Finite Group Databasebeta

← #29689112

Subgroups — #29689112

Order 15616.

1518 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840326C211yesmaximal, characteristic
2#21448411C211yesmaximal, characteristic, complemented
3#21446257C212yesmaximal, complemented
4#21446247C212yesmaximal, complemented
5#21446139C212yesmaximal, complemented
6#21446134C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445751C212yesmaximal, complemented
9C2×C61×#408C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11#16840326—24no—
12#16840326C418yescomplemented
13C2×#14519773—21no—
14C2×#14519766—21no—
15C2×#14519751C2212yes—
16C2×#14519751C2212yes—
17C2×#14519745—21no—
18C2×#14519745—21no—
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic, complemented
21C22×C4⋊Dic61C2211yescharacteristic, complemented
22C2×#14519699C2212yes—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26#16839669—21no—
27#16839665—21no—
28#16839642—21no—
29#16839642—21no—
30#16839622C2212yes—
31#16839622C2212yescomplemented
32#16839525C2212yescomplemented
33#16839525C2212yescomplemented
34#16839525C2212yescomplemented
35#16839525C2212yescomplemented
36C22×#41×C61C2211yescharacteristic
37C22×M16×C61C2211yescharacteristic, complemented
38C23×C8×C61C2211yescharacteristic, complemented
39C61×#408—24no—
40C2×C22⋊C8×C61C2212yes—
41C2×C22⋊C8×C61C2212yescomplemented
42C24×D61—21no—
43C22×D244—21no—
44C22×D244—22no—
45C22×D244—41no—
46C22×D244—41no—
47C22×D244—41no—
48C22×D244—41no—
49C22×D244C2×C414yescomplemented
50C22×D244C2312yes—
51C22×C22⋊D61—22no—
52C22×C22⋊D61—22no—
53C23×Dic61—21no—
54#14519773—28no—
55#14519766—28no—
56#14519751—24no—
57#14519751—24no—
58#14519751C2×C418yes—
59#14519751C2×C418yescomplemented
60#14519745—28no—
61#14519745—28no—
62#14519736—22no—
63#14519736—41no—
64#14519736—41no—
65#14519736C2×C414yescomplemented
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—41no—
68C2×C4⋊Dic61—41no—
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2312yes—
71C2×C4⋊Dic61C2312yes—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80#14519699—42no—
81#14519699—42no—
82#14519699—42no—
83#14519699—42no—
84#14519684—22no—
85#14519684—22no—
86C2×#41×C61—41no—
87C2×#41×C61—41no—
88C2×#41×C61—41no—
89C2×#41×C61—41no—
90C2×#41×C61C2×C414yes—
91C2×#41×C61D414yes—
92C22×Q8×C61D411yescharacteristic
93C22×D4×C61D411yescharacteristic
94C22×D4×C61D411yescharacteristic
95C22×D4×C61D411yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C23×C4×C61C2311yescharacteristic
98C23×C4×C61C2311yescharacteristic
99C2×M16×C61—22no—
100C2×M16×C61—41no—
101C2×M16×C61—41no—
102C22×C8×C61—22no—
103C22×C8×C61—41no—
104C22×C8×C61—41no—
105C22×C8×C61C2312yes—
106C22×C8×C61C2312yes—
107C22⋊C8×C61—24no—
108C22⋊C8×C61—24no—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111C24×C61C2×D411yescharacteristic
112C23×D61—22no—
113C23×D61—24no—
114C23×D61—42no—
115C23×D61—42no—
116C23×D61—82no—
117C23×D61—82no—
118#41×C61—44no—
119#41×C61—44no—
120#41×C61—44no—
121#41×C61—44no—
122C2×Q8×C61—41no—
123C2×Q8×C61—41no—
124C2×Q8×C61C22⋊C414yes—
125C2×D4×C61—41no—
126C2×D4×C61—41no—
127C2×D4×C61—41no—
128C2×D4×C61—41no—
129C2×D4×C61—41no—
130C2×D4×C61—41no—
131C2×D4×C61C22⋊C414yes—
132C2×D4×C61C22⋊C414yes—
133C2×D4×C61C22⋊C414yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C412yes—
148C22×C4×C61C22×C412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2×D412yes—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—28no—
157C2×C22⋊D61—28no—
158C2×C22⋊D61—82no—
159C2×C22⋊D61—82no—
160C2×C22⋊D61—82no—
161C2×C22⋊D61—82no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—42no—
166C22×Dic61—82no—
167C2×D244—22no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—22no—
171C2×D244—24no—
172C2×D244—22no—
173C2×D244—44no—
174C2×D244—44no—
175C2×D244—42no—
176C2×D244—42no—
177C2×D244—44no—
178C2×D244—44no—
179C2×D244C22×C414yes—
180M16×C61—44no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C2×C8×C61—44no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—82no—
