Finite Group Databasebeta

← #29689007

Subgroups — #29689007

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840341C211yesmaximal, characteristic
2#21454263C211yesmaximal, characteristic, complemented
3#21454262C211yesmaximal, characteristic, complemented
4#21446287C211yesmaximal, characteristic, complemented
5#21446286C212yesmaximal, complemented
6#21446285C211yesmaximal, characteristic, complemented
7#21446283C211yesmaximal, characteristic, complemented
8#21446279C212yesmaximal, complemented
9#21445898C212yesmaximal, complemented
10#21445744C212yesmaximal, complemented
11C61×#1900C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758224C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C2×M16×D61C2212yes—
16#16840341—22no—
17#16840341—22no—
18#16840341—22no—
19#16840341—22no—
20C2×#14519757C2211yescharacteristic
21C2×#14519756C2211yescharacteristic
22C2×#14519754C412yes—
23C2×#14519750C412yes—
24C2×#14519749C412yes—
25C2×#14519748C412yes—
26C2×#14519748C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×#14519695C2212yes—
29C2×#14519695C2212yes—
30C2×#14519694C2212yes—
31#16839730—22no—
32#16839728—22no—
33#16839727—22no—
34#16839727—22no—
35#16839726C2211yescharacteristic
36#16839725C2212yes—
37#16839725C2212yes—
38#16839725C2212yes—
39#16839531C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×#100×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#343C2212yes—
47C2×#11758222—42no—
48C2×#11758222—42no—
49C2×#11758222D412yes—
50C2×#11758222D412yes—
51C2×#11758222D412yes—
52C2×#11758222D412yes—
53C2×#11758224—22no—
54C2×#11758224—22no—
55C2×#11758224—42no—
56C2×#11758224—42no—
57C22×Dic122C2×C411yescharacteristic
58C22×Dic122C2311yescharacteristic
59C22×D244C2×C411yescharacteristic
60C22×C22⋊D61C2×C412yes—
61C22×C22⋊D61C2×C412yes—
62C22×C22⋊D61C2×C411yescharacteristic
63C22×C22⋊D61C2×C411yescharacteristic
64C23×Dic61C2×C412yes—
65C22×C4×D61C2×C412yes—
66C22×C4×D61C2311yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C2×#11758194—21no—
69C2×#11758194—42no—
70C2×#11758194C2312yes—
71C2×#11758194C2312yes—
72C2×#11758189—22no—
73M16×D61—42no—
74M16×D61—42no—
75C22×#92418—21no—
76C2×C8×D61—22no—
77#14519757—22no—
78#14519757—22no—
79#14519756—21no—
80#14519756—21no—
81#14519756—21no—
82#14519756—21no—
83#14519754—24no—
84#14519754—24no—
85#14519750—24no—
86#14519750—24no—
87#14519749—24no—
88#14519749—24no—
89#14519748—24no—
90#14519748—24no—
91#14519748—22no—
92#14519748—22no—
93C2×C4⋊Dic61C2×C412yes—
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C2×C22⋊Dic61C2×C412yes—
100C2×C22⋊Dic61C2×C412yes—
101C2×#11758198C2×C412yes—
102#14519695—42no—
103#14519695—42no—
104#14519694—42no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519689—24no—
108#14519689—24no—
109#14519689—24no—
110C22×D4×C61C2×C412yes—
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61C2311yescharacteristic
113C2×M16×C61—21no—
114C2×M16×C61—42no—
115C2×M16×C61C2312yes—
116C2×M16×C61C2312yes—
117C22×C8×C61—21no—
118#121×C61—21no—
119#121×C61—21no—
120#121×C61—21no—
121#121×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61C2×C412yes—
124#100×C61—42no—
125C22⋊C8×C61—21no—
126C22⋊C8×C61—21no—
127C24×C61C22×C412yes—
128C24×C61C22×C411yescharacteristic
129C23×D61C22×C411yescharacteristic
130C23×D61C22×C411yescharacteristic
