Finite Group Databasebeta

← #29716410

Subgroups — #29716410

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840492C211yesmaximal, characteristic, complemented
2C2×#16840050C211yesmaximal, characteristic
3C2×#16839853C211yesmaximal, characteristic
4#21450739C211yesmaximal, characteristic
5#21447151C211yesmaximal, characteristic, complemented
6#21447149C211yesmaximal, characteristic, complemented
7#21446953C211yesmaximal, characteristic
8#21446778C211yesmaximal, characteristic, complemented
9#21446753C211yesmaximal, characteristic, complemented
10#21446177C211yesmaximal, characteristic
11#21446171C211yesmaximal, characteristic
12#21446090C211yesmaximal, characteristic, complemented
13#21446088C211yesmaximal, characteristic, complemented
14#21445823C211yesmaximal, characteristic, complemented
15C61×#2023C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17#16840492—24no—
18#16840492—24no—
19C2×#14519776—21no—
20C2×#14519776C2211yescharacteristic
21C2×#14519776C2211yescharacteristic
22C2×#14519775—21no—
23C2×#14519775—21no—
24C2×#14519754—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519742—21no—
28C2×#14519742C2211yescharacteristic
29C2×#14519742C2211yescharacteristic
30C2×D4×Dic61C2211yescharacteristic
31#16840050—24no—
32#16839853—24no—
33C2×#14519709C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35C2×#14519708C2211yescharacteristic
36C2×#14519706C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519702C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519698C2211yescharacteristic
41C2×C8×Dic61C2211yescharacteristic
42#16839685—21no—
43#16839685—21no—
44#16839665C2211yescharacteristic
45#16839661C2211yescharacteristic
46#16839660C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839620—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122D411yescharacteristic
66C22×Dic122C2311yescharacteristic
67C23×Dic61—21no—
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200Q814yescomplemented
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519776—28no—
74#14519776—22no—
75#14519776—22no—
76#14519776—22no—
77#14519776D414yes—
78#14519775—28no—
79#14519775—28no—
80#14519754—44no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85#14519742—28no—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742D414yes—
90D4×Dic61—22no—
91D4×Dic61—22no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61D411yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196D411yescharacteristic
106C2×#11758196D411yescharacteristic
107C2×C22⋊Dic61—21no—
108C2×C22⋊Dic61—21no—
109C2×C22⋊Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519709—22no—
116#14519708—22no—
117#14519708—22no—
118#14519706—22no—
119#14519702—22no—
120#14519702—22no—
121#14519700—22no—
122#14519698—22no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61—21no—
127C22×D4×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C4⋊Q8×C61—22no—
131C4⋊Q8×C61—22no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C22⋊C4×C61—21no—
138C2×C42×C61C2311yescharacteristic
139#102×C61—22no—
140#102×C61—22no—
141C24×C61—21no—
142C2×Q8×C61—24no—
143C2×Q8×C61—81no—
144C2×Q8×C61—81no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×Q812yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—24no—
174C2×Dic122—24no—
175C2×Dic122—81no—
176C2×Dic122—81no—
177C2×Dic122—81no—
178C2×Dic122—81no—
179C2×Dic122SD1614yescomplemented
180C2×Dic122C2×D412yes—
181C2×Dic122C2×Q812yes—
182C2×Dic122#4112yes—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×D412yes—
196C22⋊C4×C61—42no—
197C22⋊C4×C61—81no—
198C42×C61—21no—
199C42×C61—21no—
200C22⋊Dic61—42no—
201C22⋊Dic61—42no—
202C22⋊Dic61—81no—
203#11758200—28no—
204C4⋊Dic61—21no—
205C4⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—42no—
