Finite Group Databasebeta

← #29731954

Subgroups — #29731954

Order 15616.

798 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
2C2×#16839891C211yesmaximal, characteristic, complemented
3#21451026C211yesmaximal, characteristic, complemented
4#21450739C211yesmaximal, characteristic
5#21447322C211yesmaximal, characteristic, complemented
6#21447317C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic
8#21446778C211yesmaximal, characteristic, complemented
9#21446574C211yesmaximal, characteristic, complemented
10#21446558C211yesmaximal, characteristic, complemented
11#21446554C211yesmaximal, characteristic, complemented
12#21446519C211yesmaximal, characteristic
13#21446513C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#2067C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519754—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840050—24no—
21#16839891—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839685—21no—
33#16839685—21no—
34#16839683—21no—
35#16839682—21no—
36#16839666—21no—
37#16839665C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839631—21no—
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#14519754—44no—
69C4×Dic122—22no—
70C4×Dic122—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
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#14519710—22no—
95#14519709—22no—
96#14519708—22no—
97#14519706—22no—
98#14519706D414yes—
99#14519698—22no—
100#14519698Q814yescomplemented
101#14519688—22no—
102#14519687—22no—
103C8×Dic61—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#107×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—22no—
152C2×Dic122—42no—
153C2×Dic122—81no—
154C2×Dic122—81no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C2×D412yes—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61#4112yes—
168C22⋊C4×C61—42no—
169C22⋊C4×C61—81no—
170C42×C61—21no—
171C42×C61—21no—
172C22⋊Dic61—42no—
173C22⋊Dic61—42no—
174C22⋊Dic61—81no—
175C4⋊Dic61—21no—
176C4⋊Dic61—22no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—81no—
185C4⋊Dic61C2×Q812yes—
186#11758196—22no—
187#11758196—22no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192#11758196—81no—
193#11758196—81no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61C2×D412yes—
198C4×Dic61#4112yes—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C2×D412yes—
204C2×#92418C2×Q812yes—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
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×C61C22×D411yescharacteristic
233C2×C4×C61C22×Q811yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×C4×C612−1+411yescharacteristic
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—
261C2×Dic61AGL(1,ℤ/8)12yes—
262C2×Dic61#13712yes—
263Dic122—44no—
264Dic122—84no—
265C8×C61—22no—
266C8×C61—41no—
267#92418—22no—
268#92418—22no—
269#25477—611nomaximal, Sylow
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61C2×#12111yescharacteristic
278C22×C61#54811yescharacteristic
279C22×C61D4×Q811yescharacteristic
280C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C61C2×#13711yescharacteristic
282C22×C61#57611yescharacteristic
283C22×C61#57711yescharacteristic
284C4×C61—42no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61#53311yescharacteristic
290C4×C61#54111yescharacteristic
291C4×C61#54711yescharacteristic
292C4×C61D4×Q811yescharacteristic
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61—41no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300Dic61#46814yes—
301C2×#476—611no—
302C2×#446—611no—
303#2567—611no—
304#2538—611no—
305#2088—611no—
306#2088—611no—
307#2067—611no—
308#2067D6111yescharacteristic
309#1990—611no—
310#1947—611no—
311#1943—611no—
312#1943—611no—
313#1936—611no—
314#1936—611no—
315C4×#102—611no—
316C2×C61—82no—
317C2×C61#242911yescharacteristic
318C2×C61C2×#46811yescharacteristic
319C2×C61#306211yescharacteristic
320C2×C61#336611yescharacteristic
321C2×C61#336911yescharacteristic
322C2×C61#338411yescharacteristic
323C2×C61#338511yescharacteristic
324C2×C4⋊Q8—611no—
325C2×C22⋊Q8—1221no—
326C2×C4×Q8—611no—
327C2×C4×D4—611no—
328#476—1224no—
329#446—1224no—
330C2×#107—611no—