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61—82no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—24no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—44no—
150C2×C22⋊D61—42no—
151C2×C22⋊D61—42no—
152C2×C22⋊D61—82no—
153C2×C22⋊D61C22⋊C412yes—
154C2×C22⋊D61C22⋊C412yes—
155C2×C22⋊D61C22⋊C412yes—
156C2×C22⋊D61C22⋊C412yes—
157C22×Dic61—24no—
158C22×Dic61—44no—
159C22×Dic61C22×C412yes—
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C412yes—
163#11758224—44no—
164#11758224—44no—
165#11758224—44no—
166#11758224—44no—
167#11758222—42no—
168#11758222—42no—
169#11758222—42no—
170#11758222—42no—
171#11758222—42no—
172#11758222—42no—
173#11758222—42no—
174#11758222—42no—
175C2×D244—42no—
176C2×D244C22⋊C412yes—
177C2×D244C22⋊C412yes—
178C2×C4×D61—22no—
179C2×C4×D61—22no—
180C2×C4×D61—22no—
181C2×C4×D61—22no—
182C2×C4×D61—24no—
183C2×C4×D61—82no—
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D412yes—
187C2×C4×D61C2×D412yes—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—22no—
191C2×Dic122—22no—
192C2×Dic122—42no—
193C2×Dic122C22⋊C412yes—
194C2×Dic122C22⋊C412yes—
195M16×C61—22no—
196M16×C61—22no—
197M16×C61—22no—
198M16×C61—22no—
199M16×C61—42no—
200C2×C8×C61—22no—
201C2×C8×C61—22no—
202C2×C8×C61—42no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207C22⋊Dic61—22no—
208C22⋊Dic61—22no—
209C22⋊Dic61—24no—
210#11758198—24no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—24no—
214#11758196—22no—
215#11758196—22no—
216#11758196—24no—
217#11758194—22no—
218#11758194—22no—
219#11758194—22no—
220#11758194—22no—
221#11758194—42no—
222C2×#92418—22no—
223C2×#92418—22no—
224C2×#92418—42no—
225#11758189—44no—
226C8×D61—44no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61—24no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—42no—
233C23×C61—42no—
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C2×C22⋊C411yescharacteristic
236C23×C61C23×C411yescharacteristic
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—41no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61C2×C22⋊C412yes—
244C22×D61C2×C22⋊C412yes—
245D4×C61—42no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—82no—
249D4×C61—82no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—42no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61H(5,𝔽2)12yes—
264C2×C4×C61H(5,𝔽2)12yes—
265C22⋊D61—42no—
266C22⋊D61—42no—
267C22⋊D61—42no—
268C22⋊D61—42no—
269C22⋊D61—48no—
270C22⋊D61—84no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—24no—
276C2×Dic61—48no—
277C2×Dic61—42no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—42no—
281D244—42no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—84no—
291Dic122—42no—
292Dic122—42no—
293Dic122—42no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—42no—
297C8×C61—42no—
298#92418—42no—
299#92418—42no—
300#21757—611nomaximal, Sylow
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—21no—
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—82no—
313C22×C61—84no—
314C22×C61C22×C22⋊C411yescharacteristic
315C22×C61#51812yes—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—44no—
323C2×D61—82no—
324C4×C61—21no—
325C4×C61—21no—
326C4×C61—42no—
327C4×C61#53412yes—
328C4×C61#53411yescharacteristic
329C4×C61#53411yescharacteristic
