Finite Group Databasebeta

← #29683127

Subgroups — #29683127

Order 15616.

862 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×#16839865C211yesmaximal, characteristic
2#21448458C211yesmaximal, characteristic
3#21448184C211yesmaximal, characteristic, complemented
4#21446445C211yesmaximal, characteristic
5#21446440C211yesmaximal, characteristic
6#21446408C211yesmaximal, characteristic, complemented
7#21446393C211yesmaximal, characteristic, complemented
8#21446174C211yesmaximal, characteristic
9#21446172C211yesmaximal, characteristic
10#21446075C211yesmaximal, characteristic, complemented
11#21445719C211yesmaximal, characteristic, complemented
12#103×Dic61C211yesmaximal, characteristic, complemented
13#21445633C211yesmaximal, characteristic
14#21445569C211yesmaximal, characteristic, complemented
15C61×#1865C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20#16839865C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839654—21no—
33#16839652—21no—
34#16839624—21no—
35#16839619—21no—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61—21no—
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61—21no—
42C4×C4⋊Dic61C2211yescharacteristic
43#16839608—21no—
44#16839608C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61C2×C414yes—
71#14519739—22no—
72#14519739C2×C414yes—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519711—22no—
93#14519710C2×C414yes—
94#14519701—22no—
95#14519699—22no—
96#14519699C2×C414yes—
97#14519697—22no—
98#14519687C2×C414yes—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684D411yescharacteristic
106#14519684D411yescharacteristic
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#106×C61—22no—
122#103×C61—22no—
123#103×C61C2×C414yes—
124#95×C61—21no—
125C23×D61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61#4112yes—
131C2×Q8×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—81no—
154C2×C4×D61—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C22×C412yes—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61C22×C412yes—
168C4⋊C4×C61#4112yes—
169C42×C61—24no—
170C42×C61—21no—
171C42×C61—21no—
172#11758201—28no—
173#11758198—24no—
174#11758198—81no—
175#11758198—81no—
176#11758198—81no—
177#11758198—81no—
178C4⋊Dic61—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—24no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—81no—
185C4⋊Dic61C22×C412yes—
186C4⋊Dic61#4112yes—
187C4×Dic61—24no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—24no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C4×Dic61C22×C412yes—
196C4×Dic61#4112yes—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C22×C412yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61#11714yes—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61C4×D412yes—
259C2×Dic61C4×D412yes—
260D244—22no—
261D244—22no—
262D244—41no—
263D244—41no—
264C4×D61—84no—
265C8×C61—22no—
266C8×C61—41no—
267#92418—22no—
268#92418—41no—
269#21128—611nomaximal, Sylow
270C22×C61C2×C4×D411yescharacteristic
271C22×C61#52511yescharacteristic
272C22×C61#52511yescharacteristic
273C22×C61#57611yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—24no—
285C4×C61—41no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61C2×#11711yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—42no—
298Dic61—41no—
299Dic61—42no—
300Dic61—81no—
301C2×#440—611no—
302#2319—611no—
303#2276—611no—
304#1926—611no—
305#1925—611no—
306#1921—611no—
307#1918—611no—
308#1880—611no—
309#1880—611no—
310#1865D6111yescharacteristic, complemented
311#1863—611no—
312#1776—611no—
313C4×#103—611no—
314C4×#102—611no—
315#1746—611no—
316C2×C61#229011yescharacteristic
317C2×C61#294611yescharacteristic
318C2×C61#294711yescharacteristic
319C2×C61#311911yescharacteristic
320C2×C61#312111yescharacteristic
321C2×C61#312411yescharacteristic
322C2×C61#312611yescharacteristic
323D61—82no—
324C22×SD16—611no—
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×D4—611no—
329#440—618no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#106C2×D6111yescharacteristic
