Finite Group Databasebeta

← #29699877

Subgroups — #29699877

Order 15616.

810 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448792C211yesmaximal, characteristic
2#21448776C211yesmaximal, characteristic, complemented
3#21446702C211yesmaximal, characteristic
4#21446675C211yesmaximal, characteristic
5#21446462C211yesmaximal, characteristic, complemented
6#21446455C211yesmaximal, characteristic, complemented
7#21446448C211yesmaximal, characteristic
8#21446441C211yesmaximal, characteristic
9#21445987C211yesmaximal, characteristic, complemented
10#21445834C211yesmaximal, characteristic, complemented
11#21445819C211yesmaximal, characteristic
12#21445718C211yesmaximal, characteristic, complemented
13#107×Dic61C211yesmaximal, characteristic, complemented
14#21445691C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C2×#14519711C2211yescharacteristic
17C2×#14519711C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839669C2211yescharacteristic
28#16839669C2211yescharacteristic
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839655—21no—
32#16839655—21no—
33#16839648—21no—
34#16839648—21no—
35#16839632C2211yescharacteristic
36#16839628—21no—
37#16839628C2211yescharacteristic
38#16839625—21no—
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61—21no—
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C22⋊Dic61—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C22×SD16×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#440C418yescomplemented
56C2×#107×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61D411yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80#14519711—22no—
81#14519711—41no—
82#14519709—22no—
83#14519709—41no—
84C4⋊#92418—22no—
85#14519701—41no—
86#14519701—41no—
87#14519698—41no—
88#14519698—41no—
89#14519687—22no—
90C8×Dic61—22no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C2×SD16×C61—22no—
107C2×SD16×C61—22no—
108C2×SD16×C61C2×C414yes—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C4×Q8×C61—22no—
112C4×Q8×C61C2×C414yes—
113C4×D4×C61—22no—
114C4×D4×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61C2×C414yes—
122#103×C61C2×C414yes—
123#102×C61C2×C414yes—
124#97×C61C2×C414yes—
125C24×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—24no—
142C22×C4×C61—81no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155SD16×C61—28no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C22×C412yes—
159C2×C8×C61C22×C412yes—
160C2×C8×C61#4112yes—
161C2×C8×C61#4112yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C42×C61—24no—
169C42×C61C22×C412yes—
170C42×C61#4112yes—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173C22⋊Dic61—81no—
174C22⋊Dic61—81no—
175C4⋊Dic61—22no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—41no—
185C4⋊Dic61—42no—
186C4⋊Dic61—81no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C23×C61—22no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61C22×D411yescharacteristic
209Q8×C61—22no—
210Q8×C61—22no—
211Q8×C61—41no—
212Q8×C61—41no—
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—41no—
216D4×C61—41no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
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—84no—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
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—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61#13712yes—
263C8×C61—22no—
264C8×C61#11714yes—
265#92418—41no—
266#92418—41no—
267#22819—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×C4×D411yescharacteristic
276C22×C61#54711yescharacteristic
277C22×C61#55011yescharacteristic
278C22×C61C2×#13711yescharacteristic
279C22×C61#57511yescharacteristic
280C22×C61#57611yescharacteristic
281C22×C61#57711yescharacteristic
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61C2×#11711yescharacteristic
288C4×C61#52011yescharacteristic
289C4×C61#55511yescharacteristic
290C4×C61#55611yescharacteristic
291Dic61—22no—
292Dic61—41no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299C2×#440D6111yescharacteristic
300#2357—611no—
301#2355—611no—
302#1973—611no—
303#1968—611no—
304#1928—611no—
305#1927—611no—
306#1926—611no—
307#1925—611no—
308#1847—611no—
309#1805—611no—
310#1802—611no—
311#1776—611no—
312C4×#107—611no—
313#1771—611no—
314C2×C61—82no—
315C2×C61#235411yescharacteristic
316C2×C61C4×#13711yescharacteristic
317C2×C61#294611yescharacteristic
318C2×C61#334611yescharacteristic
319C2×C61#335011yescharacteristic
320C2×C61#340011yescharacteristic
321C2×C61#340311yescharacteristic
322C22×SD16C2×D6111yescharacteristic
323C2×C4×Q8C2×D6111yescharacteristic
324C2×C4×D4C2×D6111yescharacteristic
