Finite Group Databasebeta

← #29684549

Subgroups — #29684549

Order 15616.

1142 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840389C211yesmaximal, characteristic
2C2×#16839787C211yesmaximal, characteristic
3#21448173C211yesmaximal, characteristic
4#21446776C211yesmaximal, characteristic, complemented
5#21446775C211yesmaximal, characteristic, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446773C211yesmaximal, characteristic, complemented
8#21446772C211yesmaximal, characteristic, complemented
9#21446769C211yesmaximal, characteristic, complemented
10#21446765C211yesmaximal, characteristic, complemented
11#21445537C211yesmaximal, characteristic
12#21445537C211yesmaximal, characteristic
13#21445534C211yesmaximal, characteristic
14#340×D61C211yesmaximal, characteristic
15C61×#1987C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18#16840389—24no—
19#16840389C418yescomplemented
20C2×#14519776C2211yescharacteristic
21C2×#14519775—21no—
22C2×#14519769C2211yescharacteristic
23C2×#14519768C2211yescharacteristic
24C2×#14519767—21no—
25C2×#14519755—21no—
26C2×#14519753—21no—
27C2×#14519735C2211yescharacteristic
28C2×#14519735C2211yescharacteristic
29C2×#14519734C2211yescharacteristic
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32#16839787—24no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519706C2211yescharacteristic
35C2×#14519706C2211yescharacteristic
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4×#11758196—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839613—21no—
41C4×#11758198—21no—
42#16839609—21no—
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×D61—21no—
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—22no—
67C2×#11758189—22no—
68C2×#11758189—22no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C2×C8×D61—22no—
72C2×C8×D61—22no—
73C2×C8×D61—22no—
74#14519776—22no—
75#14519776C2×C414yescomplemented
76#14519775—28no—
77#14519769—22no—
78#14519769C2×C414yescomplemented
79#14519768—22no—
80#14519768C2×C414yescomplemented
81#14519767—28no—
82#14519755—28no—
83#14519753—28no—
84#14519735—22no—
85#14519735—22no—
86#14519735C2×C414yes—
87#14519735Q814yescomplemented
88#14519734C2×C414yes—
89#14519734D414yes—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196D411yescharacteristic
102C2×#11758196D411yescharacteristic
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61C2×C414yes—
106C4⋊C4×D61Q814yescomplemented
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×#11758198D411yescharacteristic
110C2×#11758198D411yescharacteristic
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61—21no—
115C2×C4×Dic61—21no—
116C2×C4×Dic61C2311yescharacteristic
117C4⋊#92418—22no—
118#14519706—22no—
119#14519706—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61—21no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4⋊Q8×C61—22no—
127C4⋊Q8×C61C2×C414yescomplemented
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—22no—
136#95×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—24no—
139C2×Q8×C61—81no—
140C2×Q8×C61—81no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—24no—
156C2×C4×D61—24no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×Q812yes—
164C2×C4×D61C2×Q812yes—
165C2×C4×D61#4112yes—
166C2×C4×D61#4112yes—
167C2×Dic122—24no—
168C2×Dic122—81no—
169C2×Dic122—81no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C4⋊C4×C61C2×Q812yes—
184C4⋊C4×C61C2×Q812yes—
185C42×C61—81no—
186C42×C61C22×C412yes—
187C42×C61C2×D412yes—
188#11758198—24no—
189#11758198—24no—
190#11758198C22⋊C414yes—
191#11758198C22⋊C414yes—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—24no—
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C412yes—
199C4⋊Dic61C2×Q812yes—
200C4⋊Dic61C2×Q812yes—
201#11758196—24no—
202#11758196—24no—
203#11758196—24no—
204#11758196—24no—
205#11758196—24no—
206#11758196—24no—
207#11758196C22⋊C414yes—
208#11758196C22⋊C414yes—
209C4×Dic61—24no—
210C4×Dic61—24no—
211C4×Dic61—81no—
212C4×Dic61—81no—
213C4×Dic61—81no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222#11758189—44no—
223C8×D61—44no—
224C23×C61C22×D411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61C2×C22⋊C412yes—
230C22×D61C22⋊Q812yes—
231C22×D61C22⋊Q812yes—
232Q8×C61—84no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C22×Q811yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C22⋊Q812yes—
277C2×Dic61C22⋊Q812yes—
278C4×D61—84no—
279C4×D61—84no—
280C4×D61C4×Q814yes—
281C4×D61C4×Q814yes—
282C4×D61#12414yes—
283C4×D61C4⋊Q814yes—
284Dic122—84no—
285C8×C61—41no—
286C8×C61—41no—
287#92418—41no—
288#92418—41no—
289#21315—611nomaximal, Sylow
290C22×C61C22×C22⋊C411yescharacteristic
291C22×C61C2×C22⋊Q811yescharacteristic
292C22×C61C2×C22⋊Q811yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C22×C61#57811yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61#39114yes—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61C2×C4×Q811yescharacteristic
311C4×C61C2×C4×Q811yescharacteristic
312C4×C61C2×#12411yescharacteristic
313C4×C61C2×C4⋊Q811yescharacteristic
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61—42no—
318Dic61—42no—
319Dic61—42no—
320Dic61#39114yes—
321C2×#544—611no—
322C2×#424—611no—
323#2274—611no—
324#1989—611no—
325#1989—611no—
326#1989—611no—
327#1989—611no—
328#1988—611no—
