Finite Group Databasebeta

← #29685065

Subgroups — #29685065

Order 15616.

1538 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
4#21446769C212yesmaximal, complemented
5#21446744C212yesmaximal, complemented
6#21446137C211yesmaximal, characteristic, complemented
7#21446136C211yesmaximal, characteristic, complemented
8#21446134C211yesmaximal, characteristic, complemented
9#102×Dic61C212yesmaximal, complemented
10#21445633C212yesmaximal, complemented
11C61×#1874C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13#16840307—24no—
14#16840307C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519773C2211yescharacteristic
17C2×#14519763—21no—
18C2×#14519751C2211yescharacteristic
19C2×#14519749—21no—
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519737—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61—21no—
25C22×C4×Dic61C2211yescharacteristic
26#16839812—24no—
27C2×#14519706C2212yes—
28C2×#14519699C2212yes—
29C2×#14519698C2212yes—
30C2×#14519697C2212yes—
31C2×C22⋊#92418C2211yescharacteristic
32C2×C8×Dic61C2212yes—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61—21no—
35C4⋊C4×Dic61C2212yes—
36C4⋊C4×Dic61C2212yes—
37C2×#14519684—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#121×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#102×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C2×#11758224—28no—
47C2×#11758224—44no—
48C2×#11758224—44no—
49C22×Dic122—21no—
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61—21no—
56C23×Dic61D411yescharacteristic
57C23×Dic61D411yescharacteristic
58C22×C4×D61—21no—
59C22×#92418C2312yes—
60#14519776—22no—
61#14519776C2×C414yescomplemented
62#14519773—22no—
63#14519773C2×C414yescomplemented
64#14519763—28no—
65#14519751—22no—
66#14519751C2×C414yes—
67#14519749—28no—
68D4×Dic61—22no—
69D4×Dic61C2×C414yescomplemented
70#14519739—22no—
71#14519739C2×C414yescomplemented
72#14519737—28no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—24no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—44no—
84C2×C4×Dic61C2×C414yes—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D414yes—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519706—24no—
91#14519699—24no—
92#14519698—24no—
93#14519697—24no—
94C22⋊#92418—22no—
95C8×Dic61—24no—
96#14519684—22no—
97#14519684—22no—
98C22×D4×C61—21no—
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103#121×C61—22no—
104#121×C61C2×C414yes—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110#102×C61—24no—
111C22⋊C8×C61—22no—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C2×D4×C61—24no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61C22×C412yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—42no—
128C22×C4×C61—42no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—24no—
134C2×C22⋊D61—24no—
135C2×C22⋊D61—24no—
136C2×C22⋊D61—42no—
137C2×C22⋊D61—42no—
138C2×C22⋊D61—42no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C22×Dic61—24no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—24no—
146C22×Dic61—41no—
147C22×Dic61—41no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61C22⋊C414yes—
153C22×Dic61C22⋊C414yes—
154C22×Dic61D814yes—
155C22×Dic61SD1614yes—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160#11758224—432no—
161C2×D244—22no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×Dic122—24no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C22×C412yes—
178C4⋊C4×C61#4114yescomplemented
179C22⋊C4×C61—24no—
180C22⋊Dic61—24no—
181#11758198—24no—
182#11758198—24no—
183C4⋊Dic61—24no—
184C4⋊Dic61—21no—
185C4⋊Dic61—22no—
186C4⋊Dic61C22×C412yes—
187C4⋊Dic61#4114yescomplemented
188#11758196—24no—
189C4×Dic61—44no—
190C4×Dic61—44no—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22⋊C414yes—
193C4×Dic61C22×C412yes—