325#440Dic6118yescomplemented
326C2×#107—611no—
327C2×#107—611no—
328C2×#107C2×D6111yescharacteristic
329C2×#106—611no—
330C2×#106—611no—
331C2×C4⋊C8—611no—
332C2×#103—611no—
333C2×#103—611no—
334C2×#103C2×D6111yescharacteristic
335C2×#102—611no—
336C2×#102—611no—
337C2×#102C2×D6111yescharacteristic
338C2×#97—611no—
339C2×#97C2×D6111yescharacteristic
340C2×C4×C8—611no—
341#391—611no—
342#391—611no—
343#390—611no—
344#390—611no—
345#388—1221no—
346#388—1221no—
347#387—1221no—
348#387—1221no—
349#384—611no—
350#382—611no—
351#382—1221no—
352#381—1221no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—1221no—
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#336—611no—
365#336—611no—
366C61#2281911yescharacteristic, complemented, Sylow
367C22×Q8C22×D6111yescharacteristic
368C22×D4C22×D6111yescharacteristic
369C23×C4—21no—
370C2×SD16—22no—
371C2×SD16—22no—
372C2×SD16C2×Dic6114yes—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8C22×D6111yescharacteristic
376C22×C8C22×D6111yescharacteristic
377C4×Q8—22no—
378C4×Q8C2×Dic6114yes—
379C4×D4—22no—
380C4×D4C2×Dic6114yes—
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—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C4⋊C4C22×D6111yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C42—21no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—1221no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405#107—1222no—
406#107—2441no—
407#107C2×Dic6114yes—
408#106—1222no—
409#106—2441no—
410C4⋊C8—1222no—
411#103—2441no—
412#103—2441no—
413#103C2×Dic6114yes—
414#102—2441no—
415#102—2441no—
416#102C2×Dic6114yes—
417#97—1222no—
418#97C2×Dic6114yes—
419C4×C8—1222no—
420#95—611no—
421#95—611no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432C24—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C22×Dic6112yes—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C22×Dic6112yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462SD16—28no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8#1175822212yes—
472C2×C8#1175822212yes—
473C2×C8C22×Dic6112yes—
474C2×C8C22×Dic6112yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
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—2441no—
489C4⋊C4—2442no—
490C4⋊C4C22×Dic6112yes—
491C4⋊C4C22×Dic6112yes—
492C22⋊C4—24no—
493C22⋊C4—4881no—
494C22⋊C4—4881no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C42—24no—
498C42—1221no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—4881no—
503C42—4881no—
504C42—4881no—
505C42—4881no—
506C42#1175822412yes—
507C42C22×Dic6112yes—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D6111yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—84no—
538C2×C4—612no—
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—1221no—
549C2×C4—1221no—
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×C4D4×Dic6112yes—
563C2×C4D4×Dic6112yes—
564C2×C4C23×Dic6111yescharacteristic
565C2×C4C2×#1175822411yescharacteristic
566C2×C4C2×#1175822211yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4#1451982311yescharacteristic
569C2×C4#1451982411yescharacteristic
570C8—22no—
571C8—2441no—
572C8—2441no—
573C8#1451973614yes—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×D4×Dic6111yescharacteristic
582C22#1684041711yescharacteristic
583C22#1684043311yescharacteristic
584C22#137×D6111yescharacteristic
585C22#1684056111yescharacteristic
586C22#1684056611yescharacteristic
587C22#1684057211yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—1222no—
594C4—4881no—
595C4—4881no—
596C4—4881no—
597C4—4881no—
598C4—2441no—
599C4—2441no—
600C4—2441no—
601C4C2×#1451973611yescharacteristic
602C4#1684024511yescharacteristic
603C4#1684047711yescharacteristic
604C4#1684048211yescharacteristic
605C2—82no—
606C2#2144876811yescharacteristic
607C2#137×Dic6111yescharacteristic
608C2#2145465211yescharacteristic
609C2#2145830911yescharacteristic
610C2#2145836911yescharacteristic
611C2#2145887311yescharacteristic
612C2#2145890811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#22819
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. #21448776C2semidirect
5. #21446462C2semidirect
6. #21446455C2semidirect
9. #21445987C2semidirect
10. #21445834C2semidirect
12. #21445718C2semidirect
13. #107×Dic61C2semidirect
14. #21445691C2semidirect
55. C61×#440C4semidirect
325. #440Dic61semidirect
366. C61#22819semidirect

Maximal subgroups

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

subgroupindex
#214487922
#214487762
#214467022
#214466752
#214464622
#214464552
#214464482
#214464412
#214459872
#214458342
#214458192
#214457182
#107×Dic612
#214456912
C2×C61×#4402
#2281961