Finite Group Databasebeta

← #30620534

Subgroups — #30620534

Order 15616.

1224 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×#16840784C211yesmaximal, characteristic, complemented
2C2×D61⋊(C22⋊C8)C211yesmaximal, characteristic
3#21461893C212yesmaximal, complemented
4#21461892C212yesmaximal, complemented
5#21461764C212yesmaximal, complemented
6#21461711C211yesmaximal, characteristic, complemented
7#21461676C211yesmaximal, characteristic, complemented
8#21461642C211yesmaximal, characteristic
9#21461557C212yesmaximal, complemented
10#21461546C211yesmaximal, characteristic, complemented
11C4×C22⋊C4×D61C211yesmaximal, characteristic
12C22×D61⋊C8C2212yes—
13#16840784—24no—
14C2×#14519844C2212yescomplemented
15C2×C4⋊#92420C2212yescomplemented
16C2×#14519834C2212yes—
17C2×#14519834C2211yescharacteristic
18C2×#14519834C2211yescharacteristic
19C2×#14519834C2211yescharacteristic
20C2×#14519834C2211yescharacteristic
21D61⋊(C22⋊C8)—24no—
22C2×C22⋊#92420C2212yes—
23C2×C22⋊#92420C2212yes—
24C2×#14519832C2212yescomplemented
25C2×#14519832C2212yescomplemented
26#16840628C2212yescomplemented
27#16840628C2212yescomplemented
28#16840628C2212yescomplemented
29#16840628C2212yescomplemented
30C23×C4×D61C2211yescharacteristic
31C2×C22⋊C4×D61C411yescharacteristic
32C2×C22⋊C4×D61C411yescharacteristic
33C2×C42×D61C2211yescharacteristic
34C2×C42×D61C2211yescharacteristic
35C22⋊C4×Dic61C2211yescharacteristic
36C22⋊C4×Dic61C2211yescharacteristic
37C4×#11758198C411yescharacteristic
38C4×#11758198C411yescharacteristic
39C4×C22⋊Dic61C411yescharacteristic
40#16839609C2211yescharacteristic
41#16839609C2211yescharacteristic
42#95×D61C411yescharacteristic
43#95×D61C411yescharacteristic
44C4×C22⋊C4×C61C411yescharacteristic
45C2×D61⋊C8—24no—
46C2×D61⋊C8—24no—
47C2×D61⋊C8—42no—
48C22×#92420C2312yes—
49C22×#92420C2312yes—
50C22×#92420C2312yes—
51C22×#92420C2312yes—
52#14519844—42no—
53C4⋊#92420—24no—
54#14519834—22no—
55#14519834—22no—
56#14519834—24no—
57#14519834—22no—
58#14519834D414yes—
59C22⋊#92420—24no—
60C22⋊#92420—42no—
61#14519832—24no—
62#14519832—42no—
63C24×D61C2×C411yescharacteristic
64C23×Dic61C2311yescharacteristic
65C22×C4×D61—22no—
66C22×C4×D61—24no—
67C22×C4×D61—22no—
68C22×C4×D61—42no—
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61C2×C411yescharacteristic
71C22×C4×D61C2×C411yescharacteristic
72C22×C4×D61C2×C411yescharacteristic
73C22×C4×D61C2×C411yescharacteristic
74C22×C4×D61C2×C411yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C2×C22⋊Dic61C2×C411yescharacteristic
78C2×C22⋊Dic61C2×C411yescharacteristic
79C2×#11758198C2×C411yescharacteristic
80C2×#11758198C2×C411yescharacteristic
81C2×#11758198C2×C411yescharacteristic
82C2×#11758198C2×C411yescharacteristic
83C22⋊C4×D61—24no—
84C22⋊C4×D61—24no—
85C2×C4×Dic61C2×C411yescharacteristic
86C2×C4×Dic61C2×C411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C42×D61—22no—
92C42×D61—22no—
93C42×D61—42no—
94#14519684C2×C411yescharacteristic
95#14519684C2×C411yescharacteristic
96#14519684C2×C411yescharacteristic
97#14519684C2×C411yescharacteristic
98#14519684C2×C411yescharacteristic
99#14519684C2×C411yescharacteristic
100C23×C4×C61C2×C411yescharacteristic
101C2×C22⋊C4×C61C2×C411yescharacteristic
102C2×C22⋊C4×C61C2×C411yescharacteristic
103C2×C42×C61C2×C411yescharacteristic
104C2×C42×C61C2×C411yescharacteristic
105#95×C61C2×C411yescharacteristic
106#95×C61C2×C411yescharacteristic
107C24×C61C22×C411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—41no—
113C23×D61—41no—
114C23×D61M1614yes—
115C23×D61C22×C411yescharacteristic
116C23×D61C22×C411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—41no—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—22no—
130C22×Dic61C22×C411yescharacteristic
131C22×Dic61C22×C411yescharacteristic
