Finite Group Databasebeta

← #29683407

Subgroups — #29683407

Order 15616.

982 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×#16839801C211yesmaximal, characteristic
3C2×#16839716C211yesmaximal, characteristic
4#21448441C211yesmaximal, characteristic
5#21448434C211yesmaximal, characteristic
6#21446563C211yesmaximal, characteristic, complemented
7#21446559C211yesmaximal, characteristic, complemented
8#21446555C211yesmaximal, characteristic, complemented
9#21446554C211yesmaximal, characteristic, complemented
10#21446552C211yesmaximal, characteristic, complemented
11#21446551C211yesmaximal, characteristic, complemented
12#21446546C211yesmaximal, characteristic, complemented
13#21445534C211yesmaximal, characteristic
14#340×D61C211yesmaximal, characteristic
15C61×#1941C211yesmaximal, 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#16839801C418yescomplemented
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31#16839716—24no—
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839659—21no—
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839657—21no—
43#16839636—21no—
44#16839633—21no—
45#16839631—21no—
46#16839626—21no—
47#16839621—21no—
48#16839619—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758189C2×C414yes—
66C2×#11758189D414yes—
67C2×#11758189Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61Q814yes—
73#14519735—22no—
74#14519735—22no—
75#14519735—22no—
76#14519735C2×C414yes—
77#14519734—22no—
78#14519734—22no—
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×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C4⋊C4×D61—22no—
94C4⋊C4×D61—22no—
95C4⋊C4×D61—22no—
96C4⋊C4×D61C2×C414yes—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198D411yescharacteristic
100C2×#11758198D411yescharacteristic
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61C2311yescharacteristic
104#14519711—22no—
105#14519711C2×C414yescomplemented
106#14519710—22no—
107#14519710C2×C414yescomplemented
108#14519708—22no—
109#14519708C2×C414yescomplemented
110#14519688—22no—
111#14519687—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#106×C61—22no—
129#106×C61C2×C414yescomplemented
130C4×C8×C61—22no—
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×C61C22×C412yes—
163C2×C8×C61C2×D412yes—
164C2×C8×C61C2×Q812yes—
165C2×C8×C61C2×Q812yes—
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×C61—81no—
173C4⋊C4×C61—81no—
174C4⋊C4×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C42×C61—21no—
177C42×C61—21no—
178#11758198—24no—
179#11758198—22no—
180#11758198—22no—
181#11758198—42no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—81no—
189C4⋊Dic61—81no—
190C4⋊Dic61C22×C412yes—
191C4⋊Dic61C22×C412yes—
192#11758196—24no—
193#11758196—22no—
194#11758196—22no—
195#11758196—42no—
196#11758196—81no—
197#11758196—81no—
198#11758196—81no—
199#11758196—81no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—24no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C22×C412yes—
207C2×#92418C2×D412yes—
208C2×#92418C2×Q812yes—
209C2×#92418C2×Q812yes—
210#11758189C4⋊C4116yescomplemented
211C8×D61—28no—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61C4×D412yes—
218C22×D61#12112yes—
219C22×D61C4⋊Q812yes—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
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×C61—41no—
237C2×C4×C61—41no—
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—22no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
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×Dic61C4×D412yes—
263C2×Dic61#12112yes—
264C2×Dic61C4⋊Q812yes—
265C4×D61—84no—
266C4×D61—84no—
267C4×D61C2×C4⋊C414yes—
268C4×D61C4×Q814yes—
269C4×D61C22⋊Q814yes—
270C4×D61#12514yes—
271C8×C61—22no—
272C8×C61C2×C4⋊C414yes—
273#92418—22no—
274#92418C2×C4⋊C414yes—
275#21160—611nomaximal, Sylow
276C22×C61C2×C4×D411yescharacteristic
277C22×C61C2×#12111yescharacteristic
278C22×C61C2×C4⋊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—42no—
291C4×C61—42no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C22×C4⋊C411yescharacteristic
297C4×C61C2×C4×Q811yescharacteristic
298C4×C61C2×C22⋊Q811yescharacteristic
299C4×C61C2×#12511yescharacteristic
300Dic61—42no—
301Dic61—42no—
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306Dic61#38914yes—
307C2×#428—611no—
308C2×#428—611no—
309C2×#405—611no—
310#2317—611no—
311#2316—611no—
312#1944—611no—
313#1944—611no—
314#1943—611no—
315#1943—611no—
316#1942—611no—
317#1942—611no—
318#1941—611no—
319#1941D6111yescharacteristic, complemented
320#1735—611no—
321C2×#340—611no—
322C2×C61C2×#38911yescharacteristic
323C2×C61#294611yescharacteristic
324C2×C61#294711yescharacteristic
