Finite Group Databasebeta

← #29602569

Subgroups — #29602569

Order 15616.

1766 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic122C211yesmaximal, characteristic
2C2×#16839666C212yesmaximal
3C2×#16839626C211yesmaximal, characteristic
4C2×#16839613C211yesmaximal, characteristic
5#21444071C212yesmaximal
6#21444070C212yesmaximal
7#21444047C212yesmaximal
8#21444046C211yesmaximal, characteristic
9#21444043C211yesmaximal, characteristic
10#21444043C211yesmaximal, characteristic
11C61×#1434C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×C4×Dic122—24no—
14C2×C4×Dic122—24no—
15C2×C4×Dic122C418yescomplemented
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2212yes—
18C22×#11758196C2212yes—
19C22×#11758196C2211yescharacteristic
20C22×#11758196C2211yescharacteristic
21C22×C4×Dic61C2212yes—
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839666—28no—
25#16839626—24no—
26#16839613—24no—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C2×#14519684C2212yes—
36C22×C42×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2211yescharacteristic
40C2×#95×C61C2212yes—
41C22×Dic122—22no—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—22no—
45C22×Dic122C2×C414yes—
46C22×Dic122D414yes—
47C22×Dic122Q814yes—
48C23×Dic61C2312yes—
49C23×Dic61C2311yescharacteristic
50C23×Dic61C2312yes—
51C23×Dic61C2311yescharacteristic
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2311yescharacteristic
54C4×Dic122—232no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61C2×C414yes—
58C2×#11758196—24no—
59C2×#11758196—22no—
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758196—24no—
64C2×#11758196—24no—
65C2×#11758196—24no—
66C2×#11758196—24no—
67C2×#11758196—24no—
68C2×#11758196C2×C414yes—
69C2×#11758196C2×C414yes—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74C2×C4×Dic61—24no—
75C2×C4×Dic61C2×C414yes—
76C2×C4×Dic61C2×C414yes—
77C2×C4×Dic61D414yes—
78C2×C4×Dic61D414yes—
79#14519684—22no—
80#14519684—22no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84#14519684—24no—
85#14519684—24no—
86#14519684—24no—
87#14519684—24no—
88C23×C4×C61C2312yes—
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2312yes—
91C23×C4×C61C2311yescharacteristic
92C23×C4×C61C2311yescharacteristic
93C2×C42×C61—22no—
94C2×C42×C61—22no—
95C2×C42×C61C2×C414yes—
96#95×C61—22no—
97#95×C61—22no—
98#95×C61—24no—
99#95×C61—24no—
100C24×C61C2411yescharacteristic
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C22×C412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×Q812yes—
131C22×C4×C61#4114yes—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22×C412yes—
156C22×Dic61C22×C412yes—
157C22×Dic61C22×C412yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61C2×D412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×Q812yes—
166C22×Dic61C2×Q812yes—
167C22×Dic61#4114yes—
168C22×Dic61#4112yes—
169C22×Dic61#4112yes—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C2×Dic122—28no—
173C2×Dic122—28no—
174C2×Dic122—28no—
175C2×Dic122—28no—
176C2×Dic122C4⋊C4116yescomplemented
177C42×C61—28no—
178C4⋊Dic61—28no—
179#11758196—216no—
180#11758196—28no—
181#11758196—28no—
182#11758196—48no—
183C4×Dic61—28no—
184C4×Dic61—48no—
185C4×Dic61C22⋊C4116yes—
186C23×C61C23×C411yescharacteristic
