Finite Group Databasebeta

← #29699374

Subgroups — #29699374

Order 15616.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448776C211yesmaximal, characteristic, complemented
2#21448729C211yesmaximal, characteristic
3#21446511C211yesmaximal, characteristic, complemented
4#21446497C211yesmaximal, characteristic, complemented
5#21446413C211yesmaximal, characteristic, complemented
6#21446392C211yesmaximal, characteristic, complemented
7#21446155C211yesmaximal, characteristic
8#21446155C211yesmaximal, characteristic
9#21445953C211yesmaximal, characteristic, complemented
10#21445834C211yesmaximal, characteristic, complemented
11#21445819C211yesmaximal, characteristic
12#21445691C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C2×C61×#440C211yesmaximal, characteristic
16C2×D4×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519687C2211yescharacteristic
24C2×C8×Dic61C2211yescharacteristic
25#16839669C2211yescharacteristic
26#16839669C2211yescharacteristic
27#16839661C2211yescharacteristic
28#16839661C2211yescharacteristic
29#16839648—21no—
30#16839648—21no—
31#16839637—21no—
32#16839637—21no—
33#16839632C2211yescharacteristic
34#16839628—21no—
35#16839628C2211yescharacteristic
36#16839622—21no—
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41C2×#14519684—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
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
60C23×Dic61—21no—
61C23×Dic61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—41no—
67D4×Dic61—41no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C4⋊#92418—22no—
85#14519701—41no—
86#14519701—41no—
87#14519698—22no—
88#14519698—22no—
89#14519687—41no—
90C8×Dic61—41no—
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×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—21no—
150C22×Dic61—21no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—42no—
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—42no—
178C22⋊Dic61—42no—
179C4⋊Dic61—22no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—41no—
189C4⋊Dic61—42no—
190C4⋊Dic61—81no—
191C4⋊Dic61—81no—
192C4⋊Dic61—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—81no—
198C4×Dic61—81no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—41no—
216Q8×C61—41no—
217D4×C61—22no—
218D4×C61—22no—
219D4×C61—22no—
220D4×C61#11714yes—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—84no—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C4×D412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
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—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—84no—
267C2×Dic61—84no—
268C2×Dic61#13712yes—
269C8×C61—22no—
270C8×C61—22no—
271#92418—41no—
272#92418—41no—
273#22776—611nomaximal, Sylow
274C22×C61—24no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61C2×C4×D411yescharacteristic
282C22×C61#54611yescharacteristic
283C22×C61#54811yescharacteristic
284C22×C61C2×#13711yescharacteristic
285C22×C61#57511yescharacteristic
286C22×C61#57611yescharacteristic
287C22×C61#57711yescharacteristic
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61C2×#11711yescharacteristic
294C4×C61#52011yescharacteristic
295C4×C61#55211yescharacteristic
296C4×C61#55311yescharacteristic
297Dic61—22no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303Dic61—81no—
304Dic61—81no—
305C2×#440D6111yescharacteristic
306#2355—611no—
307#2349—611no—
308#1935—611no—
309#1933—611no—
310#1921—611no—
311#1918—611no—
312#1877—611no—
313#1877—611no—
314#1836—611no—
315#1805—611no—
316#1802—611no—
317#1771—611no—
318#1762—611no—
319C4×#102—611no—
320C2×C61—44no—
321C2×C61#235211yescharacteristic
322C2×C61C4×#13711yescharacteristic
323C2×C61#294611yescharacteristic
324C2×C61#331111yescharacteristic
325C2×C61#331411yescharacteristic
326C2×C61#337011yescharacteristic
327C2×C61#337211yescharacteristic
328C22×SD16C2×D6111yescharacteristic
329C2×C4×Q8C2×D6111yescharacteristic
330C2×C4×D4—611no—
331C2×C4×D4—611no—
332C2×C4×D4C2×D6111yescharacteristic
333#440Dic6118yescomplemented
334C2×#107C2×D6111yescharacteristic
335C2×C4⋊C8—611no—
336C2×#103—611no—
337C2×#103—611no—
338C2×#103C2×D6111yescharacteristic
339C2×#102—611no—
340C2×#102—611no—
341C2×#102C2×D6111yescharacteristic
342C2×#97—611no—
343C2×#97C2×D6111yescharacteristic
344C2×C4×C8—611no—
345#391—611no—
346#391—611no—
347#389—611no—
348#389—611no—
