Finite Group Databasebeta

← #29602115

Subgroups — #29602115

Order 15616.

1746 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839656C212yesmaximal
2C2×#16839622C211yesmaximal, characteristic
3C2×#16839613C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Dic61C211yesmaximal, characteristic
5#21444083C212yesmaximal, complemented
6#21444062C212yesmaximal
7#21444053C212yesmaximal, complemented
8#21444047C211yesmaximal, characteristic, complemented
9#21444044C211yesmaximal, characteristic, complemented
10#21444043C211yesmaximal, characteristic, complemented
11C61×C22⋊#95C211yesmaximal, characteristic
12C22×C4⋊Dic61C2212yes—
13C22×#11758196C2212yes—
14C22×#11758196C2212yes—
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C22⋊Dic61C2211yescharacteristic
18C22×C4×Dic61C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4×Dic61C2211yescharacteristic
21#16839656—28no—
22#16839622—24no—
23#16839613—24no—
24C4×C22⋊Dic61C418yescomplemented
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2211yescharacteristic
34C24×C4×C61C2211yescharacteristic
35C22×C22⋊C4×C61C2212yes—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C2×C4⋊Dic61—24no—
46C2×C4⋊Dic61—24no—
47C2×C4⋊Dic61—42no—
48C2×#11758196—24no—
49C2×#11758196—24no—
50C2×#11758196—24no—
51C2×#11758196—24no—
52C2×#11758196—24no—
53C2×#11758196—42no—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—24no—
57C2×C22⋊Dic61—42no—
58C2×C22⋊Dic61C2×C414yes—
59C2×C22⋊Dic61C2×C414yes—
60C2×C22⋊Dic61D414yes—
61C2×C22⋊Dic61Q814yes—
62C2×C4×Dic61—22no—
63C2×C4×Dic61—22no—
64C2×C4×Dic61—24no—
65C2×C4×Dic61—24no—
66C2×C4×Dic61—42no—
67C2×C4×Dic61—42no—
68C2×C4×Dic61C2×C414yes—
69C2×C4×Dic61C2×C414yes—
70#14519684—22no—
71#14519684—24no—
72#14519684—24no—
73#14519684—24no—
74#14519684—42no—
75#14519684—42no—
76#14519684C2×C414yes—
77#14519684C2×C414yes—
78#14519684D414yes—
79C25×C61C2311yescharacteristic
80C23×C4×C61—22no—
81C23×C4×C61—24no—
82C23×C4×C61—22no—
83C23×C4×C61—42no—
84C23×C4×C61C2×C414yes—
85C23×C4×C61C2312yes—
86C23×C4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—24no—
90C2×C22⋊C4×C61—24no—
91C2×C22⋊C4×C61—42no—
92#95×C61—22no—
93#95×C61—24no—
94#95×C61—22no—
95C24×C61—21no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—21no—
99C24×C61—22no—
100C24×C61—22no—
101C24×C61—21no—
102C24×C61—21no—
103C24×C61C22×C412yes—
104C24×C61C2×D412yes—
105C24×C61C2×Q812yes—
106C24×C61C2411yescharacteristic
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—28no—
119C22×C4×C61—28no—
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—44no—
128C22×C4×C61—48no—
129C22×C4×C61—48no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61#4114yes—
134C22×C4×C61#4114yes—
135C22×C4×C61#4112yes—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
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×Dic61C22×C412yes—
163C22×Dic61C22×C412yes—
164C22×Dic61C22×C412yes—
165C22×Dic61C22×C412yes—
166C22×Dic61C2×D414yes—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61C2×Q812yes—
170C22×Dic61C2×Q812yes—
171C22×Dic61#4114yes—
172C22⋊C4×C61—48no—
173C22⋊Dic61—28no—
174C22⋊Dic61—48no—
175C22⋊Dic61C4⋊C4116yescomplemented
176C4⋊Dic61—48no—
177#11758196—216no—
178#11758196—48no—
179C4×Dic61—28no—
180C4×Dic61—44no—
181C4×Dic61—48no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—22no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—24no—
190C23×C61—22no—
191C23×C61—24no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—24no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61C2×C4⋊C414yes—
205C23×C61C23×C411yescharacteristic
206C23×C61C22×D411yescharacteristic
