Finite Group Databasebeta

← #29647787

Subgroups — #29647787

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459653C211yesmaximal, characteristic
2#21454845C211yesmaximal, characteristic, complemented
3#21448726C211yesmaximal, characteristic, complemented
4#21444392C211yesmaximal, characteristic, complemented
5#21444387C211yesmaximal, characteristic, complemented
6#21444359C212yesmaximal, complemented
7#21444354C212yesmaximal, complemented
8#21444341C212yesmaximal, complemented
9#21444320C212yesmaximal, complemented
10C2×#16839498C211yesmaximal, characteristic, complemented
11C2×C61×#327C211yesmaximal, characteristic, complemented
12#16840464C412yescomplemented
13#16840455—22no—
14#16840447C412yescomplemented
15#16840438—22no—
16#16840418C412yescomplemented
17#16840406C412yescomplemented
18#16840357—21no—
19#16840349—21no—
20#16840321—21no—
21#16840293C2212yes—
22#16840272C2212yes—
23C2×#14519756C2211yescharacteristic
24#16840237—21no—
25C2×C4×C22⋊D61—21no—
26C2×C4×C22⋊D61C2211yescharacteristic
27#16839833—22no—
28#16839830—22no—
29#16839816C2212yescomplemented
30#16839724C2212yescomplemented
31C2×C22⋊#92418C2211yescharacteristic
32#16839659—21no—
33#16839636—21no—
34#16839635—21no—
35#16839626C2211yescharacteristic, complemented
36#16839622C2211yescharacteristic
37#16839621C2211yescharacteristic, complemented
38#16839613C2211yescharacteristic, complemented
39#95×D61C2211yescharacteristic, complemented
40#16839500C2212yescomplemented
41#16839500C2212yescomplemented
42#16839498C2212yescomplemented
43#16839498C2212yescomplemented
44#16839496C2212yes—
45#16839496C2212yes—
46C2×#121×C61C2211yescharacteristic
47C2×C22⋊C8×C61C2211yescharacteristic
48C2×#95×C61C2211yescharacteristic
49C61×#327C2212yes—
50C61×#327C2212yes—
51C2×#11758224C2×C412yescomplemented
52C22×Dic122C2×C411yescharacteristic, complemented
53C22×C22⋊D61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61C2311yescharacteristic
56C2×#11758189—22no—
57C22×#92418—21no—
58#14519772—22no—
59#14519770—22no—
60#14519770—22no—
61#14519768C2×C412yescomplemented
62#14519767—22no—
63#14519764C2×C412yescomplemented
64#14519763—22no—
65#14519761—22no—
66#14519760—22no—
67#14519759—22no—
68#14519758—22no—
69#14519758—22no—
70#14519757C2×C412yescomplemented
71#14519756—21no—
72#14519756—21no—
73#14519756C2×C412yescomplemented
74#14519756C2312yes—
75#14519754—22no—
76#14519749—22no—
77#14519749—22no—
78#14519749—22no—
79#14519748C2×C412yescomplemented
80#14519746—22no—
81#14519745—22no—
82#14519742C2×C412yescomplemented
83D4×Dic61—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61C2×C412yescomplemented
88C4×C22⋊D61—44no—
89C4×C22⋊D61C2×C412yes—
90C4×C22⋊D61C2×C412yes—
91C4×C22⋊D61D412yes—
92C4×C22⋊D61D412yes—
93#14519737—22no—
94#14519736—22no—
95#14519735C2×C412yescomplemented
96#14519733—22no—
97C2×C4⋊Dic61—21no—
98C2×C4⋊Dic61C2×C411yescharacteristic, complemented
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196D411yescharacteristic
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—22no—
106C2×C22⋊Dic61—21no—
107C2×C22⋊Dic61D411yescharacteristic
108C2×#11758198—21no—
109C2×#11758198D411yescharacteristic
110C2×C4×Dic61—21no—
111C2×C4×Dic61C2311yescharacteristic
112#14519706—22no—
113C22⋊#92418—21no—
114C22⋊#92418C2312yes—
115#14519689—22no—
116#14519687—22no—
117#14519684—21no—
118#14519684—21no—
119#14519684C2311yescharacteristic
120#14519684C2311yescharacteristic
