Finite Group Databasebeta

← #29696684

Subgroups — #29696684

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839893C211yesmaximal, characteristic, complemented
2#21448776C211yesmaximal, characteristic, complemented
3#21446793C211yesmaximal, characteristic, complemented
4#21446789C211yesmaximal, characteristic, complemented
5#21446785C211yesmaximal, characteristic, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446777C211yesmaximal, characteristic, complemented
8#21446772C211yesmaximal, characteristic, complemented
9#21445829C211yesmaximal, characteristic
10#21445829C211yesmaximal, characteristic
11#21445819C211yesmaximal, characteristic
12#21445644C211yesmaximal, characteristic
13#21445644C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C2×C61×#554C211yesmaximal, characteristic
16#16839893—24no—
17C2×C4⋊#92418C2211yescharacteristic
18C2×C4⋊#92418C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519688C2211yescharacteristic
25C2×#14519688C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839669—21no—
28#16839648—21no—
29#16839648—21no—
30#16839632C2211yescharacteristic
31#16839628C2211yescharacteristic
32#16839628C2211yescharacteristic
33C4⋊C4×Dic61—21no—
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2211yescharacteristic
36C4×C4⋊Dic61C2211yescharacteristic
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49C61×#554—24no—
50C61×#554C418yescomplemented
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C22⋊Q8×C61—21no—
55C2×C22⋊Q8×C61—21no—
56C2×C22⋊Q8×C61—21no—
57C2×C4×Q8×C61C2211yescharacteristic
58C2×C4×D4×C61C2211yescharacteristic
59C2×C4×D4×C61C2211yescharacteristic
60C2×C4×D4×C61C2211yescharacteristic
61C22×C4⋊C4×C61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4×Dic61—21no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61—21no—
75C4⋊#92418—22no—
76C4⋊#92418—22no—
77C4⋊#92418—22no—
78#14519698—22no—
79#14519698—22no—
80#14519698—22no—
81#14519698Q814yescomplemented
82#14519688—22no—
83#14519688—22no—
84C4×#92418—22no—
85#14519684—21no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C22×Q8×C61—21no—
93C22×Q8×C61D411yescharacteristic
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C23×C4×C61—21no—
97C23×C4×C61—21no—
98C4⋊Q8×C61—22no—
99C4⋊Q8×C61C2×C414yes—
100#125×C61—22no—
101#125×C61—22no—
102#125×C61C2×C414yes—
103#125×C61C2×C414yes—
104C22⋊Q8×C61—28no—
105C22⋊Q8×C61—28no—
106C22⋊Q8×C61—28no—
107C4×Q8×C61C2×C414yes—
108C4×Q8×C61D414yes—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61C2×C414yes—
113C4×D4×C61C2×C414yes—
114C4×D4×C61C2×C414yes—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—28no—
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133C24×C61—21no—
134C2×Q8×C61—24no—
135C2×Q8×C61—41no—
136C2×Q8×C61—41no—
137C2×Q8×C61—81no—
138C2×Q8×C61—81no—
139C2×Q8×C61C22⋊C414yes—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C22×C412yes—
144C2×D4×C61C2×Q812yes—
145C2×D4×C61#4112yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—24no—
150C22×C4×C61—24no—
151C22×C4×C61—24no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61—81no—
155C22×C4×C61—81no—
156C22×C4×C61—81no—
157C22×C4×C61—81no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22⋊C414yes—
178C4⋊C4×C61C22⋊C414yes—
179C4⋊C4×C61C22⋊C414yes—
180C4⋊C4×C61C22×C412yes—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—24no—
186C42×C61C22×C412yes—
187C42×C61C22×C412yes—
188C42×C61C22×C412yes—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191C42×C61C2×D412yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—81no—
201C4⋊Dic61C2×Q812yes—
202C4×Dic61—42no—
203C4×Dic61—81no—
204C4×Dic61—81no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×Q812yes—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61C22×D411yescharacteristic
225Q8×C61—44no—
226Q8×C61—84no—
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61C4×Q814yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—48no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—84no—
254C2×C4×C61—84no—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C2×C22⋊C412yes—
259C2×C4×C61C2×C22⋊C412yes—
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C612−1+411yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279#92418—22no—
280#92418—41no—
281#92418—41no—
282#92418—41no—
283#22526—611nomaximal, Sylow
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61#54811yescharacteristic
293C22×C61D4×Q811yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C22×C61#57611yescharacteristic
297C22×C61#57711yescharacteristic
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61C2×C4×Q811yescharacteristic
308C4×C61#51911yescharacteristic
309C4×C61#54811yescharacteristic
310C4×C61D4×Q811yescharacteristic
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315C2×#554D6111yescharacteristic
