Finite Group Databasebeta

← #29743138

Subgroups — #29743138

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840050C211yesmaximal, characteristic, complemented
2#21449663C211yesmaximal, characteristic, complemented
3#21448724C211yesmaximal, characteristic, complemented
4#21447342C211yesmaximal, characteristic, complemented
5#21447303C211yesmaximal, characteristic, complemented
6#21447191C211yesmaximal, characteristic, complemented
7#21447145C211yesmaximal, characteristic, complemented
8#21446547C211yesmaximal, characteristic, complemented
9#21446376C211yesmaximal, characteristic, complemented
10#21446169C211yesmaximal, characteristic
11#21446169C211yesmaximal, characteristic
12#21446082C211yesmaximal, characteristic, complemented
13#21446080C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic, complemented
15C2×C4×D8×C61C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519756—21no—
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840050—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30#16839685—21no—
31#16839674—21no—
32#16839665C2211yescharacteristic
33#16839665C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839635—21no—
39#16839635—21no—
40#16839634—21no—
41#16839627C2211yescharacteristic
42#16839620—21no—
43C4×#11758196C2211yescharacteristic
44C2×#14519684—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×D8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C4×D8×C61—24no—
57C2×#107×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—41no—
68#14519756—44no—
69D4×Dic61—22no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519710—41no—
93#14519709—22no—
94#14519709—22no—
95#14519708—22no—
96#14519706—22no—
97#14519698—22no—
98#14519698—22no—
99#14519698—22no—
100#14519698—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C22×D4×C61C2311yescharacteristic
111C23×C4×C61—21no—
112C23×C4×C61—21no—
113C2×D8×C61—22no—
114C2×D8×C61—22no—
115C2×D8×C61Q814yescomplemented
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—41no—
121C4×D4×C61—41no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#107×C61—22no—
128#102×C61—22no—
129#102×C61D414yes—
130C4×C8×C61—22no—
131C24×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C2×D4×C61C2×Q812yes—
144C2×D4×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61—81no—
152C22×C4×C61—81no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×Dic122—42no—
171C2×Dic122—42no—
172C2×Dic122—81no—
173D8×C61—28no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D412yes—
179C2×C8×C61C2×Q812yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C2×D412yes—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—42no—
202C4⋊Dic61#4112yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—42no—
206#11758196—42no—
207#11758196—81no—
208#11758196—81no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61C22×D411yescharacteristic
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61C22⋊Q814yescomplemented
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—84no—
254C2×C4×C61—84no—
255C2×C4×C61AGL(1,ℤ/8)12yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—84no—
286C2×Dic61—84no—
287C2×Dic61#13512yes—
288Dic122—84no—
289C8×C61—22no—
290C8×C61—22no—
291#92418—41no—
292#92418—41no—
293#26380—611nomaximal, Sylow
294C22×C61—24no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61#52511yescharacteristic
309C22×C61D4211yescharacteristic
310C22×C61D4×Q811yescharacteristic
311C22×C61C2×#13511yescharacteristic
312C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—42no—
317C4×C61—41no—
318C4×C61—81no—
319C4×C61C2×C22⋊Q811yescharacteristic
320C4×C61#53911yescharacteristic
321C4×C61#54711yescharacteristic
322C4×C61#55411yescharacteristic
323Dic61—22no—
324Dic61—41no—
325Dic61—41no—
326Dic61—42no—
327Dic61—41no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#476—611no—
332C2×C4×D8D6111yescharacteristic
333#2452—611no—
334#2349—611no—
335#2092—611no—
336#2085—611no—
337#2065—611no—
338#2057—611no—
339#1941—611no—
340#1916—611no—
341#1879—611no—
342#1879—611no—
343#1864—611no—
344#1864—611no—
345#1862—611no—
346C2×C61—44no—
347C2×C61—82no—
348C2×C61#244911yescharacteristic
349C2×C61#310611yescharacteristic
350C2×C61#311811yescharacteristic
351C2×C61#326411yescharacteristic
352C2×C61#328211yescharacteristic
353C2×C61#338011yescharacteristic
354C2×C61#338411yescharacteristic
355C22×D8C2×D6111yescharacteristic
356C2×C4⋊Q8—611no—
357C2×C22⋊Q8—1221no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4C2×D6111yescharacteristic
361C2×C4×D4C2×D6111yescharacteristic
362#476—1224no—
363C4×D8—24no—
364C2×#107—611no—
