Finite Group Databasebeta

← #29681251

Subgroups — #29681251

Order 15616.

1934 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
1C22×#14519753C211yesmaximal, characteristic
2C2×#117×D61C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839725C211yesmaximal, characteristic
5#21446128C214yesmaximal, complemented
6#21446123C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445539C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519753—21no—
13C2×#14519753—21no—
14C2×#14519753—24no—
15C2×#14519753C418yescomplemented
16C2×#14519753C2212yes—
17C2×#14519753C2212yes—
18C2×#14519736C2211yescharacteristic
19C2×#14519735C2212yes—
20C2×#14519734—21no—
21C2×#14519733—21no—
22#117×D61—24no—
23#117×D61—24no—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×#11758196—21no—
26C2×C4⋊C4×D61C2212yes—
27C22×#11758198—21no—
28C2×C22⋊C4×D61—21no—
29C2×C42×D61—21no—
30#16839771—24no—
31C2×#14519701C2214yescomplemented
32#16839725—24no—
33C2×#14519689C2212yes—
34C2×#14519689C2212yes—
35#16839525C2214yescomplemented
36#16839524C2214yescomplemented
37C23×Q8×C61C2211yescharacteristic
38C22×M16×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C61×#419—24no—
41C2×#103×C61C2214yescomplemented
42C24×D61D411yescharacteristic
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61C2×C414yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D414yes—
54C22×C4×D61C2311yescharacteristic
55C22×C4×D61C2311yescharacteristic
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C22×#92418C2312yes—
60#14519753—28no—
61#14519753—28no—
62#14519753—24no—
63#14519753—24no—
64#14519753C2×C418yes—
65#14519753C2×C418yescomplemented
66#14519736—22no—
67#14519736D414yes—
68#14519735—24no—
69#14519735—24no—
70#14519734—28no—
71#14519733—28no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61C2×C414yescomplemented
75C2×C4⋊Dic61C2312yes—
76C2×C4⋊Dic61C2312yes—
77C2×#11758196—28no—
78C2×#11758196—22no—
79C2×#11758196—22no—
80C2×#11758196—22no—
81C4⋊C4×D61—24no—
82C4⋊C4×D61—24no—
83C2×C22⋊Dic61—21no—
84C2×#11758198—28no—
85C2×#11758198—22no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88C22⋊C4×D61—28no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—22no—
91C42×D61—28no—
92#14519701—28no—
93#14519689—24no—
94#14519689—24no—
95C22×Q8×C61—22no—
96C22×Q8×C61—21no—
97C22×Q8×C61—22no—
98C22×Q8×C61—22no—
99C22×Q8×C61—22no—
100C22×Q8×C61—22no—
101C22×Q8×C61C2×C414yescomplemented
102C22×Q8×C61C2312yes—
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C2×M16×C61—22no—
106C2×M16×C61—22no—
107C2×M16×C61—22no—
108C22×C8×C61C2312yes—
109#117×C61—22no—
110#117×C61D414yes—
111C2×C4⋊C4×C61C2312yes—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—28no—
115C24×C61C2×D411yescharacteristic
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—42no—
120C23×D61C22⋊C414yes—
121C23×D61C2×D411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C2×Q8×C61—24no—
124C2×Q8×C61—22no—
125C2×Q8×C61—28no—
126C2×Q8×C61—28no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—42no—
132C2×Q8×C61—44no—
133C2×Q8×C61—44no—
134C2×Q8×C61—42no—
135C2×Q8×C61C22×C414yes—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
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
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—24no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—44no—
155C22×Dic61C22⋊C414yes—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×C4×D61—28no—
159C2×C4×D61—28no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—28no—
163C2×C4×D61—28no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—28no—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61C22⋊C414yes—
178C2×C4×D61C22⋊C414yes—
179C2×C4×D61C22×C412yes—
180C2×C4×D61C22×C412yes—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D412yes—
183M16×C61—44no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61C2×D414yes—
189C22⋊C4×C61—24no—
190C42×C61—24no—
191C22⋊Dic61—24no—
192#11758198—28no—
193#11758198—28no—
194#11758198—28no—
195C4⋊Dic61—24no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61C22×C414yes—