187C23×C61C22×D411yescharacteristic
188C23×C61C22×D411yescharacteristic
189C23×C61C22×D411yescharacteristic
190C23×C61C22×Q811yescharacteristic
191C23×C61C2×#4111yescharacteristic
192C23×C61C2×#4111yescharacteristic
193C23×C61C2×#4112yes—
194C23×C61C2×#4112yes—
195C23×C612−1+412yes—
196C23×C612−1+412yes—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61C2×C22⋊C414yes—
227C2×C4×C61C2×C4⋊C414yes—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C414yes—
258C2×Dic61C2×C22⋊C414yes—
259C2×Dic61C2×C4⋊C414yes—
260C2×Dic61C2×C4⋊C414yes—
261C2×Dic61#11714yes—
262C2×Dic61#12114yes—
263C2×Dic61#12114yes—
264C2×Dic61C22⋊Q814yes—
265C2×Dic61C22⋊Q814yes—
266Dic122—232no—
267#13942—611nomaximal, Sylow
268C22×C61C22×C22⋊C411yescharacteristic
269C22×C61C22×C4⋊C411yescharacteristic
270C22×C61C2×#11711yescharacteristic
271C22×C61C4×#4111yescharacteristic
272C22×C61C4×#4111yescharacteristic
273C22×C61#51911yescharacteristic
274C22×C61#51911yescharacteristic
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×C22⋊Q811yescharacteristic
278C22×C61C2×C22⋊Q811yescharacteristic
279C22×C61#52512yes—
280C22×C61#53212yes—
281C22×C61#53612yes—
282C22×C61#53612yes—
283C22×C61#54812yes—
284C22×C61#54812yes—
285C22×C61#55212yes—
286C22×C61#55212yes—
287C22×C61#55612yes—
288C22×C61#55612yes—
289C22×C61#55712yes—
290C22×C61#55712yes—
291C4×C61—24no—
292C4×C61—24no—
293C4×C61—44no—
294C4×C61—42no—
295C4×C61—82no—
296Dic61—24no—
297Dic61—24no—
298Dic61—44no—
299Dic61—42no—
300Dic61—82no—
301Dic61#38018yes—
302C22×C4×Q8—611no—
303C2×#391—612no—
304C2×#382—611no—
305C2×C4×C4⋊C4—611no—
306#1441—612no—
307#1441—612no—
308C4⋊#95—612no—
309#1434—611no—
310#1434—611no—
311#1434—611no—
312#1434D6111yescharacteristic
313C2×C61C2×#38011yescharacteristic
314C2×C61#229711yescharacteristic
315C2×C61#231011yescharacteristic
316C2×C61#235111yescharacteristic
317C2×C61#235111yescharacteristic
318C2×C61#235611yescharacteristic
319C2×C61#235611yescharacteristic
320C2×C61#253612yes—
321C2×C61#254512yes—
322C2×C61#256512yes—
323C2×C61#258112yes—
324C23×Q8—611no—
325C2×C4×Q8—618no—
326C2×C4×Q8—1224no—
327C2×C4×Q8—1224no—
328C22×C4⋊C4—611no—
329C22×C4⋊C4—611no—
330C22×C4⋊C4—611no—
331C22×C4⋊C4—612no—
332C22×C4⋊C4—612no—
333C22×C42—611no—
334C22×C42—611no—
335C22×C42—612no—
336C22×C42C2×D6111yescharacteristic
337#391—1228no—
338#382—1224no—
339C4×C4⋊C4—1224no—
340C2×#95—611no—
341C2×#95—611no—
342C2×#95—612no—
343C2×#95—612no—
344C2×#95—612no—
345C2×#95—612no—
346C2×#95—612no—
347C2×#95—612no—
348C2×#95—612no—
349C2×#95C2×D6111yescharacteristic
350C2×#95C2×D6111yescharacteristic
351C2×#95C2×D6112yes—
352C2×#95C2×D6112yes—
353C61#1394211yescharacteristic, complemented, Sylow
354C22×Q8—614no—
355C22×Q8—614no—
356C22×Q8—614no—
357C22×Q8—1222no—
358C22×Q8—1222no—
359C22×Q8—1222no—
360C22×Q8—1222no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—612no—
366C23×C4—612no—
367C23×C4C22×D6112yes—
368C23×C4C22×D6111yescharacteristic
369C23×C4C22×D6112yes—
370C23×C4C22×D6111yescharacteristic
371C23×C4C22×D6111yescharacteristic
372C4×Q8—12232no—
373C2×C4⋊C4—614no—
374C2×C4⋊C4—614no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4—1224no—
