Finite Group Databasebeta

← #29699167

Subgroups — #29699167

Order 15616.

978 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×#16839842C211yesmaximal, characteristic
2#21448767C211yesmaximal, characteristic, complemented
3#21448720C211yesmaximal, characteristic, complemented
4#21446518C211yesmaximal, characteristic, complemented
5#21446414C211yesmaximal, characteristic, complemented
6#21446408C211yesmaximal, characteristic, complemented
7#21446170C211yesmaximal, characteristic
8#21446164C211yesmaximal, characteristic
9#21445945C211yesmaximal, characteristic, complemented
10#21445818C211yesmaximal, characteristic, complemented
11#21445817C211yesmaximal, characteristic, complemented
12#21445687C211yesmaximal, characteristic, complemented
13#21445645C211yesmaximal, characteristic, complemented
14#21445643C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic, complemented
16C22×#11758199C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×#14519733—21no—
22C2×C4⋊C4×D61—21no—
23#16839842C418yescomplemented
24C2×#14519710C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839665C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839656—21no—
35#16839650—21no—
36#16839636—21no—
37#16839633—21no—
38#16839631C2211yescharacteristic
39#16839631C2211yescharacteristic
40#16839621—21no—
41#16839620—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C61×#390C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C23×Dic61—21no—
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C2×#11758199—22no—
66C2×#11758199—22no—
67C2×#11758199C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C4×D244—22no—
71C4×D244—41no—
72D4×Dic61—22no—
73D4×Dic61C2×C414yescomplemented
74#14519739—22no—
75#14519739—22no—
76#14519739—41no—
77#14519739C2×C414yes—
78#14519733—44no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
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×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519710C2×C414yescomplemented
101#14519706—22no—
102#14519699—22no—
103#14519699C2×C414yes—
104#14519698—41no—
105#14519697—22no—
106#14519687—41no—
107C8×Dic61C2×C414yescomplemented
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×D4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#102×C61—41no—
124#102×C61C2×C414yescomplemented
125#97×C61—41no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C22×C412yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—24no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×D244#4112yes—
156C2×D244#4112yes—
157C2×C4×D61—24no—
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—42no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—81no—
181C42×C61—21no—
182C42×C61—21no—
183C22⋊Dic61—24no—
184C22⋊Dic61—42no—
185C22⋊Dic61—81no—
186#11758199—28no—
187#11758198—24no—
188#11758198—42no—
189#11758198—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—42no—
200#11758196—42no—
201#11758196—81no—
202#11758196—81no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61C22×C412yes—
208C4×Dic61#4112yes—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C22×C412yes—
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—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61—22no—
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—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×C61C23×C411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—48no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C4×D412yes—
274C2×Dic61C4×D412yes—
275C2×Dic61AGL(1,ℤ/8)12yes—
276C2×Dic61AGL(1,ℤ/8)12yes—
277D244—22no—
278D244—22no—
279D244—22no—
280D244#11714yescomplemented
281C4×D61—48no—
282C4×D61—84no—
283C4×D61—84no—
284C8×C61—22no—
285C8×C61—41no—
286#92418—22no—
287#92418—41no—
288#22758—611nomaximal, Sylow
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—81no—
293C22×C61—81no—
294C22×C61—81no—
295C22×C61—81no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61#54611yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57611yescharacteristic
302C22×C61#57811yescharacteristic
303C2×D61—24no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C2×#11711yescharacteristic
316C4×C61#52011yescharacteristic
317C4×C61#55211yescharacteristic
318C4×C61#55311yescharacteristic
319Dic61—22no—
320Dic61—41no—
321Dic61—42no—
322Dic61—41no—
323Dic61—81no—
324Dic61—81no—
325Dic61#44214yes—
326C2×C4×D8—611no—
327#2354—611no—
328D4×C4⋊C4—611no—
329#1936—611no—
330#1936D6111yescharacteristic, complemented
331#1921—611no—
332#1921—611no—
333#1879—611no—
334#1879—611no—
335#1834—611no—
336#1802—611no—
337#1802—611no—
338#1770—611no—
339#1762—611no—
340#1762—611no—
341C2×C61—82no—
342C2×C61#235211yescharacteristic
