Finite Group Databasebeta

← #29681244

Subgroups — #29681244

Order 15616.

1954 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840269C211yesmaximal, characteristic
2C22×#14519735C211yesmaximal, characteristic, complemented
3C2×#16839763C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic, complemented
5#21446138C214yesmaximal, complemented
6#21446128C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445538C211yesmaximal, characteristic, complemented
9C2×C61×#417C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12#16840269—24no—
13#16840269C418yescomplemented
14C2×#14519758—21no—
15C2×#14519753C2212yes—
16C2×#14519748C2212yes—
17C2×#14519745—21no—
18C2×C4×C22⋊D61—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519735—24no—
21C2×#14519735—24no—
22C2×#14519735—22no—
23C2×#14519735C2214yes—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×#11758198—21no—
26C22×C4×Dic61—21no—
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×D61C2×C414yes—
49C22×C4×D61D412yes—
50C22×C4×D61D414yes—
51C22×C4×D61C2312yes—
52C2×#11758194—22no—
53C2×#11758194—22no—
54C2×#11758194—22no—
55C22×#92418C2312yes—
56#14519758—28no—
57#14519753—24no—
58#14519753C2×C418yescomplemented
59#14519748—24no—
60#14519748C2×C418yescomplemented
61#14519745—28no—
62C4×C22⋊D61—216no—
63#14519736—22no—
64#14519736C2×C414yes—
65#14519735—216no—
66#14519735—28no—
67#14519735—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61D414yescomplemented
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—28no—
77C2×#11758198—24no—
78C2×#11758198—22no—
79C2×C4×Dic61—28no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—24no—
82#14519701—24no—
83#14519698—24no—
84#14519689—28no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61C2×C414yes—
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61C2311yescharacteristic
94C23×C4×C61—21no—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C22×C8×C61C2312yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—22no—
103C2×C4⋊C4×C61D414yescomplemented
104C2×C4⋊C4×C61C2312yes—
105#103×C61—24no—
106#102×C61—24no—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—42no—
112C23×D61—42no—
113C23×D61C22⋊C414yes—
114C23×D61C2×D412yes—
115#41×C61—28no—
116#41×C61—28no—
117#41×C61—44no—
118#41×C61—44no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61C22×C412yes—
125C2×D4×C61—24no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—42no—
132C2×D4×C61—42no—
133C2×D4×C61C22×C412yes—
134C22×C4×C61—24no—
135C22×C4×C61—22no—
136C22×C4×C61—21no—
137C22×C4×C61—24no—
138C22×C4×C61—44no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4112yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—28no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—24no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—44no—
158C22×Dic61C22⋊C414yes—
159C22×Dic61C2×D412yes—
160C2×C4×D61—28no—
161C2×C4×D61—28no—
162C2×C4×D61—24no—
163C2×C4×D61—28no—
164C2×C4×D61—28no—
165C2×C4×D61—24no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61C22⋊C418yes—
171C2×C4×D61C22×C414yes—
172C2×C4×D61C2×D414yes—
173M16×C61—44no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C2×D414yes—
180C22⋊Dic61—24no—
181C22⋊Dic61—28no—
182#11758198—216no—
183#11758198—28no—
184C4⋊Dic61—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61C22×C414yes—
188C4⋊Dic61C2×D414yes—
189#11758196—28no—
190C4×Dic61—216no—
191C4×Dic61—24no—
192#11758194—44no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61C2×C22⋊C412yes—
204C23×C61#12012yes—
205C23×C61#12312yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—28no—
214C22×D61—48no—
215C22×D61—48no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61C2×C22⋊C414yes—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61—41no—
222Q8×C61—41no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61—48no—
226D4×C61—41no—
