Finite Group Databasebeta

← #29683527

Subgroups — #29683527

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839801C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic
3C2×#16839787C211yesmaximal, characteristic
4#21448444C211yesmaximal, characteristic
5#21448173C211yesmaximal, characteristic
6#21446590C211yesmaximal, characteristic, complemented
7#21446585C211yesmaximal, characteristic, complemented
8#21446584C211yesmaximal, characteristic, complemented
9#21446582C211yesmaximal, characteristic, complemented
10#21446579C211yesmaximal, characteristic, complemented
11#21446578C211yesmaximal, characteristic, complemented
12#21446576C211yesmaximal, characteristic, complemented
13#21445537C211yesmaximal, characteristic
14#340×D61C211yesmaximal, characteristic
15C61×#1949C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839801—24no—
24#16839799—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31#16839787C418yescomplemented
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519706C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35#16839659—21no—
36#16839659—21no—
37#16839636—21no—
38#16839636—21no—
39#16839632C2211yescharacteristic
40#16839631C2211yescharacteristic
41#16839627C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839613—21no—
47C4×#11758198—21no—
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C61×#382C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189C2×C414yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61C2×C414yes—
73#14519735—22no—
74#14519735—22no—
75#14519735—22no—
76#14519735D414yes—
77#14519734C2×C414yes—
78#14519734Q814yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196D411yescharacteristic
90C2×#11758196D411yescharacteristic
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—22no—
94C4⋊C4×D61Q814yes—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198D411yescharacteristic
98C2×#11758198D411yescharacteristic
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61C2311yescharacteristic
105#14519711—22no—
106#14519711—22no—
107#14519710—22no—
108#14519708—22no—
109#14519708—22no—
110#14519708—22no—
111C4⋊#92418C2×C414yescomplemented
112#14519706C2×C414yescomplemented
113#14519706C2×C414yescomplemented
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#107×C61—22no—
128#106×C61—22no—
129C4⋊C8×C61C2×C414yescomplemented
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133C23×D61C2×D411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C2×C4×D61—24no—
149C2×C4×D61—42no—
150C2×C4×D61—81no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61C22×C412yes—
155C2×C4×D61C2×D412yes—
156C2×C4×D61C2×Q812yes—
157C2×C4×D61C2×Q812yes—
158C2×C4×D61#4112yes—
159C2×C4×D61#4112yes—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61C22×C412yes—
165C2×C8×C61C22×C412yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×Q812yes—
174C42×C61—81no—
175C42×C61C22×C412yes—
176C42×C61C2×Q812yes—
177#11758198—24no—
178#11758198—42no—
179#11758198C4⋊C414yes—
180#11758198C4⋊C414yes—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—81no—
188C4⋊Dic61—81no—
189C4⋊Dic61C2×D412yes—
190C4⋊Dic61C2×Q812yes—
191#11758196—24no—
192#11758196—42no—
193#11758196—81no—
194#11758196—81no—
195#11758196C4⋊C414yes—
196#11758196C4⋊C414yes—
197C4×Dic61—24no—
198C4×Dic61—42no—
199C4×Dic61—81no—
200C4×Dic61—81no—
201C4×Dic61—81no—
202C4×Dic61C22×C412yes—
203C4×Dic61C2×Q812yes—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C22×C412yes—
209C2×#92418C22×C412yes—
210#11758189—28no—
211C8×D61—28no—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61C2×C4⋊C412yes—
218C22×D61#12112yes—
219C22×D61C22⋊Q812yes—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
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—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61C2×C4⋊C412yes—
237C2×C4×C61C2×C4⋊C412yes—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—22no—
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×Dic61C2×C4⋊C412yes—
261C2×Dic61C2×C4⋊C412yes—
262C2×Dic61C2×C4⋊C412yes—
263C2×Dic61#12112yes—
264C2×Dic61C22⋊Q812yes—
265C4×D61—84no—
266C4×D61—84no—
267C4×D61C4×D414yes—
268C4×D61C4×Q814yes—
269C4×D61#12514yes—
270C4×D61C4⋊Q814yes—
271C8×C61—22no—
272C8×C61—22no—
273#92418—22no—
274#92418—22no—
275#21175—611nomaximal, Sylow
276C22×C61C22×C4⋊C411yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×C22⋊Q811yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57811yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61#38914yes—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×C4×D411yescharacteristic
297C4×C61C2×C4×Q811yescharacteristic
298C4×C61C2×#12511yescharacteristic
299C4×C61C2×C4⋊Q811yescharacteristic
300Dic61—22no—
301Dic61—22no—
302Dic61—42no—
303Dic61—42no—
304Dic61—81no—
305Dic61—81no—
306Dic61#38914yes—
307C2×#428—611no—
308C2×#427—611no—
309C2×#424—611no—
310#2317—611no—
311#2274—611no—
312#1949—611no—
313#1949—611no—
314#1949—611no—
315#1949D6111yescharacteristic, complemented
316#1948—611no—
317#1948—611no—
318#1947—611no—
319#1947—611no—
320#1736—611no—
321C2×#340—611no—
322C2×C61C2×#38911yescharacteristic
323C2×C61#291211yescharacteristic
324C2×C61#291311yescharacteristic
325C2×C61#304511yescharacteristic
326C2×C61#304611yescharacteristic
327C2×C61#308111yescharacteristic
328C2×C61#308211yescharacteristic
329D61—24no—