365C2×#107C2×D6111yescharacteristic
366C2×#106—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102C2×D6111yescharacteristic
375C2×#102C2×D6111yescharacteristic
376C2×C4×C8C2×D6111yescharacteristic
377#396—1221no—
378#393—1221no—
379#390—611no—
380#390—611no—
381#389—611no—
382#389—611no—
383#389—611no—
384#389—611no—
385#385—1221no—
386#385—1221no—
387#385—1221no—
388#382—611no—
389#380—1221no—
390C4×C4⋊C4—611no—
391C2×#95—1221no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#336—611no—
397#336—611no—
398#336—611no—
399#336—611no—
400C61#2638011yescharacteristic, complemented, Sylow
401C22×Q8—1221no—
402C22×D4C22×D6111yescharacteristic
403C22×D4C22×D6111yescharacteristic
404C23×C4—21no—
405C23×C4—21no—
406C23×C4—1221no—
407C23×C4—1221no—
408C2×D8—22no—
409C2×D8—22no—
410C2×D8Dic12214yescomplemented
411C22×C8—611no—
412C22×C8—611no—
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C4⋊Q8—1222no—
416C4⋊Q8—2441no—
417C22⋊Q8—2444no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—41no—
421C4×D4—41no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—2441no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42C22×D6111yescharacteristic
450#107—22no—
451#107—2441no—
452#106—1222no—
453#106—1222no—
454#106—1222no—
455C4⋊C8—1222no—
456#102—22no—
457#102—1222no—
458#102—1222no—
459#102—1222no—
460#102—2441no—
461#102C22⋊D6114yes—
462C4×C8—22no—
463#95—611no—
464#95—611no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—21no—
472C24—21no—
473C2×Q8—4881no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4C2×Dic12212yes—
485C2×D4C2×Dic12212yes—
486C2×D4#1175822212yes—
487C2×D4C2×C22⋊D6112yes—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512C22×C4C23×D6111yescharacteristic
513D8—28no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8C2×Dic12212yes—
525C2×C8C2×C22⋊D6112yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—612no—
530C4⋊C4—1222no—
531C4⋊C4—1221no—
532C4⋊C4—1222no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—4881no—
540C4⋊C4—4881no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C4⋊C4C2×C22⋊D6112yes—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—1224no—
548C22⋊C4—4881no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C42—21no—
554C42—21no—
555C42—1221no—
556C42—1221no—
557C42—1221no—
558C42—1221no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×D4×D6111yescharacteristic
574Q8—4884no—
575D4—22no—
576D4—22no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4#1451975614yescomplemented
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—612no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1222no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—4884no—
610C2×C4—4884no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4#1451980112yes—
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4C22×Dic12211yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—22no—
631C8—2441no—
632C8—2441no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#1684027111yescharacteristic
648C22D4×C22⋊D6111yescharacteristic
649C22D4×Dic12211yescharacteristic
650C22#135×D6111yescharacteristic
651C22C2×#1451980111yescharacteristic
652C22#1684056311yescharacteristic
653C22#1684056911yescharacteristic
654C4—22no—
655C4—41no—
656C4—42no—
657C4—81no—
658C4—1222no—
659C4—4881no—
660C4—4881no—
661C4—4881no—
662C4—2441no—
663C4—2441no—
664C4—2442no—
665C4—2441no—
666C4C2×#1451975611yescharacteristic
667C4#1684035411yescharacteristic
668C4#1684041111yescharacteristic
669C4#1684047111yescharacteristic
670C2—44no—
671C2—82no—
672C2#2144962211yescharacteristic
673C2#2145587511yescharacteristic
674C2#2145599911yescharacteristic
675C2#2145729911yescharacteristic
676C2#2145752011yescharacteristic
677C2#2145868611yescharacteristic
678C2#2145871611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×D8×C61C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenterC2×C4×D8D61
hypocenterC61#26380
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16840050C2semidirect
2. #21449663C2semidirect
3. #21448724C2semidirect
4. #21447342C2semidirect
5. #21447303C2semidirect
6. #21447191C2semidirect
7. #21447145C2semidirect
8. #21446547C2semidirect
9. #21446376C2semidirect
12. #21446082C2semidirect
13. #21446080C2semidirect
14. #21446066C2semidirect
115. C2×D8×C61Q8semidirect
241. D4×C61C22⋊Q8semidirect
400. C61#26380semidirect
410. C2×D8Dic122semidirect
582. D4#14519756semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
#214496632
#214487242
#214473422
#214473032
#214471912
#214471452
#214465472
#214463762
#214461692
#214460822
#214460802
#214460662
C2×C4×D8×C612
#2638061