
593C23#1451982312yes—
594C23#1451982312yes—
595C23#1451982412yes—
596D4—24416no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—614no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1224no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1224no—
636C2×C4—1224no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2442no—
651C2×C4—2442no—
652C2×C4—2442no—
653C2×C4C4×D24414yes—
654C2×C4#1451975214yes—
655C22—4882no—
656C22—4882no—
657C22—4882no—
658C22—4882no—
659C22—4882no—
660C22—4882no—
661C22—4882no—
662C22—4882no—
663C22—2444no—
664C22—2444no—
665C22—2444no—
666C22—2444no—
667C22—2444no—
668C22—2444no—
669C22—2444no—
670C22C4×D4×D6111yescharacteristic
671C22C2×C4×D24411yescharacteristic
672C22C4×#1175822611yescharacteristic
673C22#1684023412yes—
674C22#1684024212yes—
675C22C2×#1451975211yescharacteristic
676C22#1684027411yescharacteristic
677C22#1684029911yescharacteristic
678C22#1684030911yescharacteristic
679C22#1684032411yescharacteristic
680C22#1684033012yes—
681C22#1684033111yescharacteristic
682C22#1684033111yescharacteristic
683C22#1684033811yescharacteristic
684C22#1684033912yes—
685C22#1684034512yes—
686C22#1684034812yes—
687C22#1684036112yes—
688C22#1684036412yes—
689C22#1684036812yes—
690C22#1684037812yes—
691C22#1684037812yes—
692C22#1684038312yes—
693C4—24no—
694C4—24no—
695C4—44no—
696C4—42no—
697C4—82no—
698C4—1224no—
699C4—4882no—
700C4—2442no—
701C4—2442no—
702C4—2442no—
703C2—4884no—
704C2C4×#1451975211yescharacteristic
705C2#2144847911yescharacteristic
706C2#2144850712yes—
707C2#2144853011yescharacteristic
708C2#2144860812yes—
709C2#2145156411yescharacteristic
710C2#2145157211yescharacteristic
711C2#2145168211yescharacteristic
712C2#2145170611yescharacteristic
713C2#2145185812yes—
714C2#2145196212yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1434C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenter#1434D61
hypocenterC61#14312
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
3. C2×#16839654C2semidirect
4. C2×C4×#11758196C2semidirect
5. #21444084C2semidirect
6. #21444073C2semidirect
7. #21444065C2semidirect
8. #21444063C2semidirect
10. #21444055C2semidirect
11. C61×#1434C2semidirect
25. #16839680C4semidirect
83. C2×C4×Dic61D4semidirect
356. #1434D61semidirect
396. C61#14312semidirect

Maximal subgroups

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

subgroupindex
C2×#168396802
C2×#168396642
C2×#168396542
C2×C4×#117581962
#214440842
#214440732
#214440652
#214440632
#214440592
#214440552
C61×#14342
#1431261