Finite Group Databasebeta

← #29731582

Subgroups — #29731582

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840166C211yesmaximal, characteristic, complemented
2C2×#16840040C211yesmaximal, characteristic
3#21450770C211yesmaximal, characteristic
4#21448872C211yesmaximal, characteristic, complemented
5#21447294C211yesmaximal, characteristic, complemented
6#21447290C211yesmaximal, characteristic, complemented
7#21447205C211yesmaximal, characteristic
8#21446769C211yesmaximal, characteristic, complemented
9#21446578C211yesmaximal, characteristic, complemented
10#21446574C211yesmaximal, characteristic, complemented
11#21446534C211yesmaximal, characteristic, complemented
12#21446442C211yesmaximal, characteristic
13#21446436C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#2067C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519754—21no—
19C2×#14519754—21no—
20C2×#14519742—21no—
21C2×D4×Dic61C2211yescharacteristic
22#16840166—24no—
23#16840040—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839698—21no—
35#16839698—21no—
36#16839666—21no—
37#16839665C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839644—21no—
40#16839643—21no—
41#16839632C2211yescharacteristic
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776—22no—
68#14519776—22no—
69#14519776D414yes—
70#14519754—44no—
71#14519754—44no—
72#14519742—44no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
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×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519711—22no—
97#14519711—22no—
98#14519710—22no—
99#14519710—22no—
100#14519710—22no—
101#14519706—22no—
102#14519706—22no—
103#14519698—22no—
104#14519698Q814yescomplemented
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C22⋊C4×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123C4⋊C8×C61—22no—
124#102×C61—22no—
125#102×C61—22no—
126#95×C61—21no—
127C24×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×Q812yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61—42no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×Dic122—24no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—42no—
156C2×Dic122—42no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61#4112yes—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—42no—
171C22⋊C4×C61—42no—
172C42×C61—21no—
173C42×C61—21no—
174C22⋊Dic61—42no—
175C22⋊Dic61—42no—
176C22⋊Dic61—42no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—42no—
185C4⋊Dic61C2×D412yes—
186C4⋊Dic61C2×Q812yes—
187#11758196—24no—
188#11758196—22no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196D814yes—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—42no—
197C4×Dic61—42no—
198C4×Dic61—42no—
199C4×Dic61C2×D412yes—
200C4×Dic61#4112yes—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C2×Q812yes—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61C22×D411yescharacteristic
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—41no—
218D4×C61—41no—
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—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×C61—41no—
233C2×C4×C61C2×D812yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×Q811yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
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—41no—
259C2×Dic61—84no—
260C2×Dic61#12112yes—
261C2×Dic61#12112yes—
262C2×Dic61C2×D812yes—
263C2×Dic61C2×D812yes—
264C2×Dic61AGL(1,ℤ/8)12yes—
265Dic122—48no—
266Dic122—84no—
267C8×C61—22no—
268C8×C61—41no—
269#92418—22no—
270#92418—41no—
271#25449—611nomaximal, Sylow
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C22×C61#54711yescharacteristic
281C22×C61D4×Q811yescharacteristic
282C22×C61C22×D811yescharacteristic
283C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C61#57811yescharacteristic
285C22×C61#57811yescharacteristic
286C4×C61—22no—
287C4×C61—41no—
288C4×C61—42no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#53311yescharacteristic
292C4×C61#54111yescharacteristic
293C4×C61#54811yescharacteristic
294C4×C61D4×Q811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—42no—
299Dic61—42no—
300Dic61—81no—
301Dic61—81no—
302Dic61#45914yes—
303C2×#500—611no—
304C2×#474—611no—
305#2542—611no—
306#2365—611no—
307#2083—611no—
308#2083—611no—
309#2067—611no—
310#2067D6111yescharacteristic
311#1988—611no—
312#1947—611no—
313#1947—611no—
314#1939—611no—
315#1925—611no—
316#1925—611no—
317C4×#102—611no—
318C2×C61—82no—
319C2×C61#242911yescharacteristic
320C2×C61C2×#45911yescharacteristic
321C2×C61#304411yescharacteristic
322C2×C61#333411yescharacteristic
323C2×C61#335311yescharacteristic
324C2×C61Q8×D811yescharacteristic