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—42no—
334C2×#537—611no—
335#2879—611no—
336#2879—611no—
337#1901—611no—
338#1901—612no—
339#1901—611no—
340#1900—612no—
341#1900—611no—
342#1900D6111yescharacteristic, complemented
343#1821—612no—
344#1785—612no—
345C2×C61—22no—
346C2×C61—41no—
347C2×C61—41no—
348C2×C61—82no—
349C2×C61#230811yescharacteristic
350C2×C61#288811yescharacteristic
351C2×C61#288911yescharacteristic
352D61—42no—
353D61—42no—
354C22×#41—611no—
355C22×#41—611no—
356C22×M16—611no—
357C22×M16—612no—
358C22×M16C2×D6111yescharacteristic
359#537—1222no—
360#537—1222no—
361#537—1222no—
362#537—1222no—
363C2×C22⋊Q8—611no—
364C2×C22⋊Q8—611no—
365C2×C22⋊Q8—612no—
366C2×#121—612no—
367C2×#121—612no—
368C2×#121—611no—
369C2×#121—612no—
370C2×#121C2×D6111yescharacteristic
371C22×C4⋊C4—611no—
372C2×#101—612no—
373C2×#101—612no—
374C2×#100—612no—
375C2×#100C2×D6112yes—
376#408—1222no—
377#408—1222no—
378#408—1222no—
379#408—1222no—
380#407—611no—
381#407—612no—
382#407—612no—
383#407—612no—
384#407C2×D6111yescharacteristic
385#343—612no—
386#343—612no—
387#343—612no—
388#343C2×D6112yes—
389C61#2175711yescharacteristic, complemented, Sylow
390C2×#41—612no—
391C2×#41—612no—
392C2×#41—612no—
393C2×#41—612no—
394C2×#41—1222no—
395C2×#41—1222no—
396C2×#41—2442no—
397C2×#41—2442no—
398C2×#41—2442no—
399C2×#41—2442no—
400C22×Q8—611no—
401C22×Q8—611no—
402C22×D4—611no—
403C22×D4—611no—
404C22×D4—612no—
405C22×D4—612no—
406C22×D4—611no—
407C22×D4C4×D6112yes—
408C22×D4C22×D6111yescharacteristic
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—612no—
412C23×C4—612no—
413C23×C4C22×D6111yescharacteristic
414C2×M16—21no—
415C2×M16—42no—
416C2×M16—612no—
417C2×M16—612no—
418C2×M16—1222no—
419C2×M16—1221no—
420C2×M16—2442no—
421C2×M16—2442no—
422C2×M16—2442no—
423C2×M16C22×D6112yes—
424C2×M16C22×D6112yes—
425C22×C8—21no—
426C22×C8—1222no—
427C22×C8—1221no—
428C22⋊Q8—1224no—
429C22⋊Q8—1224no—
430C22⋊Q8—1222no—
431C22⋊Q8—1222no—
432C22⋊Q8—1221no—
433C22⋊Q8—1221no—
434C22⋊Q8—1221no—
435C22⋊Q8—1221no—
436#121—21no—
437#121—21no—
438#121—21no—
439#121—21no—
440#121—1224no—
441#121—1224no—
442#121—1222no—
443#121—1222no—
444#121—1224no—
445#121—1224no—
446#121—1224no—
447#121—1224no—
448C2×C4⋊C4—612no—
449C2×C4⋊C4—612no—
450C2×C4⋊C4—612no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—1222no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—612no—
456C2×C22⋊C4—612no—
457C2×C22⋊C4—612no—
458C2×C22⋊C4C4×D6112yes—
459#101—2442no—
460#101—2442no—
461#100—42no—
462#100—2442no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8—1224no—
466C22⋊C8—1224no—
467C22⋊C8—1224no—
468C22⋊C8—1221no—
469C22⋊C8—1221no—
470C24—611no—
471C24—611no—
472C24C2×C4×D6112yes—
473C24C2×C4×D6111yescharacteristic
474#41—2442no—
475#41—2442no—
476#41—2442no—
477#41—2442no—
478#41—2442no—
479#41—2442no—
480#41—2442no—
481#41—2442no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485#41—2444no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1222no—
491C2×Q8—1222no—
492C2×Q8—2442no—
493C2×D4—22no—
494C2×D4—22no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—82no—
500C2×D4—612no—
501C2×D4—612no—
502C2×D4—612no—
503C2×D4—612no—
504C2×D4—612no—
505C2×D4—612no—
506C2×D4—1224no—
507C2×D4—1224no—
508C2×D4—4882no—
509C2×D4—2442no—
510C2×D4—2442no—