330C22×M16—611no—
331C23×C8—611no—
332C2×#117—611no—
333C2×#117—611no—
334C2×#117—611no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—611no—
337#428—1224no—
338#427—1224no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#107—611no—
342C2×#107C2×D6111yescharacteristic
343C2×#106—611no—
344C2×#106—611no—
345C2×#106—611no—
346C2×#106C2×D6111yescharacteristic
347#424—618no—
348C2×C4⋊C8—611no—
349C2×C4⋊C8—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8C2×D6111yescharacteristic
352#389—1221no—
353#389—1221no—
354#385—1221no—
355#385—1221no—
356#384—611no—
357#384—611no—
358#382—611no—
359#382C2×D6111yescharacteristic
360C4×C4⋊C4—611no—
361C4×C4⋊C4—611no—
362C4×C4⋊C4—611no—
363C4×C4⋊C4—1221no—
364C4×C4⋊C4—1221no—
365C4×C4⋊C4C2×D6111yescharacteristic
366C4×C22⋊C4—1221no—
367C4×C22⋊C4—1221no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#340—611no—
373#340—611no—
374#340C2×D6111yescharacteristic
375#340C2×D6111yescharacteristic
376C61#2117511yescharacteristic, complemented, Sylow
377C23×C4—611no—
378C23×C4—1221no—
379C23×C4—1221no—
380C2×M16—614no—
381C2×M16—1222no—
382C2×M16—1222no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—614no—
386C22×C8—1222no—
387C22×C8—1222no—
388C22×C8C22×D6111yescharacteristic
389C22×C8C22×D6111yescharacteristic
390#117—614no—
391#117—614no—
392#117—614no—
393#117—1222no—
394#117—1222no—
395#117—1222no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—614no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C42—21no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42—1221no—
427C2×C42C22×D6111yescharacteristic
428#107—22no—
429#107—1222no—
430#107—1222no—
431#107—1222no—
432#106—22no—
433#106—1222no—
434#106—1222no—
435#106—1222no—
436C4⋊C8—614no—
437C4⋊C8—614no—
438C4⋊C8—614no—
439C4⋊C8C4×D6114yescomplemented
440#95—21no—
441#95—21no—
442#95—21no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448C24—611no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—611no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—1224no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—4881no—
468C22×C4—4881no—
469C22×C4—4881no—
470C22×C4—4881no—
471C22×C4—2442no—
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4C23×D6111yescharacteristic
475M16—1228no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—1228no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8C2×C4×D6112yes—
488C2×C8C2×C4×D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—614no—
496C4⋊C4—614no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—4881no—
507C4⋊C4—4881no—
508C4⋊C4—4881no—
509C4⋊C4—4881no—
510C4⋊C4—2442no—
511C4⋊C4D4×D6112yes—
512C4⋊C4Q8×D6112yes—
513C22⋊C4—614no—
514C22⋊C4—614no—
515C22⋊C4—1224no—
516C22⋊C4—2442no—
517C42—81no—
518C42—612no—
519C42—612no—
520C42—1224no—
521C42—4881no—
522C42—4881no—
523C42—4881no—
524C42—2442no—
525C42C2×C4×D6112yes—
526C42Q8×D6112yes—
527C23—612no—
528C23—612no—
529C23—612no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—1221no—
534C23C2×D4×D6111yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
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—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—614no—
555C2×C4—614no—
556C2×C4—614no—
557C2×C4—614no—
558C2×C4—612no—
559C2×C4—612no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—4884no—
567C2×C4—4884no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C4⋊C4×D6112yes—
579C2×C4C4⋊C4×D6112yes—
580C2×C4C22×C4×D6111yescharacteristic
581C2×C4C2×D4×D6111yescharacteristic
582C2×C4C2×Q8×D6111yescharacteristic
583C2×C4C2×Q8×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4#41×D6111yescharacteristic
586C8—22no—
587C8—22no—
588C8—1222no—
589C8—1222no—
590C22—614no—
591C22—1222no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22C2×C4⋊C4×D6111yescharacteristic
598C22#121×D6111yescharacteristic
599C22C22⋊Q8×D6111yescharacteristic
600C22#1684056411yescharacteristic
601C22#1684057111yescharacteristic
602C22#1684057211yescharacteristic
603C22#1684057611yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—614no—
611C4—1222no—
612C4—1222no—
613C4—4881no—
614C4—4881no—
615C4—2442no—
616C4—2442no—
617C4C4×D4×D6111yescharacteristic
618C4C4×Q8×D6111yescharacteristic
619C4#125×D6111yescharacteristic
620C4C4⋊Q8×D6111yescharacteristic
621C2—1224no—
622C2#389×D6111yescharacteristic
623C2#2145441311yescharacteristic
624C2#2145441611yescharacteristic
625C2#2145536111yescharacteristic
626C2#2145537011yescharacteristic
627C2#2145564711yescharacteristic
628C2#2145566011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1949C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1949D61
hypocenterC61#21175
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. #21446590C2semidirect
7. #21446585C2semidirect
8. #21446584C2semidirect
9. #21446582C2semidirect
10. #21446579C2semidirect
11. #21446578C2semidirect
12. #21446576C2semidirect
15. C61×#1949C2semidirect
31. #16839787C4semidirect
111. C4⋊#92418C2×C4semidirect
112. #14519706C2×C4semidirect
113. #14519706C2×C4semidirect
129. C4⋊C8×C61C2×C4semidirect
315. #1949D61semidirect
376. C61#21175semidirect
439. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
C2×#168397992
C2×#168397872
#214484442
#214481732
#214465902
#214465852
#214465842
#214465822
#214465792
#214465782
#214465762
#214455372
#340×D612
C61×#19492
#2117561