333C2×#103—611no—
334C2×#103C2×D6111yescharacteristic
335C2×#103C2×D6111yescharacteristic
336C2×#102—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#97—611no—
340C2×C4×C8—611no—
341#390—611no—
342#390—611no—
343#389—611no—
344#389C2×D6111yescharacteristic
345#388—1221no—
346#388—1221no—
347#381—1221no—
348#380—1221no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—1221no—
354C4×C4⋊C4—1221no—
355C4×C4⋊C4—1221no—
356#376—611no—
357#376—1221no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340C2×D6111yescharacteristic
364#336—611no—
365#336—611no—
366#336—611no—
367#336C2×D6111yescharacteristic
368C61#2112811yescharacteristic, complemented, Sylow
369C22×Q8C22×D6111yescharacteristic
370C22×D4—611no—
371C23×C4—1221no—
372C2×SD16—614no—
373C2×SD16—1222no—
374C2×SD16—1222no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8—614no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×D4—614no—
386C4×D4—1222no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—21no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—614no—
414#107—1222no—
415#106—22no—
416#103—22no—
417#103—1222no—
418#103C4×D6114yes—
419#102—614no—
420#102—1222no—
421#102—1222no—
422#97—614no—
423C4×C8—1222no—
424#95—21no—
425#95—611no—
426#95—611no—
427#95—611no—
428#95—611no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—1221no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D6112yes—
439C2×Q8#1175822412yes—
440C2×Q8#1175822612yes—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1224no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—4881no—
460C22×C4—4881no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464SD16—1228no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8C2×C4×D6112yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—4881no—
492C4⋊C4C2×C4×D6112yes—
493C4⋊C4#1175822412yes—
494C22⋊C4—1224no—
495C22⋊C4—4881no—
496C22⋊C4—4881no—
497C22⋊C4—4881no—
498C22⋊C4—4881no—
499C42—24no—
500C42—21no—
501C42—21no—
502C42—612no—
503C42—612no—
504C42—1224no—
505C42—1224no—
506C42—1221no—
507C42—1221no—
508C42—4881no—
509C42—4881no—
510C42—4881no—
511C42—4881no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23C2×D4×D6111yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1451973514yes—
524D4—1222no—
525D4—1222no—
526D4—2441no—
527D4—2441no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4884no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C2×C4C2×#1175822411yescharacteristic
573C2×C4C2×#1175822611yescharacteristic
574C8—22no—
575C8—41no—
576C8—1222no—
577C8—2441no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22C4×D4×D6111yescharacteristic
586C22#1684026811yescharacteristic
587C22#1684027511yescharacteristic
588C22#1684056411yescharacteristic
589C22#1684056811yescharacteristic
590C22#1684057111yescharacteristic
591C22#1684057811yescharacteristic
592C4—24no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—1222no—
599C4—1222no—
600C4—4881no—
601C4—2442no—
602C4—2441no—
603C4—2442no—
604C4—2441no—
605C4C2×#1451973511yescharacteristic
606C4C4×#1175822411yescharacteristic
607C4#1684028911yescharacteristic
608C4#1684029811yescharacteristic
609C2—4882no—
610C2#2144825511yescharacteristic
611C2#2145464611yescharacteristic
612C2#2145466111yescharacteristic
613C2#2145602511yescharacteristic
614C2#2145605511yescharacteristic
615C2#2145608211yescharacteristic
616C2#2145610411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1865D61
hypocenterC61#21128
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. #21448184C2semidirect
6. #21446408C2semidirect
7. #21446393C2semidirect
10. #21446075C2semidirect
11. #21445719C2semidirect
12. #103×Dic61C2semidirect
14. #21445569C2semidirect
15. C61×#1865C2semidirect
20. #16839865C4semidirect
310. #1865D61semidirect
368. C61#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#168398652
#214484582
#214481842
#214464452
#214464402
#214464082
#214463932
#214461742
#214461722
#214460752
#214457192
#103×Dic612
#214456332
#214455692
C61×#18652
#2112861