Finite Group Databasebeta

← #29689571

Subgroups — #29689571

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840389C211yesmaximal, characteristic
2C2×#16839822C211yesmaximal, characteristic
3#21448424C211yesmaximal, characteristic
4#21446790C211yesmaximal, characteristic, complemented
5#21446786C211yesmaximal, characteristic, complemented
6#21446782C211yesmaximal, characteristic, complemented
7#21446778C211yesmaximal, characteristic, complemented
8#21446776C211yesmaximal, characteristic, complemented
9#21446774C211yesmaximal, characteristic, complemented
10#21446768C211yesmaximal, characteristic, complemented
11#21445913C211yesmaximal, characteristic
12#21445912C211yesmaximal, characteristic
13#21445908C211yesmaximal, characteristic
14#21445904C211yesmaximal, characteristic
15C61×#1988C211yesmaximal, characteristic, complemented
16#16840389—24no—
17#16840389C418yescomplemented
18C2×#14519776C2211yescharacteristic
19C2×#14519775—21no—
20C2×#14519769C2211yescharacteristic
21C2×#14519768C2211yescharacteristic
22C2×#14519767—21no—
23C2×#14519755—21no—
24C2×#14519753—21no—
25C2×#14519735C2211yescharacteristic
26C2×#14519734C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28#16839822—24no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839653—21no—
35#16839651—21no—
36#16839646—21no—
37#16839644—21no—
38#16839631C2211yescharacteristic
39#16839627C2211yescharacteristic
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4⋊Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776C2×C414yes—
69#14519775—28no—
70#14519769—22no—
71#14519769C2×C414yes—
72#14519768—22no—
73#14519768C2×C414yes—
74#14519767—28no—
75#14519755—28no—
76#14519753—28no—
77#14519735—22no—
78#14519735C2×C414yes—
79#14519734C2×C414yes—
80#14519734D414yes—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196D411yescharacteristic
94C2×#11758196D411yescharacteristic
95C4⋊C4×D61—22no—
96C4⋊C4×D61C2×C414yes—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198D411yescharacteristic
101C2×#11758198D411yescharacteristic
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61C2311yescharacteristic
105C4⋊#92418—22no—
106#14519689—22no—
107#14519689—22no—
108#14519687—22no—
109C8×Dic61—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×Q8×C61—21no—
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4⋊Q8×C61—22no—
119C4⋊Q8×C61C2×C414yes—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128#95×C61—21no—
129C23×D61C2×D411yescharacteristic
130C2×Q8×C61—24no—
131C2×Q8×C61—81no—
132C2×Q8×C61—81no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—24no—
152C2×C4×D61—81no—
153C2×C4×D61—81no—
154C2×C4×D61C22×C412yes—
155C2×C4×D61C2×D412yes—
156C2×Dic122—24no—
157C2×Dic122—81no—
158C2×Dic122—81no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C4⋊C4×C61—24no—
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×C61C22×C412yes—
173C4⋊C4×C61C22×C412yes—
174C42×C61—81no—
175C42×C61C22×C412yes—
176C42×C61C2×D412yes—
177#11758198—24no—
178#11758198—81no—
179#11758198—81no—
180#11758198C22⋊C414yes—
181#11758198C22⋊C414yes—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—24no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C22×C412yes—
190C4⋊Dic61C22×C412yes—
191#11758196—24no—
192#11758196—24no—
193#11758196—24no—
194#11758196—24no—
195#11758196—24no—
196#11758196—81no—
197#11758196—81no—
198#11758196C22⋊C414yes—
199#11758196C22⋊C414yes—
200C4×Dic61—24no—
201C4×Dic61—81no—
202C4×Dic61C22×C412yes—
203C4×Dic61C2×D412yes—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C23×C61C22×D411yescharacteristic
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61C2×C22⋊C412yes—
218Q8×C61—84no—
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—41no—
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×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×C4×C612−1+411yescharacteristic
241C2×C4×C612−1+411yescharacteristic
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—22no—
250C2×Dic61—22no—
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×Dic61—41no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—84no—
269Dic122—84no—
270C8×C61—41no—
271C8×C61—41no—
272#92418—41no—
273#92418—41no—
274#21843—611nomaximal, Sylow
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61#53611yescharacteristic
277C22×C61#53611yescharacteristic
278C22×C61#57611yescharacteristic
279C22×C61#57711yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—41no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61#51911yescharacteristic
296C4×C61#51911yescharacteristic
297C4×C61#53611yescharacteristic
298C4×C61#53611yescharacteristic
299Dic61—22no—
300Dic61—22no—
301Dic61—22no—
302Dic61—42no—
303Dic61—42no—
304Dic61—81no—
305Dic61—81no—
306C2×#544—611no—
307C2×#433—611no—
308#2314—611no—
309#1993—611no—
310#1992—611no—
311#1991—611no—
312#1990—611no—
313#1989—611no—
314#1989—611no—
315#1988D6111yescharacteristic, complemented
316#1987—611no—
317#1825—611no—
318#1825—611no—
319#1824—611no—
320#1823—611no—
321C2×C61#231011yescharacteristic
322C2×C61#289711yescharacteristic
323C2×C61#289811yescharacteristic
324C2×C61#324111yescharacteristic