121C22×D4×C61—21no—
122C22×D4×C61C2×C411yescharacteristic, complemented
123C23×C4×C61—21no—
124C23×C4×C61C2311yescharacteristic
125C22×C8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#121×C61C2×C412yescomplemented
129#121×C61C2312yes—
130C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
131C2×C22⋊C4×C61—21no—
132C22⋊C8×C61—21no—
133C22⋊C8×C61C2312yes—
134#95×C61—21no—
135#95×C61C2311yescharacteristic
136#95×C61C2311yescharacteristic
137C24×C61—21no—
138C24×C61C2×D411yescharacteristic
139C23×D61C2×D411yescharacteristic
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—21no—
143C2×D4×C61—42no—
144C2×D4×C61—42no—
145C2×D4×C61—81no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—42no—
151C22×C4×C61—42no—
152C22×C4×C61—81no—
153C22×C4×C61—81no—
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2411yescharacteristic
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61C22⋊C412yes—
165C2×C22⋊D61C22⋊C412yes—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175#11758224—82no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—81no—
179C2×C4×D61—81no—
180C2×C4×D61C22×C412yes—
181C2×C4×D61C2×D412yes—
182C2×Dic122—21no—
183C2×Dic122—81no—
184C2×Dic122C22×C412yes—
185C2×C8×C61—22no—
186C2×C8×C61—81no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197C22⋊Dic61—42no—
198C22⋊Dic61C22⋊C412yes—
199C22⋊Dic61C22⋊C412yes—
200#11758198—22no—
201#11758198—22no—
202#11758198—42no—
203#11758198C22⋊C412yes—
204#11758198C22⋊C412yes—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61—81no—
209C4⋊Dic61C22×C412yes—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—42no—
215#11758196—42no—
216#11758196—81no—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—42no—
222C4×Dic61C22×C412yes—
223C4×Dic61C2×D412yes—
224C2×#92418—22no—
225C2×#92418—81no—
226#11758189—82no—
227C23×C61—22no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—42no—
231C23×C61—81no—
232C23×C61—81no—
233C23×C61C2×C22⋊C411yescharacteristic
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61AGL(1,ℤ/8)12yes—
236C23×C61#13712yes—
237C23×C61C22×D411yescharacteristic
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—41no—
241C22×D61—41no—
242C22×D61C2×C22⋊C412yes—
243D4×C61—42no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—44no—
247D4×C61—81no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—84no—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C22⋊D61—24no—
268C22⋊D61—24no—
269C22⋊D61—82no—
270C22⋊D61—82no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—82no—
284C2×Dic61—82no—
285C2×Dic61—81no—
286C2×Dic61—81no—
287C2×Dic61C2×C22⋊C412yes—
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C412yes—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—84no—
293Dic122—42no—
294Dic122—81no—
295C8×C61—42no—
296#92418—42no—
297#17622—611nomaximal, Sylow
298C22×C61—21no—
299C22×C61—21no—
300C22×C61—21no—
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—162no—
310C22×C61#41712yes—
311C22×C61#41712yes—
312C22×C61C22×C22⋊C411yescharacteristic
313C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
314C22×C61C2×#13711yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—42no—
319C2×D61—42no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—21no—
323C4×C61—21no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—81no—
327Dic61—22no—
328Dic61—22no—
329Dic61—22no—
330Dic61—42no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334#3505—611no—