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×#106—611no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8C2×D6111yescharacteristic
337C2×#102—611no—
338C2×#102—611no—
339C2×#102C2×D6111yescharacteristic
340C2×#102C2×D6111yescharacteristic
341C2×#97—611no—
342C2×#97—611no—
343C2×C4×C8—611no—
344#396—1221no—
345#396—1221no—
346#395—1221no—
347#395—1221no—
348#391—1221no—
349#390—611no—
350#390C2×D6111yescharacteristic
351#389—611no—
352#389—611no—
353#389—611no—
354#389—611no—
355#389—611no—
356#389C2×D6111yescharacteristic
357#384—611no—
358#384—1221no—
359#380—1221no—
360C4×C4⋊C4—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340C2×D6111yescharacteristic
369C61#2547711yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×Q8—1221no—
372C22×D4C22×D6111yescharacteristic
373C23×C4—1221no—
374C22×C8—611no—
375C22×C8—611no—
376C22×C8C22×D6111yescharacteristic
377C22×C8C22×D6111yescharacteristic
378C4⋊Q8—1222no—
379C4⋊Q8—1222no—
380C22⋊Q8—2444no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—611no—
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—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—1221no—
412C2×C42C22×D6111yescharacteristic
413#107—22no—
414#107—1222no—
415#106—1222no—
416#106—1222no—
417C4⋊C8—22no—
418C4⋊C8—614no—
419C4⋊C8—1222no—
420#102—22no—
421#102—22no—
422#102—614no—
423#102—1222no—
424#97—1222no—
425#97—1222no—
426C4×C8—1222no—
427#95—21no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434C24—21no—
435C2×Q8—1222no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—4881no—
439C2×Q8—4881no—
440C2×Q8—2442no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4Q8×D6112yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4D4×D6111yescharacteristic
462C22×C4D4×D6111yescharacteristic
463C22×C4C23×D6111yescharacteristic
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8D4×D6112yes—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—81no—
482C4⋊C4—612no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4#1175822412yes—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—4881no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—21no—
508C42—21no—
509C42—612no—
510C42—612no—
511C42—1221no—
512C42—1221no—
513C42—2442no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D6111yescharacteristic
522Q8—4884no—
523Q8—2444no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—4884no—
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×C4—2441no—
567C2×C4—2441no—
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×Q8×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C2×C4#1451982411yescharacteristic
573C2×C4#1451982411yescharacteristic
574C8—22no—
575C8—41no—
576C8—1222no—
577C8—1222no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#121×D6111yescharacteristic
586C22#1684041811yescharacteristic
587C22#1684042511yescharacteristic
588C22AGL(1,ℤ/8)×D6111yescharacteristic
589C22#137×D6111yescharacteristic
590C22#1684056311yescharacteristic
591C22#1684056911yescharacteristic
592C4—41no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—614no—
598C4—1222no—
599C4—4881no—
600C4—4881no—
601C4—4881no—
602C4—2442no—
603C4—2441no—
604C4—2441no—
605C4#1684031411yescharacteristic
606C4#1684036911yescharacteristic
607C4#1684041211yescharacteristic
608C4#1684042511yescharacteristic
609C2—82no—
610C2#2144941111yescharacteristic
611C2#468×D6111yescharacteristic
612C2#2145549811yescharacteristic
613C2#2145852111yescharacteristic
614C2#2145855411yescharacteristic
615C2#2145872111yescharacteristic
616C2#2145872411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2067C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2067D61
hypocenterC61#25477
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×#16839891C2semidirect
3. #21451026C2semidirect
5. #21447322C2semidirect
6. #21447317C2semidirect
8. #21446778C2semidirect
9. #21446574C2semidirect
10. #21446558C2semidirect
11. #21446554C2semidirect
100. #14519698Q8semidirect
369. C61#25477semidirect

Maximal subgroups

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

subgroupindex
C2×#168400502
C2×#168398912
#214510262
#214507392
#214473222
#214473172
#214472032
#214467782
#214465742
#214465582
#214465542
#214465192
#214465132
#102×Dic612
C61×#20672
#2547761