325C2×C61#324211yescharacteristic
326C2×C61#324311yescharacteristic
327C2×C61#324411yescharacteristic
328D61—42no—
329#544—618no—
330#544—1224no—
331C2×C4⋊Q8—611no—
332C2×C4⋊Q8—1221no—
333C2×C4⋊Q8C2×D6111yescharacteristic
334C2×#125—611no—
335C2×#125—611no—
336C2×#125—1221no—
337C2×C22⋊Q8—1221no—
338C2×C22⋊Q8—1221no—
339C2×#117—611no—
340C2×#117—611no—
341C22×C4⋊C4—611no—
342#433—1224no—
343C2×C4⋊C8—611no—
344C2×C4⋊C8C2×D6111yescharacteristic
345C2×C22⋊C8—611no—
346C2×C22⋊C8—611no—
347C2×#97—611no—
348C2×C4×C8—611no—
349#388—1221no—
350#388—1221no—
351#387—1221no—
352#387—1221no—
353#384—611no—
354#382—611no—
355#380—611no—
356#380—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4C2×D6111yescharacteristic
359#340—611no—
360#340—611no—
361#340—611no—
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#340—611no—
371#340C2×D6111yescharacteristic
372#340C2×D6111yescharacteristic
373#340C2×D6111yescharacteristic
374#340C2×D6111yescharacteristic
375C61#2184311yescharacteristic, complemented, Sylow
376C22×Q8—21no—
377C22×Q8—1221no—
378C23×C4—611no—
379C23×C4—1221no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8C22×D6111yescharacteristic
383C22×C8C22×D6111yescharacteristic
384C4⋊Q8—22no—
385C4⋊Q8—614no—
386C4⋊Q8—1222no—
387C4⋊Q8—1228no—
388C4⋊Q8C4×D6114yes—
389#125—614no—
390#125—614no—
391#125—1222no—
392#125—1228no—
393#125—1222no—
394C22⋊Q8—1228no—
395C22⋊Q8—1228no—
396#117—614no—
397#117—614no—
398#117—614no—
399#117—1222no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—614no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1222no—
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⋊C4—1221no—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—21no—
427C2×C42—611no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42C22×D6111yescharacteristic
431C4⋊C8—22no—
432C4⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435#97—1222no—
436C4×C8—1222no—
437#95—21no—
438#95—1221no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443C24—611no—
444C2×Q8—24no—
445C2×Q8—81no—
446C2×Q8—81no—
447C2×Q8—1224no—
448C2×Q8—4881no—
449C2×Q8—4881no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—4881no—
469C22×C4—4881no—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4C23×D6111yescharacteristic
473C2×C8—21no—
474C2×C8—21no—
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—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—614no—
493C4⋊C4—614no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1224no—
499C4⋊C4—1224no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1224no—
503C4⋊C4—1224no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—4881no—
509C4⋊C4—4881no—
510C4⋊C4C2×C4×D6112yes—
511C4⋊C4C2×C4×D6112yes—
512C22⋊C4—614no—
513C22⋊C4—614no—
514C22⋊C4—1224no—
515C22⋊C4—4881no—
516C22⋊C4—4881no—
517C42—81no—
518C42—612no—
519C42—612no—
520C42—1224no—
521C42—4881no—
522C42C2×C4×D6112yes—
523C42D4×D6112yes—
524C23—612no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23C2×D4×D6111yescharacteristic
532Q8—84no—
533Q8—4884no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
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—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—4884no—
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×C4—2441no—
575C2×C4—2441no—
576C2×C4C22⋊C4×D6112yes—
577C2×C4C22⋊C4×D6112yes—
578C2×C4C22×C4×D6111yescharacteristic
579C2×C4C2×D4×D6111yescharacteristic
580C2×C4#1451982411yescharacteristic
581C2×C4#1451982411yescharacteristic
582C2×C4#1451982511yescharacteristic
583C2×C4#1451982511yescharacteristic
584C8—41no—
585C8—41no—
586C8—2441no—
587C8—2441no—
588C22—1222no—
589C22—1222no—
590C22—1222no—
591C22—2441no—
592C22—2441no—
593C22—2441no—
594C22—2441no—
595C22C2×C22⋊C4×D6111yescharacteristic
596C22#1684033511yescharacteristic
597C22#1684033511yescharacteristic
598C22#1684056311yescharacteristic
599C22#1684056811yescharacteristic
600C22#1684057111yescharacteristic
601C22#1684057811yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—1222no—
609C4—1222no—
610C4—1222no—
611C4—4881no—
612C4—4881no—
613C4—2442no—
614C4—2442no—
615C4#1684023811yescharacteristic
616C4#1684023811yescharacteristic
617C4#1684033211yescharacteristic
618C4#1684033711yescharacteristic
619C2—2442no—
620C2#2144840211yescharacteristic
621C2#2145434011yescharacteristic
622C2#2145435111yescharacteristic
623C2#2145707911yescharacteristic
624C2#2145709411yescharacteristic
625C2#2145709911yescharacteristic
626C2#2145712211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1988D61
hypocenterC61#21843
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
4. #21446790C2semidirect
5. #21446786C2semidirect
6. #21446782C2semidirect
7. #21446778C2semidirect
8. #21446776C2semidirect
9. #21446774C2semidirect
10. #21446768C2semidirect
15. C61×#1988C2semidirect
17. #16840389C4semidirect
315. #1988D61semidirect
375. C61#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#168403892
C2×#168398222
#214484242
#214467902
#214467862
#214467822
#214467782
#214467762
#214467742
#214467682
#214459132
#214459122
#214459082
#214459042
C61×#19882
#2184361