Finite Group Databasebeta

← #29683044

Subgroups — #29683044

Order 15616.

1042 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×#16839872C211yesmaximal, characteristic
2#21448283C211yesmaximal, characteristic, complemented
3#21446446C211yesmaximal, characteristic, complemented
4#21446439C211yesmaximal, characteristic, complemented
5#21446404C211yesmaximal, characteristic, complemented
6#21446397C211yesmaximal, characteristic, complemented
7#21446167C211yesmaximal, characteristic
8#21446167C211yesmaximal, characteristic
9#21446080C211yesmaximal, characteristic, complemented
10#21446066C211yesmaximal, characteristic, complemented
11#21445717C211yesmaximal, characteristic, complemented
12#21445647C211yesmaximal, characteristic, complemented
13#21445644C211yesmaximal, characteristic, complemented
14#21445568C211yesmaximal, characteristic, complemented
15C42×D4×C61C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519734—21no—
19#16839872C418yescomplemented
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839653—21no—
36#16839653—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×#11758198—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839517C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×D4×C61C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C22×C42×C61—21no—
53C4×C4⋊C4×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C4×C22⋊C4×C61—21no—
57C4×C22⋊C4×C61—21no—
58C4×C22⋊C4×C61—21no—
59C43×C61C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758199—22no—
63C2×#11758199—22no—
64C2×#11758199C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C4×D244—22no—
70C4×D244C2×C414yescomplemented
71#14519734—44no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519710—22no—
84#14519708—22no—
85#14519708C2×C414yescomplemented
86#14519698—22no—
87#14519698—22no—
88#14519698C2×C414yes—
89#14519697—22no—
90#14519697—22no—
91#14519697C2×C414yescomplemented
92#14519688—22no—
93#14519688C2×C414yescomplemented
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C22×D4×C61C2311yescharacteristic
98C23×C4×C61—21no—
99C23×C4×C61—21no—
100C23×C4×C61—21no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61C2×C414yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61—21no—
116C2×C42×C61—44no—
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C22×C412yes—
133C2×D4×C61#4112yes—
134C2×D4×C61#4112yes—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C412yes—
170C4⋊C4×C61#4112yes—
171C4⋊C4×C61#4112yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61—41no—
183C42×C61—42no—
184C42×C61—42no—
185C42×C61—42no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188#11758199—28no—
189#11758198—24no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C412yes—
200#11758196—42no—
201#11758196—42no—
202C4×Dic61—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C22×C412yes—
214C2×#92418C22×C412yes—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61#11714yescomplemented
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—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—48no—
252C2×C4×C61—48no—
253C2×C4×C61—48no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C4×D412yes—
261C2×C4×C61C4×D412yes—
262C2×C4×C61AGL(1,ℤ/8)12yes—
263C2×C4×C61AGL(1,ℤ/8)12yes—
264C2×C4×C61C23×C411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279D244—22no—
280D244—22no—
281D244—41no—
282D244—41no—
283C4×D61—48no—
284#92418—22no—
285#92418—22no—
286#92418—41no—
287#92418—41no—
288#21122—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61#52511yescharacteristic
298C22×C61#52511yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57511yescharacteristic
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—22no—
312C4×C61—22no—
313C4×C61—22no—
314C4×C61—42no—
315C4×C61—41no—
316C4×C61—42no—
317C4×C61—41no—
318C4×C61—42no—
319C4×C61C2×#11711yescharacteristic
320C4×C61C4×#4111yescharacteristic
321C4×C61#52911yescharacteristic
322C4×C61#52911yescharacteristic
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326C2×#442—611no—
327#2293—611no—
328C42×D4D6111yescharacteristic, complemented
329#1926—611no—
330#1925—611no—
331#1920—611no—
332#1919—611no—
333#1879—611no—
334#1879—611no—
335#1864—611no—
336#1862—611no—
337#1776—611no—
338#1762—611no—
339#1762—611no—
340#1745—611no—
341C2×C61—44no—
342C2×C61#229011yescharacteristic
343C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
344C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
345C2×C61#310911yescharacteristic
346C2×C61#311111yescharacteristic
347C2×C61#311411yescharacteristic
348C2×C61#311511yescharacteristic
349D61—82no—
350C22×D8—611no—
351C2×C4×D4—611no—
352C2×C4×D4C2×D6111yescharacteristic
353C2×C4×D4C2×D6111yescharacteristic
