Finite Group Databasebeta

← #29729558

Subgroups — #29729558

Order 15616.

848 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×#16839977C211yesmaximal, characteristic
2C2×C8×Dic122C211yesmaximal, characteristic, complemented
3#21451302C211yesmaximal, characteristic, complemented
4#21451173C211yesmaximal, characteristic, complemented
5#21447215C211yesmaximal, characteristic, complemented
6#21447194C211yesmaximal, characteristic, complemented
7#21447135C211yesmaximal, characteristic, complemented
8#21447128C211yesmaximal, characteristic
9#21446682C211yesmaximal, characteristic
10#21446674C211yesmaximal, characteristic, complemented
11#21446630C211yesmaximal, characteristic, complemented
12#21446587C211yesmaximal, characteristic, complemented
13#21446500C211yesmaximal, characteristic, complemented
14#21446493C211yesmaximal, characteristic, complemented
15C4×#102×C61C211yesmaximal, characteristic, complemented
16C22×#11758200C2211yescharacteristic
17C22×#11758199C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839977—24no—
21C8×Dic122—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839698—21no—
36#16839689—21no—
37#16839675—21no—
38#16839674—21no—
39#16839667C2211yescharacteristic
40#16839667C2211yescharacteristic
41#16839665C2211yescharacteristic
42#16839665C2211yescharacteristic
43#16839663C2211yescharacteristic
44#16839663C2211yescharacteristic
45#16839658C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839653—21no—
48#16839653—21no—
49#16839635—21no—
50#16839635—21no—
51#16839628C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C2×#11758199—22no—
68C2×#11758199—22no—
69C2×#11758199—41no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519765—22no—
73#14519765—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196D411yescharacteristic
82C2×#11758196D411yescharacteristic
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710—41no—
96#14519708—41no—
97C4⋊#92418D414yes—
98#14519706—22no—
99#14519706—22no—
100#14519702—22no—
101#14519702—41no—
102#14519700—22no—
103#14519699—22no—
104#14519699—41no—
105#14519697—22no—
106C8×Dic61—22no—
107C8×Dic61—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#102×C61—22no—
129#102×C61—22no—
130C4×C8×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—81no—
145C22×C4×C61—81no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—22no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61#4112yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C22⋊C4×C61—42no—
184C42×C61—81no—
185C42×C61C2×D412yes—
186C42×C61#4112yes—
187C22⋊Dic61—42no—
188C22⋊Dic61—42no—
189C22⋊Dic61—81no—
190C22⋊Dic61—81no—
191#11758200—44no—
192#11758199—44no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197#11758198—81no—
198#11758198—81no—
199C4⋊Dic61—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204#11758196—22no—
205#11758196—22no—
206#11758196—81no—
207#11758196—81no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D412yes—
217C2×#92418C2×D412yes—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61C22×D411yescharacteristic
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
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×C61—84no—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
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×Dic61—41no—
278C2×Dic61#13512yes—
279C2×Dic61#13512yes—
280D244—41no—
281D244—41no—
282D244—41no—
283D244—41no—
284Dic122—28no—
285Dic122—41no—
286Dic122—41no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293#25296—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×#12111yescharacteristic
302C22×C61#54811yescharacteristic
303C22×C61#55011yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61#57611yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61#52911yescharacteristic
321C4×C61#54211yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#54811yescharacteristic
324Dic61—22no—
325Dic61—22no—
326Dic61—41no—
327Dic61—41no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#463—611no—
332C2×C8×Q8—611no—
333#2592—611no—
334#2580—611no—
335#2068—611no—
336#2065—611no—
337#2055—611no—
338#2054—611no—
339#1970—611no—
340#1968—611no—
341#1958—611no—
342#1949—611no—
343#1934—611no—
344#1933—611no—
345C4×#102D6111yescharacteristic, complemented
346C2×C61—82no—
347C2×C61#242611yescharacteristic
348C2×C61#305011yescharacteristic
349C2×C61#306211yescharacteristic
350C2×C61#336011yescharacteristic
351C2×C61#336111yescharacteristic
