Finite Group Databasebeta

← #29699251

Subgroups — #29699251

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839844C211yesmaximal, characteristic
2#21448764C211yesmaximal, characteristic, complemented
3#21448720C211yesmaximal, characteristic, complemented
4#21446503C211yesmaximal, characteristic, complemented
5#21446501C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446168C211yesmaximal, characteristic
8#21446166C211yesmaximal, characteristic
9#21445945C211yesmaximal, characteristic, complemented
10#21445818C211yesmaximal, characteristic, complemented
11#21445818C211yesmaximal, characteristic, complemented
12#21445680C211yesmaximal, characteristic, complemented
13C4×#14519697C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#1918C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519734—21no—
22C2×C4⋊C4×D61—21no—
23#16839844C418yescomplemented
24C2×#14519706C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839653—21no—
38#16839651—21no—
39#16839636—21no—
40#16839633—21no—
41#16839631C2211yescharacteristic
42#16839631C2211yescharacteristic
43#16839621—21no—
44#16839621—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C2×D488—22no—
67C2×D488—22no—
68C2×D488C2×C414yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×D244—22no—
72C4×D244—41no—
73#14519739—22no—
74#14519739—22no—
75#14519739—22no—
76#14519739—41no—
77#14519739C2×C414yes—
78#14519739C2×C414yescomplemented
79#14519734—44no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C4⋊C4×D61—44no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519706—22no—
102#14519699—22no—
103#14519699—41no—
104#14519697—22no—
105#14519697—41no—
106#14519697C2×C414yes—
107#14519697C2×C414yescomplemented
108C4×#92418—41no—
109C8×Dic61—41no—
110C8×Dic61C2×C414yescomplemented
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124#107×C61C2×C414yescomplemented
125#95×C61—21no—
126#95×C61—21no—
127C23×D61—21no—
128C23×D61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244C22×C412yes—
151C2×D244C22×C412yes—
152C2×D244#4112yes—
153C2×D244#4112yes—
154C2×C4×D61—24no—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C22×C412yes—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C22×C412yes—
180C4⋊C4×C61C22×C412yes—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—42no—
184#11758198—24no—
185#11758198—24no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—81no—
191#11758198—81no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—42no—
197#11758196—22no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201#11758196—81no—
202#11758196D814yescomplemented
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C4×Dic61C22×C412yes—
207C4×Dic61#4112yes—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C22×C412yes—
214D488—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
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—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
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×C61C2×D812yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×C4×C612−1+411yescharacteristic
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61C4×D412yes—
268C2×Dic61C4×D412yes—
269C2×Dic61C2×D812yes—
270C2×Dic61C2×D812yes—
271C2×Dic61#13512yes—
272D244—22no—
273D244—22no—
274D244—22no—
275D244—22no—
276D244—22no—
277D244—41no—
278D244—41no—
279D244#11714yescomplemented
280C4×D61—48no—
281C4×D61—48no—
282C4×D61—84no—
283C4×D61—84no—
284C8×C61—22no—
285C8×C61—41no—
286#92418—22no—
287#92418—41no—
288#22766—611nomaximal, Sylow
289C22×C61C2×C4×D411yescharacteristic
290C22×C61#54611yescharacteristic
291C22×C61#54811yescharacteristic
292C22×C61C22×D811yescharacteristic
293C22×C61C2×#13511yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×#11711yescharacteristic
317C4×C61#52011yescharacteristic
318C4×C61#55211yescharacteristic
319C4×C61#55311yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325Dic61C4×D814yes—
326C2×C4×D8—611no—
327#2354—611no—
328D4×C4⋊C4—611no—
329#1934—611no—
330#1934—611no—
331#1918—611no—
332#1918D6111yescharacteristic, complemented
333#1879—611no—
334#1879—611no—
335#1834—611no—
336#1802—611no—
337#1802—611no—
338C8×C4⋊C4—611no—
339C4×#102—611no—
340C4×#102—611no—
341C2×C61#235211yescharacteristic
342C2×C61C2×C4×D811yescharacteristic
343C2×C61#295211yescharacteristic
344C2×C61#329411yescharacteristic
345C2×C61#330311yescharacteristic
346C2×C61#335811yescharacteristic
347C2×C61#336011yescharacteristic
348D61—44no—
349D61—82no—
350C22×D8—611no—
351C2×C4×D4—611no—
352C2×C4×D4—611no—
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×#117—1221no—
