Finite Group Databasebeta

← #29683093

Subgroups — #29683093

Order 15616.

890 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
1#21448470C211yesmaximal, characteristic, complemented
2#21448180C211yesmaximal, characteristic
3#21446462C211yesmaximal, characteristic, complemented
4#21446455C211yesmaximal, characteristic, complemented
5#21446413C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446155C211yesmaximal, characteristic
8#21446155C211yesmaximal, characteristic
9#21446082C211yesmaximal, characteristic
10#21446068C211yesmaximal, characteristic
11#21445716C211yesmaximal, characteristic, complemented
12C4×#14519701C211yesmaximal, characteristic, complemented
13C4×#14519698C211yesmaximal, characteristic
14#21445572C211yesmaximal, characteristic, complemented
15C2×C61×#440C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×C4×#92418C2211yescharacteristic
25#16839669C2211yescharacteristic
26#16839669C2211yescharacteristic
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839648—21no—
30#16839648—21no—
31#16839630—21no—
32#16839628—21no—
33C4⋊C4×Dic61C2211yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37C22⋊C4×Dic61—21no—
38C22⋊C4×Dic61—21no—
39C4×C22⋊Dic61—21no—
40#16839608C2211yescharacteristic
41C2×#14519684—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839516C2211yescharacteristic
49#16839516C2211yescharacteristic
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
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67D4×Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2311yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82#14519711—22no—
83#14519709—22no—
84#14519701—22no—
85#14519701—22no—
86#14519698—22no—
87#14519698—22no—
88C4×#92418—22no—
89#14519684—21no—
90#14519684—21no—
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#14519684D411yescharacteristic
100#14519684D411yescharacteristic
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×C414yescomplemented
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×C414yescomplemented
122#103×C61C2×C414yescomplemented
123#102×C61C2×C414yes—
124#97×C61C2×C414yescomplemented
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×C61C22×C412yes—
136C2×D4×C61#4112yes—
137C2×D4×C61#4112yes—
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—24no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—81no—
154C22×Dic61—81no—
155C22×Dic61—81no—
156C22×Dic61—81no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161SD16×C61—28no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61C22×C412yes—
167C2×C8×C61C22×C412yes—
168C4⋊C4×C61—24no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C412yes—
172C4⋊C4×C61C22×C412yes—
173C22⋊C4×C61—24no—
174C42×C61—24no—
175C42×C61C22×C412yes—
176C42×C61#4112yes—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—81no—
186C4⋊Dic61—81no—
187C4⋊Dic61#4112yes—
188C4⋊Dic61#4112yes—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—81no—
194C4×Dic61—81no—
195C4×Dic61—81no—
196C4×Dic61—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C22×D411yescharacteristic
211Q8×C61—22no—
212Q8×C61—22no—
213Q8×C61—41no—
214Q8×C61—41no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61—22no—
218D4×C61#11714yes—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
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—84no—
233C2×C4×C61C4×D412yes—
234C2×C4×C61C4×D412yes—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—84no—
266C2×Dic61—84no—
267C8×C61—22no—
268C8×C61—22no—
269#92418—41no—
270#92418—41no—
271#21125—611nomaximal, Sylow
272C22×C61—24no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61C2×C4×D411yescharacteristic
280C22×C61#52511yescharacteristic
281C22×C61#52511yescharacteristic
282C22×C61#57611yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57711yescharacteristic
285C22×C61#57811yescharacteristic
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61C2×#11711yescharacteristic
292C4×C61C4×#4111yescharacteristic
293C4×C61#52911yescharacteristic
294C4×C61#52911yescharacteristic
295Dic61—41no—
296Dic61—42no—
297Dic61—41no—
298Dic61—41no—
299Dic61—42no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303C2×#440D6111yescharacteristic
304#2320—611no—
305#2275—611no—
306#1928—611no—
307#1927—611no—
308#1921—611no—
309#1918—611no—
310#1877—611no—
311#1877—611no—
312#1864—611no—
313#1862—611no—
314#1776—611no—
315C4×#103—611no—
316C4×#102—611no—
317#1746—611no—
318C2×C61—44no—
319C2×C61#229011yescharacteristic
320C2×C61#294611yescharacteristic
321C2×C61#294711yescharacteristic
322C2×C61#311811yescharacteristic
323C2×C61#312011yescharacteristic
324C2×C61#312311yescharacteristic
325C2×C61#312411yescharacteristic
326C22×SD16C2×D6111yescharacteristic
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×D4—611no—
329C2×C4×D4—611no—