199C4⋊Dic61C2×D414yes—
200#11758196—28no—
201#11758196—28no—
202#11758196—28no—
203C4×Dic61—28no—
204C4×Dic61—24no—
205#11758194—44no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61C2×C22⋊C412yes—
212C23×C61#12012yes—
213C23×C61#12312yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—28no—
223C22×D61—28no—
224C22×D61—44no—
225C22×D61—48no—
226C22×D61—48no—
227C22×D61C2×C22⋊C412yes—
228C22×D61C2×C22⋊C412yes—
229C22×D61#13714yes—
230Q8×C61—24no—
231Q8×C61—24no—
232Q8×C61—42no—
233Q8×C61—42no—
234Q8×C61—48no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—48no—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C4×D414yes—
244C2×C4×C61C4×D414yes—
245C2×C4×C61#12012yes—
246C2×C4×C61#12114yes—
247C2×C4×C61#12114yes—
248C2×C4×C61#12312yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—28no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—48no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—44no—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—28no—
277C4×D61—28no—
278C4×D61—416no—
279C8×C61—42no—
280#92418—42no—
281#20911—611nomaximal, Sylow
282C22×C61—24no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—24no—
286C22×C61#38614yes—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61C2×#12011yescharacteristic
289C22×C61C2×#12311yescharacteristic
290C22×C61C2×#13712yes—
291C22×C61#57512yes—
292C2×D61—24no—
293C2×D61—22no—
294C2×D61—24no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—416no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61#41914yescomplemented
303C4×C61—44no—
304C4×C61—44no—
305C4×C61#38614yes—
306C4×C61C2×C4×D411yescharacteristic
307C4×C61C2×C4×D411yescharacteristic
308C4×C61C2×#12111yescharacteristic
309C4×C61C2×#12111yescharacteristic
310Dic61—22no—
311Dic61—22no—
312Dic61—44no—
313Dic61—44no—
314C22×C22⋊Q8—611no—
315C22×#117—611no—
316C2×#419—611no—
317C2×#419D6111yescharacteristic, complemented
318C2×#407—611no—
319#1873—614no—
320#1872—614no—
321#1736—611no—
322#1736—611no—
323C2×C61—24no—
324C2×C61C2×#38611yescharacteristic
325C2×C61C2×#41911yescharacteristic
326C2×C61#289611yescharacteristic
327C2×C61#300712yes—
328C2×C61#309512yes—
329D61—24no—
330D61—42no—
331C23×Q8C2×D6111yescharacteristic
332C24×C4—611no—
333C22×M16—611no—
334C22×M16C2×D6111yescharacteristic
335C2×C22⋊Q8—618no—
336C2×C22⋊Q8—612no—
337C2×C22⋊Q8—612no—
338C2×C22⋊Q8—1221no—
339C2×C22⋊Q8—1224no—
340C2×C22⋊Q8—1221no—
341C2×#117—611no—
342C2×#117—612no—
343C2×#117—1221no—
344C2×#117—1221no—
345C2×#117—1224no—
346C2×#117—1224no—
347C2×#117C2×D6111yescharacteristic
348C22×C4⋊C4—611no—
349C22×C4⋊C4—612no—
350C22×C4⋊C4—1221no—
351C22×C22⋊C4—1221no—
352C22×C22⋊C4—1221no—
353C22×C42—1221no—
354#419—24no—
355#419—1224no—
356C2×#103—614no—
357C2×#103C2×D6114yescomplemented
358#407—1224no—
359C2×C22⋊C8—612no—
360C2×C22⋊C8—612no—
361#340—614no—
362#340—614no—
363C61#2091111yescharacteristic, complemented, Sylow
364C25—611no—
365C22×Q8—21no—
366C22×Q8—22no—
367C22×Q8—22no—
368C22×Q8—22no—
369C22×Q8—22no—
370C22×Q8—22no—
371C22×Q8C4×D6114yescomplemented
372C22×Q8C22×D6112yes—
373C23×C4—21no—
374C23×C4—611no—
375C23×C4—614no—
376C23×C4—611no—
377C23×C4—614no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—1222no—
385C23×C4—1222no—
386C23×C4—1221no—
387C23×C4C22×D6111yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—1222no—
392C2×M16—1222no—
393C2×M16—1222no—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396C22⋊Q8—618no—
397C22⋊Q8—618no—
398C22⋊Q8—1224no—
399C22⋊Q8—1224no—
400C22⋊Q8—1228no—
401C22⋊Q8—1228no—
402#117—22no—
403#117—614no—
404#117—1228no—
405#117—1228no—
406#117—1224no—
407#117—1222no—
408#117—1224no—
409#117C22⋊D6114yes—
410C2×C4⋊C4—614no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—1228no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1224no—
421C2×C4⋊C4C22×D6112yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1228no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1228no—
428C2×C22⋊C4—1221no—
429C2×C42—21no—
430C2×C42—1228no—
431C2×C42—1222no—
432C2×C42—1221no—
433#103—28no—