356C22×C4⋊C4—1221no—
357C4×D8—618no—
358C2×#107C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×C4×C8—611no—
367C2×C4×C8—611no—
368C2×C4×C8—611no—
369#390—611no—
370#390—611no—
371#389—611no—
372#389C2×D6111yescharacteristic
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#385—1221no—
377#384—611no—
378#384—611no—
379#380—1221no—
380#380—1221no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4C2×D6111yescharacteristic
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#336—611no—
393#336C2×D6111yescharacteristic
394C61#2276611yescharacteristic, complemented, Sylow
395C22×D4—611no—
396C22×D4—611no—
397C23×C4—1221no—
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×D8—614no—
402C2×D8—1222no—
403C2×D8—1222no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C4×D4—614no—
409C4×D4—614no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—2441no—
415C4×D4—2441no—
416#117—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⋊C4—2444no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
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×C42—21no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107C4×D6114yescomplemented
446C4⋊C8—1222no—
447#102—614no—
448#102—614no—
449#102—1222no—
450#102—1222no—
451#102—2441no—
452#102—2441no—
453C4×C8—614no—
454C4×C8—2441no—
455C4×C8—2441no—
456#95—21no—
457#95—21no—
458#95—611no—
459#95—611no—
460#95—1221no—
461#95—1221no—
462C24—1221no—
463C24—1221no—
464C2×D4—612no—
465C2×D4—612no—
466C2×D4—612no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501D8—1228no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—612no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8C2×C4×D6112yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—614no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4C2×C4×D6112yes—
533C4⋊C4C2×C4×D6112yes—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—612no—
546C42—612no—
547C42—1221no—
548C42—1221no—
549C42—2442no—
550C23—1222no—
551C23—1222no—
552C23—1222no—
553C23—1222no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C2×D4×D6111yescharacteristic
565D4—614no—
566D4—1222no—
567D4—1222no—
568D4—1222no—
569D4—1222no—
570D4—1222no—
571D4—2441no—
572D4—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
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—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—4884no—
603C2×C4—4884no—
604C2×C4—2448no—
605C2×C4—2448no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4D8×D6112yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#1451982311yescharacteristic
618C2×C4#1451982511yescharacteristic
619C8—22no—
620C8—41no—
621C8—1222no—
622C8—2441no—
623C22—1224no—
624C22—4881no—
625C22—4881no—
626C22—4881no—
627C22—4881no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C4×D4×D6111yescharacteristic
638C22#1684040411yescharacteristic
639C22#1684042211yescharacteristic
640C22C2×D8×D6111yescharacteristic
641C22#135×D6111yescharacteristic
642C22#1684057111yescharacteristic
643C22#1684057111yescharacteristic
644C4—22no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—614no—
651C4—1222no—
652C4—1222no—
653C4—4881no—
654C4—4881no—
655C4—2441no—
656C4#117×D6111yescharacteristic
657C4#1684024411yescharacteristic
658C4#1684044411yescharacteristic
659C4#1684046311yescharacteristic
660C2—4882no—
661C2—2444no—
662C2#2144874311yescharacteristic
663C2C4×D8×D6111yescharacteristic
664C2#2145469911yescharacteristic
665C2#2145763011yescharacteristic
666C2#2145774511yescharacteristic
667C2#2145845311yescharacteristic
668C2#2145847111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1918C2
socleC23×C61C22×D4
prime radicalC22×D488C22
cosocleC22×C4×C61C24
hypercenter#1918D61
hypocenterC61#22766
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #21448764C2semidirect
3. #21448720C2semidirect
4. #21446503C2semidirect
5. #21446501C2semidirect
6. #21446392C2semidirect
9. #21445945C2semidirect
10. #21445818C2semidirect
11. #21445818C2semidirect
12. #21445680C2semidirect
13. C4×#14519697C2semidirect
14. #21445633C2semidirect
15. C61×#1918C2semidirect
23. #16839844C4semidirect
78. #14519739C2×C4semidirect
107. #14519697C2×C4semidirect
110. C8×Dic61C2×C4semidirect
124. #107×C61C2×C4semidirect
202. #11758196D8semidirect
279. D244#117semidirect
332. #1918D61semidirect
394. C61#22766semidirect
445. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398442
#214487642
#214487202
#214465032
#214465012
#214463922
#214461682
#214461662
#214459452
#214458182
#214456802
C4×#145196972
#214456332
C61×#19182
#2276661