352C2×C61#339411yescharacteristic
353C2×C61#339611yescharacteristic
354D61—161no—
355C22×SD16—611no—
356C22×D8—611no—
357C2×#124—611no—
358C2×C4×Q8—611no—
359C2×C4×D4C2×D6111yescharacteristic
360#463—1224no—
361C8×Q8—1224no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C4⋊C8—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102C2×D6111yescharacteristic
374C2×#102C2×D6111yescharacteristic
375C2×C4×C8—611no—
376C2×C4×C8—611no—
377C2×C4×C8C2×D6111yescharacteristic
378#399—1221no—
379#397—1221no—
380#393—1221no—
381#393—1221no—
382#391—611no—
383#391—611no—
384#390—611no—
385#390—611no—
386#390—611no—
387#390—611no—
388#389—611no—
389#389—611no—
390#388—1221no—
391#388—1221no—
392#385—1221no—
393#385—1221no—
394#382—611no—
395C4×C4⋊C4C2×D6111yescharacteristic
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#340C2×D6111yescharacteristic
400C61#2529611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8—611no—
403C22×D4—611no—
404C22×D4C22×D6111yescharacteristic
405C23×C4—21no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—2441no—
409C2×D8—1222no—
410C2×D8—1222no—
411C2×D8—2441no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416#124—1222no—
417#124—1222no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—21no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42C22×D6111yescharacteristic
447#107—2441no—
448#106—2441no—
449C4⋊C8—614no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—1222no—
453#103—1222no—
454#103—2441no—
455#102—22no—
456#102—22no—
457#102—1222no—
458#102—1222no—
459#102—2441no—
460C4×C8—22no—
461C4×C8—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—21no—
473C24—1221no—
474C2×Q8—1222no—
475C2×Q8—1222no—
476C2×Q8—1222no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4#1175822212yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512D8—2444no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8#1175822212yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C22⋊C4—42no—
541C22⋊C4—4881no—
542C22⋊C4—4881no—
543C22⋊C4—4881no—
544C22⋊C4—4881no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—81no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42#1175822212yes—
557C42D4×D6112yes—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C2×D4×D6111yescharacteristic
573Q8—1228no—
574Q8—2441no—
575Q8—2441no—
576Q8—2441no—
577Q8—2441no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—2441no—
583D4—2441no—
584D4—2441no—
585D4—2441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—84no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
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#1451975012yes—
622C2×C4#1451975012yes—
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982411yescharacteristic
629C8—22no—
630C8—41no—
631C8—1222no—
632C8—1222no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—4881no—
647C22C2×#1451975011yescharacteristic
648C22#1684042111yescharacteristic
649C22#1684043311yescharacteristic
650C22#135×D6111yescharacteristic
651C22#135×D6111yescharacteristic
652C22#1684056311yescharacteristic
653C22#1684056811yescharacteristic
654C4—22no—
655C4—22no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—1222no—
660C4—1222no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—2441no—
665C4—2441no—
666C4#1684029611yescharacteristic
667C4#1684038311yescharacteristic
668C4#1684040811yescharacteristic
669C4#1684042111yescharacteristic
670C2—82no—
671C2—9761no—
672C2#2144938511yescharacteristic
673C2#2145539911yescharacteristic
674C2#2145549711yescharacteristic
675C2#2145846911yescharacteristic
676C2#2145848011yescharacteristic
677C2#2145881611yescharacteristic
678C2#2145883111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#102×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
hypercenterC4×#102D61
hypocenterC61#25296
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
2. C2×C8×Dic122C2semidirect
3. #21451302C2semidirect
4. #21451173C2semidirect
5. #21447215C2semidirect
6. #21447194C2semidirect
7. #21447135C2semidirect
10. #21446674C2semidirect
11. #21446630C2semidirect
12. #21446587C2semidirect
13. #21446500C2semidirect
14. #21446493C2semidirect
15. C4×#102×C61C2semidirect
345. C4×#102D61semidirect
400. C61#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#168399772
C2×C8×Dic1222
#214513022
#214511732
#214472152
#214471942
#214471352
#214471282
#214466822
#214466742
#214466302
#214465872
#214465002
#214464932
C4×#102×C612
#2529661