384C2×C4⋊C4—1224no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1224no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—614no—
391C2×C42—614no—
392C2×C42—614no—
393C2×C42—614no—
394C2×C42—1222no—
395C2×C42—1222no—
396C2×C42—1224no—
397C2×C42—1224no—
398C2×C42—1224no—
399C2×C42C4×D6114yes—
400#95—24no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—1224no—
405#95—1224no—
406#95—1224no—
407#95—1224no—
408#95—1224no—
409#95—1224no—
410#95—1222no—
411#95—1222no—
412#95—1224no—
413C24C23×D6111yescharacteristic
414C2×Q8—6116no—
415C2×Q8—1228no—
416C2×Q8—1228no—
417C2×Q8—1228no—
418C2×Q8—1228no—
419C2×Q8—1228no—
420C2×Q8—1228no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1222no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4C2×C4×D6112yes—
483C22×C4C2×C4×D6112yes—
484C22×C4C2×C4×D6112yes—
485C22×C4#1175822214yes—
486C22×C4D4×D6112yes—
487C22×C4D4×D6112yes—
488C22×C4D4×D6112yes—
489C22×C4Q8×D6112yes—
490C4⋊C4—1228no—
491C4⋊C4—1228no—
492C4⋊C4—1228no—
493C4⋊C4—12216no—
494C4⋊C4—2448no—
495C42—28no—
496C42—6116no—
497C42—1228no—
498C42—2448no—
499C23C22×C4×D6111yescharacteristic
500C23C2×D4×D6111yescharacteristic
501C23C2×D4×D6111yescharacteristic
502C23C2×D4×D6111yescharacteristic
503C23C2×Q8×D6111yescharacteristic
504C23#41×D6111yescharacteristic
505C23#41×D6112yes—
506C23#41×D6111yescharacteristic
507C23C2×#1175822212yes—
508C23#1451982412yes—
509C23#1451982412yes—
510Q8—12232no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—614no—
541C2×C4—614no—
542C2×C4—614no—
543C2×C4—614no—
544C2×C4—614no—
545C2×C4—614no—
546C2×C4—614no—
547C2×C4—614no—
548C2×C4—614no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4C22⋊C4×D6114yes—
579C2×C4C4⋊C4×D6114yes—
580C22C2×C22⋊C4×D6111yescharacteristic
581C22C2×C4⋊C4×D6111yescharacteristic
582C22#117×D6111yescharacteristic
583C22#1684023011yescharacteristic
584C22C4×#1175822211yescharacteristic
585C22#1684023911yescharacteristic
586C22#1684023911yescharacteristic
587C22#121×D6111yescharacteristic
588C22#121×D6111yescharacteristic
589C22C22⋊Q8×D6111yescharacteristic
590C22C22⋊Q8×D6111yescharacteristic
591C22#1684027112yes—
592C22#1684030912yes—
593C22#1684033412yes—
594C22#1684033412yes—
595C22#1684042012yes—
596C22#1684042112yes—
597C22#1684044512yes—
598C22#1684044812yes—
599C22#1684047912yes—
600C22#1684048012yes—
601C22#1684048912yes—
602C22#1684049012yes—
603C4—24no—
604C4—24no—
605C4—42no—
606C4—44no—
607C4—82no—
608C4—618no—
609C4—1224no—
610C4—1224no—
611C4—4882no—
612C4—2442no—
613C4—2444no—
614C2#380×D6111yescharacteristic
615C2#2144830511yescharacteristic
616C2#2144840411yescharacteristic
617C2#2144873611yescharacteristic
618C2#2144873911yescharacteristic
619C2#2144877911yescharacteristic
620C2#2144878511yescharacteristic
621C2#2145072712yes—
622C2#2145079712yes—
623C2#2145100812yes—
624C2#2145118412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1434C2
socle, prime radical, cosocleC24×C61C24
hypercenter#1434D61
hypocenterC61#13942
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
15. C2×C4×Dic122C4semidirect
176. C2×Dic122C4⋊C4semidirect
353. C61#13942semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1222
C2×#168396662
C2×#168396262
C2×#168396132
#214440712
#214440702
#214440472
#214440462
#214440432
C61×#14342
#1394261