Finite Group Databasebeta

← #29681237

Subgroups — #29681237

Order 15616.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C22×C4⋊C4×D61C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic, complemented
5#21446133C214yesmaximal, complemented
6#21446123C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C23×C4×D61—21no—
11C23×C4×D61C2211yescharacteristic
12C22×#11758194C2211yescharacteristic
13#16840269—24no—
14#16840269C418yescomplemented
15C2×#14519758—21no—
16C2×#14519753C2212yes—
17C2×#14519748C2212yes—
18C2×#14519745—21no—
19C2×C4×C22⋊D61—22no—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758196—21no—
23C2×C4⋊C4×D61—24no—
24C2×C4⋊C4×D61—24no—
25C2×C4⋊C4×D61—22no—
26C2×C4⋊C4×D61C2214yes—
27#16839763—24no—
28C2×#14519701C2212yescomplemented
29C2×#14519698C2212yescomplemented
30#16839725—24no—
31C2×#14519689C2214yes—
32#16839525C2214yescomplemented
33#16839524C2214yescomplemented
34C22×#41×C61C2211yescharacteristic
35C22×M16×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#417—24no—
38C2×#103×C61C2212yescomplemented
39C2×#102×C61C2212yescomplemented
40C24×D61D411yescharacteristic
41C22×C22⋊D61—22no—
42C23×Dic61—21no—
43C23×Dic61D411yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—28no—
49C22×C4×D61—24no—
50C22×C4×D61—48no—
51C22×C4×D61C2×C414yes—
52C22×C4×D61D412yes—
53C22×C4×D61D414yes—
54C22×C4×D61C2312yes—
55C2×#11758194—22no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C22×#92418C2312yes—
59#14519758—28no—
60#14519753—24no—
61#14519753C2×C418yescomplemented
62#14519748—24no—
63#14519748C2×C418yescomplemented
64#14519745—28no—
65C4×C22⋊D61—216no—
66#14519736—22no—
67#14519736C2×C414yescomplemented
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D414yescomplemented
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—28no—
74C2×#11758196—24no—
75C2×#11758196—22no—
76C4⋊C4×D61—216no—
77C4⋊C4×D61—28no—
78C4⋊C4×D61—28no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—22no—
81C2×#11758198—22no—
82C2×C4×Dic61—21no—
83#14519701—24no—
84#14519698—24no—
85#14519689—28no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61C2×C414yescomplemented
92C22×Q8×C61C2311yescharacteristic
93C22×D4×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C2×M16×C61—22no—
101C22×C8×C61C2312yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—22no—
104C2×C4⋊C4×C61D414yescomplemented
105C2×C4⋊C4×C61C2312yes—
106#103×C61—24no—
107#102×C61—24no—
108C24×C61—21no—
109C24×C61C2×D411yescharacteristic
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—24no—
113C23×D61—24no—
114C23×D61C22⋊C414yes—
115C23×D61C2×D412yes—
116#41×C61—28no—
117#41×C61—28no—
118#41×C61—44no—
119#41×C61—44no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61C22×C412yescomplemented
126C2×D4×C61—24no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—42no—
133C2×D4×C61—42no—
134C2×D4×C61C22×C412yescomplemented
135C22×C4×C61—24no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—24no—
139C22×C4×C61—44no—
140C22×C4×C61—42no—
141C22×C4×C61—42no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4112yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—28no—
151C2×C22⋊D61—44no—
152C2×C22⋊D61—44no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—24no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—44no—
159C22×Dic61C22⋊C414yes—
160C22×Dic61C2×D412yes—
161C2×C4×D61—28no—
162C2×C4×D61—28no—
163C2×C4×D61—24no—
164C2×C4×D61—28no—
165C2×C4×D61—28no—
166C2×C4×D61—24no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—216no—
172C2×C4×D61—432no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61C22⋊C418yes—
176C2×C4×D61C22×C414yes—
177C2×C4×D61C2×D414yes—
178M16×C61—44no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61C2×D414yes—
185C22⋊Dic61—24no—
186C22⋊Dic61—28no—
187#11758198—28no—
188C4⋊Dic61—24no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191C4⋊Dic61C22×C414yescomplemented
192C4⋊Dic61C2×D414yes—
193#11758196—216no—
194#11758196—28no—
195C4×Dic61—24no—
196#11758194—44no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61C2×C22⋊C412yes—
208C23×C61#12012yes—
209C23×C61#12312yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—28no—
212C22×D61—216no—