325C2×C61#338611yescharacteristic
326C2×C4⋊Q8—611no—
327C2×C4⋊Q8—611no—
328C2×C22⋊Q8—1221no—
329C2×C22⋊Q8—1221no—
330C2×C4×Q8—1221no—
331C2×C4×D4—611no—
332#500—1224no—
333#474—1224no—
334C2×#107—611no—
335C2×#107—611no—
336C2×#107—611no—
337C2×#107—611no—
338C2×#107—611no—
339C2×#107C2×D6111yescharacteristic
340C2×C4⋊C8—611no—
341C2×C4⋊C8—611no—
342C2×C4⋊C8C2×D6111yescharacteristic
343C2×#102—611no—
344C2×#102—611no—
345C2×#102C2×D6111yescharacteristic
346C2×#102C2×D6111yescharacteristic
347C2×C4×C8—611no—
348#399—1221no—
349#399—1221no—
350#391—1221no—
351#390—611no—
352#390C2×D6111yescharacteristic
353#389—611no—
354#389C2×D6111yescharacteristic
355#387—1221no—
356#387—1221no—
357#384—611no—
358#384—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340C2×D6111yescharacteristic
369#336—611no—
370#336—611no—
371C61#2544911yescharacteristic, complemented, Sylow
372C22×Q8—1221no—
373C22×Q8—1221no—
374C22×D4C22×D6111yescharacteristic
375C23×C4—1221no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8C22×D6111yescharacteristic
379C22×C8C22×D6111yescharacteristic
380C4⋊Q8—614no—
381C4⋊Q8—1222no—
382C4⋊Q8—1222no—
383C4⋊Q8—1222no—
384C22⋊Q8—2444no—
385C22⋊Q8—2444no—
386C4×Q8—2444no—
387C4×D4—1222no—
388C4×D4—1222no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
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×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417#107—22no—
418#107—1222no—
419#107—1222no—
420#107—1222no—
421#107—1222no—
422#107—1222no—
423C4⋊C8—22no—
424C4⋊C8—1222no—
425C4⋊C8—1222no—
426#102—22no—
427#102—22no—
428#102—614no—
429#102—1222no—
430C4×C8—1222no—
431#95—21no—
432#95—611no—
433#95—611no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—21no—
439C2×Q8—1224no—
440C2×Q8—2442no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×Q8—2442no—
444C2×Q8—2442no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4Q8×D6112yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—612no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8#1175822412yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—614no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—1222no—
490C4⋊C4—1224no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4D4×D6112yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—2442no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—21no—
510C42—21no—
511C42—612no—
512C42—612no—
513C42—1221no—
514C42—1221no—
515C42—2442no—
516C42—2442no—
517C42—2442no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D6111yescharacteristic
526Q8—4884no—
527Q8—2448no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
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—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
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×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4D8×D6112yes—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×Q8×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982411yescharacteristic
577C2×C4#1451982411yescharacteristic
578C8—22no—
579C8—41no—
580C8—1222no—
581C8—2441no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22#121×D6111yescharacteristic
590C22#1684041211yescharacteristic
591C22#1684042511yescharacteristic
592C22C2×D8×D6111yescharacteristic
593C22AGL(1,ℤ/8)×D6111yescharacteristic
594C22#1684057611yescharacteristic
595C22#1684057811yescharacteristic
596C4—22no—
597C4—42no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—614no—
602C4—1222no—
603C4—1222no—
604C4—4881no—
605C4—4881no—
606C4—2442no—
607C4—2441no—
608C4—2442no—
609C4#1684031411yescharacteristic
610C4#1684036911yescharacteristic
611C4#1684041811yescharacteristic
612C4#1684042511yescharacteristic
613C2—82no—
614C2#2144941111yescharacteristic
615C2#459×D6111yescharacteristic
616C2#2145534711yescharacteristic
617C2#2145816311yescharacteristic
618C2#2145839811yescharacteristic
619C2#2145866911yescharacteristic
620C2#2145873511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2067C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2067D61
hypocenterC61#25449
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
1. C2×#16840166C2semidirect
4. #21448872C2semidirect
5. #21447294C2semidirect
6. #21447290C2semidirect
8. #21446769C2semidirect
9. #21446578C2semidirect
10. #21446574C2semidirect
11. #21446534C2semidirect
104. #14519698Q8semidirect
371. C61#25449semidirect

Maximal subgroups

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

subgroupindex
C2×#168401662
C2×#168400402
#214507702
#214488722
#214472942
#214472902
#214472052
#214467692
#214465782
#214465742
#214465342
#214464422
#214464362
#102×Dic612
C61×#20672
#2544961