211C4⋊Dic61C2×D412yes—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—22no—
218#11758196—22no—
219#11758196—42no—
220#11758196—42no—
221#11758196—81no—
222C4×Dic61—24no—
223C4×Dic61—24no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—24no—
227C4×Dic61C2×D412yes—
228C4×Dic61C2×D412yes—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418C2×Q812yes—
235C23×C61—22no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61—41no—
242C23×C61C22×D411yescharacteristic
243Q8×C61—84no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C2×SD1612yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×Q811yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—84no—
289C2×Dic61#12012yes—
290C2×Dic61#12012yes—
291C2×Dic61C2×SD1612yes—
292C2×Dic61C2×SD1612yes—
293C2×Dic61#13712yes—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122—28no—
298Dic122—84no—
299Dic122—84no—
300Dic122C22⋊Q814yescomplemented
301C8×C61—41no—
302C8×C61—41no—
303#92418—22no—
304#92418—41no—
305#24329—611nomaximal, Sylow
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61—81no—
313C22×C61C2×#12011yescharacteristic
314C22×C61#54811yescharacteristic
315C22×C61D4×Q811yescharacteristic
316C22×C61C22×SD1611yescharacteristic
317C22×C61C2×#13711yescharacteristic
318C22×C61#57611yescharacteristic
319C22×C61#57711yescharacteristic
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61C2×C22⋊Q811yescharacteristic
326C4×C61#53611yescharacteristic
327C4×C61#54811yescharacteristic
328C4×C61D4×Q811yescharacteristic
329Dic61—24no—
330Dic61—22no—
331Dic61—22no—
332Dic61—41no—
333Dic61—42no—
334Dic61—81no—
335Dic61—81no—
336Dic61#44814yes—
337C2×Q82—611no—
338C2×#476—611no—
339C2×C4×SD16—611no—
340#2538—611no—
341#2058—611no—
342#2058—611no—
343#2023—611no—
344#2023D6111yescharacteristic
345#1990—611no—
346#1984—611no—
347#1880—611no—
348#1880—611no—
349#1865—611no—
350#1865—611no—
351#1803—611no—
352C2×C61—82no—
353C2×C61#240911yescharacteristic
354C2×C61C2×#44811yescharacteristic
355C2×C61#301211yescharacteristic
356C2×C61#336211yescharacteristic
357C2×C61#336911yescharacteristic
358C2×C61Q8×SD1611yescharacteristic
359C2×C61#338511yescharacteristic
360C22×SD16—611no—
361Q82—1224no—
362Q82—1224no—
363C2×C4⋊Q8—611no—
364C2×C4⋊Q8—611no—
365C2×C4⋊Q8—1221no—
366C2×C4⋊Q8—1221no—
367C2×C4⋊Q8—1221no—
368C2×C4⋊Q8C2×D6111yescharacteristic
369C2×C22⋊Q8—1221no—
370C2×C4×Q8—611no—
371C2×C4×Q8—611no—
372C2×C4×Q8—611no—
373C2×C4×Q8—611no—
374C2×C4×Q8—1221no—
375C2×C4×D4—611no—
376#476—1224no—
377C4×SD16—1224no—
378C2×#106—611no—
379C2×#106—611no—
380C2×#106—611no—
381C2×C4⋊C8—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#102—611no—
386C2×#102C2×D6111yescharacteristic
387C2×#102C2×D6111yescharacteristic
388C2×C4×C8—611no—
389#396—1221no—
390#396—1221no—
391#390—611no—
392#390C2×D6111yescharacteristic
393#389—611no—
394#389—611no—
395#384—611no—
396#380—1221no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340C2×D6111yescharacteristic
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405#336—611no—
406#336C2×D6111yescharacteristic
407C61#2432911yescharacteristic, complemented, Sylow
408C22×Q8—21no—
409C22×Q8—611no—
410C22×Q8—611no—
411C22×Q8—1221no—
412C22×Q8—1221no—
413C22×D4C22×D6111yescharacteristic
414C23×C4—1221no—
415C2×SD16—614no—
416C2×SD16—1222no—
417C2×SD16—1222no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422C4⋊Q8—22no—
423C4⋊Q8—22no—
424C4⋊Q8—614no—
425C4⋊Q8—1222no—
426C4⋊Q8—1222no—
427C4⋊Q8—1228no—
428C4⋊Q8—1228no—
429C4⋊Q8—1228no—
430C4⋊Q8—1222no—
431C22⋊Q8—2444no—
432C4×Q8—614no—
433C4×Q8—1222no—
434C4×Q8—1222no—
435C4×Q8—1222no—
436C4×Q8—1222no—
437C4×Q8—1222no—
438C4×Q8—1222no—
439C4×Q8—1228no—
440C4×Q8—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—1221no—
