Finite Group Databasebeta

← #29729552

Subgroups — #29729552

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×#16839978C211yesmaximal, characteristic
2#21451307C211yesmaximal, characteristic, complemented
3#21451177C211yesmaximal, characteristic, complemented
4#21447213C211yesmaximal, characteristic, complemented
5#21447192C211yesmaximal, characteristic, complemented
6#21447137C211yesmaximal, characteristic, complemented
7#21447130C211yesmaximal, characteristic
8#21446685C211yesmaximal, characteristic
9#21446673C211yesmaximal, characteristic, complemented
10#21446630C211yesmaximal, characteristic, complemented
11#21446589C211yesmaximal, characteristic, complemented
12#21446502C211yesmaximal, characteristic, complemented
13#21446495C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic, complemented
15C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×D488C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16839978—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31#16839697—21no—
32#16839688—21no—
33#16839673—21no—
34#16839673—21no—
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839652—21no—
44#16839652—21no—
45#16839633—21no—
46#16839633—21no—
47#16839628C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C8×Q8×C61—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190—41no—
68C2×D488—22no—
69C2×D488—22no—
70C2×D488—41no—
71#14519765—22no—
72#14519765—22no—
73C4×D244—22no—
74C4×D244—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94#14519711—41no—
95#14519709—41no—
96#14519702—22no—
97#14519702—22no—
98#14519702—41no—
99#14519699—22no—
100#14519699—22no—
101#14519699—22no—
102#14519699—22no—
103#14519699—41no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×Q8×C61D411yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127C4⋊C8×C61—22no—
128C4⋊C8×C61D414yes—
129C4×C8×C61—22no—
130C4×C8×C61—22no—
131C4×C8×C61—22no—
132C23×D61—21no—
133C23×D61—21no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244#4112yes—
162C2×C4×D61—42no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
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—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D412yes—
181C2×C8×C61C2×D412yes—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C2×D412yes—
191C42×C61#4112yes—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213#11758196—81no—
214#11758196—81no—
215C4×Dic61—81no—
216D488—44no—
217#11758190—44no—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233Q8×C61—28no—
234C2×C4×C61—21no—
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—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61#12112yes—
249C2×C4×C61#12112yes—
250C2×C4×C61#13512yes—
251C2×C4×C61#13512yes—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
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—41no—
275C2×Dic61—41no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282D244—41no—
283D244—41no—
284C4×D61—84no—
285Dic122—41no—
286Dic122—41no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—22no—
290C8×C61—22no—
291C8×C61—22no—
292C8×C61—41no—
293#25296—611nomaximal, Sylow
294C22×C61C2×#12111yescharacteristic
295C22×C61#54811yescharacteristic
296C22×C61#55011yescharacteristic
297C22×C61C2×#13511yescharacteristic
298C22×C61C2×#13511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57711yescharacteristic
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
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—22no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61#52911yescharacteristic
322C4×C61#54211yescharacteristic
323C4×C61#54611yescharacteristic
324C4×C61#54811yescharacteristic
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#463—611no—
332C2×C8×Q8D6111yescharacteristic, complemented
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×#102—611no—
346C2×C61#242611yescharacteristic
347C2×C61#305011yescharacteristic
348C2×C61#306211yescharacteristic
349C2×C61#336011yescharacteristic
350C2×C61#336111yescharacteristic
351C2×C61#339411yescharacteristic
352C2×C61#339611yescharacteristic
353D61—82no—
354D61—161no—
355C22×SD16—611no—
356C22×D8—611no—
357C2×#124—611no—
358C2×C4×Q8C2×D6111yescharacteristic
359C2×C4×D4—611no—
360#463—1224no—
361C8×Q8—24no—
362C2×#107—611no—
363C2×#106—611no—
364C2×C4⋊C8C2×D6111yescharacteristic
365C2×C4⋊C8C2×D6111yescharacteristic
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#102—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×C4×C8C2×D6111yescharacteristic