316C2×#446—611no—
317#2355—611no—
318#1993—611no—
319#1992—611no—
320#1991—611no—
321#1989—611no—
322#1989—611no—
323#1988—611no—
324#1804—611no—
325#1804—611no—
326#1802—611no—
327#1762—611no—
328#1762—611no—
329C4×#102—611no—
330C2×C61—44no—
331C2×C61Q8×C22⋊C411yescharacteristic
332C2×C61#289611yescharacteristic
333C2×C61#289711yescharacteristic
334C2×C61#337011yescharacteristic
335C2×C61#337611yescharacteristic
336C2×C61#338811yescharacteristic
337C2×C61#338911yescharacteristic
338#554—24no—
339#554Dic6118yescomplemented
340C2×C4⋊Q8C2×D6111yescharacteristic
341C2×#125C2×D6111yescharacteristic
342C2×#125C2×D6111yescharacteristic
343C2×C22⋊Q8—21no—
344C2×C22⋊Q8—21no—
345C2×C22⋊Q8—21no—
346C2×C4×Q8C2×D6111yescharacteristic
347C2×C4×D4C2×D6111yescharacteristic
348C2×C4×D4C2×D6111yescharacteristic
349C2×C4×D4C2×D6111yescharacteristic
350C22×C4⋊C4—21no—
351#446—1224no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#97—611no—
360C2×#97—611no—
361C2×C4×C8—611no—
362#391—1221no—
363#387—1221no—
364#387—1221no—
365#384—611no—
366#382—611no—
367#382—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—1221no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384C61#2252611yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8C22⋊D6111yescharacteristic
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—611no—
395C4⋊Q8—22no—
396C4⋊Q8C2×Dic6114yes—
397#125—22no—
398#125—22no—
399#125C2×Dic6114yes—
400#125C2×Dic6114yes—
401C22⋊Q8—28no—
402C22⋊Q8—28no—
403C22⋊Q8—28no—
404C4×Q8C2×Dic6114yes—
405C4×Q8C22⋊D6114yes—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—22no—
409C4×D4C2×Dic6114yes—
410C4×D4C2×Dic6114yes—
411C4×D4C2×Dic6114yes—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—28no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22⋊D6111yescharacteristic
426C2×C4⋊C4C22⋊D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—1222no—
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442#102—614no—
443#102—1222no—
444#102—1222no—
445#102—1222no—
446#97—1222no—
447#97—1222no—
448C4×C8—1222no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—21no—
457C2×Q8—24no—
458C2×Q8—41no—
459C2×Q8—41no—
460C2×Q8—81no—
461C2×Q8—81no—
462C2×Q8C22⋊Dic6114yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4Q8×D6112yes—
467C2×D4#1175822612yes—
468C2×D4C22×Dic6112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—612no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—4881no—
521C4⋊C4—2442no—
522C4⋊C4C22⋊Dic6114yes—
523C4⋊C4C22⋊Dic6114yes—
524C4⋊C4C22⋊Dic6114yes—
525C4⋊C4C22×Dic6112yes—
526C4⋊C4C22×Dic6112yes—
527C4⋊C4C22×Dic6112yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C42—4881no—
532C42—4881no—
533C42—2442no—
534C42C22×Dic6112yes—
535C42C22×Dic6112yes—
536C42C22×Dic6112yes—
537C42C2×C22⋊D6112yes—
538C42C2×C22⋊D6112yes—
539C42C2×C22⋊D6112yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C22×C22⋊D6111yescharacteristic
548Q8—44no—
549Q8—84no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4Q8×Dic6114yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—1222no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C2×C22⋊Dic6112yes—
591C2×C4C2×C22⋊Dic6112yes—
592C2×C4C2×C22⋊Dic6112yes—
593C2×C4C2×C22⋊Dic6112yes—
594C2×C4C2×C22⋊Dic6112yes—
595C2×C4C2×C22⋊Dic6112yes—
596C2×C4C23×Dic6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4C2×#1175822611yescharacteristic
600C2×C4#1451982511yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—1222no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C22×C22⋊Dic6111yescharacteristic
614C22#1684042311yescharacteristic
615C22Q8×C22⋊D6111yescharacteristic
616C22#1684056011yescharacteristic
617C22#1684056011yescharacteristic
618C22#1684056511yescharacteristic
619C22#1684057311yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—4881no—
630C4—4881no—
631C4—4881no—
632C4—2442no—
633C4C2×Q8×Dic6111yescharacteristic
634C4#1684024011yescharacteristic
635C4#1684042311yescharacteristic
636C4Q8×C22⋊D6111yescharacteristic
637C2—44no—
638C2Q8×C22⋊Dic6111yescharacteristic
639C2#2145433811yescharacteristic
640C2#2145434511yescharacteristic
641C2#2145857211yescharacteristic
642C2#2145864611yescharacteristic
643C2#2145876411yescharacteristic
644C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#554D61
hypocenterC61#22526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#16839893C2semidirect
2. #21448776C2semidirect
3. #21446793C2semidirect
4. #21446789C2semidirect
5. #21446785C2semidirect
6. #21446777C2semidirect
7. #21446777C2semidirect
8. #21446772C2semidirect
50. C61×#554C4semidirect
81. #14519698Q8semidirect
339. #554Dic61semidirect
384. C61#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#168398932
#214487762
#214467932
#214467892
#214467852
#214467772
#214467722
#214458292
#214458192
#214456442
C4×#145196982
C2×C61×#5542
#2252661