Finite Group Databasebeta

← #29732081

Subgroups — #29732081

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840060C211yesmaximal, characteristic
2C2×#16839889C211yesmaximal, characteristic, complemented
3#21451206C211yesmaximal, characteristic, complemented
4#21450739C211yesmaximal, characteristic
5#21447322C211yesmaximal, characteristic, complemented
6#21447307C211yesmaximal, characteristic, complemented
7#21447189C211yesmaximal, characteristic
8#21446786C211yesmaximal, characteristic, complemented
9#21446579C211yesmaximal, characteristic, complemented
10#21446563C211yesmaximal, characteristic, complemented
11#21446544C211yesmaximal, characteristic, complemented
12#21446513C211yesmaximal, characteristic
13#21446505C211yesmaximal, characteristic
14#21445646C211yesmaximal, characteristic
15C61×#2065C211yesmaximal, characteristic
16C2×#14519768C2211yescharacteristic
17C2×#14519754—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840060—24no—
21#16839889—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C4×#92418C2211yescharacteristic
32#16839695—21no—
33#16839685—21no—
34#16839685—21no—
35#16839683—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839659—21no—
42#16839658—21no—
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×#106×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122D411yescharacteristic
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519768—22no—
66#14519768—22no—
67#14519754—44no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61D411yescharacteristic
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×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519711—22no—
95#14519708—22no—
96#14519708—22no—
97#14519706—22no—
98#14519706D414yes—
99#14519698—22no—
100#14519698Q814yescomplemented
101#14519687—22no—
102#14519687—22no—
103C4×#92418—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110C22×D4×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#106×C61—22no—
121C4⋊C8×C61—22no—
122#102×C61—22no—
123#102×C61—22no—
124#95×C61—21no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C2×Q812yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—42no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—41no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61C2×D412yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61#4112yes—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—81no—
167C42×C61—21no—
168C42×C61—21no—
169C22⋊Dic61—42no—
170C22⋊Dic61—42no—
171C22⋊Dic61—81no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—42no—
181C4⋊Dic61—42no—
182C4⋊Dic61—42no—
183C4⋊Dic61C2×Q812yes—
184#11758196—22no—
185#11758196—22no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—81no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—81no—
195C4×Dic61C2×D412yes—
196C4×Dic61#4112yes—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418C2×D412yes—
202C2×#92418C2×Q812yes—
203C23×C61—22no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C22×D411yescharacteristic
211D4×C61—22no—
212D4×C61—22no—
213D4×C61—41no—
214D4×C61—41no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
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×C61AGL(1,ℤ/8)12yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
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—
261Dic122—44no—
262C8×C61—22no—
263C8×C61—41no—
264#92418—22no—
265#92418—22no—
266#25486—611nomaximal, Sylow
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61C2×#12111yescharacteristic
275C22×C61#54811yescharacteristic
276C22×C61D4×Q811yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61#57511yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C4×C61—22no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#53311yescharacteristic
287C4×C61#54111yescharacteristic
288C4×C61#54711yescharacteristic
289C4×C61D4×Q811yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—42no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298C2×#478—611no—
299C2×#446—611no—
300#2584—611no—
301#2538—611no—
302#2088—611no—
303#2086—611no—
304#2065—611no—
305#2065D6111yescharacteristic
306#1992—611no—
307#1948—611no—
308#1944—611no—
309#1941—611no—
310#1936—611no—
311#1934—611no—
312#1762—611no—
313C2×C61—82no—
314C2×C61#242911yescharacteristic
315C2×C61#305511yescharacteristic
316C2×C61#306011yescharacteristic
317C2×C61#336811yescharacteristic
318C2×C61#337611yescharacteristic
319C2×C61#338411yescharacteristic
320C2×C61#338811yescharacteristic
321C2×#125—611no—
322C2×C22⋊Q8—1221no—
323C2×C4×Q8—611no—
324C2×C4×D4—611no—
325#478—1224no—
326#446—1224no—
327C2×#107—611no—
328C2×#106—611no—
329C2×#106—611no—
330C2×#106C2×D6111yescharacteristic
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#102—611no—
335C2×#102—611no—
336C2×#102C2×D6111yescharacteristic
337C2×#102C2×D6111yescharacteristic
338C2×#97—611no—
339C2×#97—611no—
340C2×C4×C8—611no—
341#398—1221no—
342#396—1221no—
343#396—1221no—
344#395—1221no—
345#390—611no—
346#390C2×D6111yescharacteristic
347#389—611no—
348#389—611no—
349#389—611no—
350#389—611no—
351#389—611no—
352#389—1221no—
353#389—1221no—
354#389C2×D6111yescharacteristic
355#384—611no—
356#380—1221no—
357C4×C4⋊C4—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340C2×D6111yescharacteristic
366C61#2548611yescharacteristic, complemented, Sylow
367C22×Q8—611no—
368C22×D4C22×D6111yescharacteristic
369C23×C4—1221no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—1222no—
375#125—1222no—
376C22⋊Q8—2444no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×D4—1222no—
380C4×D4—1222no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1221no—
409C2×C42C22×D6111yescharacteristic
410#107—1222no—
411#106—22no—
412#106—1222no—
413#106—1222no—
414C4⋊C8—22no—
415C4⋊C8—614no—
416C4⋊C8—1222no—
417#102—22no—
418#102—22no—
419#102—614no—
420#102—1222no—
421#97—1222no—
422#97—1222no—
423C4×C8—1222no—
424#95—21no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431C24—21no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—2441no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4Q8×D6112yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4D4×D6111yescharacteristic
456C22×C4D4×D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8D4×D6112yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—81no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4#1175822412yes—
497C22⋊C4—42no—
498C22⋊C4—81no—
499C22⋊C4—4881no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—612no—
505C42—612no—
506C42—1221no—
507C42—1221no—
508C42—4881no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×D4×D6111yescharacteristic
517Q8—2444no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—4884no—
550C2×C4—2441no—
551C2×C4—2441no—
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#1451980412yes—
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4#1451982411yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—41no—
570C8—1222no—
571C8—1222no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22#121×D6111yescharacteristic
580C22#1684041811yescharacteristic
581C22#1684042511yescharacteristic
582C22C2×#1451980411yescharacteristic
583C22#1684055911yescharacteristic
584C22#1684057211yescharacteristic
585C22#1684057211yescharacteristic
586C4—22no—
587C4—41no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—1222no—
592C4—1222no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—2442no—
597C4—2441no—
598C4—2441no—
599C4#1684031411yescharacteristic
600C4#1684036911yescharacteristic
601C4#1684041211yescharacteristic
602C4#1684042511yescharacteristic
603C2—82no—
604C2#2144941111yescharacteristic
605C2#2145543111yescharacteristic
606C2#2145547911yescharacteristic
607C2#2145855211yescharacteristic
608C2#2145865011yescharacteristic
609C2#2145871711yescharacteristic
610C2#2145876811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2065C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2065D61
hypocenterC61#25486
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
2. C2×#16839889C2semidirect
3. #21451206C2semidirect
5. #21447322C2semidirect
6. #21447307C2semidirect
8. #21446786C2semidirect
9. #21446579C2semidirect
10. #21446563C2semidirect
11. #21446544C2semidirect
100. #14519698Q8semidirect
366. C61#25486semidirect

Maximal subgroups

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

subgroupindex
C2×#168400602
C2×#168398892
#214512062
#214507392
#214473222
#214473072
#214471892
#214467862
#214465792
#214465632
#214465442
#214465132
#214465052
#214456462
C61×#20652
#2548661