349#387—1221no—
350#387—1221no—
351#385—1221no—
352#385—1221no—
353#384—611no—
354#382—611no—
355#382—1221no—
356#380—1221no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—1221no—
361C2×#95—1221no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#340—611no—
369#340—611no—
370#336—611no—
371#336—611no—
372C61#2277611yescharacteristic, complemented, Sylow
373C22×Q8C22×D6111yescharacteristic
374C22×D4C22×D6111yescharacteristic
375C23×C4—21no—
376C23×C4—1221no—
377C23×C4—1221no—
378C2×SD16—22no—
379C2×SD16—22no—
380C2×SD16C2×Dic6114yes—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8C22×D6111yescharacteristic
384C22×C8C22×D6111yescharacteristic
385C4×Q8—22no—
386C4×Q8C2×Dic6114yescomplemented
387C4×D4—22no—
388C4×D4—1222no—
389C4×D4—1222no—
390C4×D4—2441no—
391C4×D4—2441no—
392C4×D4C2×Dic6114yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C4⋊C4C22×D6111yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#107C2×Dic6114yescomplemented
416C4⋊C8—1222no—
417#103—2441no—
418#103—2441no—
419#103C2×Dic6114yescomplemented
420#102—1222no—
421#102—1222no—
422#102C2×Dic6114yes—
423#97—2441no—
424#97C2×Dic6114yescomplemented
425C4×C8—2441no—
426#95—611no—
427#95—611no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C22×Dic6112yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4#1175822212yes—
449C2×D4#1175822212yes—
450C2×D4C22×Dic6112yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—4881no—
466C22×C4—4881no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8C22×Dic6112yes—
486C2×C8C22×Dic6112yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—1222no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—4881no—
500C4⋊C4—4881no—
501C4⋊C4—4881no—
502C4⋊C4—2441no—
503C4⋊C4—2442no—
504C4⋊C4C22×Dic6112yes—
505C4⋊C4C22×Dic6112yes—
506C22⋊C4—24no—
507C22⋊C4—2442no—
508C22⋊C4—2442no—
509C42—24no—
510C42—1221no—
511C42—1221no—
512C42—1221no—
513C42—1221no—
514C42—4881no—
515C42—4881no—
516C42#1175822412yes—
517C42C22×Dic6112yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D6111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4#1451973614yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
548C2×C4—612no—
549C2×C4—1221no—
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—1221no—
559C2×C4—1221no—
560C2×C4—4884no—
561C2×C4—4884no—
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×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4D4×Dic6112yes—
575C2×C4D4×Dic6112yes—
576C2×C4C23×Dic6111yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822211yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C2×C4#1451982311yescharacteristic
581C2×C4#1451982411yescharacteristic
582C8—22no—
583C8—22no—
584C8—2441no—
585C8—2441no—
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×D4×Dic6111yescharacteristic
594C22#1684040811yescharacteristic
595C22#1684042111yescharacteristic
596C22#137×D6111yescharacteristic
597C22#1684056111yescharacteristic
598C22#1684056611yescharacteristic
599C22#1684057211yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—1222no—
606C4—4881no—
607C4—4881no—
608C4—4881no—
609C4—4881no—
610C4—2441no—
611C4—2441no—
612C4—2441no—
613C4C2×#1451973611yescharacteristic
614C4#1684024511yescharacteristic
615C4#1684045211yescharacteristic
616C4#1684045811yescharacteristic
617C2—44no—
618C2#2144874811yescharacteristic
619C2#137×Dic6111yescharacteristic
620C2#2145465211yescharacteristic
621C2#2145786011yescharacteristic
622C2#2145790511yescharacteristic
623C2#2145857311yescharacteristic
624C2#2145860111yescharacteristic
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#22776
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. #21448776C2semidirect
3. #21446511C2semidirect
4. #21446497C2semidirect
5. #21446413C2semidirect
6. #21446392C2semidirect
9. #21445953C2semidirect
10. #21445834C2semidirect
12. #21445691C2semidirect
55. C61×#440C4semidirect
112. C4×Q8×C61C2×C4semidirect
121. #107×C61C2×C4semidirect
122. #103×C61C2×C4semidirect
124. #97×C61C2×C4semidirect
333. #440Dic61semidirect
372. C61#22776semidirect
386. C4×Q8C2×Dic61semidirect
415. #107C2×Dic61semidirect
419. #103C2×Dic61semidirect
424. #97C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214487762
#214487292
#214465112
#214464972
#214464132
#214463922
#214461552
#214459532
#214458342
#214458192
#214456912
#214456462
#102×Dic612
C2×C61×#4402
#2277661