Finite Group Databasebeta

← #29721314

Subgroups — #29721314

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840040C211yesmaximal, characteristic, complemented
2C2×#16839817C211yesmaximal, characteristic, complemented
3#21454170C211yesmaximal, characteristic, complemented
4#21448335C211yesmaximal, characteristic
5#21447251C211yesmaximal, characteristic, complemented
6#21447249C211yesmaximal, characteristic, complemented
7#21447007C211yesmaximal, characteristic
8#21447006C211yesmaximal, characteristic
9#21446769C211yesmaximal, characteristic, complemented
10#21446581C211yesmaximal, characteristic, complemented
11#21446562C211yesmaximal, characteristic, complemented
12#21446559C211yesmaximal, characteristic, complemented
13#21446519C211yesmaximal, characteristic, complemented
14#21445817C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519748C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519737—21no—
21C2×C42×D61C2211yescharacteristic
22#16840040—24no—
23#16839817—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839666—21no—
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839657—21no—
39#16839656—21no—
40#16839638—21no—
41#16839631C2211yescharacteristic
42#16839621—21no—
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C22⋊C4×Dic61—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×C22⋊D61—21no—
63C23×Dic61—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—22no—
71#14519755—44no—
72#14519748—22no—
73#14519748—22no—
74D4×Dic61—22no—
75D4×Dic61—22no—
76#14519737—44no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—22no—
96C42×D61—22no—
97#14519711—22no—
98#14519710—22no—
99#14519709—22no—
100#14519708—22no—
101#14519706—22no—
102#14519706D414yescomplemented
103#14519698—22no—
104#14519698—22no—
105#14519689—22no—
106#14519689—22no—
107#14519688—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×D4×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C42×C61C2311yescharacteristic
124#102×C61—22no—
125#102×C61—22no—
126#97×C61—22no—
127#95×C61—21no—
128C24×C61—21no—
129C23×D61C2×D411yescharacteristic
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61#4112yes—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C2×C22⋊D61—24no—
143C2×C22⋊D61—42no—
144C2×C22⋊D61—42no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—22no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—41no—
167C2×Dic122—42no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D412yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C42×C61—21no—
186C42×C61—21no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61#4112yes—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—42no—
208C4×Dic61C2×D412yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61#13712yes—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—41no—
234D4×C61—41no—
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—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C22⋊D61—48no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—84no—
275C2×Dic61#12112yes—
276C2×Dic61#12112yes—
277C2×Dic61AGL(1,ℤ/8)12yes—
278C2×Dic61AGL(1,ℤ/8)12yes—
279C4×D61—44no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—44no—
285Dic122—84no—
286C8×C61—22no—
287C8×C61—41no—
288#92418—22no—
289#92418—41no—
290#24709—611nomaximal, Sylow
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61C2×#12111yescharacteristic
299C22×C61#52511yescharacteristic
300C22×C61#53611yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C22×C61#57511yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C2×D61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61—81no—
317C4×C61#54711yescharacteristic
318C4×C61#54811yescharacteristic
319C4×C61#55211yescharacteristic
320C4×C61#55611yescharacteristic
321Dic61—22no—
322Dic61—42no—
323Dic61—41no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327Dic61#46914yes—
328C2×#474—611no—
329C2×#432—611no—
330#2857—611no—
331C4×#121—611no—
332#2076—611no—
333#2076—611no—
334#2034—611no—
335#2034—611no—
336#1988—611no—
337#1948—611no—
338#1944—611no—
339#1944—611no—
340#1936—611no—
341#1936D6111yescharacteristic, complemented
342#1802—611no—
343C2×C61—82no—
344C2×C61#241511yescharacteristic
345C2×C61C2×#46911yescharacteristic
346C2×C61#305611yescharacteristic
347C2×C61#311011yescharacteristic
348C2×C61#311511yescharacteristic
349C2×C61#323511yescharacteristic
350C2×C61#323911yescharacteristic
351D61—42no—
352C2×C4⋊Q8—611no—
353C2×C22⋊Q8—1221no—
354C2×#121—611no—