458C2×C4⋊C4—1221no—
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—1221no—
470C2×C42—1221no—
471C2×C42—1221no—
472C2×C42C22×D6111yescharacteristic
473#106—1222no—
474#106—1222no—
475#106—1222no—
476C4⋊C8—1222no—
477#103—1222no—
478#103—1222no—
479#103—1222no—
480#102—22no—
481#102—22no—
482#102—1222no—
483C4×C8—1222no—
484#95—1221no—
485#95—1221no—
486C24—21no—
487C2×Q8—24no—
488C2×Q8—81no—
489C2×Q8—81no—
490C2×Q8—614no—
491C2×Q8—612no—
492C2×Q8—612no—
493C2×Q8—612no—
494C2×Q8—1222no—
495C2×Q8—1222no—
496C2×Q8—1224no—
497C2×Q8—1224no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—4881no—
502C2×Q8—4881no—
503C2×Q8—4881no—
504C2×Q8—4881no—
505C2×D4—21no—
506C2×D4—21no—
507C2×D4—21no—
508C2×D4—21no—
509C2×D4—21no—
510C2×D4Q8×D6112yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528SD16—1228no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—612no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—612no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—1222no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—4881no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C4⋊C4—2442no—
564C4⋊C4D4×D6112yes—
565C4⋊C4D4×D6112yes—
566C22⋊C4—42no—
567C22⋊C4—81no—
568C22⋊C4—4881no—
569C22⋊C4—2442no—
570C22⋊C4—2442no—
571C42—21no—
572C42—21no—
573C42—612no—
574C42—612no—
575C42—1224no—
576C42—1224no—
577C42—1224no—
578C42—1221no—
579C42—1221no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23C2×D4×D6111yescharacteristic
588Q8—84no—
589Q8—614no—
590Q8—1228no—
591Q8—1222no—
592Q8—1222no—
593Q8—1222no—
594Q8—4884no—
595Q8—4884no—
596D4—22no—
597D4—22no—
598D4—41no—
599D4—41no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—612no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—612no—
618C2×C4—612no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—4884no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4SD16×D6112yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×Q8×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#1451982411yescharacteristic
645C2×C4#1451982411yescharacteristic
646C8—41no—
647C8—41no—
648C8—1222no—
649C8—2441no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22#120×D6111yescharacteristic
658C22#1684042011yescharacteristic
659C22#1684042511yescharacteristic
660C22C2×SD16×D6111yescharacteristic
661C22#137×D6111yescharacteristic
662C22#1684056311yescharacteristic
663C22#1684057211yescharacteristic
664C4—22no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—81no—
669C4—614no—
670C4—1222no—
671C4—1222no—
672C4—1224no—
673C4—4881no—
674C4—4881no—
675C4—2441no—
676C4—2442no—
677C4C22⋊Q8×D6111yescharacteristic
678C4#1684033211yescharacteristic
679C4#1684042011yescharacteristic
680C4#1684042511yescharacteristic
681C2—82no—
682C2#2144917711yescharacteristic
683C2#448×D6111yescharacteristic
684C2#2145511811yescharacteristic
685C2#2145848711yescharacteristic
686C2#2145855611yescharacteristic
687C2#2145867411yescharacteristic
688C2#2145872411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2023C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2023D61
hypocenterC61#24329
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
1. C2×#16840492C2semidirect
5. #21447151C2semidirect
6. #21447149C2semidirect
8. #21446778C2semidirect
9. #21446753C2semidirect
12. #21446090C2semidirect
13. #21446088C2semidirect
14. #21445823C2semidirect
70. C2×#11758200Q8semidirect
179. C2×Dic122SD16semidirect
300. Dic122C22⋊Q8semidirect
407. C61#24329semidirect

Maximal subgroups

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

subgroupindex
C2×#168404922
C2×#168400502
C2×#168398532
#214507392
#214471512
#214471492
#214469532
#214467782
#214467532
#214461772
#214461712
#214460902
#214460882
#214458232
C61×#20232
#2432961