Finite Group Databasebeta

← #29701818

Subgroups — #29701818

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840050C211yesmaximal, characteristic
2#21448905C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21446778C211yesmaximal, characteristic, complemented
5#21446579C211yesmaximal, characteristic, complemented
6#21446545C211yesmaximal, characteristic, complemented
7#21446544C211yesmaximal, characteristic, complemented
8#21446398C211yesmaximal, characteristic, complemented
9#21446395C211yesmaximal, characteristic, complemented
10#21446078C211yesmaximal, characteristic
11#21445680C211yesmaximal, characteristic, complemented
12#102×Dic61C211yesmaximal, characteristic
13#102×Dic61C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#1864C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C22×#11758196—21no—
20#16840050C418yescomplemented
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C4×#92418C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839667—21no—
31#16839666—21no—
32#16839660C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839657—21no—
36#16839635—21no—
37#16839634—21no—
38#16839631C2211yescharacteristic
39#16839620—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44#16839613—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519776—22no—
66#14519776C2×C414yescomplemented
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61—41no—
70D4×Dic61C2×C414yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—44no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519709C2×C414yescomplemented
92#14519708—41no—
93#14519708C2×C414yescomplemented
94#14519706C2×C414yescomplemented
95#14519698C2×C414yes—
96#14519698Q814yescomplemented
97C4×#92418—41no—
98C8×Dic61—22no—
99C8×Dic61—41no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C23×C4×C61—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×D4×C61—22no—
113C4×D4×C61—41no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#106×C61—41no—
121#102×C61—22no—
122#102×C61C2×C414yes—
123#95×C61—21no—
124C24×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C22×C412yes—
129C2×D4×C61C2×Q812yes—
130C2×D4×C61#4112yes—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—81no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—81no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—24no—
155C2×Dic122—81no—
156C2×Dic122—81no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61C22×C412yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61C22×C412yes—
170C22⋊C4×C61—42no—
171C42×C61—21no—
172C42×C61—21no—
173C22⋊Dic61—24no—
174C22⋊Dic61—42no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C2×Q812yes—
183#11758196—24no—
184#11758196—22no—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—81no—
189#11758196—81no—
190#11758196SD1614yes—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—42no—
194C4×Dic61—81no—
195C4×Dic61—81no—
196C4×Dic61C22×C412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C22×C412yes—
203C2×#92418C2×Q812yes—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—22no—
215D4×C61C4×Q814yes—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—84no—
231C2×C4×C61C2×SD1612yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×Q811yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—48no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—84no—
259C2×Dic61C4×D412yes—
260C2×Dic61C4×D412yes—
261C2×Dic61C2×SD1612yes—
262C2×Dic61C2×SD1612yes—
263C2×Dic61#13512yes—
264Dic122—84no—
265C8×C61—22no—
266C8×C61—41no—
267#92418—22no—
268#92418—22no—
269#22963—611nomaximal, Sylow
270C22×C61—24no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61C2×C4×D411yescharacteristic
278C22×C61#54611yescharacteristic
279C22×C61D4×Q811yescharacteristic
280C22×C61C22×SD1611yescharacteristic
281C22×C61C2×#13511yescharacteristic
282C22×C61#57611yescharacteristic
283C22×C61#57711yescharacteristic
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C2×C4×Q811yescharacteristic
290C4×C61#52011yescharacteristic
291C4×C61#55211yescharacteristic
292C4×C61#55411yescharacteristic
293Dic61—22no—
294Dic61—22no—
295Dic61—42no—
296Dic61—42no—
297Dic61—42no—
298Dic61—41no—
299Dic61—81no—
300Dic61C4×SD1614yes—
301C2×#476—611no—
302#2369—611no—
303D4×C4⋊C4—611no—
304#1990—611no—
305#1948—611no—
306#1941—611no—
307#1941—611no—
308#1919—611no—
309#1919—611no—
310#1864—611no—
311#1864D6111yescharacteristic
312C8×C4⋊C4—611no—
313C4×#102—611no—
314C4×#102—611no—
315C4×#102—611no—
316C2×C61—44no—
317C2×C61#236011yescharacteristic