227D4×C61—41no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—48no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—44no—
234C2×C4×C61—48no—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C4×D414yes—
237C2×C4×C61C4×D414yes—
238C2×C4×C61#12012yes—
239C2×C4×C61#12114yes—
240C2×C4×C61#12114yes—
241C2×C4×C61#12312yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—416no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—28no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—48no—
255C2×Dic61—44no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61C2×C22⋊C414yes—
259C4×D61—24no—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—24no—
263C4×D61—216no—
264C8×C61—42no—
265#92418—42no—
266#20910—611nomaximal, Sylow
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—44no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61#38614yes—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×#12311yescharacteristic
280C22×C61#57512yes—
281C22×C61#57512yes—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—24no—
285C2×D61—416no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C4×C61—44no—
291C4×C61—42no—
292C4×C61#38614yes—
293C4×C61C2×C4×D411yescharacteristic
294C4×C61C2×C4×D411yescharacteristic
295C4×C61C2×#12111yescharacteristic
296C4×C61C2×#12111yescharacteristic
297Dic61—24no—
298Dic61—44no—
299Dic61—44no—
300C2×#525—611no—
301C22×#117—611no—
302C2×#417—611no—
303C2×#417D6111yescharacteristic, complemented
304C2×#407—611no—
305#1875—614no—
306#1873—614no—
307#1736—611no—
308#1736—611no—
309C2×C61—24no—
310C2×C61—82no—
311C2×C61C2×#38611yescharacteristic
312C2×C61#289612yes—
313C2×C61#300511yescharacteristic
314C2×C61#300511yescharacteristic
315C2×C61#309312yes—
316D61—44no—
317C22×#41C2×D6111yescharacteristic
318C24×C4—611no—
319C22×M16—611no—
320C22×M16C2×D6111yescharacteristic
321#525—618no—
322#525—1224no—
323C2×#123—1221no—
324C2×C22⋊Q8—612no—
325C2×#121—612no—
326C2×#120—1221no—
327C2×C4×D4—1222no—
328C2×#117—611no—
329C2×#117—614no—
330C2×#117—1224no—
331C2×#117—1222no—
332C2×#117—1224no—
333C22×C4⋊C4—611no—
334C22×C4⋊C4C2×D6111yescharacteristic
335C22×C22⋊C4—1221no—
336C22×C42—1221no—
337#417—24no—
338#417—1224no—
339C2×#103—612no—
340C2×#103C2×D6112yescomplemented
341C2×#102—612no—
342C2×#102C2×D6112yescomplemented
343#407—1224no—
344C2×C22⋊C8—614no—
345#340—614no—
346#340—614no—
347C61#2091011yescharacteristic, complemented, Sylow
348C25—611no—
349C2×#41—22no—
350C2×#41—22no—
351C2×#41—22no—
352C2×#41—22no—
353C2×#41—22no—
354C2×#41C4×D6114yes—
355C22×Q8C22×D6111yescharacteristic
356C22×D4—21no—
357C22×D4—1222no—
358C22×D4C22×D6111yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—611no—
362C23×C4—614no—
363C23×C4—612no—
364C23×C4—614no—
365C23×C4—612no—
366C23×C4—1222no—
367C23×C4—1222no—
368C23×C4—1222no—
369C23×C4—1222no—
370C23×C4—1221no—
371C23×C4C22×D6111yescharacteristic
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—1222no—
376C2×M16—1222no—
377C2×M16—1222no—
378C22×C8—612no—
379C22×C8C22×D6112yes—
380#123—1228no—
381C22⋊Q8—618no—
382C22⋊Q8—1224no—
383#121—618no—
384#121—1224no—
385#120—1228no—
386C4×D4—12216no—
387#117—614no—
388#117—1228no—
389#117—12216no—
390#117—1222no—
391#117—1228no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—614no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4C22⋊D6114yescomplemented
401C2×C4⋊C4C22×D6112yes—
402C2×C22⋊C4—1224no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1228no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1221no—
407C2×C42—1224no—
408C2×C42—1228no—
409C2×C42—1221no—
410#103—24no—
411#103—1224no—
412#102—24no—
413#102—1224no—
414C22⋊C8—1228no—
415C24—21no—
416C24—612no—
417C24—614no—
418C24—1222no—
419C24—1222no—
420C24—2442no—
421C24—2442no—
422C24D4×D6111yescharacteristic
423#41—28no—
424#41—28no—
425#41—44no—
426#41—44no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8C2×C4×D6112yes—
