Finite Group Databasebeta

← #29680024

Subgroups — #29680024

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21445908C212yesmaximal
5#21445825C211yesmaximal, characteristic, complemented
6#21445824C211yesmaximal, characteristic, complemented
7#21445821C211yesmaximal, characteristic, complemented
8#21445752C212yesmaximal
9#103×Dic61C212yesmaximal, complemented
10#21445638C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C2×Q8×Dic61C2211yescharacteristic
14C2×C4×Dic122—21no—
15C2×C4×Dic122C2211yescharacteristic
16C2×#14519742—21no—
17C2×#14519742C2211yescharacteristic
18C4×Q8×D61—24no—
19C4×Q8×D61C418yescomplemented
20C2×#14519735—21no—
21C2×C4⋊C4×D61—21no—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61—21no—
24C2×C42×D61C2211yescharacteristic
25#16839814—24no—
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519689C2212yes—
31C2×C8×Dic61C2212yes—
32#16839632C2211yescharacteristic
33#16839627—21no—
34#16839619—21no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C2×Q8×D61—22no—
49C2×Q8×D61—22no—
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61C2×C414yes—
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59C22×#92418C2312yes—
60Q8×Dic61—22no—
61Q8×Dic61C2×C414yes—
62C4×Dic122—22no—
63C4×Dic122—28no—
64C4×Dic122C2×C414yes—
65#14519742—28no—
66#14519742—22no—
67#14519742C2×C414yes—
68#14519735—44no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C4⋊C4×D61—28no—
78C4⋊C4×D61—22no—
79C4⋊C4×D61C2×C414yes—
80C2×#11758198—22no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61—28no—
89C42×D61C2×C414yes—
90C42×D61D414yes—
91C4⋊#92418—22no—
92#14519702—24no—
93#14519701—24no—
94#14519700—24no—
95#14519689—24no—
96C8×Dic61—24no—
97#14519684—22no—
98#14519684—22no—
99C22×Q8×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101C4×Q8×C61—22no—
102C4×Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2312yes—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C4⋊C8×C61—22no—
111#103×C61—24no—
112C23×D61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C2×Q8×C61#4112yes—
118C2×Q8×C61#4112yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C2×D411yescharacteristic
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131Q8×D61—28no—
132Q8×D61—28no—
133Q8×D61—28no—
134Q8×D61—28no—
135C2×C4×D61—24no—
136C2×C4×D61—24no—
137C2×C4×D61—24no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—24no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61C22⋊C414yes—
148C2×C4×D61C22⋊C414yes—
149C2×C4×D61SD1614yes—
150C2×C4×D61Q1614yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D412yes—
153C2×Dic122—24no—
154C2×Dic122—24no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—24no—
159C2×Dic122C22×C412yes—
160C2×Dic122#4112yes—
161C2×Dic122#4112yes—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—24no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61C22×C412yes—
172C42×C61—24no—
173C42×C61C22×C412yes—
174C42×C61C2×D412yes—
175#11758198—44no—
176C4⋊Dic61—21no—
177C4⋊Dic61—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—42no—
181C4⋊Dic61—42no—
182C4⋊Dic61C22×C412yes—
183#11758196—24no—
184#11758196—24no—
185#11758196—24no—
186C4×Dic61—24no—
187C4×Dic61—24no—
188C4×Dic61—24no—
189C4×Dic61—44no—
190C4×Dic61C22⋊C414yes—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22×C412yes—
193C4×Dic61C2×D412yes—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C23×C61C22×D411yescharacteristic
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61C2×C22⋊C412yes—
203C22×D61C2×SD1612yes—
204C22×D61C2×Q1612yes—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—22no—
208Q8×C61#11714yescomplemented
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×SD1612yes—
223C2×C4×C61C2×Q1612yes—
224C2×C4×C61C23×C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×SD1612yes—
247C2×Dic61C2×Q1612yes—
248C4×D61—216no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—44no—
254C4×D61—48no—
255C4×D61#103116yescomplemented
256Dic122—216no—
257Dic122—22no—
258Dic122—22no—
259Dic122—22no—
260Dic122#11714yescomplemented
261C8×C61—42no—
262#92418—42no—
263#20734—611nomaximal, Sylow
264C22×C61C22×C22⋊C411yescharacteristic
265C22×C61#52511yescharacteristic
266C22×C61#52511yescharacteristic
267C22×C61C22×SD1611yescharacteristic
268C22×C61C22×Q1611yescharacteristic
269C22×C61#57511yescharacteristic
270C22×C61#57511yescharacteristic
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61C2×#10314yes—
278C4×C61—22no—
279C4×C61—24no—
280C4×C61—42no—
281C4×C61—82no—
282C4×C61C2×#10314yes—
283C4×C61C2×#11711yescharacteristic
284C4×C61C2×#11711yescharacteristic
285C4×C61#52511yescharacteristic
286C4×C61#52511yescharacteristic
287Dic61—22no—
288Dic61—22no—
289Dic61—24no—
290Dic61—42no—
291Dic61—82no—
292Dic61C2×#10314yes—
293C22×C4×Q8—611no—
294C2×#431—611no—
295#2285—611no—
296#1824—612no—
297#1803—611no—
298#1803—611no—
299#1803—611no—
300#1803D6111yescharacteristic, complemented
301#1787—612no—
302C4×#103—612no—
303C4×#103—612no—
304C2×C61#228011yescharacteristic
305C2×C61C22×#10311yescharacteristic
306C2×C61#289611yescharacteristic
307C2×C61#310311yescharacteristic
308C2×C61#310311yescharacteristic
309C2×C61#310411yescharacteristic
310C2×C61#310411yescharacteristic
311D61—24no—
312C23×Q8—611no—