213C22×D61—28no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—28no—
219C22×D61—216no—
220C22×D61C2×C22⋊C414yes—
221C22×D61AGL(1,ℤ/8)14yes—
222C22×D61#13714yes—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—48no—
230D4×C61—41no—
231D4×C61—41no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—48no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—44no—
238C2×C4×C61—48no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C4×D414yes—
241C2×C4×C61C4×D414yes—
242C2×C4×C61#12012yes—
243C2×C4×C61#12114yes—
244C2×C4×C61#12114yes—
245C2×C4×C61#12312yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C22⋊D61—416no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—28no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—48no—
259C2×Dic61—44no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61C2×C22⋊C414yes—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—216no—
268C4×D61—432no—
269C8×C61—42no—
270#92418—42no—
271#20909—611nomaximal, Sylow
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—82no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61#38614yes—
282C22×C61C22×C22⋊C411yescharacteristic
283C22×C61C2×#12011yescharacteristic
284C22×C61C2×#12311yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)12yes—
286C22×C61C2×#13712yes—
287C2×D61—232no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—24no—
294C2×D61#41718yescomplemented
295C4×C61—44no—
296C4×C61—42no—
297C4×C61#38614yes—
298C4×C61C2×C4×D411yescharacteristic
299C4×C61C2×C4×D411yescharacteristic
300C4×C61C2×#12111yescharacteristic
301C4×C61C2×#12111yescharacteristic
302Dic61—24no—
303Dic61—44no—
304Dic61—44no—
305C2×#525—611no—
306C23×C4⋊C4—611no—
307C2×#417—611no—
308C2×#417D6111yescharacteristic, complemented
309C2×#407—611no—
310#1874—614no—
311#1872—614no—
312#1736—611no—
313#1736—611no—
314C2×C61—24no—
315C2×C61—82no—
316C2×C61C2×#38611yescharacteristic
317C2×C61C2×#41712yes—
318C2×C61#300311yescharacteristic
319C2×C61#300411yescharacteristic
320C2×C61#309212yes—
321D61—28no—
322C22×#41C2×D6111yescharacteristic
323C24×C4—611no—
324C24×C4—1221no—
325C22×M16—611no—
326C22×M16C2×D6111yescharacteristic
327#525—618no—
328#525—1224no—
329C2×#123—1221no—
330C2×C22⋊Q8—612no—
331C2×#121—612no—
332C2×#120—1221no—
333C2×C4×D4—1222no—
334C2×#117—611no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—614no—
337C22×C4⋊C4—1224no—
338C22×C4⋊C4—1222no—
339C22×C4⋊C4—1221no—
340C22×C4⋊C4—1224no—
341C22×C4⋊C4C2×D6111yescharacteristic
342#417—24no—
343#417—1224no—
344C2×#103—612no—
345C2×#103C2×D6112yescomplemented
346C2×#102—612no—
347C2×#102C2×D6112yescomplemented
348#407—1224no—
349C2×C22⋊C8—614no—
350#340—614no—
351#340—614no—
352C61#2090911yescharacteristic, complemented, Sylow
353C25—611no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41C4×D6114yescomplemented
360C22×Q8C22×D6111yescharacteristic
361C22×D4—21no—
362C22×D4—1222no—
363C22×D4C22×D6111yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—611no—
367C23×C4—614no—
368C23×C4—612no—
369C23×C4—614no—
370C23×C4—612no—
371C23×C4—1224no—
372C23×C4—1228no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—1222no—
376C23×C4—1222no—
377C23×C4—1221no—
378C23×C4—2448no—
379C23×C4C22×D6111yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1222no—
384C2×M16—1222no—
385C2×M16—1222no—
386C22×C8—612no—
387C22×C8C22×D6112yes—
388#123—1228no—
389C22⋊Q8—618no—
390C22⋊Q8—1224no—
391#121—618no—
392#121—1224no—
393#120—1228no—
394C4×D4—12216no—
395#117—614no—
396#117—1222no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—614no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1228no—
403C2×C4⋊C4—12216no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1228no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1228no—
410C2×C4⋊C4C22⋊D6114yescomplemented
411C2×C4⋊C4C22×D6112yes—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1221no—
415C2×C42—1221no—
416#103—24no—
417#103—1224no—
418#102—24no—
419#102—1224no—
420C22⋊C8—1228no—
421C24—21no—
422C24—612no—
423C24—614no—
424C24—1224no—
425C24—1224no—
426C24—1222no—
427C24—1222no—
428C24D4×D6111yescharacteristic
429#41—28no—
430#41—28no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D6112yescomplemented