335#2973—611no—
336#2349—611no—
337#1520—611no—
338#1519—611no—
339#1513—612no—
340#1512—612no—
341#1509—612no—
342#1504—612no—
343C2×#328—611no—
344C2×#327D6111yescharacteristic, complemented
345C2×C61—21no—
346C2×C61—21no—
347C2×C61—22no—
348C2×C61—22no—
349C2×C61—82no—
350C2×C61#288311yescharacteristic
351C2×C61C2×#41711yescharacteristic
352C2×C61#290111yescharacteristic
353D61—42no—
354#553—612no—
355#553—1222no—
356#552—612no—
357#551—1222no—
358#548—612no—
359#546—612no—
360#539—1221no—
361#538—1221no—
362#534—1221no—
363#529—612no—
364#525—612no—
365C2×C22⋊Q8—611no—
366C2×#121C2×D6111yescharacteristic
367#518—1221no—
368C2×C4×D4—611no—
369C2×C4×D4—1221no—
370#435—1222no—
371#435—1222no—
372#432—612no—
373#407—612no—
374C2×C22⋊C8—611no—
375C2×C22⋊C8C2×D6111yescharacteristic
376#389—1221no—
377#385—1221no—
378#385—1221no—
379#382—611no—
380#380—611no—
381#380—611no—
382C4×C4⋊C4—611no—
383C2×#95—611no—
384C2×#95C2×D6111yescharacteristic
385#328—612no—
386#328—612no—
387#328—612no—
388#328—612no—
389#327—612no—
390#327—612no—
391#327C2×D6112yes—
392#327C2×D6112yes—
393C61#1762211yescharacteristic, complemented, Sylow
394C2×#41—612no—
395C22×Q8—611no—
396C22×D4—21no—
397C22×D4—611no—
398C22×D4C4×D6111yescharacteristic, complemented
399C23×C4—21no—
400C23×C4—611no—
401C23×C4—1221no—
402C23×C4C22×D6111yescharacteristic
403C2×M16—1222no—
404C22×C8—21no—
405C22×C8—1221no—
406#126—1222no—
407#126—1222no—
408#126—1222no—
409#125—612no—
410#125—1222no—
411#124—612no—
412#124—1222no—
413#123—1222no—
414#123—1222no—
415#123—1222no—
416#123—1222no—
417#123—1222no—
418C22⋊Q8—612no—
419C22⋊Q8—612no—
420C22⋊Q8—612no—
421C22⋊Q8—1222no—
422C22⋊Q8—1221no—
423C22⋊Q8—1221no—
424#121—21no—
425#121—21no—
426#121—612no—
427#121—1222no—
428#121—1222no—
429#121—1222no—
430#121C4×D6112yescomplemented
431#121C22×D6112yes—
432#120—1222no—
433#120—1222no—
434C4×Q8—612no—
435C4×D4—612no—
436C4×D4—612no—
437C4×D4—612no—
438C4×D4—612no—
439C4×D4—612no—
440C4×D4—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4—1222no—
445C4×D4—2444no—
446#117—612no—
447#117—1222no—
448#117—1222no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4C4×D6111yescharacteristic, complemented
457C2×C22⋊C4—21no—
458C2×C22⋊C4—611no—
459C2×C22⋊C4—611no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C42—611no—
465C2×C42—1221no—
466C4⋊C8—1222no—
467C22⋊C8—21no—
468C22⋊C8—612no—
469C22⋊C8—1222no—
470C22⋊C8—1221no—
471C22⋊C8C22×D6112yes—
472#97—1222no—
473#95—21no—
474#95—611no—
475#95—611no—
476#95—1221no—
477#95—1221no—
478#95C22×D6111yescharacteristic
479#95C22×D6111yescharacteristic
480C24—21no—
481C24—611no—
482C24D4×D6111yescharacteristic
483#41—4882no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×Q8—4881no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—81no—
493C2×D4—612no—
494C2×D4—612no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4C2×C4×D6112yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—4881no—
519C22×C4—4881no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4C2×C4×D6111yescharacteristic
523C22×C4C2×C4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528M16—4882no—
529C2×C8—22no—
530C2×C8—81no—
531C2×C8—1222no—
532C2×C8—4881no—
533C4⋊C4—21no—
534C4⋊C4—612no—
535C4⋊C4—612no—
536C4⋊C4—612no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1222no—
544C4⋊C4—4881no—
