Finite Group Databasebeta

← #30620527

Subgroups — #30620527

Order 15616.

1212 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840779C211yesmaximal, characteristic, complemented
2C2×#16840732C211yesmaximal, characteristic
3#21461892C212yesmaximal, complemented
4#21461888C212yesmaximal, complemented
5#21461766C212yesmaximal, complemented
6#21461710C211yesmaximal, characteristic, complemented
7#21461673C211yesmaximal, characteristic, complemented
8#21461641C211yesmaximal, characteristic
9#21461561C212yesmaximal, complemented
10#21461545C211yesmaximal, characteristic, complemented
11C2×C22⋊C4×Dic61C211yesmaximal, characteristic
12C22×#11758217C2212yes—
13#16840779—24no—
14C2×#14519844C2212yescomplemented
15C2×C4⋊#92420C2212yescomplemented
16#16840732—24no—
17C2×C22⋊#92420C2212yes—
18C2×C22⋊#92420C2211yescharacteristic
19C2×C22⋊#92420C2212yes—
20C2×C22⋊#92420C2211yescharacteristic
21C2×C22⋊#92420C2211yescharacteristic
22C2×C22⋊#92420C2212yes—
23C2×C22⋊#92420C2211yescharacteristic
24C2×#14519832C2212yescomplemented
25C2×C4×#92420C2212yescomplemented
26#16840628C2212yescomplemented
27#16840628C2212yescomplemented
28#16840628C2212yescomplemented
29#16840628C2212yescomplemented
30C24×Dic61C2211yescharacteristic
31C22×C22⋊Dic61C411yescharacteristic
32C22×C4×Dic61C2211yescharacteristic
33C22×C4×Dic61C2211yescharacteristic
34C22⋊C4×Dic61C411yescharacteristic
35C22⋊C4×Dic61C411yescharacteristic
36C22⋊C4×Dic61C411yescharacteristic
37C22⋊C4×Dic61C411yescharacteristic
38C22⋊C4×Dic61C2211yescharacteristic
39C22⋊C4×Dic61C2211yescharacteristic
40C22⋊C4×Dic61C2211yescharacteristic
41C22⋊C4×Dic61C2211yescharacteristic
42C2×#14519684C411yescharacteristic
43C2×#14519684C411yescharacteristic
44C22×C22⋊C4×C61C411yescharacteristic
45C2×#11758217—24no—
46C2×#11758217—24no—
47C2×#11758217—42no—
48C22×#92420C2312yes—
49C22×#92420C2312yes—
50C22×#92420C2312yes—
51C22×#92420C2312yes—
52#14519844—24no—
53C4⋊#92420—42no—
54C22⋊#92420—22no—
55C22⋊#92420—22no—
56C22⋊#92420—24no—
57C22⋊#92420—22no—
58C22⋊#92420—24no—
59C22⋊#92420—42no—
60C22⋊#92420D414yes—
61#14519832—42no—
62C4×#92420—24no—
63C23×Dic61—22no—
64C23×Dic61—24no—
65C23×Dic61—22no—
66C23×Dic61—42no—
67C23×Dic61C2×C411yescharacteristic
68C23×Dic61C2×C411yescharacteristic
69C23×Dic61C2×C411yescharacteristic
70C23×Dic61C2×C411yescharacteristic
71C23×Dic61C2×C411yescharacteristic
72C23×Dic61C2311yescharacteristic
73C23×Dic61C2311yescharacteristic
74C23×Dic61C2311yescharacteristic
75C2×C22⋊Dic61—24no—
76C2×C22⋊Dic61C2×C411yescharacteristic
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×C22⋊Dic61C2×C411yescharacteristic
80C2×C4×Dic61—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—42no—
83C2×C4×Dic61C2×C411yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519684C2×C411yescharacteristic
92#14519684C2×C411yescharacteristic
93#14519684C2×C411yescharacteristic
94#14519684C2×C411yescharacteristic
95#14519684C2×C411yescharacteristic
96#14519684C2×C411yescharacteristic
97#14519684C2×C411yescharacteristic
98#14519684C2×C411yescharacteristic
99C25×C61C2×C411yescharacteristic
100C23×C4×C61C2×C411yescharacteristic
101C23×C4×C61C2×C411yescharacteristic
102C2×C22⋊C4×C61—24no—
103C2×C22⋊C4×C61C2×C411yescharacteristic
104C2×C22⋊C4×C61C2×C411yescharacteristic
105C2×C22⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2×C411yescharacteristic
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61—41no—
113C24×C61—41no—
114C24×C61C22×C411yescharacteristic
115C24×C61C22×C411yescharacteristic
116C24×C61C22×C411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—21no—
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—21no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61—41no—
149C22×Dic61—41no—
