Finite Group Databasebeta

← #29713070

Subgroups — #29713070

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840095C211yesmaximal, characteristic, complemented
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21450136C211yesmaximal, characteristic
4#21447062C212yesmaximal, complemented
5#21447019C212yesmaximal, complemented
6#21446243C211yesmaximal, characteristic, complemented
7#21446112C211yesmaximal, characteristic
8#21446073C212yesmaximal, complemented
9#21445754C211yesmaximal, characteristic, complemented
10#21445754C212yesmaximal, complemented
11C2×D4×Q8×C61C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16840095—24no—
15C2×#14519710C2212yes—
16C2×#14519701—24no—
17C2×#14519701C2212yes—
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2212yes—
25#16839701—21no—
26#16839698—21no—
27#16839660C2212yes—
28#16839642—21no—
29#16839634—21no—
30#16839622C2211yescharacteristic
31#16839622C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C23×Q8×C61C2211yescharacteristic
35C23×Q8×C61C2211yescharacteristic
36D4×Q8×C61—24no—
37D4×Q8×C61—24no—
38C2×C4⋊Q8×C61—21no—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C22⋊Q8×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61—21no—
46C2×C4×D4×C61C2211yescharacteristic
47C23×Dic61—21no—
48C22×#92418—22no—
49C22×#92418—42no—
50C22×#92418C2312yes—
51C22×#92418C2312yes—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×C4⋊Dic61C2312yes—
57C2×#11758196—22no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—22no—
60#14519710—24no—
61#14519701—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—24no—
65#14519701—42no—
66#14519701—42no—
67C22⋊#92418—22no—
68C22⋊#92418—42no—
69#14519684—21no—
70#14519684—22no—
71#14519684—22no—
72C22×Q8×C61—22no—
73C22×Q8×C61—22no—
74C22×Q8×C61—21no—
75C22×Q8×C61—21no—
76C22×Q8×C61—22no—
77C22×Q8×C61—24no—
78C22×Q8×C61—24no—
79C22×Q8×C61—42no—
80C22×Q8×C61—42no—
81C22×Q8×C61D414yes—
82C22×Q8×C61C2311yescharacteristic
83C22×Q8×C61C2311yescharacteristic
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C4⋊Q8×C61—28no—
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C22⋊Q8×C61—28no—
94C22⋊Q8×C61—28no—
95C22⋊Q8×C61—22no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61D414yes—
99C4×Q8×C61—22no—
100C4×Q8×C61—22no—
101C4×D4×C61—28no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117C24×C61C2×D411yescharacteristic
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—24no—
120C2×Q8×C61—24no—
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—28no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—44no—
131C2×Q8×C61—44no—
132C2×Q8×C61—44no—
133C2×Q8×C61—48no—
134C2×Q8×C61—48no—
135C2×Q8×C61—44no—
136C2×Q8×C61—44no—
137C2×Q8×C61C2×D412yes—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61C2×D412yes—
140C2×Q8×C61#4114yes—
141C2×D4×C61—22no—
142C2×D4×C61—22no—
143C2×D4×C61Q1614yes—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61—24no—
148C22×C4×C61—24no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—44no—
157C22×C4×C61—44no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—42no—
168C22×Dic61—82no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—24no—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—24no—
182C42×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—42no—
185C22⋊Dic61—82no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—42no—
191#11758196—82no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—44no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61#12012yes—
206C23×C61C2×Q1612yes—
207C23×C61C2×Q1612yes—
208C23×C61#13712yes—
209C23×C61C22×D411yescharacteristic
210Q8×C61—22no—
211Q8×C61—24no—
212Q8×C61—22no—
213Q8×C61—22no—
214Q8×C61—24no—
215Q8×C61—48no—
216Q8×C61—48no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—42no—
220Q8×C61#12314yes—
221D4×C61—28no—
222C2×C4×C61—21no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—48no—
238C2×C4×C61—48no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—42no—
245C2×C4×C61#12012yes—
246C2×C4×C61C2×Q1612yes—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4112yes—
250C2×C4×C61H(5,𝔽2)12yes—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—84no—
258#92418—42no—
259#92418—42no—
260#24042—611nomaximal, Sylow
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—24no—
265C22×C61—22no—
266C22×C61—24no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61#45114yes—
271C22×C61C2×#12011yescharacteristic
272C22×C61#54612yes—
273C22×C61C22×Q1611yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61#57712yes—
276C4×C61—22no—
277C4×C61—24no—
278C4×C61—22no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61C2×#12311yescharacteristic
285C4×C61#53411yescharacteristic
286C4×C61#54612yes—
287Dic61—82no—
288Dic61—82no—
289C2×D4×Q8D6111yescharacteristic
290C2×#484—611no—
291C22×#103—611no—
292#2487—611no—
293#2043—612no—
294#2036—612no—
295#1892—611no—
296#1869—611no—
297#1863—612no—
298#1787—612no—
