Finite Group Databasebeta

← #29731850

Subgroups — #29731850

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840040C211yesmaximal, characteristic
2C2×#16839886C211yesmaximal, characteristic, complemented
3#21450806C211yesmaximal, characteristic, complemented
4#21450739C211yesmaximal, characteristic
5#21447311C211yesmaximal, characteristic, complemented
6#21447307C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic
8#21446778C211yesmaximal, characteristic, complemented
9#21446574C211yesmaximal, characteristic, complemented
10#21446553C211yesmaximal, characteristic, complemented
11#21446549C211yesmaximal, characteristic, complemented
12#21446505C211yesmaximal, characteristic
13#21446499C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#2067C211yesmaximal, characteristic
16C2×#14519776—21no—
17C2×#14519776C2211yescharacteristic
18C2×#14519754—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21#16840040—24no—
22#16839886—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839696—21no—
34#16839695—21no—
35#16839685—21no—
36#16839685—21no—
37#16839666—21no—
38#16839665C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839631C2211yescharacteristic
45#16839620—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122D411yescharacteristic
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—22no—
68#14519776—44no—
69#14519754—44no—
70C4×Dic122—22no—
71C4×Dic122—22no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519711—22no—
96#14519710—22no—
97#14519710—22no—
98#14519706—22no—
99#14519706D414yes—
100#14519698—22no—
101#14519698Q814yescomplemented
102C4×#92418—22no—
103C8×Dic61—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×D4×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—22no—
122C4⋊C8×C61—22no—
123#102×C61—22no—
124#102×C61—22no—
125#95×C61—21no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C2×Q812yes—
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×Dic61—21no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122D814yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C2×D412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C4⋊C4×C61#4112yes—
169C22⋊C4×C61—42no—
170C22⋊C4×C61—81no—
171C42×C61—21no—
172C42×C61—21no—
173C22⋊Dic61—42no—
174C22⋊Dic61—42no—
175C22⋊Dic61—81no—
176C4⋊Dic61—21no—
177C4⋊Dic61—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61C2×Q812yes—
187#11758196—22no—
188#11758196—22no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—81no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—42no—
198C4×Dic61C2×D412yes—
199C4×Dic61#4112yes—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D412yes—
205C2×#92418C2×Q812yes—
206C23×C61—22no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61C22×D411yescharacteristic
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
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—41no—
231C2×C4×C61—41no—
232C2×C4×C61C2×D812yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
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—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—84no—
259C2×Dic61#12112yes—
260C2×Dic61#12112yes—
261C2×Dic61C2×D812yes—
262C2×Dic61C2×D812yes—
263C2×Dic61#13512yes—
264Dic122—28no—
265Dic122—84no—
266C8×C61—22no—
267C8×C61—41no—
268#92418—22no—
269#92418—22no—
270#25469—611nomaximal, Sylow
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61C2×#12111yescharacteristic
279C22×C61#54811yescharacteristic
280C22×C61D4×Q811yescharacteristic
281C22×C61C22×D811yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57811yescharacteristic
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61#53311yescharacteristic
291C4×C61#54111yescharacteristic
292C4×C61#54711yescharacteristic
293C4×C61D4×Q811yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—42no—
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301Dic61#46614yes—
302C2×#474—611no—
303C2×C8×Q8—611no—
304#2546—611no—
305#2538—611no—
306#2086—611no—
307#2086—611no—
308#2067—611no—
309#2067D6111yescharacteristic
310#1990—611no—
311#1947—611no—
312#1942—611no—
313#1942—611no—
314#1934—611no—
315#1934—611no—
316C4×#102—611no—
317C2×C61—82no—
318C2×C61#242911yescharacteristic
319C2×C61C2×#46611yescharacteristic