150C22×Dic61—48no—
151C22×Dic61—48no—
152C22×Dic61M1614yescomplemented
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C411yescharacteristic
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C2×D412yes—
161C22×Dic61#4114yes—
162C22×Dic61#4112yes—
163C22×Dic61C2411yescharacteristic
164#11758217—48no—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420—22no—
176C2×#92420C2×D414yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61M1614yescomplemented
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—22no—
192C4×Dic61—22no—
193C4×Dic61—41no—
194C4×Dic61—41no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—28no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—44no—
213C23×C61—44no—
214C23×C61—44no—
215C23×C61—44no—
216C23×C61—44no—
217C23×C61—44no—
218C23×C61#11712yes—
219C23×C61C4×D412yes—
220C23×C61C2×M1612yes—
221C23×C61C2×M1612yes—
222C23×C61#13112yes—
223C23×C61C23×C411yescharacteristic
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—44no—
235C2×C4×C61—44no—
236C2×C4×C61C2×M1612yes—
237C2×C4×C61C2×M1612yes—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—28no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262C2×Dic61—44no—
263C2×Dic61—44no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61#11712yes—
267C2×Dic61C4×D412yes—
268C2×Dic61C2×M1612yes—
269C2×Dic61C2×M1612yes—
270C2×Dic61C22×D411yescharacteristic
271C2×Dic61C2×#4112yes—
272C2×Dic61C2×#4111yescharacteristic
273C2×Dic61H(5,𝔽2)12yes—
274#92420—24no—
275#92420—42no—
276#92420—42no—
277#92420—42no—
278#22553—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—416no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61#43514yes—
301C22×C61C2×#11711yescharacteristic
302C22×C61C2×C4×D411yescharacteristic
303C22×C61#51811yescharacteristic
304C22×C61C22×M1611yescharacteristic
305C22×C61C22×M1611yescharacteristic
306C22×C61C2×#13111yescharacteristic
307C22×C61#56811yescharacteristic
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312Dic61—22no—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61#54612yes—
320Dic61#55112yes—
321C2×#431—611no—
322C2×#407—611no—
323C2×C4×C22⋊C4—611no—
324#1924—612no—
325#1923—612no—
326#1851—612no—
327#1817—611no—
328#1797—611no—
329#1779—611no—
330#1753—612no—
331#1749—611no—
332C2×C61—24no—
333C2×C61—42no—
334C2×C61—42no—
335C2×C61#234211yescharacteristic
336C2×C61#292011yescharacteristic
337C2×C61#292411yescharacteristic
338C2×C61C2×#43511yescharacteristic
339C2×C61D4×M1611yescharacteristic
340C2×C61#297411yescharacteristic
341C2×C61#297511yescharacteristic
342C24×C4—611no—
343C22×M16—612no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4#692311yescharacteristic
346C22×C42—611no—
347C22×C42—611no—
348#431—1224no—
349C2×C4⋊C8—612no—
350C2×C4⋊C8—612no—
351#407—1224no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—611no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—612no—
358C2×C22⋊C8—612no—
359C2×#97—612no—
360C2×C4×C8—612no—
361C4×C22⋊C4—611no—
362C4×C22⋊C4—611no—
363C4×C22⋊C4—611no—
364C4×C22⋊C4—611no—
365C4×C22⋊C4—611no—
366C4×C22⋊C4—611no—
367C4×C22⋊C4—611no—
368C4×C22⋊C4—611no—
369C2×#95—611no—
370C2×#95—611no—
371#336—612no—
372#336—612no—
373#336—612no—
374#336—612no—
375C61#2255311yescharacteristic, complemented, Sylow
376C25C2×#692311yescharacteristic
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—1224no—
386C23×C4—1222no—
387C23×C4—1222no—
388C23×C4—2442no—
389C23×C4C2×#692311yescharacteristic
390C23×C4C2×#692311yescharacteristic
391C2×M16—1224no—
392C2×M16—1224no—
393C2×M16—2442no—
394C22×C8—612no—
395C22×C8—612no—
396C22×C8—612no—
397C22×C8—612no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4C2×#692311yescharacteristic
405C2×C22⋊C4C2×#692311yescharacteristic