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—1228no—
442C2×D4—2444no—
443C2×D4—2444no—
444C2×D4C2×C4×D6112yes—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—612no—
453C22×C4—614no—
454C22×C4—614no—
455C22×C4—614no—
456C22×C4—618no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1228no—
464C22×C4—1228no—
465C22×C4—1224no—
466C22×C4—1228no—
467C22×C4—1228no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1224no—
472C22×C4—2444no—
473C22×C4C2×C4×D6112yes—
474C22×C4#1175822212yes—
475C22×C4D4×D6112yes—
476C22×C4D4×D6112yes—
477C22×C4D4×D6111yescharacteristic
478C22×C4#1175822412yes—
479C22×C4C2×C22⋊D6112yes—
480C22×C4C23×D6111yescharacteristic
481M16—44no—
482M16—2444no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—614no—
492C4⋊C4—614no—
493C4⋊C4—1228no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1224no—
497C4⋊C4C2×C22⋊D6114yes—
498C22⋊C4—1228no—
499C22⋊C4—12216no—
500C22⋊C4—1228no—
501C22⋊C4—1224no—
502C42—12216no—
503C42—1224no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—614no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1222no—
516C23—1222no—
517C23—1222no—
518C23—1228no—
519C23—2448no—
520C23—2448no—
521C23—2444no—
522C23—2444no—
523C23C22⋊C4×D6112yes—
524C23#1451974512yes—
525C23#1451975912yes—
526C23C2×D4×D6111yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531D4—22no—
532D4—22no—
533D4—48no—
534D4—41no—
535D4—41no—
536D4—24416no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—614no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1228no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—12216no—
553C2×C4—1224no—
554C2×C4—1224no—
555C2×C4—2448no—
556C2×C4—2442no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C22⋊C4×D6112yes—
560C2×C4#1451973714yes—
561C2×C4C4×C22⋊D6114yes—
562C2×C4#1451974512yes—
563C2×C4#1451974814yes—
564C2×C4#1451975014yes—
565C2×C4#1451975912yes—
566C2×C4C22×C4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C22×C22⋊D6111yescharacteristic
570C2×C4C2×#1175822411yescharacteristic
571C2×C4C2×#1175822211yescharacteristic
572C8—42no—
573C8—2442no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—42no—
579C22—44no—
580C22—82no—
581C22—81no—
582C22—81no—
583C22—1224no—
584C22—1224no—
585C22—1224no—
586C22—24416no—
587C22—2442no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22#1683963814yes—
592C22C2×C22⋊C4×D6111yescharacteristic
593C22C2×#1451974511yescharacteristic
594C22C2×#1451975911yescharacteristic
595C22#1684055812yes—
596C22#1684056112yes—
597C4—42no—
598C4—44no—
599C4—1224no—
600C4—2444no—
601C4—2444no—
602C4#1683963814yes—
603C4C2×#1451973711yescharacteristic
604C4C2×C4×C22⋊D6111yescharacteristic
605C4C2×#1451974811yescharacteristic
606C4C2×#1451975011yescharacteristic
607C2—24no—
608C2—82no—
609C2—2444no—
610C2C2×#1683963811yescharacteristic
611C2#2145433512yes—
612C2#2145505811yescharacteristic
613C2#2145506011yescharacteristic
614C2#2145574912yes—
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#20910
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×#14519735C2semidirect
3. C2×#16839763C2semidirect
4. C2×#16839725C2semidirect
5. #21446138C2semidirect
6. #21446128C2semidirect
7. #21445539C2semidirect
8. #21445538C2semidirect
9. C2×C61×#417C2semidirect
13. #16840269C4semidirect
28. C2×#14519701C22semidirect
29. C2×#14519698C22semidirect
32. #16839525C22semidirect
33. #16839524C22semidirect
38. C2×#103×C61C22semidirect
39. C2×#102×C61C22semidirect
58. #14519753C2×C4semidirect
60. #14519748C2×C4semidirect
70. C2×C4⋊Dic61D4semidirect
103. C2×C4⋊C4×C61D4semidirect
303. C2×#417D61semidirect
340. C2×#103C2×D61semidirect
342. C2×#102C2×D61semidirect
347. C61#20910semidirect
400. C2×C4⋊C4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C22×#145197352
C2×#168397632
C2×#168397252
#214461382
#214461282
#214455392
#214455382
C2×C61×#4172
#2091061