Finite Group Databasebeta

← #29779749

Subgroups — #29779749

Order 15616.

742 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
1#21454178C211yesmaximal, characteristic, complemented
2#124×Dic61C211yesmaximal, characteristic
3#21447388C211yesmaximal, characteristic, complemented
4#21447387C211yesmaximal, characteristic, complemented
5#21447055C211yesmaximal, characteristic
6#21446518C211yesmaximal, characteristic
7#21446513C211yesmaximal, characteristic
8#21446511C211yesmaximal, characteristic, complemented
9#21446506C211yesmaximal, characteristic, complemented
10#21445822C211yesmaximal, characteristic, complemented
11#21445822C211yesmaximal, characteristic, complemented
12#21445817C211yesmaximal, characteristic
13#21445817C211yesmaximal, characteristic
14#21445671C211yesmaximal, characteristic, complemented
15C61×#2042C211yesmaximal, characteristic
16C2×#14519767—21no—
17C2×Q8×Dic61C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519706C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839669C2211yescharacteristic
28#16839665C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839659—21no—
32#16839658C2211yescharacteristic
33#16839657—21no—
34#16839655—21no—
35#16839655—21no—
36#16839631C2211yescharacteristic
37#16839631C2211yescharacteristic
38#16839628C2211yescharacteristic
39#16839626—21no—
40C22⋊C4×Dic61—21no—
41C22⋊C4×Dic61—21no—
42C42×Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52C2×#124×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519767—44no—
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519706—22no—
90#14519706—22no—
91#14519701—22no—
92#14519701—41no—
93#14519698—22no—
94#14519698—41no—
95#14519687—41no—
96#14519687—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61C2311yescharacteristic
105C22×D4×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#124×C61—22no—
109#124×C61—22no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#103×C61—22no—
117#103×C61—41no—
118#102×C61—22no—
119#102×C61—41no—
120#95×C61—21no—
121C24×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61#4112yes—
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×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—42no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C42×C61—21no—
165C42×C61—21no—
166C22⋊Dic61—42no—
167C22⋊Dic61—42no—
168C22⋊Dic61—42no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—42no—
178C4⋊Dic61—42no—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C4×Dic61—41no—
190C4×Dic61—41no—
191C4×Dic61—42no—
192C4×Dic61#4112yes—
193C4×Dic61#4112yes—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61C22×D411yescharacteristic
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—41no—
211Q8×C61—41no—
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61H(5,𝔽2)11yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
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—
256C2×Dic61—41no—
257C2×Dic61—84no—
258C2×Dic61#12412yes—
259C2×Dic61#12412yes—
260C2×Dic61AGL(1,ℤ/8)12yes—
261C2×Dic61#13712yes—
262C8×C61—41no—
263C8×C61—41no—
264#92418—41no—
265#92418—41no—
266#29131—611nomaximal, Sylow
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61#52511yescharacteristic
275C22×C61C2×#12411yescharacteristic
276C22×C61#53711yescharacteristic
277C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57511yescharacteristic
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61#55211yescharacteristic
287C4×C61#55211yescharacteristic
288C4×C61#55311yescharacteristic
289C4×C61#55311yescharacteristic
290Dic61—22no—
291Dic61—41no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297Dic61#48914yes—
298#2859—611no—
299C4×#124—611no—
300#2101—611no—
301#2101—611no—
302#2042—611no—
303#2042D6111yescharacteristic
304#1936—611no—
305#1936—611no—
306#1935—611no—
307#1935—611no—
308#1803—611no—
309#1803—611no—
310#1802—611no—
311#1802—611no—
312#1767—611no—
313C2×C61—82no—
314C2×C61#256211yescharacteristic
315C2×C61#311511yescharacteristic
316C2×C61#311611yescharacteristic
317C2×C61C2×#48911yescharacteristic
318C2×C61#314211yescharacteristic