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357C22×C42—611no—
358#474—1224no—
359#432—1224no—
360C2×#107—611no—
361C2×#107—611no—
362C2×#106—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102C2×D6111yescharacteristic
369C2×#102C2×D6111yescharacteristic
370C2×C22⋊C8—611no—
371C2×C22⋊C8—611no—
372C2×#97—611no—
373C2×#97C2×D6111yescharacteristic
374#391—1221no—
375#390C2×D6111yescharacteristic
376#389—611no—
377#389—611no—
378#389—1221no—
379#389C2×D6111yescharacteristic
380#388—1221no—
381#386—1221no—
382#384—611no—
383#380—611no—
384#380—611no—
385#380—611no—
386#380—1221no—
387C4×C4⋊C4—611no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#340C2×D6111yescharacteristic
397C61#2470911yescharacteristic, complemented, Sylow
398C22×Q8—1221no—
399C22×D4—1221no—
400C22×D4C22×D6111yescharacteristic
401C23×C4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—1221no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411C22⋊Q8—2444no—
412#121—1222no—
413#121—1222no—
414C4×D4—1222no—
415C4×D4—1222no—
416C4×D4—2444no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1222no—
441C2×C42—1222no—
442C2×C42—1221no—
443C2×C42C22×D6111yescharacteristic
444#107—1222no—
445#107—1222no—
446#106—1222no—
447#106—1222no—
448C4⋊C8—614no—
449C4⋊C8—1222no—
450#102—22no—
451#102—22no—
452#102—1222no—
453#102—1222no—
454C22⋊C8—1222no—
455C22⋊C8—1222no—
456#97—22no—
457#97—1222no—
458#95—21no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465C24—21no—
466C24—611no—
467C2×Q8—2442no—
468C2×Q8—2442no—
469C2×Q8—2442no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—1224no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4#1175822212yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—2441no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4D4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4C23×D6111yescharacteristic
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—612no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8D4×D6112yes—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—612no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—1224no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C42—21no—
547C42—21no—
548C42—612no—
549C42—612no—
550C42—1222no—
551C42—1222no—
552C42—2442no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—612no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23C2×D4×D6111yescharacteristic
568Q8—4884no—
569D4—22no—
570D4—22no—
571D4—41no—
572D4—41no—
573D4—2448no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—4884no—
600C2×C4—2444no—
601C2×C4—2444no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4C2×#1175822211yescharacteristic
621C2×C4#1451982411yescharacteristic
622C2×C4#1451982411yescharacteristic
623C8—22no—
624C8—41no—
625C8—1222no—
626C8—2441no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22—2441no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22#121×D6111yescharacteristic
642C22#1684027111yescharacteristic
643C22#1684033211yescharacteristic
644C22AGL(1,ℤ/8)×D6111yescharacteristic
645C22AGL(1,ℤ/8)×D6111yescharacteristic
646C22#137×D6111yescharacteristic
647C22#1684055811yescharacteristic
648C4—41no—
649C4—41no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—614no—
654C4—1222no—
655C4—4881no—
656C4—4881no—
657C4—4881no—
658C4—2441no—
659C4—2442no—
660C4#1684041011yescharacteristic
661C4#1684042111yescharacteristic
662C4#1684044511yescharacteristic
663C4#1684047911yescharacteristic
664C2—82no—
665C2—2442no—
666C2#2144924511yescharacteristic
667C2#469×D6111yescharacteristic
668C2#2145544411yescharacteristic
669C2#2145591611yescharacteristic
670C2#2145596611yescharacteristic
671C2#2145701211yescharacteristic
672C2#2145704511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenter#1936D61
hypocenterC61#24709
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
1. C2×#16840040C2semidirect
2. C2×#16839817C2semidirect
3. #21454170C2semidirect
5. #21447251C2semidirect
6. #21447249C2semidirect
9. #21446769C2semidirect
10. #21446581C2semidirect
11. #21446562C2semidirect
12. #21446559C2semidirect
13. #21446519C2semidirect
14. #21445817C2semidirect
15. C61×#1936C2semidirect
102. #14519706D4semidirect
341. #1936D61semidirect
397. C61#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#168400402
C2×#168398172
#214541702
#214483352
#214472512
#214472492
#214470072
#214470062
#214467692
#214465812
#214465622
#214465592
#214465192
#214458172
C61×#19362
#2470961