545C4⋊C4—4881no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4C2×C4×D6112yes—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—612no—
552C22⋊C4—612no—
553C22⋊C4—612no—
554C22⋊C4—612no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—1222no—
562C22⋊C4—1222no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—612no—
566C42—612no—
567C42—1222no—
568C42—1222no—
569C42—2442no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—81no—
575C23—81no—
576C23—612no—
577C23—1221no—
578C23—1221no—
579C23—2441no—
580C23—2441no—
581C23C22⋊C4×D6111yescharacteristic
582C23C22⋊C4×D6111yescharacteristic
583C23#1451980112yes—
584C23#1451980612yes—
585C23C2×D4×D6111yescharacteristic
586Q8—4881no—
587Q8—2442no—
588D4—42no—
589D4—44no—
590D4—42no—
591D4—42no—
592D4—81no—
593D4—1224no—
594D4—1224no—
595D4—4882no—
596D4—4882no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—84no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—4884no—
620C2×C4—4882no—
621C2×C4—4882no—
622C2×C4—4881no—
623C2×C4—4881no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4C22⋊C4×D6111yescharacteristic
635C2×C4C22⋊C4×D6111yescharacteristic
636C2×C4C22×C4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C8—42no—
639C8—2442no—
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—162no—
652C22—1222no—
653C22—1222no—
654C22—1222no—
655C22—2442no—
656C22—2442no—
657C22#1683976012yes—
658C22#1683976012yes—
659C22C2×C22⋊C4×D6111yescharacteristic
660C22C2×#1451980111yescharacteristic
661C22C2×#1451980611yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—42no—
667C4—42no—
668C4—81no—
669C4—1222no—
670C4—1222no—
671C4—1222no—
672C4—4881no—
673C4—4881no—
674C4—2442no—
675C4—2442no—
676C2—21no—
677C2—22no—
678C2—21no—
679C2—22no—
680C2—82no—
681C2—2442no—
682C2#2145427711yescharacteristic
683C2C2×#1683976011yescharacteristic
684C2#2145436311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#327C2
socleC22×C61C22×C22⋊C4
prime radical#21459653C2
cosocleC22×C4×C61C24
hypercenterC2×#327D61
hypocenterC61#17622
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #21454845C2semidirect
3. #21448726C2semidirect
4. #21444392C2semidirect
5. #21444387C2semidirect
6. #21444359C2semidirect
7. #21444354C2semidirect
8. #21444341C2semidirect
9. #21444320C2semidirect
10. C2×#16839498C2semidirect
11. C2×C61×#327C2semidirect
12. #16840464C4semidirect
14. #16840447C4semidirect
16. #16840418C4semidirect
17. #16840406C4semidirect
29. #16839816C22semidirect
30. #16839724C22semidirect
35. #16839626C22semidirect
37. #16839621C22semidirect
38. #16839613C22semidirect
39. #95×D61C22semidirect
40. #16839500C22semidirect
41. #16839500C22semidirect
42. #16839498C22semidirect
43. #16839498C22semidirect
51. C2×#11758224C2×C4semidirect
52. C22×Dic122C2×C4semidirect
61. #14519768C2×C4semidirect
63. #14519764C2×C4semidirect
70. #14519757C2×C4semidirect
73. #14519756C2×C4semidirect
79. #14519748C2×C4semidirect
82. #14519742C2×C4semidirect
87. D4×Dic61C2×C4semidirect
95. #14519735C2×C4semidirect
98. C2×C4⋊Dic61C2×C4semidirect
122. C22×D4×C61C2×C4semidirect
128. #121×C61C2×C4semidirect
130. C2×C4⋊C4×C61C2×C4semidirect
344. C2×#327D61semidirect
393. C61#17622semidirect
398. C22×D4C4×D61semidirect
430. #121C4×D61semidirect
456. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214596532
#214548452
#214487262
#214443922
#214443872
#214443592
#214443542
#214443412
#214443202
C2×#168394982
C2×C61×#3272
#1762261