Finite Group Databasebeta

← #29605334

Subgroups — #29605334

Order 15616.

1758 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839681C211yesmaximal, characteristic
2C2×#16839643C212yesmaximal
3C2×C4×#11758196C211yesmaximal, characteristic
4#21444090C211yesmaximal, characteristic
5#21444078C211yesmaximal, characteristic
6#21444069C211yesmaximal, characteristic
7#21444052C212yesmaximal
8#21444052C212yesmaximal
9#21444051C212yesmaximal
10#21444046C211yesmaximal, characteristic
11C22×C4×Q8×C61C211yesmaximal, characteristic
12C22×#11758196C2212yes—
13C22×#11758196C2211yescharacteristic
14C22×#11758196C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839681C418yescomplemented
18#16839643—28no—
19C4×#11758196—24no—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2211yescharacteristic
30C2×#14519684C2211yescharacteristic
31C2×#14519684C2211yescharacteristic
32C2×#14519684C2212yes—
33C23×Q8×C61C2211yescharacteristic
34C2×C4×Q8×C61—28no—
35C2×C4×Q8×C61—24no—
36C22×C4⋊C4×C61C2211yescharacteristic
37C22×C4⋊C4×C61C2212yes—
38C22×C42×C61C2212yes—
39C22×C42×C61C2211yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2312yes—
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46C2×#11758196—22no—
47C2×#11758196—22no—
48C2×#11758196—24no—
49C2×#11758196—22no—
50C2×#11758196—24no—
51C2×#11758196—22no—
52C2×#11758196C2×C418yes—
53C2×#11758196C2×C414yes—
54C2×#11758196Q814yescomplemented
55C2×C4×Dic61—22no—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—24no—
60#14519684—22no—
61#14519684—22no—
62#14519684—24no—
63#14519684—24no—
64#14519684—24no—
65#14519684—24no—
66#14519684—24no—
67#14519684—22no—
68#14519684—24no—
69#14519684—24no—
70#14519684—24no—
71#14519684C2×C414yes—
72#14519684C2×C418yes—
73C22×Q8×C61—24no—
74C22×Q8×C61—22no—
75C22×Q8×C61—22no—
76C22×Q8×C61—24no—
77C22×Q8×C61C2×C414yes—
78C23×C4×C61C2312yes—
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C4×Q8×C61—232no—
84C2×C4⋊C4×C61—24no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—24no—
87C2×C4⋊C4×C61—24no—
88C2×C4⋊C4×C61D418yes—
89C2×C42×C61—24no—
90C2×C42×C61—24no—
91C2×C42×C61—24no—
92C2×C42×C61—24no—
93C2×C42×C61D414yes—
94C24×C61C2411yescharacteristic
95C2×Q8×C61—28no—
96C2×Q8×C61—28no—
97C2×Q8×C61—216no—
98C2×Q8×C61—28no—
99C2×Q8×C61—216no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
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×C61C22×C414yes—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C2×D414yes—
120C22×C4×C61C2×D414yes—
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D414yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×Q812yes—
126C22×C4×C61#4112yes—
127C22×C4×C61#4114yes—
128C22×C4×C61#4112yes—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
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×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C22×C414yes—
159C22×Dic61C22×C412yes—
160C22×Dic61C22×C412yes—
161C22×Dic61C2×Q812yes—
162C22×Dic61C2×Q812yes—
163C22×Dic61#4112yes—
164C22×Dic61#4112yes—
165C4⋊C4×C61—28no—
166C4⋊C4×C61—216no—
167C42×C61—216no—
168C42×C61—28no—
169#11758196—28no—
170#11758196—216no—
171#11758196—28no—
172C4×Dic61—28no—
173C4×Dic61—44no—
174C23×C61C23×C411yescharacteristic
175C23×C61C22×D411yescharacteristic
176C23×C61C22×D412yes—
177C23×C61C22×Q811yescharacteristic
178C23×C61C2×#4111yescharacteristic
179C23×C61C2×#4112yes—