511C2×D4—2442no—
512C2×D4—2444no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—612no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—1224no—
524C22×C4—1224no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—4882no—
530C22×C4—2444no—
531C22×C4C2×C4×D6111yescharacteristic
532C22×C4C2×C4×D6112yes—
533C22×C4D4×D6111yescharacteristic
534C22×C4D4×D6111yescharacteristic
535C22×C4D4×D6111yescharacteristic
536C22×C4D4×D6111yescharacteristic
537C22×C4C23×D6111yescharacteristic
538M16—22no—
539M16—22no—
540M16—22no—
541M16—22no—
542M16—42no—
543M16—1222no—
544M16—1222no—
545M16—1222no—
546M16—1222no—
547M16—2444no—
548M16—2442no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—42no—
552C2×C8—1222no—
553C2×C8—1222no—
554C2×C8—2444no—
555C2×C8—2442no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—1224no—
559C4⋊C4—1224no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C4⋊C4—1222no—
563C4⋊C4—1222no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C23—22no—
571C23—22no—
572C23—24no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—612no—
578C23—612no—
579C23—1222no—
580C23—1222no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23C22⋊C4×D6111yescharacteristic
586C23C22⋊C4×D6111yescharacteristic
587C23C22×C4×D6111yescharacteristic
588Q8—2442no—
589Q8—2442no—
590Q8—2442no—
591Q8—2441no—
592Q8—2441no—
593D4—42no—
594D4—41no—
595D4—41no—
596D4—82no—
597D4—82no—
598D4—4884no—
599D4—2442no—
600D4—2442no—
601D4—2442no—
602D4—2442no—
603D4—2442no—
604D4—2442no—
605D4—2448no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1224no—
630C2×C4—4884no—
631C2×C4—2448no—
632C2×C4—2442no—
633C2×C4C22⋊C4×D6111yescharacteristic
634C2×C4C22⋊C4×D6111yescharacteristic
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4#1451982212yes—
638C2×C4#1451982312yes—
639C8—42no—
640C8—42no—
641C8—2442no—
642C8—2442no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—82no—
655C22—84no—
656C22—1222no—
657C22—1222no—
658C22—1222no—
659C22—1222no—
660C22—1222no—
661C22—1222no—
662C22—4882no—
663C22—2444no—
664C22C2×C22⋊C4×D6111yescharacteristic
665C22#1684023412yes—
666C4—21no—
667C4—21no—
668C4—42no—
669C4—1222no—
670C4—1222no—
671C4—2442no—
672C4—2442no—
673C4#1684032012yes—
674C4#1684032211yescharacteristic
675C4#1684032311yescharacteristic
676C2—22no—
677C2—41no—
678C2—41no—
679C2—82no—
680C2—2442no—
681C2—2442no—
682C2#2144839311yescharacteristic
683C2#2145429511yescharacteristic
684C2#2145430011yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1900C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840341C2
cosocleC22×C4×C61C24
hypercenter#1900D61
hypocenterC61#21757
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1900

Complements

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

NH ≅ G/Ntype
2. #21454263C2semidirect
3. #21454262C2semidirect
4. #21446287C2semidirect
5. #21446286C2semidirect
6. #21446285C2semidirect
7. #21446283C2semidirect
8. #21446279C2semidirect
9. #21445898C2semidirect
10. #21445744C2semidirect
11. C61×#1900C2semidirect
342. #1900D61semidirect
389. C61#21757semidirect

Maximal subgroups

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

subgroupindex
C2×#168403412
#214542632
#214542622
#214462872
#214462862
#214462852
#214462832
#214462792
#214458982
#214457442
C61×#19002
#2175761