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—1228no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4C2×C4×D6112yescomplemented
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—612no—
459C22×C4—614no—
460C22×C4—614no—
461C22×C4—614no—
462C22×C4—618no—
463C22×C4—12216no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1228no—
471C22×C4—1228no—
472C22×C4—1224no—
473C22×C4—1228no—
474C22×C4—1228no—
475C22×C4—1224no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1224no—
479C22×C4—2448no—
480C22×C4—24432no—
481C22×C4—2448no—
482C22×C4—2444no—
483C22×C4C2×C4×D6112yes—
484C22×C4#1175822212yes—
485C22×C4D4×D6112yes—
486C22×C4D4×D6112yes—
487C22×C4D4×D6111yescharacteristic
488C22×C4#1175822412yes—
489C22×C4C2×C22⋊D6112yes—
490C22×C4C23×D6111yescharacteristic
491M16—44no—
492M16—2444no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—614no—
502C4⋊C4—614no—
503C4⋊C4—1228no—
504C4⋊C4—12216no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1224no—
508C4⋊C4C2×C22⋊D6114yes—
509C22⋊C4—1228no—
510C22⋊C4—1228no—
511C22⋊C4—1224no—
512C42—1224no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—614no—
521C23—614no—
522C23—614no—
523C23—12216no—
524C23—1228no—
525C23—12216no—
526C23—1228no—
527C23—1222no—
528C23—1222no—
529C23—1222no—
530C23—1222no—
531C23—1228no—
532C23C22⋊C4×D6112yes—
533C23#1451974512yes—
534C23#1451975912yes—
535C23C2×D4×D6111yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—24416no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—614no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1228no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—12216no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—24432no—
565C2×C4—2448no—
566C2×C4—2444no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4C22⋊C4×D6112yes—
570C2×C4#1451973714yes—
571C2×C4C4×C22⋊D6114yes—
572C2×C4#1451974512yes—
573C2×C4#1451974814yes—
574C2×C4#1451975014yes—
575C2×C4#1451975912yes—
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C2×C4C2×#1175822411yescharacteristic
581C2×C4C2×#1175822211yescharacteristic
582C8—42no—
583C8—2442no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—42no—
589C22—44no—
590C22—82no—
591C22—81no—
592C22—81no—
593C22—618no—
594C22—12232no—
595C22—1224no—
596C22—1224no—
597C22—1224no—
598C22—1224no—
599C22—1224no—
600C22—1224no—
601C22#1683963814yes—
602C22C2×C22⋊C4×D6111yescharacteristic
603C22C2×#1451974511yescharacteristic
604C22C2×#1451975911yescharacteristic
605C22AGL(1,ℤ/8)×D6112yes—
606C22#137×D6112yes—
607C4—42no—
608C4—44no—
609C4—1224no—
610C4—2444no—
611C4—2444no—
612C4#1683963814yes—
613C4C2×#1451973711yescharacteristic
614C4C2×C4×C22⋊D6111yescharacteristic
615C4C2×#1451974811yescharacteristic
616C4C2×#1451975011yescharacteristic
617C2—24no—
618C2—82no—
619C2—1228no—
620C2C2×#1683963811yescharacteristic
621C2#417×D6112yes—
622C2#2145504411yescharacteristic
623C2#2145505111yescharacteristic
624C2#2145574012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenterC2×#417D61
hypocenterC61#20909
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D61C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839725C2semidirect
5. #21446133C2semidirect
6. #21446123C2semidirect
7. #21445539C2semidirect
8. #21445538C2semidirect
9. C2×C61×#417C2semidirect
14. #16840269C4semidirect
28. C2×#14519701C22semidirect
29. C2×#14519698C22semidirect
32. #16839525C22semidirect
33. #16839524C22semidirect
38. C2×#103×C61C22semidirect
39. C2×#102×C61C22semidirect
61. #14519753C2×C4semidirect
63. #14519748C2×C4semidirect
67. #14519736C2×C4semidirect
70. C2×C4⋊Dic61D4semidirect
91. C2×#41×C61C2×C4semidirect
104. C2×C4⋊C4×C61D4semidirect
125. C2×Q8×C61C22×C4semidirect
134. C2×D4×C61C22×C4semidirect
191. C4⋊Dic61C22×C4semidirect
294. C2×D61#417semidirect
308. C2×#417D61semidirect
345. C2×#103C2×D61semidirect
347. C2×#102C2×D61semidirect
352. C61#20909semidirect
359. C2×#41C4×D61semidirect
410. C2×C4⋊C4C22⋊D61semidirect
438. C2×Q8C2×C4×D61semidirect
450. C2×D4C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C22×C4⋊C4×D612
C2×#168397632
C2×#168397252
#214461332
#214461232
#214455392
#214455382
C2×C61×#4172
#2090961