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C2411yescharacteristic
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—48no—
154C2×C4×D61—48no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×C4×D61C2×D412yes—
162C2×C4×D61#4114yes—
163C2×C4×D61#4112yes—
164C2×#92420—22no—
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×#92420C2×D414yes—
176D61⋊C8—48no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61—22no—
180C42×C61—41no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198M1614yes—
187C4×Dic61—22no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—41no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61C2×M1612yes—
202C23×C61C23×C411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—28no—
212C22×D61—28no—
213C22×D61—44no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61#11712yes—
219C22×D61C4×D412yes—
220C22×D61C2×M1612yes—
221C22×D61C2×M1612yes—
222C22×D61C2×M1612yes—
223C22×D61#13112yes—
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—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—44no—
240C2×C4×C61#11712yes—
241C2×C4×C61C4×D412yes—
242C2×C4×C61C2×M1612yes—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61C2×M1612yes—
260C2×Dic61C22×D411yescharacteristic
261C2×Dic61C2×#4112yes—
262C2×Dic61C2×#4111yescharacteristic
263C2×Dic61H(5,𝔽2)12yes—
264C4×D61—24no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—44no—
269C4×D61—44no—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—44no—
273C4×D61—44no—
274#92420—24no—
275#92420—42no—
276#92420—42no—
277#92420—42no—
278#22554—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61C2×#11711yescharacteristic
287C22×C61C2×C4×D411yescharacteristic
288C22×C61#51811yescharacteristic
289C22×C61C22×M1611yescharacteristic
290C22×C61C22×M1611yescharacteristic
291C22×C61C2×#13111yescharacteristic
292C22×C61#56811yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—416no—
306C2×D61#43214yescomplemented
307C2×D61#43514yes—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314Dic61—22no—
315Dic61—41no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61#54612yes—
320Dic61#55112yes—
321C2×#432—611no—
322C22×C22⋊C8—611no—
323C2×C4×C22⋊C4—611no—
324#1924—612no—
325#1924—612no—
326#1850—612no—
327#1817—611no—
328#1797—611no—
329#1779—611no—
330#1752—612no—
331#1749—611no—
332C2×C61—42no—
333C2×C61#234211yescharacteristic
334C2×C61C2×#43211yescharacteristic
335C2×C61#292411yescharacteristic
336C2×C61C2×#43511yescharacteristic
337C2×C61D4×M1611yescharacteristic
338C2×C61#297311yescharacteristic
339C2×C61#297411yescharacteristic
340D61—24no—
341D61—24no—
342C24×C4—611no—
343C23×C8—612no—
344C22×C22⋊C4—611no—
345C22×C22⋊C4—611no—
346C22×C42—611no—
347C22×C42—611no—
348#432—1224no—
349C2×C4⋊C8—612no—
350C2×C4⋊C8—612no—
351C2×C22⋊C8—611no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—611no—
355C2×C22⋊C8—612no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—612no—
358C2×C22⋊C8—1224no—
359C2×#97—612no—
360C2×#97—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#692311yescharacteristic
369C2×#95—611no—
370C2×#95—611no—
371#336—612no—
372#336—612no—
373#336—612no—
374#336—612no—
375C61#2255411yescharacteristic, complemented, Sylow
376C25—611no—
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—611no—
386C23×C4—1224no—
387C23×C4—1222no—
388C23×C4—1222no—
389C23×C4—2442no—
390C23×C4C2×#692311yescharacteristic
391C22×C8—612no—
392C22×C8—612no—
393C22×C8—612no—
394C22×C8—612no—
395C22×C8—1224no—
396C22×C8—1224no—
397C22×C8—2442no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—611no—
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—1224no—
405C2×C22⋊C4—1224no—
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—1222no—
415C2×C42—1222no—