376C2×C4×C8C2×D6111yescharacteristic
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⋊C4—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400C61#2529611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8D24411yescharacteristic
403C22×D4—611no—
404C22×D4—611no—
405C23×C4—1221no—
406C2×SD16—1222no—
407C2×SD16—1222no—
408C2×SD16—2441no—
409C2×D8—1222no—
410C2×D8—1222no—
411C2×D8—2441no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416#124—1222no—
417#124—1222no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×D4—1222no—
421C4×D4—1222no—
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⋊C4D24411yescharacteristic
430C2×C4⋊C4D24411yescharacteristic
431C2×C4⋊C4D24411yescharacteristic
432C2×C22⋊C4—1221no—
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—1221no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6111yescharacteristic
446C2×C42C22×D6111yescharacteristic
447#107—2441no—
448#106—2441no—
449C4⋊C8—22no—
450C4⋊C8—22no—
451C4⋊C8C22⋊D6114yes—
452#103—1222no—
453#103—1222no—
454#103—2441no—
455#102—1222no—
456#102—1222no—
457#102—1222no—
458#102—1222no—
459#102—2441no—
460C4×C8—22no—
461C4×C8—22no—
462C4×C8—22no—
463#95—1221no—
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—1221no—
473C24—1221no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—4881no—
502C22×C4—4881no—
503C22×C4—2442no—
504C22×C4C2×D24411yescharacteristic
505C22×C4C2×D24411yescharacteristic
506C22×C4C2×D24411yescharacteristic
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C23×D6111yescharacteristic
511SD16—2444no—
512D8—2444no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8#1175822612yes—
523C2×C8C2×C22⋊D6112yes—
524C2×C8C2×C22⋊D6112yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—4881no—
539C4⋊C4—4881no—
540C22⋊C4—4881no—
541C22⋊C4—4881no—
542C22⋊C4—4881no—
543C22⋊C4—4881no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C22⋊C4—2442no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—4881no—
556C42#1175822212yes—
557C42C2×C22⋊D6112yes—
558C23—1222no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23C22×D24411yescharacteristic
573Q8—28no—
574Q8—2441no—
575Q8—2441no—
576Q8—2441no—
577Q8—2441no—
578D4—1222no—
579D4—1222no—
580D4—2441no—
581D4—2441no—
582D4—2441no—
583D4—2441no—
584D4—2441no—
585D4—2441no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—4884no—
601C2×C4—2441no—
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×C4—2441no—
618C2×C4—2441no—
619C2×C4#1451975112yes—
620C2×C4#1451975112yes—
621C2×C4#1451979912yes—
622C2×C4#1451979912yes—
623C2×C4C22×C22⋊D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4C2×#1175822611yescharacteristic
627C2×C4#1451982311yescharacteristic
628C2×C4#1451982511yescharacteristic
629C8—22no—
630C8—22no—
631C8—22no—
632C8—41no—
633C22—4881no—
634C22—4881no—
635C22—4881no—
636C22—4881no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22C2×#1451975111yescharacteristic
648C22#1684042411yescharacteristic
649C22#1684043411yescharacteristic
650C22C2×#1451979911yescharacteristic
651C22C2×#1451979911yescharacteristic
652C22#1684056711yescharacteristic
653C22#1684057411yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—4881no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4#1684029911yescharacteristic
667C4#1684038411yescharacteristic
668C4#1684040011yescharacteristic
669C4#1684042311yescharacteristic
670C2—9761no—
671C2—4882no—
672C2#2144937911yescharacteristic
673C2#2145540011yescharacteristic
674C2#2145549911yescharacteristic
675C2#2145847011yescharacteristic
676C2#2145848111yescharacteristic
677C2#2145881711yescharacteristic
678C2#2145883211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenterC2×C8×Q8D61
hypocenterC61#25296
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #21451307C2semidirect
3. #21451177C2semidirect
4. #21447213C2semidirect
5. #21447192C2semidirect
6. #21447137C2semidirect
9. #21446673C2semidirect
10. #21446630C2semidirect
11. #21446589C2semidirect
12. #21446502C2semidirect
13. #21446495C2semidirect
14. C4×#14519699C2semidirect
15. C2×C8×Q8×C61C2semidirect
332. C2×C8×Q8D61semidirect
400. C61#25296semidirect

Maximal subgroups

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

subgroupindex
C2×#168399782
#214513072
#214511772
#214472132
#214471922
#214471372
#214471302
#214466852
#214466732
#214466302
#214465892
#214465022
#214464952
C4×#145196992
C2×C8×Q8×C612
#2529661