Finite Group Databasebeta

← #29683136

Subgroups — #29683136

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×#16839864C211yesmaximal, characteristic
2#21448463C211yesmaximal, characteristic
3#21448182C211yesmaximal, characteristic, complemented
4#21446447C211yesmaximal, characteristic
5#21446440C211yesmaximal, characteristic
6#21446412C211yesmaximal, characteristic, complemented
7#21446391C211yesmaximal, characteristic, complemented
8#21446173C211yesmaximal, characteristic
9#21446173C211yesmaximal, characteristic
10#21446090C211yesmaximal, characteristic, complemented
11#21446076C211yesmaximal, characteristic, complemented
12#21445716C211yesmaximal, characteristic, complemented
13C4×#14519702C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1746C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839864C418yescomplemented
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839651—21no—
36#16839651—21no—
37#16839624—21no—
38#16839621—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839613—21no—
44#16839613—21no—
45#16839613—21no—
46#16839607—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#97×C61C2211yescharacteristic
54C2×C4×C8×C61C2211yescharacteristic
55C61×#376C2211yescharacteristic
56C61×#336C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758190—22no—
64C2×#11758190—22no—
65C2×#11758190C2×C414yes—
66C4×Dic122—22no—
67C4×Dic122—22no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71C4×Dic122C2×C414yes—
72C4×D244—22no—
73C4×D244C2×C414yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97#14519711—22no—
98#14519711C2×C414yes—
99#14519709—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702C2×C414yes—
103#14519699—22no—
104#14519699—22no—
105#14519699C2×C414yes—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#97×C61C2×C414yes—
120C4×C8×C61—22no—
121#95×C61D411yescharacteristic
122#95×C61D411yescharacteristic
123#95×C61D411yescharacteristic
124#95×C61D411yescharacteristic
125C23×D61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—24no—
147C2×C4×D61—81no—
148C2×C4×D61—81no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122C22×C412yes—
153C2×Dic122#4112yes—
154C2×Dic122#4112yes—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C22×C412yes—
166C2×C8×C61C22×C412yes—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61—21no—
171C42×C61C22×C412yes—
172C42×C61#4112yes—
173#11758198—24no—
174#11758198—81no—
175#11758198—81no—
176#11758198—81no—
177#11758198—81no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22×C412yes—
187C4⋊Dic61C22×C412yes—
188C4⋊Dic61#4112yes—
189C4⋊Dic61#4112yes—
190#11758196—24no—
191#11758196—24no—
192#11758196—24no—
193#11758196—81no—
194#11758196—81no—
195C4×Dic61—24no—
196C4×Dic61—24no—
197C4×Dic61—24no—
198C4×Dic61—81no—
199C4×Dic61—81no—
200C4×Dic61—81no—
201C4×Dic61—81no—
202#11758190—28no—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256D244—22no—
257D244—22no—
258D244—41no—
259D244—41no—
260C4×D61—84no—
261Dic122—22no—
262Dic122—22no—
263Dic122—22no—
264Dic122#11714yes—
265C8×C61—22no—
266C8×C61—22no—
267C8×C61—41no—
268C8×C61—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—22no—
285C4×C61—22no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×#11711yescharacteristic
291C4×C61C4×#4111yescharacteristic
292C4×C61#52911yescharacteristic
293C4×C61#52911yescharacteristic
294Dic61—24no—
295Dic61—42no—
296Dic61—42no—
297Dic61—42no—
298Dic61—42no—
299Dic61—81no—
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#1865—611no—
311#1863—611no—
312#1776—611no—
313C4×#103—611no—
314C4×#102—611no—
315#1746D6111yescharacteristic, complemented
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×Q8—611no—
328C2×C4×D4—611no—
329#440—618no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#106—611no—
333C2×#103—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#102—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#97C2×D6111yescharacteristic
340C2×C4×C8C2×D6111yescharacteristic
341#390—611no—
342#390—611no—
343#389—611no—
344#389—611no—
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—1221no—
357#376C2×D6111yescharacteristic