330C2×C4×D4C2×D6111yescharacteristic
331#440Dic6118yescomplemented
332C2×#107—611no—
333C2×#107C2×D6111yescharacteristic
334C2×#106—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#103C2×D6111yescharacteristic
338C2×#102—611no—
339C2×#102—611no—
340C2×#102C2×D6111yescharacteristic
341C2×#97C2×D6111yescharacteristic
342C2×C4×C8—611no—
343#391—611no—
344#391—611no—
345#389—611no—
346#389—611no—
347#387—1221no—
348#387—1221no—
349#383—1221no—
350#382—1221no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—611no—
353C4×C4⋊C4—611no—
354C4×C4⋊C4—611no—
355C4×C22⋊C4—1221no—
356C4×C22⋊C4—1221no—
357C4×C22⋊C4—1221no—
358#376—611no—
359C2×#95—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370C61#2112511yescharacteristic, complemented, Sylow
371C22×Q8C22×D6111yescharacteristic
372C22×D4C22×D6111yescharacteristic
373C23×C4—21no—
374C23×C4—1221no—
375C23×C4—1221no—
376C2×SD16—22no—
377C2×SD16—22no—
378C2×SD16C2×Dic6114yes—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8C22×D6111yescharacteristic
382C22×C8C22×D6111yescharacteristic
383C4×Q8—22no—
384C4×Q8C2×Dic6114yescomplemented
385C4×D4—22no—
386C4×D4—1222no—
387C4×D4—1222no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4C2×Dic6114yes—
391C2×C4⋊C4—21no—
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⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—21no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42—1221no—
409C2×C42—1221no—
410C2×C42C22×D6111yescharacteristic
411#107—1222no—
412#107C2×Dic6114yescomplemented
413#106—1222no—
414#103—1222no—
415#103—1222no—
416#103C2×Dic6114yescomplemented
417#102—1222no—
418#102—1222no—
419#102C2×Dic6114yes—
420#97C2×Dic6114yescomplemented
421C4×C8—1222no—
422#95—611no—
423#95—611no—
424#95—611no—
425#95—611no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic6112yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4#1175822212yes—
447C2×D4#1175822212yes—
448C2×D4C22×Dic6112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—4881no—
466C22×C4—4881no—
467C22×C4—4881no—
468C22×C4—4881no—
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4C23×D6111yescharacteristic
472SD16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8C22×Dic6112yes—
484C2×C8C22×Dic6112yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—4881no—
497C4⋊C4—4881no—
498C4⋊C4C22×Dic6112yes—
499C4⋊C4C22×Dic6112yes—
500C22⋊C4—24no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—24no—
504C42—1221no—
505C42—1221no—
506C42—1221no—
507C42—1221no—
508C42—4881no—
509C42—4881no—
510C42—4881no—
511C42—4881no—
512C42#1175822412yes—
513C42C22×Dic6112yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D6111yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1451973614yes—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4884no—
559C2×C4—4884no—
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×C4—2441no—
569C2×C4—2441no—
570C2×C4D4×Dic6112yes—
571C2×C4D4×Dic6112yes—
572C2×C4C23×Dic6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4C2×#1175822211yescharacteristic
577C2×C4C2×#1175822211yescharacteristic
578C8—22no—
579C8—22no—
580C8—2441no—
581C8—2441no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×D4×Dic6111yescharacteristic
590C22#1684027111yescharacteristic
591C22#1684027111yescharacteristic
592C22#1684056611yescharacteristic
593C22#1684057111yescharacteristic
594C22#1684057211yescharacteristic
595C22#1684057811yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—4881no—
602C4—4881no—
603C4—2441no—
604C4—2442no—
605C4—2441no—
606C4—2441no—
607C4—2442no—
608C4—2441no—
609C4C2×#1451973611yescharacteristic
610C4#41×Dic6111yescharacteristic
611C4#1684029211yescharacteristic
612C4#1684029211yescharacteristic
613C2—44no—
614C2#2144826411yescharacteristic
615C2#2145465211yescharacteristic
616C2#2145466311yescharacteristic
617C2#2145600711yescharacteristic
618C2#2145603711yescharacteristic
619C2#2145607011yescharacteristic
620C2#2145608511yescharacteristic
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#21125
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
1. #21448470C2semidirect
3. #21446462C2semidirect
4. #21446455C2semidirect
5. #21446413C2semidirect
6. #21446392C2semidirect
11. #21445716C2semidirect
12. C4×#14519701C2semidirect
14. #21445572C2semidirect
55. C61×#440C4semidirect
112. C4×Q8×C61C2×C4semidirect
121. #107×C61C2×C4semidirect
122. #103×C61C2×C4semidirect
124. #97×C61C2×C4semidirect
331. #440Dic61semidirect
370. C61#21125semidirect
384. C4×Q8C2×Dic61semidirect
412. #107C2×Dic61semidirect
416. #103C2×Dic61semidirect
420. #97C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214484702
#214481802
#214464622
#214464552
#214464132
#214463922
#214461552
#214460822
#214460682
#214457162
C4×#145197012
C4×#145196982
#214455722
C2×C61×#4402
#2112561