Finite Group Databasebeta

← #30620523

Subgroups — #30620523

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×#16840778C211yesmaximal, characteristic, complemented
2C2×#16840732C211yesmaximal, characteristic
3#21461893C212yesmaximal, complemented
4#21461889C212yesmaximal, complemented
5#21461766C212yesmaximal, complemented
6#21461709C211yesmaximal, characteristic, complemented
7#21461675C211yesmaximal, characteristic, complemented
8#21461641C211yesmaximal, characteristic
9#21461561C212yesmaximal, complemented
10#21461545C211yesmaximal, characteristic, complemented
11C2×C22⋊C4×Dic61C211yesmaximal, characteristic
12C22×#11758217C2212yes—
13#16840778—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—42no—
53C4⋊#92420—24no—
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—41no—
121C22×C4×C61—41no—
122C22×C4×C61M1614yescomplemented
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—22no—
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—22no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61—41no—
149C22×Dic61—41no—
150C22×Dic61—41no—
151C22×Dic61—48no—
152C22×Dic61—48no—
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×C61—22no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61M1614yescomplemented
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—28no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
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—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—21no—
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—44no—
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—41no—
452C22×C4—41no—
453C22×C4—611no—
454C22×C4—614no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
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—2441no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2441no—
488C22×C4—2448no—
489C22×C4—2448no—
490C22×C4#1175821714yescomplemented
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—22no—
512C22⋊C4—614no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C42—1221no—
517C42—1222no—
518C42—1221no—
519C42—1222no—
520C42—1221no—
521C42—1221no—
522C42—1222no—
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—28no—
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—21no—
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—1221no—
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—2442no—
597C2×C4—2442no—
598C2×C4—2444no—
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#1684079314yes—
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×#1684079311yescharacteristic
658C2#2146255711yescharacteristic
659C2#2146267811yescharacteristic
660C2#2146268111yescharacteristic
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×#16840778C2semidirect
3. #21461893C2semidirect
4. #21461889C2semidirect
5. #21461766C2semidirect
6. #21461709C2semidirect
7. #21461675C2semidirect
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
122. C22×C4×C61M16semidirect
184. C22⋊Dic61M16semidirect
375. C61#22553semidirect
490. C22×C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407782
C2×#168407322
#214618932
#214618892
#214617662
#214617092
#214616752
#214616412
#214615612
#214615452
C2×C22⋊C4×Dic612
#2255361