191C22⋊Dic61—82no—
192#11758198—28no—
193#11758198—28no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—42no—
200C4⋊Dic61C22×C414yes—
201C4×Dic61—24no—
202C4×Dic61—42no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61C2×C22⋊C412yes—
216C23×C61C2×C22⋊C412yes—
217C23×C61C2×C22⋊C412yes—
218C23×C61C22×D411yescharacteristic
219C22×D61—24no—
220C22×D61—48no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—42no—
227C22×D61—88no—
228C22×D61—88no—
229C22×D61—84no—
230C22×D61—84no—
231Q8×C61—44no—
232D4×C61—44no—
233D4×C61—44no—
234D4×C61—44no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C22⋊D61—88no—
263C22⋊D61—88no—
264C2×Dic61—24no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—84no—
274D244—24no—
275D244—24no—
276D244—48no—
277D244—42no—
278D244—42no—
279C8×C61—42no—
280C8×C61—42no—
281#21770—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—24no—
286C22×C61—24no—
287C22×C61—24no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61C22×C22⋊C411yescharacteristic
298C22×C61#53411yescharacteristic
299C22×C61#53411yescharacteristic
300C22×C61#57612yes—
301C22×C61#57712yes—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C2×D61—816no—
306C2×D61—82no—
307C2×D61—82no—
308C2×D61—82no—
309C2×D61—82no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61#51811yescharacteristic
314C4×C61#51811yescharacteristic
315C4×C61#53411yescharacteristic
316C4×C61#53411yescharacteristic
317Dic61—44no—
318Dic61—82no—
319Dic61—82no—
320C2×#535—611no—
321C2×#408D6111yescharacteristic, complemented
322#2311—611no—
323#1895—612no—
324#1893—612no—
325#1875—612no—
326#1874—612no—
327#1825—612no—
328#1786—612no—
329C2×C61—42no—
330C2×C61—42no—
331C2×C61—42no—
332C2×C61#230811yescharacteristic
333C2×C61#289712yes—
334C2×C61#319812yes—
335C2×C61#319911yescharacteristic
336C2×C61#320011yescharacteristic
337D61—84no—
338C22×#41C2×D6111yescharacteristic
339C23×D4—611no—
340C22×M16C2×D6111yescharacteristic, complemented
341C23×C8C2×D6111yescharacteristic, complemented
342#535—618no—
343#535—1224no—
344C2×#127—1221no—
345C2×#124—1221no—
346C2×#121—612no—
347C2×#121—612no—
348C2×#120—1221no—
349C2×#120—1221no—
350C2×#117—611no—
351C2×#117—611no—
352C22×C4⋊C4—611no—
353C2×#102—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#102—612no—
357#408—24no—
358C2×C22⋊C8C2×D6112yes—
359C2×C22⋊C8C2×D6112yescomplemented
360#391—1221no—
361#390—1221no—
362#386—1221no—
363#386—1221no—
364#380—612no—
365#380—612no—
366#340—612no—
367#340—612no—
368#340—612no—
369#340—612no—
370C61#2177011yescharacteristic, complemented, Sylow
371C25—1221no—
372C2×#41—41no—
373C2×#41—41no—
374C2×#41—41no—
375C2×#41—41no—
376C2×#41C4×D6114yes—
377C2×#41C22⋊D6114yes—
378C22×Q8D24411yescharacteristic
379C22×D4—614no—
380C22×D4—612no—
381C22×D4—1222no—
382C22×D4—1222no—
383C22×D4—1221no—
384C22×D4—1222no—
385C22×D4—2441no—
386C22×D4—2441no—
387C22×D4—2441no—
388C22×D4—2441no—
389C22×D4D24411yescharacteristic
390C22×D4D24411yescharacteristic
391C22×D4D24411yescharacteristic
392C23×C4—1221no—
393C23×C4C22×D6111yescharacteristic
394C23×C4C22×D6111yescharacteristic
395C23×C4C22×D6111yescharacteristic
396C2×M16—22no—
397C2×M16—41no—
398C2×M16—41no—
399C22×C8—22no—
400C22×C8—41no—
401C22×C8—41no—
402C22×C8C22×D6112yes—
403C22×C8C22×D6112yes—
404#127—1228no—
405#124—1228no—
406#121—618no—
407#121—618no—
408#121—1224no—
409#121—1224no—
410#120—1228no—
411#120—1228no—
412#117—614no—
413#117—1222no—
414#117—2441no—
415#117—2441no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—2441no—
421C2×C4⋊C4—2441no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430#102—2442no—
431#102—2442no—
432#102—2442no—
433#102—2442no—
434C22⋊C8—24no—
435C22⋊C8—24no—
436#95—1222no—
437#95—1222no—
438C24—1224no—
439C24—1222no—
440C24—4882no—
441C24—4882no—
442C24—2442no—
443C24—2442no—
444C24C2×D24411yescharacteristic
445C24C2×D24411yescharacteristic
446C24C2×D24411yescharacteristic
447#41—44no—
448#41—44no—
