Finite Group Databasebeta

← #29602190

Subgroups — #29602190

Order 15616.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839622C211yesmaximal, characteristic
2C2×#16839608C211yesmaximal, characteristic, complemented
3#21444076C212yesmaximal, complemented
4#21444068C212yesmaximal
5#21444056C212yesmaximal
6#21444053C212yesmaximal, complemented
7#21444046C211yesmaximal, characteristic, complemented
8#21444046C211yesmaximal, characteristic, complemented
9#21444046C211yesmaximal, characteristic, complemented
10#21444042C211yesmaximal, characteristic
11C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
12C22×C4⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C4×Dic61C2212yes—
16#16839622—24no—
17#16839608—24no—
18C2×#14519684C2211yescharacteristic
19C2×#14519684C2211yescharacteristic
20C2×#14519684C2211yescharacteristic
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C24×C4×C61C2211yescharacteristic
31C22×C22⋊C4×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C4×C22⋊C4×C61C418yescomplemented
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C2×C4⋊Dic61—24no—
43C2×C4⋊Dic61—24no—
44C2×C4⋊Dic61—42no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61—42no—
51C2×C4×Dic61—24no—
52C2×C4×Dic61—24no—
53C2×C4×Dic61—24no—
54#14519684—22no—
55#14519684—22no—
56#14519684—24no—
57#14519684—24no—
58#14519684—22no—
59#14519684—24no—
60#14519684—24no—
61#14519684—42no—
62#14519684—42no—
63#14519684—42no—
64#14519684—42no—
65#14519684D414yes—
66C25×C61C2311yescharacteristic
67C23×C4×C61—22no—
68C23×C4×C61—24no—
69C23×C4×C61—22no—
70C23×C4×C61—42no—
71C23×C4×C61C2×C414yes—
72C23×C4×C61C2311yescharacteristic
73C23×C4×C61C2311yescharacteristic
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2311yescharacteristic
77C23×C4×C61C2311yescharacteristic
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—24no—
81C2×C22⋊C4×C61C2×C414yes—
82C2×C22⋊C4×C61C2×C414yes—
83C2×C42×C61—22no—
84C2×C42×C61—22no—
85C2×C42×C61—42no—
86C2×C42×C61C2×C414yes—
87C2×C42×C61C2×C414yes—
88#95×C61C2×C414yes—
89#95×C61C2×C414yes—
90C24×C61—21no—
91C24×C61—22no—
92C24×C61—22no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61—21no—
97C24×C61—21no—
98C24×C61C22×C412yes—
99C24×C61#4112yes—
100C24×C61#4112yes—
101C24×C61C2411yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—28no—
125C22×C4×C61—28no—
126C22×C4×C61—44no—
127C22×C4×C61—48no—
128C22×C4×C61—48no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61#4114yes—
137C22×C4×C61#4114yes—
138C22×C4×C61#4112yes—
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—22no—
154C22×Dic61—22no—
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×Dic61C2×D414yes—
166C22×Dic61#4114yes—
167C22⋊C4×C61—28no—
168C22⋊C4×C61—28no—
169C42×C61—28no—
170C42×C61—44no—
171C22⋊Dic61—48no—
172C22⋊Dic61—48no—
173C4⋊Dic61—48no—
174C4×Dic61—216no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—22no—
184C23×C61—22no—
185C23×C61—24no—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—24no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61#11714yes—
198C23×C61C23×C411yescharacteristic
199C23×C61C22×D411yescharacteristic
200C23×C61C2×#4112yes—
201C23×C61C2×#4112yes—
202C23×C61C2×#4111yescharacteristic
203C23×C61C2×#4111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C23×C61C2×#4112yes—
206C23×C61H(5,𝔽2)12yes—
207C23×C61H(5,𝔽2)11yescharacteristic
208C23×C612−1+411yescharacteristic
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—416no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61#11714yes—
240C2×C4×C61#11714yes—
241C2×C4×C61C4×D414yes—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
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—
257C2×Dic61—42no—
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—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270#13898—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—44no—
278C22×C61—44no—
279C22×C61—44no—
280C22×C61—44no—
281C22×C61—42no—
282C22×C61C2×#11711yescharacteristic
283C22×C61C2×#11711yescharacteristic
284C22×C61C2×#11711yescharacteristic
285C22×C61C2×C4×D411yescharacteristic
286C22×C61C4×#4111yescharacteristic
287C22×C61#51811yescharacteristic
288C22×C61#52011yescharacteristic
289C22×C61#52512yes—
290C22×C61#52912yes—
291C22×C61#52912yes—
292C22×C61#52912yes—
293C22×C61#54012yes—
294C22×C61#54212yes—
295C22×C61#54612yes—
296C22×C61#54611yescharacteristic
297C22×C61#54811yescharacteristic
298C22×C61#55112yes—
299C22×C61#55112yes—
300C22×C61#55212yes—
301C22×C61#55211yescharacteristic
302C22×C61#55311yescharacteristic
303C22×C61#55312yes—
304C22×C61#55512yes—
305C4×C61—24no—
306C4×C61—24no—
307C4×C61—24no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—42no—
311Dic61—44no—
312Dic61—44no—
313Dic61—82no—
314Dic61—82no—
315C2×#380—611no—
316C2×C4×C22⋊C4D6111yescharacteristic
317C2×#376—611no—
318#1442—612no—
319#1440—612no—
320#1437—612no—
321#1436—612no—
322#1434—611no—
323#1434—611no—
324#1434—611no—
325C22⋊#95—611no—