416C2×C42—2442no—
417C2×C42C2×#692311yescharacteristic
418C2×C42C2×#692311yescharacteristic
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—1224no—
429#97—2442no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95C2×#692311yescharacteristic
437#95C2×#692311yescharacteristic
438C24—611no—
439C24—611no—
440C24—614no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1222no—
445C24—2441no—
446C24—2441no—
447C24C22×#692311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—614no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1221no—
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—1221no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1221no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2448no—
488C22×C4—2448no—
489C22×C4C22×#692311yescharacteristic
490C22×C4C22×#692311yescharacteristic
491C22×C4C22×#692311yescharacteristic
492C22×C4C22×#692311yescharacteristic
493C22×C4C22×#692311yescharacteristic
494C22×C4C22×#692311yescharacteristic
495C2×C8—614no—
496C2×C8—1222no—
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—2448no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—614no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C42—22no—
517C42—41no—
518C42—1221no—
519C42—1222no—
520C42—1221no—
521C42—1222no—
522C42—1221no—
523C42—1221no—
524C42—1222no—
525C42—2441no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—612no—
533C23—612no—
534C23—612no—
535C23—612no—
536C23—612no—
537C23—612no—
538C23—1228no—
539C23—1228no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—2444no—
549C23—2444no—
550C23—2444no—
551C23—2444no—
552C23—2444no—
553C23D61⋊M1612yes—
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—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—612no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1224no—
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—2444no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4#1451985612yes—
603C2×C4D4×#692312yes—
604C2×C4D61⋊M1612yes—
605C8—1224no—
606C8—2442no—
607C8—2442no—
608C8—2442no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—614no—
617C22—614no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22—1222no—
630C22—24416no—
631C22C2×#1451985611yescharacteristic
632C22C2×D4×#692311yescharacteristic
633C22#1684082911yescharacteristic
634C22C2×D61⋊M1611yescharacteristic
635C22C2×D61⋊M1611yescharacteristic
636C22D61⋊#13111yescharacteristic
637C22#1684084111yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—612no—
645C4—612no—
646C4—1222no—
647C4—2441no—
648C4—2441no—
649C4—2441no—
650C4—2441no—
651C2—42no—
652C2—1224no—
653C2—1224no—
654C2#2146232511yescharacteristic
655C2D61⋊#43211yescharacteristic
656C2#2146251211yescharacteristic
657C2D61⋊#43511yescharacteristic
658C2#2146255811yescharacteristic
659C2#2146267111yescharacteristic
660C2#2146267711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC4×C22⋊C4×C61C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC4×C22⋊C4#6923
hypocenterC61#22554
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840784C2semidirect
3. #21461893C2semidirect
4. #21461892C2semidirect
5. #21461764C2semidirect
6. #21461711C2semidirect
7. #21461676C2semidirect
9. #21461557C2semidirect
10. #21461546C2semidirect
14. C2×#14519844C22semidirect
15. C2×C4⋊#92420C22semidirect
24. C2×#14519832C22semidirect
25. C2×#14519832C22semidirect
26. #16840628C22semidirect
27. #16840628C22semidirect
28. #16840628C22semidirect
29. #16840628C22semidirect
306. C2×D61#432semidirect
375. C61#22554semidirect

Maximal subgroups

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

subgroupindex
C2×#168407842
C2×D61⋊(C22⋊C8)2
#214618932
#214618922
#214617642
#214617112
#214616762
#214616422
#214615572
#214615462
C4×C22⋊C4×D612
#2255461