194C4×Dic61C22×C412yes—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D412yes—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C2×C22⋊C412yes—
212C23×C61C2×D812yes—
213C23×C61C2×SD1612yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222D4×C61—24no—
223D4×C61—48no—
224D4×C61—42no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—48no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4112yes—
240C2×C4×C61C2×#4112yes—
241C22⋊D61—48no—
242C22⋊D61—48no—
243C22⋊D61—48no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—44no—
251C2×Dic61—48no—
252C2×Dic61—42no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—48no—
257C2×Dic61#102116yescomplemented
258C2×Dic61C2×C22⋊C412yes—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×D812yes—
263C2×Dic61C2×D812yes—
264C2×Dic61C2×SD1612yes—
265C2×Dic61C2×SD1612yes—
266D244—24no—
267D244—42no—
268C4×D61—48no—
269Dic122—48no—
270C8×C61—42no—
271#92418—42no—
272#21386—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61C2×#10214yes—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#52511yescharacteristic
289C22×C61#52511yescharacteristic
290C22×C61C22×D811yescharacteristic
291C22×C61C22×SD1611yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57511yescharacteristic
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C2×D61—81no—
301C4×C61—22no—
302C4×C61—44no—
303C4×C61—42no—
304C4×C61C4×#4112yes—
305C4×C61#53212yes—
306Dic61—22no—
307Dic61—22no—
308Dic61—44no—
309Dic61—42no—
310Dic61C2×#10214yes—
311Dic61C2×#10214yes—
312C2×#532—611no—
313C2×#431—611no—
314C2×C4×C4⋊C4—611no—
315#1988—612no—
316#1982—612no—
317#1874—611no—
318#1874—611no—
319#1874—611no—
320#1874D6111yescharacteristic, complemented
321C4×#102—612no—
322C4×#102—612no—
323C2×C61—24no—
324C2×C61—82no—
325C2×C61#229711yescharacteristic
326C2×C61C22×#10211yescharacteristic
327C2×C61#289611yescharacteristic
328C2×C61#310011yescharacteristic
329C2×C61#310011yescharacteristic
330C2×C61#310311yescharacteristic
331C2×C61#310311yescharacteristic
332D61—82no—
333C22×#41—1221no—
334#532—618no—
335#532—1224no—
336C2×C4⋊Q8—611no—
337C2×#127—611no—
338C2×#124—1221no—
339C2×#121—611no—
340C2×#121—1221no—
341C2×#121C2×D6111yescharacteristic
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4—1221no—
345C22×C4⋊C4—611no—
346C22×C4⋊C4C2×D6111yescharacteristic
347C22×C42—611no—
348C22×C42—1221no—
349#431—1224no—
350C2×C4⋊C8—612no—
351C2×#102—612no—
352C2×#102—612no—
353C2×#102—612no—
354C2×#102C2×D6112yes—
355C2×C22⋊C8—611no—
356C2×C22⋊C8C2×D6111yescharacteristic
357C2×C4×C8—612no—
358C4×C4⋊C4—612no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—1221no—
361C4×C4⋊C4—1221no—
362C2×#95—1221no—
363#340—612no—
364#340—612no—
365#340—612no—
366#340C2×D6112yes—
367C61#2138611yescharacteristic, complemented, Sylow
368C2×#41—1228no—
369C2×#41—2444no—
370C2×#41—2444no—
371C22×Q8—1221no—
372C22×D4—21no—
373C22×D4—611no—
374C22×D4—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4C22×D6111yescharacteristic
378C23×C4—21no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—1221no—
382C23×C4—1221no—
383C23×C4—1221no—
384C23×C4C22×D6111yescharacteristic
385C22×C8—612no—
386C22×C8C22×D6112yes—
387C4⋊Q8—614no—
388C4⋊Q8—1222no—
389#127—614no—
390#127—1222no—
391#124—1228no—
392#121—22no—
393#121—614no—
394#121—1228no—
395#121—1222no—
396#121C4×D6114yes—
397C4×D4—614no—
398C4×D4—614no—
399C4×D4—1228no—
400C4×D4—1222no—
401C4×D4—1222no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—24no—
404C2×C4⋊C4—612no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6112yes—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—614no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—614no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—1222no—
422C2×C42—1222no—
423C2×C42—2444no—
424C4⋊C8—1224no—
425#102—24no—
426#102—1224no—
