Finite Group Databasebeta

← #29748074

Subgroups — #29748074

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450187C211yesmaximal, characteristic, complemented
2#21448831C211yesmaximal, characteristic
3#21447135C211yesmaximal, characteristic, complemented
4#21447006C211yesmaximal, characteristic
5#21446953C211yesmaximal, characteristic, complemented
6#21446823C211yesmaximal, characteristic
7#21446765C211yesmaximal, characteristic, complemented
8#21446495C211yesmaximal, characteristic, complemented
9#21446451C211yesmaximal, characteristic, complemented
10#21446425C211yesmaximal, characteristic, complemented
11#21446397C211yesmaximal, characteristic, complemented
12#21446153C211yesmaximal, characteristic, complemented
13#21446125C211yesmaximal, characteristic, complemented
14#21445904C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×C4×C22⋊D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30#16839702—21no—
31#16839675—21no—
32#16839668—21no—
33#16839668C2211yescharacteristic
34#16839667C2211yescharacteristic
35#16839665C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839656—21no—
38#16839653—21no—
39#16839638—21no—
40#16839635—21no—
41#16839619C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C2×#11758200—22no—
65C2×#11758200—41no—
66C2×#11758200—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—41no—
71#14519757—22no—
72#14519757—41no—
73#14519748—22no—
74#14519748—22no—
75C4×C22⋊D61—44no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—41no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198D411yescharacteristic
91C2×#11758198D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519708—22no—
95#14519706—41no—
96#14519702—22no—
97#14519702—41no—
98#14519700—22no—
99#14519698—41no—
100#14519689—41no—
101#14519689—41no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4×D4×C61—22no—
116C4×D4×C61—41no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#102×C61—22no—
124#102×C61—22no—
125C4×C8×C61—41no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61C2×D411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61#4112yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—81no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—24no—
145C2×C22⋊D61—42no—
146C2×C22⋊D61—42no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—81no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61#4112yes—
183C4⋊C4×C61#4112yes—
184C22⋊C4×C61—42no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—81no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190C22⋊Dic61—81no—
191#11758200—44no—
192#11758198—41no—
193#11758198—42no—
194#11758198—81no—
195#11758198SD1614yescomplemented
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—42no—
202C4⋊Dic61#4112yes—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61C2×SD1612yes—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
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—84no—
249C2×C4×C61C2×SD1612yes—
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C22⋊D61—48no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
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—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61C2×SD1612yes—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—84no—
283Dic122—41no—
284Dic122—41no—
285Dic122—41no—
286Dic122—41no—
287Dic122—84no—
288C8×C61—41no—
289C8×C61—41no—
290#92418—41no—
291#92418—41no—
292#26722—611nomaximal, Sylow
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61#52511yescharacteristic
301C22×C61#54611yescharacteristic
302C22×C61#54611yescharacteristic
303C22×C61C22×SD1611yescharacteristic
304C22×C61#57511yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57811yescharacteristic
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C4×C61—22no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61#52911yescharacteristic
320C4×C61#53811yescharacteristic
321C4×C61#55511yescharacteristic
322C4×C61#55911yescharacteristic
323Dic61—22no—
324Dic61—42no—
325Dic61—41no—
326Dic61—41no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330#2492—611no—
331#2361—611no—
332#2055—611no—
333#2034—611no—
334#2023—611no—
335#2000—611no—
336#1987—611no—
337#1933—611no—
338#1927—611no—
339#1923—611no—
340#1919—611no—
341#1877—611no—
342#1872—611no—
343#1823—611no—
344C4×#102D6111yescharacteristic, complemented
345C2×C61—82no—
346C2×C61#245911yescharacteristic
347C2×C61#310111yescharacteristic
348C2×C61#312411yescharacteristic
349C2×C61#329511yescharacteristic
350C2×C61#329811yescharacteristic
351C2×C61#331311yescharacteristic
352C2×C61#332211yescharacteristic
353D61—42no—
354C22×SD16—611no—
355C2×C4⋊Q8—611no—
356C2×C22⋊Q8—611no—