329#1988—611no—
330#1987—611no—
331#1987D6111yescharacteristic, complemented
332#1736—611no—
333#1736—611no—
334#1735—611no—
335C2×#340—611no—
336C2×C61C2×#39111yescharacteristic
337C2×C61#289711yescharacteristic
338C2×C61#289811yescharacteristic
339C2×C61#308111yescharacteristic
340C2×C61#308211yescharacteristic
341C2×C61#308311yescharacteristic
342C2×C61#308411yescharacteristic
343D61—24no—
344C22×M16—611no—
345C23×C8—611no—
346#544—618no—
347#544—1224no—
348C2×C4⋊Q8—611no—
349C2×C4⋊Q8—1221no—
350C2×C4⋊Q8C2×D6111yescharacteristic
351C2×#125—611no—
352C2×#125—611no—
353C2×#125—1221no—
354C2×C22⋊Q8—1221no—
355C2×C22⋊Q8—1221no—
356C2×#117—611no—
357C2×#117—611no—
358C2×#117—611no—
359C22×C4⋊C4—611no—
360C22×C4⋊C4—611no—
361#424—1224no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8C2×D6111yescharacteristic
366C4×C4⋊C4—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—1221no—
370C4×C4⋊C4—1221no—
371C4×C4⋊C4C2×D6111yescharacteristic
372C4×C22⋊C4—1221no—
373C4×C22⋊C4—1221no—
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—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389#340C2×D6111yescharacteristic
390C61#2131511yescharacteristic, complemented, Sylow
391C22×Q8—21no—
392C22×Q8—1221no—
393C23×C4—611no—
394C23×C4—1221no—
395C23×C4—1221no—
396C2×M16—1222no—
397C2×M16—1222no—
398C2×M16—1222no—
399C22×C8—611no—
400C22×C8—611no—
401C22×C8—1222no—
402C22×C8—1222no—
403C22×C8—1222no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—22no—
407C4⋊Q8—614no—
408C4⋊Q8—1222no—
409C4⋊Q8—1228no—
410C4⋊Q8C4×D6114yescomplemented
411#125—614no—
412#125—614no—
413#125—1222no—
414#125—1228no—
415#125—1222no—
416C22⋊Q8—1228no—
417C22⋊Q8—1228no—
418#117—614no—
419#117—614no—
420#117—614no—
421#117—614no—
422#117—1222no—
423#117—1222no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—614no—
432C2×C4⋊C4—614no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458C4⋊C8—22no—
459C4⋊C8—1222no—
460C4⋊C8—1222no—
461C4⋊C8—1222no—
462#95—21no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466C24—611no—
467C2×Q8—24no—
468C2×Q8—81no—
469C2×Q8—81no—
470C2×Q8—1224no—
471C2×Q8—4881no—
472C2×Q8—4881no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—1224no—
487C22×C4—1224no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—4881no—
493C22×C4—4881no—
494C22×C4—4881no—
495C22×C4—4881no—
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499M16—2444no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—2444no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—614no—
519C4⋊C4—614no—
520C4⋊C4—612no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C4⋊C4—1224no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4C2×C4×D6112yes—
536C4⋊C4C2×C4×D6112yes—
537C4⋊C4Q8×D6112yes—
538C4⋊C4Q8×D6112yes—
539C22⋊C4—614no—
540C22⋊C4—614no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C42—81no—
544C42—612no—
545C42—612no—
546C42—1224no—
547C42—1224no—
548C42—4881no—
549C42—4881no—
550C42—4881no—
551C42C2×C4×D6112yes—
552C42D4×D6112yes—
553C23—612no—
554C23—612no—
555C23—612no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23C2×D4×D6111yescharacteristic
561Q8—84no—
562Q8—4884no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—614no—
584C2×C4—614no—
585C2×C4—614no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—4884no—
597C2×C4—4884no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4C22⋊C4×D6112yes—
607C2×C4C22⋊C4×D6112yes—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×Q8×D6111yescharacteristic
611C2×C4C2×Q8×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C8—41no—
615C8—41no—
616C8—2441no—
617C8—2441no—
618C22—614no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22C2×C22⋊C4×D6111yescharacteristic
626C22C22⋊Q8×D6111yescharacteristic
627C22C22⋊Q8×D6111yescharacteristic
628C22#1684056611yescharacteristic
629C22#1684056811yescharacteristic
630C22#1684057211yescharacteristic
631C22#1684057511yescharacteristic
632C4—22no—
633C4—22no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—614no—
639C4—1222no—
640C4—1222no—
641C4—2442no—
642C4—2442no—
643C4—2442no—
644C4—2442no—
645C4C4×Q8×D6111yescharacteristic
646C4C4×Q8×D6111yescharacteristic
647C4#124×D6111yescharacteristic
648C4C4⋊Q8×D6111yescharacteristic
649C2—1224no—
650C2#391×D6111yescharacteristic
651C2#2145434411yescharacteristic
652C2#2145434711yescharacteristic
653C2#2145564511yescharacteristic
654C2#2145565611yescharacteristic
655C2#2145566711yescharacteristic
656C2#2145567611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1987D61
hypocenterC61#21315
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
4. #21446776C2semidirect
5. #21446775C2semidirect
6. #21446774C2semidirect
7. #21446773C2semidirect
8. #21446772C2semidirect
9. #21446769C2semidirect
10. #21446765C2semidirect
15. C61×#1987C2semidirect
19. #16840389C4semidirect
75. #14519776C2×C4semidirect
78. #14519769C2×C4semidirect
80. #14519768C2×C4semidirect
87. #14519735Q8semidirect
106. C4⋊C4×D61Q8semidirect
127. C4⋊Q8×C61C2×C4semidirect
331. #1987D61semidirect
390. C61#21315semidirect
410. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403892
C2×#168397872
#214481732
#214467762
#214467752
#214467742
#214467732
#214467722
#214467692
#214467652
#214455372
#214455342
#340×D612
C61×#19872
#2131561