427#102—1224no—
428#102—1224no—
429C22⋊C8—22no—
430C22⋊C8—1222no—
431C4×C8—1224no—
432#95—1222no—
433#95—1222no—
434C24—21no—
435C24—1221no—
436C24D4×D6111yescharacteristic
437#41—24432no—
438C2×Q8—1224no—
439C2×Q8—2442no—
440C2×Q8—2442no—
441C2×D4—24no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—612no—
448C2×D4—1224no—
449C2×D4—1224no—
450C2×D4—1224no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—2442no—
455C2×D4—2442no—
456C2×D4—2442no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4C2×C4×D6112yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—614no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1222no—
480C22×C4—1221no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4C2×C4×D6112yes—
490C22×C4D4×D6112yes—
491C22×C4D4×D6111yescharacteristic
492C22×C4C23×D6111yescharacteristic
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—612no—
503C4⋊C4—614no—
504C4⋊C4—1224no—
505C4⋊C4—1224no—
506C4⋊C4—1221no—
507C4⋊C4—1222no—
508C4⋊C4C2×C4×D6112yes—
509C4⋊C4#1175822414yescomplemented
510C22⋊C4—24no—
511C22⋊C4—1224no—
512C22⋊C4—1224no—
513C22⋊C4—1224no—
514C42—612no—
515C42—612no—
516C42—612no—
517C42—612no—
518C42—614no—
519C42—614no—
520C42—2444no—
521C42—2444no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—1222no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23C22⋊C4×D6112yes—
541C23D8×D6112yes—
542C23SD16×D6112yes—
543C23C2×D4×D6111yescharacteristic
544Q8—2448no—
545D4—24no—
546D4—48no—
547D4—42no—
548D4—1224no—
549D4—2448no—
550D4—2448no—
551D4—2448no—
552D4—2442no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—6116no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—1224no—
579C2×C4—2448no—
580C2×C4—2448no—
581C2×C4—2444no—
582C2×C4—2448no—
583C2×C4—2442no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2442no—
587C2×C4C22⋊C4×D6112yes—
588C2×C4C22×C4×D6111yescharacteristic
589C2×C4C2×D4×D6111yescharacteristic
590C2×C4#41×D6112yes—
591C2×C4C2×#1175822412yes—
592C8—42no—
593C8—2442no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—4881no—
608C22—4881no—
609C22—4881no—
610C22—4881no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22#102×D6114yes—
615C22C2×C22⋊C4×D6111yescharacteristic
616C22#1684026811yescharacteristic
617C22#1684027011yescharacteristic
618C22C2×D8×D6111yescharacteristic
619C22C2×SD16×D6111yescharacteristic
620C22#1684055811yescharacteristic
621C22#1684056111yescharacteristic
622C4—22no—
623C4—44no—
624C4—42no—
625C4—614no—
626C4—614no—
627C4—1222no—
628C4—1222no—
629C4—2444no—
630C4—2442no—
631C4C4×#1175822412yes—
632C4#1684030712yes—
633C2—24no—
634C2—82no—
635C2—4882no—
636C2#2144830411yescharacteristic
637C2C2×#102×D6111yescharacteristic
638C2#2145433511yescharacteristic
639C2#2145581411yescharacteristic
640C2#2145581611yescharacteristic
641C2#2145584511yescharacteristic
642C2#2145584711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1874C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
cosocleC22×C4×C61C24
hypercenter#1874D61
hypocenterC61#21386
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
2. C2×#16839812C2semidirect
3. C2×C4⋊C4×Dic61C2semidirect
4. #21446769C2semidirect
5. #21446744C2semidirect
6. #21446137C2semidirect
7. #21446136C2semidirect
8. #21446134C2semidirect
9. #102×Dic61C2semidirect
10. #21445633C2semidirect
11. C61×#1874C2semidirect
14. #16840307C4semidirect
61. #14519776C2×C4semidirect
63. #14519773C2×C4semidirect
69. D4×Dic61C2×C4semidirect
71. #14519739C2×C4semidirect
178. C4⋊C4×C61#41semidirect
187. C4⋊Dic61#41semidirect
257. C2×Dic61#102semidirect
320. #1874D61semidirect
367. C61#21386semidirect
509. C4⋊C4#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403072
C2×#168398122
C2×C4⋊C4×Dic612
#214467692
#214467442
#214461372
#214461362
#214461342
#102×Dic612
#214456332
C61×#18742
#2138661