318C2×C61C2×C4×SD1611yescharacteristic
319C2×C61#294611yescharacteristic
320C2×C61#329511yescharacteristic
321C2×C61#330311yescharacteristic
322C2×C61Q8×SD1611yescharacteristic
323C2×C61#338011yescharacteristic
324C2×C4⋊Q8—611no—
325C2×C4×D4—611no—
326C2×C4×D4—611no—
327C2×C4×D4C2×D6111yescharacteristic
328C22×C4⋊C4—1221no—
329#476—618no—
330C2×#106—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×#106C2×D6111yescharacteristic
334C2×C4⋊C8—611no—
335C2×#102—611no—
336C2×#102—611no—
337C2×#102C2×D6111yescharacteristic
338C2×#102C2×D6111yescharacteristic
339C2×C4×C8—611no—
340C2×C4×C8—611no—
341C2×C4×C8—611no—
342#391—1221no—
343#391—1221no—
344#389—611no—
345#389—611no—
346#389—611no—
347#389—1221no—
348#389C2×D6111yescharacteristic
349#385—1221no—
350#385—1221no—
351#384—611no—
352#380—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—1221no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340C2×D6111yescharacteristic
366#336—611no—
367#336C2×D6111yescharacteristic
368C61#2296311yescharacteristic, complemented, Sylow
369C22×Q8—1221no—
370C22×D4C22×D6111yescharacteristic
371C23×C4—21no—
372C23×C4—1221no—
373C23×C4—1221no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—614no—
379C4⋊Q8—1222no—
380C4×D4—22no—
381C4×D4—41no—
382C4×D4—614no—
383C4×D4—1222no—
384C4×D4—1222no—
385C4×D4—2441no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—2444no—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#106—41no—
413#106—614no—
414#106—614no—
415#106—2441no—
416C4⋊C8—614no—
417#102—22no—
418#102—614no—
419#102—614no—
420#102C4×D6114yes—
421C4×C8—1222no—
422C4×C8—2441no—
423C4×C8—2441no—
424#95—21no—
425#95—611no—
426#95—611no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—21no—
434C2×Q8—1224no—
435C2×Q8—4881no—
436C2×Q8—4881no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×C4×D6112yes—
441C2×D4#1175822412yes—
442C2×D4Q8×D6112yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—4881no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8C2×C4×D6112yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—614no—
485C4⋊C4—612no—
486C4⋊C4—612no—
487C4⋊C4—612no—
488C4⋊C4—1222no—
489C4⋊C4—1224no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4C2×C4×D6112yes—
501C22⋊C4—42no—
502C22⋊C4—1224no—
503C22⋊C4—2442no—
504C42—21no—
505C42—21no—
506C42—612no—
507C42—612no—
508C42—1221no—
509C42—1221no—
510C42—4881no—
511C42—4881no—
512C42—2442no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D6111yescharacteristic
521Q8—4884no—
522D4—22no—
523D4—22no—
524D4—22no—
525D4#1451974214yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—4884no—
557C2×C4—2448no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4SD16×D6112yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4C2×Q8×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C2×C4#1451982211yescharacteristic
573C2×C4#1451982411yescharacteristic
574C8—22no—
575C8—41no—
576C8—1222no—
577C8—1222no—
578C22—24no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C4×D4×D6111yescharacteristic
586C22#1684039911yescharacteristic
587C22#1684042511yescharacteristic
588C22C2×SD16×D6111yescharacteristic
589C22#135×D6111yescharacteristic
590C22#1684056611yescharacteristic
591C22#1684057211yescharacteristic
592C4—22no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—614no—
598C4—1222no—
599C4—1222no—
600C4—4881no—
601C4—2442no—
602C4—2441no—
603C4—2442no—
604C4—2442no—
605C4C2×#1451974211yescharacteristic
606C4#1684024111yescharacteristic
607C4#1684044911yescharacteristic
608C4#1684046611yescharacteristic
609C2—44no—
610C2#2144881911yescharacteristic
611C2C4×SD16×D6111yescharacteristic
612C2#2145464811yescharacteristic
613C2#2145763611yescharacteristic
614C2#2145774011yescharacteristic
615C2#2145867411yescharacteristic
616C2#2145868111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1864D61
hypocenterC61#22963
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #21448905C2semidirect
4. #21446778C2semidirect
5. #21446579C2semidirect
6. #21446545C2semidirect
7. #21446544C2semidirect
8. #21446398C2semidirect
9. #21446395C2semidirect
11. #21445680C2semidirect
20. #16840050C4semidirect
66. #14519776C2×C4semidirect
91. #14519709C2×C4semidirect
93. #14519708C2×C4semidirect
94. #14519706C2×C4semidirect
96. #14519698Q8semidirect
368. C61#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
#214489052
D4×#117581962
#214467782
#214465792
#214465452
#214465442
#214463982
#214463952
#214460782
#214456802
#102×Dic612
C4×#145196982
C61×#18642
#2296361