434#103—1228no—
435C22⋊C8—1224no—
436C22⋊C8—1224no—
437C24—611no—
438C24—611no—
439C24—614no—
440C24—1224no—
441C24—1222no—
442C24—1222no—
443C24—2442no—
444C24D4×D6111yescharacteristic
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—28no—
452C2×Q8—28no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8C2×C4×D6114yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—614no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—612no—
469C22×C4—612no—
470C22×C4—614no—
471C22×C4—614no—
472C22×C4—1228no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1228no—
484C22×C4—1228no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1228no—
488C22×C4—1228no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1224no—
494C22×C4—2444no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4C2×C4×D6112yes—
498C22×C4#1175822212yes—
499C22×C4D4×D6112yes—
500C22×C4D4×D6112yes—
501C22×C4D4×D6111yescharacteristic
502C22×C4#1175822412yes—
503C22×C4C2×C22⋊D6112yes—
504C22×C4C23×D6111yescharacteristic
505M16—44no—
506M16—2444no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C4⋊C4—22no—
514C4⋊C4—614no—
515C4⋊C4—614no—
516C4⋊C4—1228no—
517C4⋊C4—1228no—
518C4⋊C4—1228no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1224no—
522C4⋊C4C2×C22⋊D6114yes—
523C22⋊C4—24no—
524C22⋊C4—1228no—
525C22⋊C4—1228no—
526C22⋊C4—1228no—
527C22⋊C4—1224no—
528C42—24no—
529C42—1228no—
530C42—1224no—
531C23—22no—
532C23—22no—
533C23—612no—
534C23—614no—
535C23—612no—
536C23—1228no—
537C23—1221no—
538C23—1221no—
539C23—1222no—
540C23—1221no—
541C23—1221no—
542C23—1222no—
543C23—1222no—
544C23—1228no—
545C23—2444no—
546C23—2448no—
547C23—2448no—
548C23C22⋊C4×D6112yes—
549C23#1451974512yes—
550C23#1451975912yes—
551C23C2×D4×D6111yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—42no—
555Q8—42no—
556Q8—48no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1228no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1228no—
580C2×C4—1228no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—24416no—
584C2×C4—2448no—
585C2×C4—2442no—
586C2×C4—2444no—
587C2×C4—2442no—
588C2×C4C22⋊C4×D6112yes—
589C2×C4#1451973714yes—
590C2×C4C4×C22⋊D6114yes—
591C2×C4#1451974512yes—
592C2×C4#1451974814yes—
593C2×C4#1451975014yes—
594C2×C4#1451975912yes—
595C2×C4C22×C4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C22×C22⋊D6111yescharacteristic
599C2×C4C2×#1175822411yescharacteristic
600C2×C4C2×#1175822211yescharacteristic
601C8—42no—
602C8—2442no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—22no—
607C22—614no—
608C22—1222no—
609C22—1222no—
610C22—1224no—
611C22—1224no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—24416no—
616C22—2442no—
617C22—2442no—
618C22#1683963814yes—
619C22C2×C22⋊C4×D6111yescharacteristic
620C22C2×#1451974511yescharacteristic
621C22C2×#1451975911yescharacteristic
622C22#137×D6112yes—
623C22#1684056112yes—
624C4—44no—
625C4—44no—
626C4—1222no—
627C4—1222no—
628C4—2444no—
629C4—2444no—
630C4#1683963814yes—
631C4C2×#1451973711yescharacteristic
632C4C2×C4×C22⋊D6111yescharacteristic
633C4C2×#1451974811yescharacteristic
634C4C2×#1451975011yescharacteristic
635C2—24no—
636C2—1224no—
637C2—2442no—
638C2C2×#1683963811yescharacteristic
639C2#419×D6111yescharacteristic
640C2#2145433711yescharacteristic
641C2#2145507512yes—
642C2#2145576512yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#20911
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21446128C2semidirect
6. #21446123C2semidirect
7. #21445540C2semidirect
8. #21445539C2semidirect
9. C2×C61×#419C2semidirect
15. C2×#14519753C4semidirect
31. C2×#14519701C22semidirect
35. #16839525C22semidirect
36. #16839524C22semidirect
41. C2×#103×C61C22semidirect
65. #14519753C2×C4semidirect
74. C2×C4⋊Dic61C2×C4semidirect
101. C22×Q8×C61C2×C4semidirect
302. C2×D61#419semidirect
317. C2×#419D61semidirect
357. C2×#103C2×D61semidirect
363. C61#20911semidirect
371. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197532
C2×#117×D612
C2×#168397712
C2×#168397252
#214461282
#214461232
#214455402
#214455392
C2×C61×#4192
#2091161