319C2×C61#325111yescharacteristic
320C2×C61#325211yescharacteristic
321C2×#125—1221no—
322C2×#124C2×D6111yescharacteristic
323C2×C4×Q8—611no—
324C2×C4×D4—611no—
325C2×C4⋊C8—611no—
326C2×C4⋊C8—611no—
327C2×#103—611no—
328C2×#103—611no—
329C2×#103C2×D6111yescharacteristic
330C2×#103C2×D6111yescharacteristic
331C2×#102—611no—
332C2×#102—611no—
333C2×#102C2×D6111yescharacteristic
334C2×#102C2×D6111yescharacteristic
335C2×#97—611no—
336C2×#97—611no—
337#391—611no—
338#390—611no—
339#389—611no—
340#389—611no—
341#389—611no—
342#389—1221no—
343#389—1221no—
344#389C2×D6111yescharacteristic
345#388—1221no—
346#388—1221no—
347#384—611no—
348#384—611no—
349#382—611no—
350#382—1221no—
351C4×C22⋊C4—1221no—
352C4×C22⋊C4—1221no—
353C43—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#336—611no—
363#336C2×D6111yescharacteristic
364C61#2913111yescharacteristic, complemented, Sylow
365C22×Q8C22×D6111yescharacteristic
366C22×D4C22×D6111yescharacteristic
367C23×C4—1221no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8C22×D6111yescharacteristic
371C22×C8C22×D6111yescharacteristic
372#125—2444no—
373#124—22no—
374#124—22no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×D4—1222no—
378C4×D4—1222no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407C4⋊C8—1222no—
408C4⋊C8—1222no—
409#103—22no—
410#103—41no—
411#103—1222no—
412#103—2441no—
413#102—22no—
414#102—41no—
415#102—1222no—
416#102—2441no—
417#97—2441no—
418#97—2441no—
419#95—21no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8#1175822412yes—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4#1175822412yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4D4×D6111yescharacteristic
455C22×C4D4×D6111yescharacteristic
456C22×C4C23×D6111yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—42no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C42—21no—
496C42—21no—
497C42—612no—
498C42—612no—
499C42—1222no—
500C42—1222no—
501C42—1221no—
502C42—1221no—
503C42—2441no—
504C42—2441no—
505C42—2442no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C2×D4×D6111yescharacteristic
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
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—2441no—
562C2×C4#41×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4C2×#1175822411yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4#1451982211yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—41no—
569C8—41no—
570C8—2441no—
571C8—2441no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22#1684026811yescharacteristic
580C22#124×D6111yescharacteristic
581C22#1684034111yescharacteristic
582C22AGL(1,ℤ/8)×D6111yescharacteristic
583C22#137×D6111yescharacteristic
584C22#1684055911yescharacteristic
585C22#1684055911yescharacteristic
586C4—41no—
587C4—41no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—614no—
592C4—1222no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—4881no—
597C4—2441no—
598C4—2441no—
599C4#1684044911yescharacteristic
600C4#1684044911yescharacteristic
601C4#1684045511yescharacteristic
602C4#1684045511yescharacteristic
603C2—82no—
604C2#2145097911yescharacteristic
605C2#2145596311yescharacteristic
606C2#2145597411yescharacteristic
607C2#489×D6111yescharacteristic
608C2#2145621611yescharacteristic
609C2#2145717811yescharacteristic
610C2#2145718911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2042C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2042D61
hypocenterC61#29131
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2042

Complements

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

NH ≅ G/Ntype
1. #21454178C2semidirect
3. #21447388C2semidirect
4. #21447387C2semidirect
8. #21446511C2semidirect
9. #21446506C2semidirect
10. #21445822C2semidirect
11. #21445822C2semidirect
14. #21445671C2semidirect
364. C61#29131semidirect

Maximal subgroups

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

subgroupindex
#214541782
#124×Dic612
#214473882
#214473872
#214470552
#214465182
#214465132
#214465112
#214465062
#214458222
#214458172
#214456712
C61×#20422
#2913161