320C2×C61#306311yescharacteristic
321C2×C61#335811yescharacteristic
322C2×C61#336411yescharacteristic
323C2×C61Q8×D811yescharacteristic
324C2×C61#338111yescharacteristic
325C2×C4⋊Q8—611no—
326C2×C4⋊Q8—1221no—
327C2×C22⋊Q8—1221no—
328C2×C4×Q8—611no—
329C2×C4×D4—611no—
330#474—1224no—
331C8×Q8—1224no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107C2×D6111yescharacteristic
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339C2×#102—611no—
340C2×#102—611no—
341C2×#102C2×D6111yescharacteristic
342C2×#102C2×D6111yescharacteristic
343C2×C4×C8—611no—
344C2×C4×C8—611no—
345C2×C4×C8—611no—
346#398—1221no—
347#398—1221no—
348#396—1221no—
349#396—1221no—
350#391—1221no—
351#390—611no—
352#390C2×D6111yescharacteristic
353#389—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#389—611no—
358#389C2×D6111yescharacteristic
359#384—611no—
360#380—1221no—
361C4×C4⋊C4—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340C2×D6111yescharacteristic
370C61#2546911yescharacteristic, complemented, Sylow
371C22×Q8—611no—
372C22×Q8—1221no—
373C22×D4C22×D6111yescharacteristic
374C23×C4—1221no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C4⋊Q8—1222no—
380C4⋊Q8—1222no—
381C4⋊Q8—2444no—
382C22⋊Q8—2444no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×D4—1222no—
386C4×D4—1222no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#107—22no—
416#107—1222no—
417#107—1222no—
418#107—1222no—
419C4⋊C8—22no—
420C4⋊C8—614no—
421C4⋊C8—1222no—
422#102—22no—
423#102—22no—
424#102—614no—
425#102—1222no—
426C4×C8—1222no—
427C4×C8—1222no—
428C4×C8—1222no—
429#95—21no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—614no—
438C2×Q8—1222no—
439C2×Q8—1222no—
440C2×Q8—2442no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4Q8×D6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
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×C8D4×D6112yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—612no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—4881no—
496C4⋊C4—2442no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4#1175822412yes—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—4881no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—21no—
510C42—21no—
511C42—612no—
512C42—612no—
513C42—1221no—
514C42—1221no—
515C42—2442no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D6111yescharacteristic
524Q8—1228no—
525Q8—4884no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—4884no—
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×C4—2441no—
568C2×C4—2441no—
569C2×C4D8×D6112yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4#1451982411yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1222no—
579C8—1222no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#121×D6111yescharacteristic
588C22#1684041811yescharacteristic
589C22#1684042511yescharacteristic
590C22C2×D8×D6111yescharacteristic
591C22#135×D6111yescharacteristic
592C22#1684057211yescharacteristic
593C22#1684057811yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—614no—
600C4—1222no—
601C4—1222no—
602C4—4881no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2442no—
607C4#1684031411yescharacteristic
608C4#1684036911yescharacteristic
609C4#1684041211yescharacteristic
610C4#1684042511yescharacteristic
611C2—82no—
612C2#2144941111yescharacteristic
613C2#466×D6111yescharacteristic
614C2#2145550511yescharacteristic
615C2#2145844911yescharacteristic
616C2#2145849911yescharacteristic
617C2#2145866911yescharacteristic
618C2#2145868911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2067C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2067D61
hypocenterC61#25469
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. C2×#16839886C2semidirect
3. #21450806C2semidirect
5. #21447311C2semidirect
6. #21447307C2semidirect
8. #21446778C2semidirect
9. #21446574C2semidirect
10. #21446553C2semidirect
11. #21446549C2semidirect
101. #14519698Q8semidirect
370. C61#25469semidirect

Maximal subgroups

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

subgroupindex
C2×#168400402
C2×#168398862
#214508062
#214507392
#214473112
#214473072
#214472032
#214467782
#214465742
#214465532
#214465492
#214465052
#214464992
#102×Dic612
C61×#20672
#2546961