313C2×C4×Q8—611no—
314C2×C4×Q8—618no—
315C2×C4×Q8—611no—
316C2×C4×Q8—611no—
317C2×C4×Q8—1221no—
318C2×C4×Q8—1221no—
319C2×C4×Q8—1224no—
320C2×C4×Q8C2×D6111yescharacteristic
321C2×#117—1221no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—1221no—
324C22×C42—611no—
325C22×C42—1221no—
326#431—1224no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#103—612no—
330C2×#103—612no—
331C2×#103—612no—
332C2×#103C2×D6112yes—
333C2×C22⋊C8—612no—
334C2×C4×C8—612no—
335#384—611no—
336#384C2×D6111yescharacteristic
337#382—1221no—
338#380—612no—
339#380—1221no—
340C4×C4⋊C4—612no—
341C4×C4⋊C4—1221no—
342#340—612no—
343#340—612no—
344#340—612no—
345#340C2×D6112yes—
346C61#2073411yescharacteristic, complemented, Sylow
347C22×Q8—614no—
348C22×Q8—611no—
349C22×Q8—1221no—
350C22×Q8—1222no—
351C22×Q8—1222no—
352C22×Q8—1222no—
353C22×Q8—1222no—
354C22×Q8—1222no—
355C22×Q8C22×D6111yescharacteristic
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—1221no—
360C23×C4—1221no—
361C22×C8—612no—
362C22×C8C22×D6112yes—
363C4×Q8—22no—
364C4×Q8—614no—
365C4×Q8—614no—
366C4×Q8—614no—
367C4×Q8—1228no—
368C4×Q8—1222no—
369C4×Q8—1228no—
370C4×Q8—1222no—
371C4×Q8—1222no—
372C4×Q8C4×D6114yes—
373#117—2444no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—614no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1228no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6112yes—
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C22⋊C4—1222no—
391C2×C42—21no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—614no—
396C2×C42—614no—
397C2×C42—1228no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42—1222no—
402C2×C42C22×D6111yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—1222no—
405#103—24no—
406#103—1224no—
407#103—1224no—
408#103—1224no—
409C22⋊C8—1224no—
410C4×C8—1224no—
411#95—1222no—
412#95—1222no—
413C24—611no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—1224no—
421C2×Q8—1224no—
422C2×Q8—1224no—
423C2×Q8—1228no—
424C2×Q8—1228no—
425C2×Q8—1228no—
426C2×Q8—1228no—
427C2×Q8—1221no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8C2×C4×D6112yes—
431C2×Q8#1175822412yes—
432C2×Q8#1175822612yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—611no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—614no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—614no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1222no—
456C22×C4—2441no—
457C22×C4—2441no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—612no—
476C4⋊C4—1224no—
477C4⋊C4—1224no—
478C4⋊C4—1224no—
479C4⋊C4—1221no—
480C4⋊C4—1224no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4C2×C4×D6112yes—
486C22⋊C4—2444no—
487C42—24no—
488C42—612no—
489C42—612no—
490C42—614no—
491C42—614no—
492C42—1224no—
493C42—1224no—
494C42—1224no—
495C42—2444no—
496C42C2×C4×D6112yes—
497C42D4×D6112yes—
498C23—612no—
499C23—612no—
500C23—612no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23C2×D4×D6111yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—614no—
510Q8—12216no—
511Q8—1222no—
512Q8—1222no—
513Q8—1222no—
514Q8#1451973514yescomplemented
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—6116no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—12216no—
533C2×C4—1222no—
534C2×C4—1221no—
535C2×C4—1222no—
536C2×C4—1221no—
537C2×C4—1222no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—2444no—
546C2×C4—2448no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C22⋊C4×D6112yes—
553C2×C4C22⋊C4×D6112yes—
554C2×C4SD16×D6112yes—
555C2×C4Q16×D6112yes—
556C2×C4C22×C4×D6111yescharacteristic
557C2×C4C2×D4×D6111yescharacteristic
558C2×C4#41×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4C2×#1175822411yescharacteristic
561C2×C4C2×#1175822611yescharacteristic
562C8—42no—
563C8—2442no—
564C22—614no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22—1222no—
570C22—1222no—
571C22C2×C22⋊C4×D6111yescharacteristic
572C22#1684027211yescharacteristic
573C22#1684027511yescharacteristic
574C22C2×SD16×D6111yescharacteristic
575C22C2×Q16×D6111yescharacteristic
576C22#1684055811yescharacteristic
577C22#1684056111yescharacteristic
578C4—24no—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—614no—
583C4—1222no—
584C4—1224no—
585C4—1222no—
586C4—4882no—
587C4—2442no—
588C4#103×D6114yes—
589C4#117×D6111yescharacteristic
590C4C2×#1451973511yescharacteristic
591C4#1684027211yescharacteristic
592C4#1684027511yescharacteristic
593C2—1224no—
594C2#2144820111yescharacteristic
595C2C2×#103×D6111yescharacteristic
596C2#2145433511yescharacteristic
597C2#2145584911yescharacteristic
598C2#2145585211yescharacteristic
599C2#2145585711yescharacteristic
600C2#2145586011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#20734
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #21445825C2semidirect
6. #21445824C2semidirect
7. #21445821C2semidirect
9. #103×Dic61C2semidirect
10. #21445638C2semidirect
11. C61×#1803C2semidirect
19. C4×Q8×D61C4semidirect
208. Q8×C61#117semidirect
255. C4×D61#103semidirect
260. Dic122#117semidirect
300. #1803D61semidirect
346. C61#20734semidirect
514. Q8#14519735semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
C2×#168398142
#214482242
#214459082
#214458252
#214458242
#214458212
#214457522
#103×Dic612
#214456382
C61×#18032
#2073461