325C2×C61#306211yescharacteristic
326C2×C61#306311yescharacteristic
327C2×C61#316311yescharacteristic
328C2×C61#316411yescharacteristic
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—618no—
338#428—1224no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#107—611no—
342C2×#107—611no—
343C2×#106—611no—
344C2×#106—611no—
345C2×#106C2×D6111yescharacteristic
346C2×#106C2×D6111yescharacteristic
347#405—1224no—
348C2×#97—611no—
349C2×#97—611no—
350C2×C4×C8—611no—
351C2×C4×C8C2×D6111yescharacteristic
352#389—611no—
353#389—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#389—611no—
358#389—1221no—
359#389—1221no—
360#389C2×D6111yescharacteristic
361#389C2×D6111yescharacteristic
362#385—1221no—
363#385—1221no—
364#384—1221no—
365#382—1221no—
366#380—1221no—
367#380—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#2116011yescharacteristic, complemented, Sylow
377C23×C4—611no—
378C23×C4—1221no—
379C23×C4—1221no—
380C2×M16—614no—
381C2×M16—614no—
382C2×M16—614no—
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—1222no—
392#117—1222no—
393#117—1222no—
394#117—1222no—
395#117—1222no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—614no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1222no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—611no—
425C2×C42—1221no—
426C2×C42—1221no—
427C2×C42C22×D6111yescharacteristic
428#107—614no—
429#107—614no—
430#107—1222no—
431#107—1222no—
432#106—22no—
433#106—614no—
434#106—1222no—
435#106C4×D6114yescomplemented
436#97—1222no—
437#97—1222no—
438C4×C8—22no—
439C4×C8—1222no—
440#95—21no—
441#95—21no—
442#95—1221no—
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—6116no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—612no—
479C2×C8—612no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—1228no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8C2×C4×D6112yes—
486C2×C8D4×D6112yes—
487C2×C8Q8×D6112yes—
488C2×C8Q8×D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1224no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—4881no—
509C4⋊C4—4881no—
510C4⋊C4—4881no—
511C4⋊C4—4881no—
512C4⋊C4—4881no—
513C4⋊C4—4881no—
514C4⋊C4—2442no—
515C4⋊C4C2×C4×D6112yes—
516C4⋊C4C2×C4×D6112yes—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1224no—
520C22⋊C4—2442no—
521C42—21no—
522C42—21no—
523C42—1224no—
524C42—1221no—
525C42—1221no—
526C42—2442no—
527C23—612no—
528C23—612no—
529C23—612no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—1221no—
534C23C2×D4×D6111yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
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—41no—
552C2×C4—41no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—614no—
557C2×C4—614no—
558C2×C4—614no—
559C2×C4—614no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—4884no—
569C2×C4—4884no—
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×C4—2441no—
579C2×C4—2441no—
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—614no—
588C8—1222no—
589C8C4⋊C4×D6114yes—
590C22—614no—
591C22—1222no—
592C22—1222no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—1222no—
597C22C4×D4×D6111yescharacteristic
598C22#121×D6111yescharacteristic
599C22C4⋊Q8×D6111yescharacteristic
600C22#1684056411yescharacteristic
601C22#1684056811yescharacteristic
602C22#1684056811yescharacteristic
603C22#1684057611yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—614no—
611C4—4881no—
612C4—4881no—
613C4—2442no—
614C4—2442no—
615C4—2441no—
616C4—2441no—
617C4C2×C4⋊C4×D6111yescharacteristic
618C4C4×Q8×D6111yescharacteristic
619C4C22⋊Q8×D6111yescharacteristic
620C4#125×D6111yescharacteristic
621C2—1224no—
622C2#389×D6111yescharacteristic
623C2#2145464611yescharacteristic
624C2#2145465511yescharacteristic
625C2#2145549211yescharacteristic
626C2#2145550311yescharacteristic
627C2#2145634411yescharacteristic
628C2#2145635611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1941C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1941D61
hypocenterC61#21160
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
6. #21446563C2semidirect
7. #21446559C2semidirect
8. #21446555C2semidirect
9. #21446554C2semidirect
10. #21446552C2semidirect
11. #21446551C2semidirect
12. #21446546C2semidirect
15. C61×#1941C2semidirect
24. #16839801C4semidirect
105. #14519711C2×C4semidirect
107. #14519710C2×C4semidirect
109. #14519708C2×C4semidirect
129. #106×C61C2×C4semidirect
210. #11758189C4⋊C4semidirect
319. #1941D61semidirect
376. C61#21160semidirect
435. #106C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
C2×#168397162
#214484412
#214484342
#214465632
#214465592
#214465552
#214465542
#214465522
#214465512
#214465462
#214455342
#340×D612
C61×#19412
#2116061