180C23×C61C2×#4111yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C612−1+412yes—
184C23×C612−1+412yes—
185Q8×C61—232no—
186C2×C4×C61—24no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—24no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—24no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61C4×D414yes—
212C2×C4×C61C4×D418yes—
213C2×C4×C61#12014yes—
214C2×C4×C61C22⋊Q814yes—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—22no—
220C2×Dic61—22no—
221C2×Dic61—24no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61—24no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61C4×Q814yescomplemented
251#14174—611nomaximal, Sylow
252C22×C61C2×C4×D411yescharacteristic
253C22×C61C2×C4×D412yes—
254C22×C61C2×C4×Q811yescharacteristic
255C22×C61C4×#4111yescharacteristic
256C22×C61#51912yes—
257C22×C61C2×#12011yescharacteristic
258C22×C61C2×C22⋊Q811yescharacteristic
259C22×C61#52511yescharacteristic
260C22×C61#52511yescharacteristic
261C22×C61#52512yes—
262C22×C61#52911yescharacteristic
263C22×C61#53311yescharacteristic
264C22×C61#53611yescharacteristic
265C22×C61#53611yescharacteristic
266C22×C61#54112yes—
267C22×C61#54712yes—
268C22×C61#54712yes—
269C22×C61#54812yes—
270C22×C61D4×Q812yes—
271C22×C61#55212yes—
272C22×C61#55612yes—
273C22×C61#55612yes—
274C22×C61#55712yes—
275C4×C61—24no—
276C4×C61—24no—
277C4×C61—28no—
278C4×C61—24no—
279C4×C61—44no—
280Dic61—24no—
281Dic61—42no—
282Dic61—44no—
283Dic61—42no—
284Dic61—42no—
285Dic61—82no—
286C22×C4×Q8D6111yescharacteristic
287C2×#395—611no—
288C2×#387—612no—
289C2×C4×C4⋊C4—611no—
290#1446—611no—
291#1442—611no—
292#1441—611no—
293#1436—612no—
294#1436—612no—
295#1436—612no—
296#1434—611no—
297C2×C61C4×#12011yescharacteristic
298C2×C61C4×C22⋊Q811yescharacteristic
299C2×C61#232711yescharacteristic
300C2×C61#236212yes—
301C2×C61#236512yes—
302C2×C61#240611yescharacteristic
303C2×C61#240911yescharacteristic
304C2×C61#246212yes—
305C2×C61#246912yes—
306C2×C61#258111yescharacteristic
307C2×C61#259911yescharacteristic
308C23×Q8C2×D6111yescharacteristic
309C2×C4×Q8—28no—
310C2×C4×Q8—24no—
311C22×C4⋊C4—612no—
312C22×C4⋊C4—611no—
313C22×C4⋊C4—611no—
314C22×C4⋊C4C2×D6111yescharacteristic
315C22×C4⋊C4C2×D6112yes—
316C22×C42—611no—
317C22×C42—611no—
318C22×C42C2×D6112yes—
319C22×C42C2×D6111yescharacteristic
320#395—618no—
321#387—1228no—
322C4×C4⋊C4—1224no—
323C2×#95—611no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—612no—
327C2×#95—611no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95—611no—
332C2×#95—612no—
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—611no—
336C61#1417411yescharacteristic, complemented, Sylow
337C22×Q8—24no—
338C22×Q8—22no—
339C22×Q8—24no—
340C22×Q8—22no—
341C22×Q8C4×D6114yes—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—612no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—612no—
348C23×C4C22×D6112yes—
349C23×C4C22×D6112yes—
350C23×C4C22×D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352C23×C4C22×D6111yescharacteristic
353C4×Q8—232no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—614no—
359C2×C4⋊C4—618no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—1222no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4C22⋊D6118yes—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—24no—
372C2×C42—1222no—
373C2×C42—1222no—
374C2×C42—1222no—
375C2×C42—1222no—
376C2×C42—1224no—
377C2×C42C22⋊D6114yes—
378#95—618no—