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#336C2×D6111yescharacteristic
365#336C2×D6111yescharacteristic
366#336C2×D6111yescharacteristic
367#336C2×D6111yescharacteristic
368C61#2112811yescharacteristic, complemented, Sylow
369C22×Q8—611no—
370C22×D4—611no—
371C23×C4—1221no—
372C2×SD16—614no—
373C2×SD16—1222no—
374C2×SD16—1222no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C4×Q8—614no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×D4—614no—
386C4×D4—1222no—
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×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411C2×C42C22×D6111yescharacteristic
412C2×C42C22×D6111yescharacteristic
413#107—614no—
414#107—1222no—
415#106—1222no—
416#103—614no—
417#103—1222no—
418#103—1222no—
419#102—614no—
420#102—1222no—
421#102—1222no—
422#97C4×D6114yes—
423C4×C8—22no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95D24411yescharacteristic
431#95D24411yescharacteristic
432#95D24411yescharacteristic
433#95D24411yescharacteristic
434C24—1221no—
435C2×Q8—612no—
436C2×Q8—612no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1224no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—4881no—
456C22×C4—4881no—
457C22×C4C2×D24411yescharacteristic
458C22×C4C2×D24411yescharacteristic
459C22×C4C2×D24411yescharacteristic
460C22×C4C2×D24411yescharacteristic
461C22×C4C2×D24411yescharacteristic
462C22×C4C2×D24411yescharacteristic
463C22×C4C23×D6111yescharacteristic
464SD16—1228no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8C2×C4×D6112yes—
476C2×C8C2×C4×D6112yes—
477C4⋊C4—612no—
478C4⋊C4—612no—
479C4⋊C4—612no—
480C4⋊C4—612no—
481C4⋊C4—1224no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1224no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C22⋊C4—1224no—
495C22⋊C4—4881no—
496C22⋊C4—4881no—
497C22⋊C4—4881no—
498C22⋊C4—4881no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—1224no—
504C42—1224no—
505C42—1224no—
506C42—4881no—
507C42—4881no—
508C42—4881no—
509C42—4881no—
510C42C2×C4×D6112yes—
511C42#1175822212yes—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23C22×D24411yescharacteristic
520Q8—614no—
521Q8—1222no—
522Q8—1222no—
523Q8—1222no—
524D4—1222no—
525D4—1222no—
526D4—2441no—
527D4—2441no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—4884no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
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×C4C4×D24412yes—
567C2×C4C4×D24412yes—
568C2×C4C22×C4×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4#41×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4C2×#1175822211yescharacteristic
574C8—22no—
575C8—22no—
576C8—41no—
577C8—41no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22C2×C4×D24411yescharacteristic
586C22#1684027411yescharacteristic
587C22#1684027411yescharacteristic
588C22#1684056711yescharacteristic
589C22#1684057411yescharacteristic
590C22#1684057411yescharacteristic
591C22#1684057911yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—1224no—
599C4—4881no—
600C4—4881no—
601C4—2442no—
602C4—2442no—
603C4—2442no—
604C4—2442no—
605C4#117×D6111yescharacteristic
606C4#1684023011yescharacteristic
607C4#1684029011yescharacteristic
608C4#1684029011yescharacteristic
609C2—4882no—
610C2#2144826311yescharacteristic
611C2#2145465311yescharacteristic
612C2#2145466411yescharacteristic
613C2#2145602011yescharacteristic
614C2#2145605011yescharacteristic
615C2#2145608311yescharacteristic
616C2#2145611111yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1746C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1746D61
hypocenterC61#21128
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
3. #21448182C2semidirect
6. #21446412C2semidirect
7. #21446391C2semidirect
10. #21446090C2semidirect
11. #21446076C2semidirect
12. #21445716C2semidirect
13. C4×#14519702C2semidirect
15. C61×#1746C2semidirect
21. #16839864C4semidirect
315. #1746D61semidirect
368. C61#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
#214484632
#214481822
#214464472
#214464402
#214464122
#214463912
#214461732
#214460902
#214460762
#214457162
C4×#145197022
C4×#145196992
C61×#17462
#2112861