357C2×#121—611no—
358C2×C4×D4—1221no—
359C2×C4×D4C2×D6111yescharacteristic
360C22×C4⋊C4—611no—
361C2×#106—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102C2×D6111yescharacteristic
368C2×#102C2×D6111yescharacteristic
369C2×C22⋊C8—611no—
370C2×C22⋊C8—611no—
371C2×C4×C8C2×D6111yescharacteristic
372#400—1221no—
373#393—1221no—
374#391—611no—
375#391—611no—
376#391—1221no—
377#390—611no—
378#389—611no—
379#388—1221no—
380#388—1221no—
381#386—1221no—
382#385—1221no—
383#380—611no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4C2×D6111yescharacteristic
386C4×C22⋊C4—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395#336—611no—
396#336—611no—
397C61#2672211yescharacteristic, complemented, Sylow
398C22×Q8—611no—
399C22×Q8—1221no—
400C22×D4—1221no—
401C22×D4C22×D6111yescharacteristic
402C23×C4—21no—
403C23×C4—611no—
404C23×C4—1221no—
405C2×SD16—1222no—
406C2×SD16—2441no—
407C2×SD16—2441no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C4⋊Q8—1222no—
413C4⋊Q8—2441no—
414C22⋊Q8—1222no—
415C22⋊Q8—2441no—
416#121—1222no—
417#121—1222no—
418C4×D4—22no—
419C4×D4—41no—
420C4×D4—2444no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#106—1222no—
446C4⋊C8—2441no—
447#103—1222no—
448#103—1222no—
449#103—2441no—
450#102—22no—
451#102—22no—
452#102—2441no—
453C22⋊C8—2441no—
454C22⋊C8—2441no—
455C4×C8—41no—
456#95—21no—
457#95—611no—
458#95—611no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466C24—21no—
467C24—611no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—4881no—
475C2×Q8—2442no—
476C2×Q8—2442no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—1224no—
483C2×D4—2442no—
484C2×D4—2442no—
485C2×D4#1175822212yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—81no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512C2×C8—21no—
513C2×C8—21no—
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—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—612no—
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#1175822212yes—
539C4⋊C4#1175822212yes—
540C22⋊C4—42no—
541C22⋊C4—614no—
542C22⋊C4—1224no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—2441no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C42—21no—
549C42—21no—
550C42—81no—
551C42—1221no—
552C42—1221no—
553C42—2442no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—612no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23C2×D4×D6111yescharacteristic
569Q8—4884no—
570Q8—2441no—
571Q8—2441no—
572Q8—2441no—
573Q8—2441no—
574D4—22no—
575D4—22no—
576D4—41no—
577D4—41no—
578D4—2448no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—4884no—
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×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4SD16×D6112yes—
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822211yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4#1451982211yescharacteristic
626C2×C4#1451982311yescharacteristic
627C8—41no—
628C8—41no—
629C8—2441no—
630C8—2441no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22—2441no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22#1684027111yescharacteristic
646C22#1684040211yescharacteristic
647C22#1684040811yescharacteristic
648C22C2×SD16×D6111yescharacteristic
649C22#1684055911yescharacteristic
650C22#1684056311yescharacteristic
651C22#1684057611yescharacteristic
652C4—22no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—81no—
657C4—1222no—
658C4—4881no—
659C4—4881no—
660C4—4881no—
661C4—2441no—
662C4—2441no—
663C4—2442no—
664C4#1684029611yescharacteristic
665C4#1684034511yescharacteristic
666C4#1684047511yescharacteristic
667C4#1684049511yescharacteristic
668C2—82no—
669C2—2442no—
670C2#2144975611yescharacteristic
671C2#2145582511yescharacteristic
672C2#2145607711yescharacteristic
673C2#2145763911yescharacteristic
674C2#2145768611yescharacteristic
675C2#2145788111yescharacteristic
676C2#2145802511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
hypercenterC4×#102D61
hypocenterC61#26722
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #21450187C2semidirect
3. #21447135C2semidirect
5. #21446953C2semidirect
7. #21446765C2semidirect
8. #21446495C2semidirect
9. #21446451C2semidirect
10. #21446425C2semidirect
11. #21446397C2semidirect
12. #21446153C2semidirect
13. #21446125C2semidirect
14. #21445904C2semidirect
15. C4×#102×C61C2semidirect
195. #11758198SD16semidirect
344. C4×#102D61semidirect
397. C61#26722semidirect

Maximal subgroups

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

subgroupindex
#214501872
#214488312
#214471352
#214470062
#214469532
#214468232
#214467652
#214464952
#214464512
#214464252
#214463972
#214461532
#214461252
#214459042
C4×#102×C612
#2672261