449#41—44no—
450#41—44no—
451C2×Q8—41no—
452C2×Q8—41no—
453C2×Q8#1175819814yes—
454C2×D4—41no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—614no—
461C2×D4—1228no—
462C2×D4—1228no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4—1222no—
468C2×D4—1224no—
469C2×D4—4882no—
470C2×D4—4882no—
471C2×D4—4882no—
472C2×D4—4882no—
473C2×D4—2444no—
474C2×D4—2444no—
475C2×D4—2442no—
476C2×D4—2442no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4#1175819814yes—
480C2×D4#1175819814yes—
481C2×D4#1175819814yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—4882no—
498C22×C4—2442no—
499C22×C4C2×C4×D6112yes—
500C22×C4C2×C4×D6112yes—
501C22×C4C2×C4×D6112yes—
502C22×C4C2×D24411yescharacteristic
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4C2×C22⋊D6112yes—
506C22×C4C2×C22⋊D6112yes—
507C22×C4C2×C22⋊D6112yes—
508C22×C4C23×D6111yescharacteristic
509M16—44no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—44no—
517C4⋊C4—614no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—2442no—
524C22⋊C4—1224no—
525C22⋊C4—1228no—
526C22⋊C4—1228no—
527C22⋊C4—4882no—
528C22⋊C4—4882no—
529C22⋊C4—2442no—
530C42—1224no—
531C42—2442no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—1224no—
545C23—4888no—
546C23—4888no—
547C23—4884no—
548C23—4884no—
549C23—2448no—
550C23—2442no—
551C23—2442no—
552C23—2442no—
553C23—2442no—
554C23—2442no—
555C23—2442no—
556C23C2×#1175819812yes—
557C23C2×#1175819812yes—
558C23C2×#1175819812yes—
559C23C22×D24411yescharacteristic
560Q8—44no—
561D4—44no—
562D4—44no—
563D4—44no—
564D4—1224no—
565D4—1224no—
566D4—4888no—
567D4—4888no—
568D4—2448no—
569D4—2442no—
570D4—2442no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—1224no—
590C2×C4—4884no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4C2×#1175819812yes—
600C2×C4C2×#1175819812yes—
601C2×C4C2×#1175819812yes—
602C2×C4C22×C4×D6111yescharacteristic
603C2×C4C22×C22⋊D6111yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982211yescharacteristic
606C2×C4#1451982211yescharacteristic
607C2×C4#1451982211yescharacteristic
608C8—42no—
609C8—42no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—48816no—
626C22—4882no—
627C22—4882no—
628C22—4882no—
629C22—4882no—
630C22—2444no—
631C22—2444no—
632C22—2444no—
633C22C22×#1175819811yescharacteristic
634C22#1684032411yescharacteristic
635C22#1684032411yescharacteristic
636C22#1684056712yes—
637C22#1684057412yes—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—4882no—
642C4—4882no—
643C4—2444no—
644C4#1684023511yescharacteristic
645C4#1684023511yescharacteristic
646C4#1684032111yescharacteristic
647C4#1684032111yescharacteristic
648C2—42no—
649C2—42no—
650C2—42no—
651C2—4884no—
652C2#2144839411yescharacteristic
653C2#2145434612yes—
654C2#2145667312yes—
655C2#2145668411yescharacteristic
656C2#2145669211yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#408C2
socleC23×C61C22×D4
prime radicalC2×#16840326C2
cosocleC22×C4×C61C24
hypercenterC2×#408D61
hypocenterC61#21770
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. #21448411C2semidirect
3. #21446257C2semidirect
4. #21446247C2semidirect
5. #21446139C2semidirect
6. #21446134C2semidirect
7. #21445915C2semidirect
8. #21445751C2semidirect
9. C2×C61×#408C2semidirect
12. #16840326C4semidirect
20. C2×#14519736C22semidirect
21. C22×C4⋊Dic61C22semidirect
31. #16839622C22semidirect
32. #16839525C22semidirect
33. #16839525C22semidirect
34. #16839525C22semidirect
35. #16839525C22semidirect
37. C22×M16×C61C22semidirect
38. C23×C8×C61C22semidirect
41. C2×C22⋊C8×C61C22semidirect
49. C22×D244C2×C4semidirect
59. #14519751C2×C4semidirect
65. #14519736C2×C4semidirect
321. C2×#408D61semidirect
340. C22×M16C2×D61semidirect
341. C23×C8C2×D61semidirect
359. C2×C22⋊C8C2×D61semidirect
370. C61#21770semidirect

Maximal subgroups

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

subgroupindex
C2×#168403262
#214484112
#214462572
#214462472
#214461392
#214461342
#214459152
#214457512
C2×C61×#4082
#2177061