207C23×C61C22×D411yescharacteristic
208C23×C61C22×Q811yescharacteristic
209C23×C61C2×#4112yes—
210C23×C61C2×#4112yes—
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4112yes—
213C23×C61H(5,𝔽2)12yes—
214C23×C61H(5,𝔽2)11yescharacteristic
215C23×C612−1+411yescharacteristic
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—24no—
226C2×C4×C61—24no—
227C2×C4×C61—416no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61#11714yes—
240C2×C4×C61C4×D414yes—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—22no—
250C2×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61C2×C4⋊C414yes—
275C2×Dic61C2×C4⋊C414yes—
276C2×Dic61#11714yes—
277#13891—611nomaximal, Sylow
278C22×C61—24no—
279C22×C61—24no—
280C22×C61—24no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—44no—
285C22×C61—44no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—42no—
289C22×C61C22×C4⋊C411yescharacteristic
290C22×C61C2×#11711yescharacteristic
291C22×C61C2×#11711yescharacteristic
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C4×#4111yescharacteristic
294C22×C61#51811yescharacteristic
295C22×C61#52011yescharacteristic
296C22×C61#52512yes—
297C22×C61#52912yes—
298C22×C61#53212yes—
299C22×C61#53312yes—
300C22×C61#54012yes—
301C22×C61#54212yes—
302C22×C61D4211yescharacteristic
303C22×C61#54612yes—
304C22×C61#54612yes—
305C22×C61#54711yescharacteristic
306C22×C61D4×Q811yescharacteristic
307C22×C61#55112yes—
308C22×C61#55312yes—
309C22×C61#55412yes—
310C22×C61#55411yescharacteristic
311C22×C61#55612yes—
312C4×C61—24no—
313C4×C61—24no—
314C4×C61—44no—
315C4×C61—82no—
316Dic61—24no—
317Dic61—24no—
318Dic61—24no—
319Dic61—42no—
320Dic61—44no—
321Dic61—82no—
322C2×#388—612no—
323C2×#380—611no—
324C2×C4×C4⋊C4—611no—
325C2×C4×C22⋊C4—611no—
326#1444—612no—
327#1439—612no—
328#1436—612no—
329C4⋊#95—611no—
330#1434—611no—
331#1434—611no—
332C22⋊#95D6111yescharacteristic
333C2×C61—44no—
334C2×C61#228511yescharacteristic
335C2×C61#229011yescharacteristic
336C2×C61#231911yescharacteristic
337C2×C61#234211yescharacteristic
338C2×C61#234711yescharacteristic
339C2×C61D4×C4⋊C411yescharacteristic
340C2×C61#234911yescharacteristic
341C2×C61#246412yes—
342C2×C61#246512yes—
343C2×C61#252112yes—
344C2×C61#252212yes—
345C24×C4C2×D6111yescharacteristic
346C22×C4⋊C4—612no—
347C22×C4⋊C4—612no—
348C22×C4⋊C4—612no—
349C22×C22⋊C4—611no—
350C22×C22⋊C4—611no—
351C22×C22⋊C4—612no—
352C22×C22⋊C4C2×D6112yes—
353C22×C42—611no—
354C22×C42—611no—
355C22×C42—612no—
356#388—1228no—
357#380—1224no—
358C4×C4⋊C4—1224no—
359C4×C22⋊C4—618no—
360C2×#95—611no—
361C2×#95—611no—
362C2×#95—611no—
363C2×#95—611no—
364C2×#95—612no—
365C2×#95—612no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C2×#95C2×D6111yescharacteristic
370C2×#95C2×D6111yescharacteristic
371C2×#95C2×D6112yes—
372C61#1389111yescharacteristic, complemented, Sylow
373C25C22×D6111yescharacteristic
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—22no—
377C23×C4—42no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—612no—
382C23×C4—611no—
383C23×C4—612no—
384C23×C4C4×D6114yes—
385C23×C4C22×D6112yes—
386C23×C4C22×D6112yes—
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6111yescharacteristic
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—1224no—
391C2×C4⋊C4—1224no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4—1224no—
395C2×C4⋊C4—1224no—
396C2×C4⋊C4—2442no—
397C2×C4⋊C4—2442no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—42no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—614no—
404C2×C22⋊C4—614no—
405C2×C22⋊C4—1224no—
406C2×C22⋊C4—1224no—
407C2×C22⋊C4—1224no—