354C2×C4×D4C2×D6111yescharacteristic
355C2×#117—1221no—
356C22×C42—21no—
357#442—618no—
358C2×#107—611no—
359C2×#106—611no—
360C2×#106—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#97—611no—
368C2×#97—611no—
369#390—611no—
370#390—611no—
371#389—611no—
372#389—611no—
373#388—1221no—
374#388—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4C2×D6111yescharacteristic
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C22⋊C4—21no—
380C4×C22⋊C4—21no—
381C4×C22⋊C4—21no—
382C4×C22⋊C4—1221no—
383C43C2×D6111yescharacteristic
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336—611no—
391#336—611no—
392#336—611no—
393#336—611no—
394C61#2112211yescharacteristic, complemented, Sylow
395C22×D4—611no—
396C22×D4C22×D6111yescharacteristic
397C23×C4—21no—
398C23×C4—21no—
399C23×C4—21no—
400C23×C4—1221no—
401C2×D8—614no—
402C2×D8—1222no—
403C2×D8—1222no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—611no—
408C4×D4—22no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—614no—
414C4×D4—1222no—
415C4×D4C4×D6114yes—
416#117—2444no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—44no—
437C2×C42—1221no—
438C2×C42C22⋊D6111yescharacteristic
439C2×C42C22⋊D6111yescharacteristic
440C2×C42C22⋊D6111yescharacteristic
441C2×C42C22⋊D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#106—614no—
447#106—1222no—
448#102—614no—
449#102—614no—
450#102—1222no—
451#102—1222no—
452#102—1222no—
453#102—1222no—
454#97—614no—
455#97—1222no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C24—1221no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4C2×C4×D6112yes—
474C2×D4#1175822212yes—
475C2×D4#1175822212yes—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—1221no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501D8—1228no—
502C2×C8—612no—
503C2×C8—612no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—612no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4C2×C4×D6112yes—
526C4⋊C4#1175822212yes—
527C4⋊C4#1175822212yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—1224no—
532C22⋊C4—2442no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—22no—
541C42—22no—
542C42—41no—
543C42—41no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—2442no—
548C42C2×C4×D6112yes—
549C42#1175822212yes—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1222no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C22×C22⋊D6111yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572D4#1451973414yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—1222no—
600C2×C4—2448no—
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×C4C4×C22⋊D6112yes—
610C2×C4C4×C22⋊D6112yes—
611C2×C4#1451980212yes—
612C2×C4#1451980312yes—
613C2×C4C22×C4×D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4C2×#1175822211yescharacteristic
617C2×C4C2×#1175822211yescharacteristic
618C2×C4C2×#1175822211yescharacteristic
619C8—1222no—
620C8—1222no—
621C8—2441no—
622C8—2441no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—4881no—
631C22—4881no—
632C22—4881no—
633C22—4881no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C2×C4×C22⋊D6111yescharacteristic
638C22#1684027311yescharacteristic
639C22#1684027311yescharacteristic
640C22C2×#1451980211yescharacteristic
641C22C2×#1451980311yescharacteristic
642C22#1684056011yescharacteristic
643C22#1684056011yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—4881no—
654C4—4881no—
655C4—2442no—
656C4C2×#1451973411yescharacteristic
657C4C4×#1175822211yescharacteristic
658C4#1684029611yescharacteristic
659C4#1684029611yescharacteristic
660C2—44no—
661C2—4882no—
662C2#2144826211yescharacteristic
663C2C4×#1451980211yescharacteristic
664C2C4×#1451980311yescharacteristic
665C2#2145591011yescharacteristic
666C2#2145592811yescharacteristic
667C2#2145595711yescharacteristic
668C2#2145597011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC42×D4×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenterC42×D4D61
hypocenterC61#21122
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #21448283C2semidirect
3. #21446446C2semidirect
4. #21446439C2semidirect
5. #21446404C2semidirect
6. #21446397C2semidirect
9. #21446080C2semidirect
10. #21446066C2semidirect
11. #21445717C2semidirect
12. #21445647C2semidirect
13. #21445644C2semidirect
14. #21445568C2semidirect
15. C42×D4×C61C2semidirect
19. #16839872C4semidirect
70. C4×D244C2×C4semidirect
85. #14519708C2×C4semidirect
91. #14519697C2×C4semidirect
93. #14519688C2×C4semidirect
233. D4×C61#117semidirect
328. C42×D4D61semidirect
394. C61#21122semidirect
572. D4#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398722
#214482832
#214464462
#214464392
#214464042
#214463972
#214461672
#214460802
#214460662
#214457172
#214456472
#214456442
#214455682
C42×D4×C612
#2112261