299#1787—611no—
300C2×C61—24no—
301C2×C61—42no—
302C2×C61#240511yescharacteristic
303C2×C61C2×#45111yescharacteristic
304C2×C61#301011yescharacteristic
305C2×C61#329912yes—
306C2×C61#332012yes—
307C23×Q8C2×D6111yescharacteristic
308C23×Q8C2×D6111yescharacteristic
309C23×C8—611no—
310D4×Q8—24no—
311D4×Q8—24no—
312C2×C4⋊Q8—21no—
313C2×C4⋊Q8C2×D6111yescharacteristic
314C2×C22⋊Q8—21no—
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8C2×D6111yescharacteristic
317C2×C22⋊Q8C2×D6111yescharacteristic
318C2×C4×Q8C2×D6111yescharacteristic
319C2×C4×D4—21no—
320C2×C4×D4C2×D6111yescharacteristic
321C22×C4⋊C4—611no—
322#484—1224no—
323C2×#107—612no—
324C2×#103—612no—
325C2×#103—612no—
326C2×#103—612no—
327C2×#103—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×#103—1224no—
331C2×C22⋊C8—611no—
332C2×C22⋊C8—612no—
333#400—1221no—
334#399—1221no—
335#389—612no—
336#386—1221no—
337#385—1221no—
338#380—611no—
339#380—612no—
340#340—612no—
341#336—612no—
342C61#2404211yescharacteristic, complemented, Sylow
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—42no—
351C22×Q8—42no—
352C22×Q8C22⋊D6114yes—
353C22×Q8C22×D6111yescharacteristic
354C22×Q8C22×D6111yescharacteristic
355C22×Q8C22×D6111yescharacteristic
356C22×D4C22⋊D6111yescharacteristic
357C23×C4—21no—
358C23×C4—21no—
359C23×C4—1221no—
360C23×C4C22×D6111yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C22×C8—612no—
363C22×C8—612no—
364C22×C8—1222no—
365C22×C8—2442no—
366C4⋊Q8—28no—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C22⋊Q8—28no—
370C22⋊Q8—28no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8C22⋊D6114yes—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×D4—28no—
378C4×D4—22no—
379C4×D4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4C22⋊D6111yescharacteristic
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4C22⋊D6111yescharacteristic
398C2×C22⋊C4C22⋊D6111yescharacteristic
399C2×C42—21no—
400C2×C42C22×D6111yescharacteristic
401#107—1224no—
402#103—1224no—
403#103—1224no—
404#103—1224no—
405#103—1224no—
406#103—2442no—
407#103—2442no—
408C22⋊C8—1222no—
409C22⋊C8—2442no—
410#95—1222no—
411#95—1221no—
412#95—1222no—
413C24C2×C22⋊D6111yescharacteristic
414C24C2×C22⋊D6111yescharacteristic
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8#1175822414yes—
434C2×Q8C2×C22⋊D6112yes—
435C2×Q8C2×C22⋊D6112yes—
436C2×Q8C2×C22⋊D6112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#1175820214yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—4882no—
457C22×C4—2442no—
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6112yes—
461C22×C4C2×C22⋊D6112yes—
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C23×D6111yescharacteristic
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2444no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—4882no—
484C4⋊C4—2442no—
485C4⋊C4C2×C22⋊D6112yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—4882no—
491C22⋊C4—2442no—
492C42—24no—
493C42—21no—
494C42—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—22no—
501C23#1451974612yes—
502C23C2×#1175820212yes—
503C23C2×#1175820212yes—
504C23#1451980812yes—
505C23C22×C22⋊D6111yescharacteristic
506Q8—22no—
507Q8—24no—
508Q8—22no—
509Q8—22no—
510Q8—24no—
511Q8—48no—
512Q8—48no—
513Q8—41no—
514Q8—41no—
515Q8—42no—
516Q8#1451975814yes—
517D4—28no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—4884no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4#1451974612yes—
549C2×C4C2×#1175820212yes—
550C2×C4C22×C22⋊D6111yescharacteristic
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4C2×#1175822412yes—
553C2×C4#1451982212yes—
554C8—2442no—
555C8—2442no—
556C22—22no—
557C22—24no—
558C22—24no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—41no—
563C22—42no—
564C22—41no—
565C22#1683991814yes—
566C22C2×#1451974611yescharacteristic
567C22#1684040612yes—
568C22C22×#1175820211yescharacteristic
569C22C2×#1451980811yescharacteristic
570C22#1684057012yes—
571C4—22no—
572C4—22no—
573C4—24no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—42no—
579C4—4882no—
580C4—4882no—
581C4C2×#1451975811yescharacteristic
582C4#1684032111yescharacteristic
583C4#1684040612yes—
584C2—24no—
585C2—42no—
586C2#2144914211yescharacteristic
587C2C2×#1683991811yescharacteristic
588C2#2145510511yescharacteristic
589C2#2145769512yes—
590C2#2145799012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×D4×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×D4×Q8D61
hypocenterC61#24042
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840095C2semidirect
2. C22×#14519701C2semidirect
4. #21447062C2semidirect
5. #21447019C2semidirect
6. #21446243C2semidirect
8. #21446073C2semidirect
9. #21445754C2semidirect
10. #21445754C2semidirect
342. C61#24042semidirect

Maximal subgroups

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

subgroupindex
C2×#168400952
C22×#145197012
#214501362
#214470622
#214470192
#214462432
#214461122
#214460732
#214457542
C2×D4×Q8×C612
#2404261