379#95—614no—
380#95—1224no—
381#95—1224no—
382#95—1224no—
383#95—1222no—
384#95—1224no—
385#95—1222no—
386#95—1222no—
387#95—1224no—
388#95—1224no—
389#95—1224no—
390#95—1224no—
391C24C23×D6111yescharacteristic
392C2×Q8—28no—
393C2×Q8—216no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—216no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—612no—
417C22×C4—614no—
418C22×C4—612no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1222no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1221no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4C2×C4×D6114yes—
451C22×C4C2×C4×D6112yes—
452C22×C4#1175822214yes—
453C22×C4#1175822212yes—
454C22×C4Q8×D6112yes—
455C22×C4#1175822612yes—
456C22×C4C2×C22⋊D6114yes—
457C22×C4C2×C22⋊D6114yes—
458C22×C4C2×C22⋊D6112yes—
459C22×C4C2×C22⋊D6114yes—
460C22×C4C2×C22⋊D6112yes—
461C22×C4C2×C22⋊D6112yes—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—1228no—
465C4⋊C4—12216no—
466C4⋊C4—1228no—
467C42—216no—
468C42—28no—
469C42—1228no—
470C42—2444no—
471C23C22×C4×D6111yescharacteristic
472C23C22×C22⋊D6111yescharacteristic
473C23C22×C22⋊D6112yes—
474C23C2×Q8×D6111yescharacteristic
475C23#41×D6111yescharacteristic
476C23#41×D6111yescharacteristic
477C23C2×#1175822212yes—
478C23C2×#1175822211yescharacteristic
479C23C2×#1175822611yescharacteristic
480C23#1451982412yes—
481C23#1451982512yes—
482Q8—232no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—614no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1224no—
516C2×C4—1222no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1224no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4C4×C22⋊D6114yes—
545C2×C4C4×C22⋊D6118yes—
546C2×C4#1451974614yes—
547C2×C4#1451975314yes—
548C22C2×C4×C22⋊D6111yescharacteristic
549C22C2×C4×C22⋊D6112yes—
550C22C4×Q8×D6111yescharacteristic
551C22C4×#1175822611yescharacteristic
552C22#1684023812yes—
553C22C2×#1451974611yescharacteristic
554C22C2×#1451975311yescharacteristic
555C22#1684026911yescharacteristic
556C22#1684027312yes—
557C22#1684027311yescharacteristic
558C22#1684029911yescharacteristic
559C22#1684031611yescharacteristic
560C22#1684033311yescharacteristic
561C22#1684033611yescharacteristic
562C22#1684037012yes—
563C22#1684041112yes—
564C22#1684041112yes—
565C22#1684042312yes—
566C22Q8×C22⋊D6112yes—
567C22#1684045312yes—
568C22#1684048412yes—
569C22#1684048512yes—
570C22#1684049012yes—
571C4—28no—
572C4—24no—
573C4—24no—
574C4—24no—
575C4—44no—
576C4—1224no—
577C4—4882no—
578C4—2442no—
579C4—2442no—
580C4—2444no—
581C4—2442no—
582C2C4×#1451974611yescharacteristic
583C2C4×#1451975311yescharacteristic
584C2#2144853511yescharacteristic
585C2#2144884212yes—
586C2#2144887312yes—
587C2#2144915011yescharacteristic
588C2#2144917811yescharacteristic
589C2#2144979512yes—
590C2#2144989412yes—
591C2#2145118511yescharacteristic
592C2#2145139011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×C4×Q8×C61C2
socle, prime radical, cosocleC24×C61C24
hypercenterC22×C4×Q8D61
hypocenterC61#14174
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
17. #16839681C4semidirect
54. C2×#11758196Q8semidirect
250. C2×Dic61C4×Q8semidirect
336. C61#14174semidirect

Maximal subgroups

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

subgroupindex
C2×#168396812
C2×#168396432
C2×C4×#117581962
#214440902
#214440782
#214440692
#214440522
#214440512
#214440462
C22×C4×Q8×C612
#1417461