343C2×C61C2×#44211yescharacteristic
344C2×C61#295111yescharacteristic
345C2×C61#332511yescharacteristic
346C2×C61#333011yescharacteristic
347C2×C61#336611yescharacteristic
348C2×C61#336711yescharacteristic
349D61—44no—
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×#107—611no—
359C2×C4⋊C8—611no—
360C2×#102—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102C2×D6111yescharacteristic
365C2×#102C2×D6111yescharacteristic
366C2×#97—611no—
367C2×#97C2×D6111yescharacteristic
368C2×C4×C8—611no—
369#390—611no—
370#390C2×D6111yescharacteristic
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⋊C4—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394C61#2275811yescharacteristic, complemented, Sylow
395C22×D4—611no—
396C22×D4C22×D6111yescharacteristic
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—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—2444no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
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—611no—
441C2×C42—611no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—614no—
446C4⋊C8—1222no—
447#102—41no—
448#102—614no—
449#102—1222no—
450#102—1222no—
451#102—2441no—
452#102C4×D6114yescomplemented
453#97—41no—
454#97—2441no—
455C4×C8—614no—
456#95—21no—
457#95—611no—
458#95—611no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C24—1221no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—612no—
470C2×D4—612no—
471C2×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4C2×C4×D6112yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1221no—
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—42no—
520C4⋊C4—612no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4C2×C4×D6112yes—
534C22⋊C4—42no—
535C22⋊C4—81no—
536C22⋊C4—1224no—
537C22⋊C4—1224no—
538C22⋊C4—4881no—
539C22⋊C4—2442no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C42—21no—
543C42—21no—
544C42—612no—
545C42—612no—
546C42—1221no—
547C42—1221no—
548C42—2442no—
549C42—2442no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—1222no—
558C23—1222no—
559C23—1222no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23C2×D4×D6111yescharacteristic
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—614no—
570D4—1222no—
571D4—1222no—
572D4—1222no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
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—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1222no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—4884no—
602C2×C4—4884no—
603C2×C4—2448no—
604C2×C4—2448no—
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×C4C22×C4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4#41×D6111yescharacteristic
617C2×C4#1451982211yescharacteristic
618C2×C4#1451982411yescharacteristic
619C8—22no—
620C8—41no—
621C8—1222no—
622C8—2441no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—1224no—
631C22—2442no—
632C22—2442no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C4×D4×D6111yescharacteristic
638C22#1684040111yescharacteristic
639C22#1684042011yescharacteristic
640C22AGL(1,ℤ/8)×D6111yescharacteristic
641C22AGL(1,ℤ/8)×D6111yescharacteristic
642C22#1684056611yescharacteristic
643C22#1684057811yescharacteristic
644C4—42no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—614no—
650C4—1222no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4—2441no—
655C4—2441no—
656C4#117×D6111yescharacteristic
657C4#1684024111yescharacteristic
658C4#1684044811yescharacteristic
659C4#1684045711yescharacteristic
660C2—82no—
661C2—2444no—
662C2#2144874711yescharacteristic
663C2#442×D6111yescharacteristic
664C2#2145469111yescharacteristic
665C2#2145805611yescharacteristic
666C2#2145811911yescharacteristic
667C2#2145852311yescharacteristic
668C2#2145853411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC22×#11758199C22
cosocleC22×C4×C61C24
hypercenter#1936D61
hypocenterC61#22758
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
2. #21448767C2semidirect
3. #21448720C2semidirect
4. #21446518C2semidirect
5. #21446414C2semidirect
6. #21446408C2semidirect
9. #21445945C2semidirect
10. #21445818C2semidirect
11. #21445817C2semidirect
12. #21445687C2semidirect
13. #21445645C2semidirect
14. #21445643C2semidirect
15. C61×#1936C2semidirect
23. #16839842C4semidirect
73. D4×Dic61C2×C4semidirect
100. #14519710C2×C4semidirect
107. C8×Dic61C2×C4semidirect
124. #102×C61C2×C4semidirect
280. D244#117semidirect
330. #1936D61semidirect
394. C61#22758semidirect
452. #102C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398422
#214487672
#214487202
#214465182
#214464142
#214464082
#214461702
#214461642
#214459452
#214458182
#214458172
#214456872
#214456452
#214456432
C61×#19362
#2275861