406C2×C22⋊C4C2×#692311yescharacteristic
407C2×C22⋊C4C2×#692311yescharacteristic
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—1222no—
417C2×C42—1222no—
418C2×C42—2442no—
419C4⋊C8—1224no—
420C4⋊C8—2442no—
421C22⋊C8—614no—
422C22⋊C8—1224no—
423C22⋊C8—1224no—
424C22⋊C8—1222no—
425C22⋊C8—1222no—
426C22⋊C8—1222no—
427C22⋊C8—2442no—
428#97—2442no—
429C4×C8—1224no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95—611no—
437#95—611no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—41no—
444C24—41no—
445C24C22×#692311yescharacteristic
446C24C22×#692311yescharacteristic
447C24C22×#692311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—611no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
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—2441no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2441no—
488C22×C4—2441no—
489C22×C4—2448no—
490C22×C4—2448no—
491C22×C4C22×#692311yescharacteristic
492C22×C4C22×#692311yescharacteristic
493C22×C4C22×#692311yescharacteristic
494C22×C4C22×#692311yescharacteristic
495M16—2448no—
496C2×C8—614no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—1222no—
507C2×C8—1222no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4#1175821714yescomplemented
516C42—1221no—
517C42—1222no—
518C42—1221no—
519C42—1222no—
520C42—1221no—
521C42—1222no—
522C42—1221no—
523C42—1222no—
524C42—2441no—
525C42—2441no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—28no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23#1451985612yes—
550C23D4×#692312yes—
551C23C2×#1175821712yes—
552C23D61⋊M1612yes—
553C23#1451986412yes—
554C23C23×#692311yescharacteristic
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—612no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1228no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2444no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2444no—
603C2×C4C2×#1175821712yes—
604C2×C4C2×#1175821712yes—
605C8—1224no—
606C8—2442no—
607C8—2442no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—416no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22#1684079414yes—
631C22C2×#1451985611yescharacteristic
632C22C2×D4×#692311yescharacteristic
633C22#1684082911yescharacteristic
634C22C22×#1175821711yescharacteristic
635C22C2×D61⋊M1611yescharacteristic
636C22C2×#1451986411yescharacteristic
637C22#1684084011yescharacteristic
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—612no—
643C4—612no—
644C4—1222no—
645C4—1222no—
646C4—1222no—
647C4—1222no—
648C4—2441no—
649C4—2441no—
650C4—2441no—
651C2—24no—
652C2—42no—
653C2—42no—
654C2#2146232511yescharacteristic
655C2#2146249511yescharacteristic
656C2#2146250811yescharacteristic
657C2C2×#1684079411yescharacteristic
658C2D4×#1175821711yescharacteristic
659C2#2146267511yescharacteristic
660C2#2146268311yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC22×C22⋊C4×C61C4
prime radicalC25×C61C2×C4
cosocleC22×Dic61C24
hypercenterC22×C22⋊C4#6923
hypocenterC61#22553
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840779C2semidirect
3. #21461892C2semidirect
4. #21461888C2semidirect
5. #21461766C2semidirect
6. #21461710C2semidirect
7. #21461673C2semidirect
9. #21461561C2semidirect
10. #21461545C2semidirect
14. C2×#14519844C22semidirect
15. C2×C4⋊#92420C22semidirect
24. C2×#14519832C22semidirect
25. C2×C4×#92420C22semidirect
26. #16840628C22semidirect
27. #16840628C22semidirect
28. #16840628C22semidirect
29. #16840628C22semidirect
152. C22×Dic61M16semidirect
180. C22⋊C4×C61M16semidirect
375. C61#22553semidirect
515. C22⋊C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407792
C2×#168407322
#214618922
#214618882
#214617662
#214617102
#214616732
#214616412
#214615612
#214615452
C2×C22⋊C4×Dic612
#2255361