408C2×C22⋊C4—2442no—
409C2×C42—614no—
410C2×C42—614no—
411C2×C42—1222no—
412C2×C42—1222no—
413C2×C42—1224no—
414C2×C42—1224no—
415C2×C42—2442no—
416C2×C42—2442no—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—614no—
421#95—614no—
422#95—614no—
423#95—1224no—
424#95—1224no—
425#95—1224no—
426#95—1222no—
427#95—2442no—
428#95—2442no—
429C24—21no—
430C24—22no—
431C24—21no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24C2×C4×D6112yes—
438C24D4×D6112yes—
439C24Q8×D6112yes—
440C24C23×D6111yescharacteristic
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—614no—
472C22×C4—612no—
473C22×C4—614no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4C2×C4×D6112yes—
501C22×C4C2×C4×D6112yes—
502C22×C4#1175822214yes—
503C22×C4D4×D6112yes—
504C22×C4#1175822414yes—
505C22×C4#1175822412yes—
506C4⋊C4—12216no—
507C4⋊C4—2448no—
508C4⋊C4—2448no—
509C22⋊C4—48no—
510C22⋊C4—6116no—
511C22⋊C4—1228no—
512C22⋊C4—2448no—
513C42—1228no—
514C42—2444no—
515C42—2448no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C4⋊C4×D6114yes—
539C23C22×C4×D6111yescharacteristic
540C23C2×D4×D6111yescharacteristic
541C23C2×D4×D6111yescharacteristic
542C23C2×Q8×D6111yescharacteristic
543C23#41×D6112yes—
544C23C2×#1175822412yes—
545C23C2×#1175822411yescharacteristic
546C23C2×#1175822212yes—
547C23#1451982212yes—
548C23#1451982311yescharacteristic
549C23#1451982411yescharacteristic
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—614no—
574C2×C4—614no—
575C2×C4—614no—
576C2×C4—1224no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1222no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1224no—
594C2×C4—1224no—
595C2×C4—1224no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4#1451973314yes—
610C2×C4#1451973714yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22C2×C4⋊C4×D6111yescharacteristic
623C22#117×D6111yescharacteristic
624C22C2×#1451973311yescharacteristic
625C22C2×#1451973711yescharacteristic
626C22C4×#1175822211yescharacteristic
627C22#1684023511yescharacteristic
628C22#1684024211yescharacteristic
629C22#1684027112yes—
630C22#1684029412yes—
631C22#1684030712yes—
632C22#1684031412yes—
633C22#1684036412yes—
634C22#1684038012yes—
635C22#1684039611yescharacteristic
636C22#1684039912yes—
637C22#1684040212yes—
638C22#1684041211yescharacteristic
639C22#1684042511yescharacteristic
640C22#1684044312yes—
641C22#1684045812yes—
642C22#1684046712yes—
643C22#1684047011yescharacteristic
644C22#1684048212yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—82no—
649C4—1224no—
650C4—1224no—
651C4—1224no—
652C4—4882no—
653C4—2442no—
654C4—2444no—
655C2—44no—
656C2#2144822111yescharacteristic
657C2#2144826511yescharacteristic
658C2#2144845911yescharacteristic
659C2#2144867011yescharacteristic
660C2#2144871011yescharacteristic
661C2#2144871511yescharacteristic
662C2#2144872611yescharacteristic
663C2#2144982912yes—
664C2#2144984412yes—
665C2#2145052612yes—
666C2#2145054112yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C22⋊#95C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22⋊#95D61
hypocenterC61#13891
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16839613C2semidirect
5. #21444083C2semidirect
7. #21444053C2semidirect
8. #21444047C2semidirect
9. #21444044C2semidirect
10. #21444043C2semidirect
24. C4×C22⋊Dic61C4semidirect
175. C22⋊Dic61C4⋊C4semidirect
372. C61#13891semidirect

Maximal subgroups

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

subgroupindex
C2×#168396562
C2×#168396222
C2×#168396132
C2×C4×C22⋊Dic612
#214440832
#214440622
#214440532
#214440472
#214440442
#214440432
C61×C22⋊#952
#1389161