326C2×C61—44no—
327C2×C61#228311yescharacteristic
328C2×C61#229011yescharacteristic
329C2×C61#231911yescharacteristic
330C2×C61#234211yescharacteristic
331C2×C61#234611yescharacteristic
332C2×C61#235211yescharacteristic
333C2×C61#235311yescharacteristic
334C2×C61#247612yes—
335C2×C61#247712yes—
336C2×C61#253012yes—
337C2×C61#253112yes—
338C24×C4C2×D6111yescharacteristic
339C22×C4⋊C4—612no—
340C22×C22⋊C4—612no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4C2×D6111yescharacteristic
343C22×C22⋊C4C2×D6111yescharacteristic
344C22×C42—612no—
345C22×C42C2×D6111yescharacteristic
346C22×C42C2×D6111yescharacteristic
347#380—1224no—
348C4×C22⋊C4Dic6118yescomplemented
349#376—1224no—
350C2×#95—611no—
351C2×#95—611no—
352C2×#95—611no—
353C2×#95—612no—
354C2×#95—611no—
355C2×#95—612no—
356C2×#95—612no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C2×#95C2×D6111yescharacteristic
363C2×#95C2×D6111yescharacteristic
364C61#1389811yescharacteristic, complemented, Sylow
365C25C22×D6111yescharacteristic
366C23×C4—24no—
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—612no—
371C23×C4—612no—
372C23×C4—612no—
373C23×C4—612no—
374C23×C4C2×Dic6114yes—
375C23×C4C22×D6111yescharacteristic
376C23×C4C22×D6111yescharacteristic
377C23×C4C22×D6111yescharacteristic
378C23×C4C22×D6111yescharacteristic
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4—2442no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—1224no—
390C2×C22⋊C4—1224no—
391C2×C22⋊C4—2442no—
392C2×C22⋊C4—2442no—
393C2×C22⋊C4C2×Dic6114yes—
394C2×C22⋊C4C2×Dic6114yes—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—42no—
398C2×C42—1224no—
399C2×C42—1224no—
400C2×C42—1224no—
401C2×C42C2×Dic6114yes—
402C2×C42C2×Dic6114yes—
403#95—614no—
404#95—1224no—
405#95—1224no—
406#95—1224no—
407#95—1224no—
408#95—1222no—
409#95—1222no—
410#95—1222no—
411#95—2442no—
412#95—2442no—
413#95—2442no—
414#95—2442no—
415#95C2×Dic6114yes—
416#95C2×Dic6114yes—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—21no—
424C24—21no—
425C24#1175822212yes—
426C24#1175822212yes—
427C24C22×Dic6112yes—
428C24C23×D6111yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—614no—
457C22×C4—614no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4#1175822214yes—
485C22×C4#1175822214yes—
486C22×C4D4×D6112yes—
487C22×C4#1175822412yes—
488C22×C4C22×Dic6112yes—
489C22×C4C22×Dic6112yes—
490C22×C4C22×Dic6112yes—
491C22×C4C22×Dic6112yes—
492C22×C4C22×Dic6112yes—
493C22×C4C22×Dic6112yes—
494C4⋊C4—2448no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—2448no—
498C22⋊C4—2448no—
499C42—28no—
500C42—44no—
501C42—12216no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#1451973614yes—
525C23C23×Dic6111yescharacteristic
526C23C2×D4×D6111yescharacteristic
527C23#41×D6112yes—
528C23C2×#1175822411yescharacteristic
529C23C2×#1175822212yes—
530C23C2×#1175822211yescharacteristic
531C23C2×#1175822211yescharacteristic
532C23C2×#1175822212yes—
533C23#1451982212yes—
534C23#1451982311yescharacteristic
535C23#1451982411yescharacteristic
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—1224no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1224no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4#1451973614yes—
595C2×C4#1451973614yes—
596C2×C4D4×Dic6114yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22C2×#1451973611yescharacteristic
609C22C2×#1451973611yescharacteristic
610C22C2×#1451973611yescharacteristic
611C22C2×D4×Dic6111yescharacteristic
612C22#41×Dic6111yescharacteristic
613C22#1684023711yescharacteristic
614C22#1684024511yescharacteristic
615C22#1684027112yes—
616C22#1684029212yes—
617C22#1684029612yes—
618C22#1684029612yes—
619C22#1684036312yes—
620C22#1684037712yes—
621C22#1684040212yes—
622C22#1684040811yescharacteristic
623C22#1684042111yescharacteristic
624C22#1684043812yes—
625C22#1684044112yes—
626C22#1684045112yes—
627C22#1684045211yescharacteristic
628C22#1684045811yescharacteristic
629C22#1684046112yes—
630C22#1684047512yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—4882no—
638C4—4882no—
639C4—2444no—
640C4—2444no—
641C2—44no—
642C2#2144821711yescharacteristic
643C2#2144826411yescharacteristic
644C2#2144846211yescharacteristic
645C2#2144866811yescharacteristic
646C2#2144870411yescharacteristic
647C2#2144874811yescharacteristic
648C2#2144875911yescharacteristic
649C2#2145000512yes—
650C2#2145002012yes—
651C2#2145065112yes—
652C2#2145066612yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13898
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839608C2semidirect
3. #21444076C2semidirect
6. #21444053C2semidirect
7. #21444046C2semidirect
8. #21444046C2semidirect
9. #21444046C2semidirect
35. C4×C22⋊C4×C61C4semidirect
348. C4×C22⋊C4Dic61semidirect
364. C61#13898semidirect

Maximal subgroups

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

subgroupindex
C2×#168396222
C2×#168396082
#214440762
#214440682
#214440562
#214440532
#214440462
#214440422
C2×C4×C22⋊C4×C612
#1389861