Finite Group Databasebeta

← #30615913

Subgroups — #30615913

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840773C211yesmaximal, characteristic
2C2×#16840755C211yesmaximal, characteristic
3C2×(C2×#6923)⋊Q8C211yesmaximal, characteristic
4#117×#6923C211yesmaximal, characteristic
5#21461899C214yesmaximal, complemented
6#21461898C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic
8#21461481C211yesmaximal, characteristic
9#419×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12C22×C4⋊#6923C2211yescharacteristic
13C22×C4×#6923C2211yescharacteristic
14#16840773—24no—
15C2×#14519846C2212yes—
16C2×#14519845C2212yes—
17#16840755—24no—
18C2×#14519840C2212yes—
19C2×#14519840C2212yes—
20(C2×#6923)⋊Q8—21no—
21(C2×#6923)⋊Q8—21no—
22(C2×#6923)⋊Q8C2214yes—
23C4⋊C4×#6923C2212yes—
24C4⋊C4×#6923C2212yes—
25C22⋊C4×#6923—22no—
26#16840698—21no—
27C2×(C2×#6923)⋊C4—22no—
28C42×#6923—21no—
29#16840635C2214yes—
30(C2×#6923)⋊C8C2214yes—
31C22×Q8×D61C2211yescharacteristic
32C2×M16×D61C2211yescharacteristic
33#117×D61C2211yescharacteristic
34#16839772C411yescharacteristic, complemented
35#16839771C411yescharacteristic, complemented
36#16839770C411yescharacteristic, complemented
37#16839769C414yescomplemented
38#103×D61C2214yes—
39C61×#419C411yescharacteristic, complemented
40C23×#6923—21no—
41C23×#6923D412yes—
42C2×#11758213—21no—
43D61⋊M16—22no—
44D61⋊M16—22no—
45C2×#11758217—21no—
46C2×D61⋊C8C2312yes—
47#14519856—22no—
48#14519856D414yes—
49C2×C4⋊#6923—22no—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923C2312yes—
53C2×C4⋊#6923C2311yescharacteristic
54C2×C4⋊#6923C2311yescharacteristic
55C2×C22⋊#6923—21no—
56C2×C4×#6923—21no—
57C2×C4×#6923—22no—
58C2×C4×#6923—22no—
59C2×C4×#6923—22no—
60C2×C4×#6923—22no—
61C2×C4×#6923D411yescharacteristic
62C2×C4×#6923D411yescharacteristic
63C2×C4×#6923C2312yes—
64#14519846—24no—
65#14519845—24no—
66#14519840—24no—
67#14519840—24no—
68(C2×#6923)⋊C4—24no—
69(C2×#6923)⋊C4—24no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72C22×Dic122—21no—
73C22×Dic122C2×C411yescharacteristic
74C2×Q8×D61—21no—
75C2×Q8×D61—21no—
76C2×Q8×D61—22no—
77C2×Q8×D61—22no—
78C2×Q8×D61—22no—
79C2×Q8×D61—22no—
80C2×Q8×D61C2×C412yes—
81C2×Q8×D61C2312yes—
82C22×C4×D61—21no—
83C22×C4×D61C2311yescharacteristic
84C2×#11758194C2×C411yescharacteristic
85C2×#11758189C2×C412yes—
86M16×D61—22no—
87M16×D61—22no—
88C2×C8×D61C2312yes—
89#14519736C2×C411yescharacteristic
90#14519735C2×C412yes—
91#14519734—21no—
92#14519733—21no—
93C4⋊C4×D61C2312yes—
94C22⋊C4×D61—21no—
95C42×D61—21no—
96#14519702C2×C414yescomplemented
97#14519701C2×C414yescomplemented
98#14519700C2×C414yescomplemented
99C22×Q8×C61C2×C411yescharacteristic
100C2×M16×C61C2×C411yescharacteristic
101#117×C61C2×C411yescharacteristic
102#103×C61C2×C414yescomplemented
103C23×D61C2×D411yescharacteristic
104C22×#6923—24no—
105C22×#6923—21no—
106C22×#6923—21no—
107C22×#6923—24no—
108C22×#6923—22no—
109C22×#6923—42no—
110C22×#6923—44no—
111C22×#6923—42no—
112C22×#6923C2×D412yes—
113C22×#6923C2×D412yes—
114C2×Q8×C61—21no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61C22×C412yes—
118C22×C4×C61—21no—
119C22×C4×C61C22×C411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61#4111yescharacteristic
122Q8×D61—24no—
123Q8×D61—24no—
124Q8×D61—22no—
125Q8×D61—22no—
126Q8×D61—22no—
127Q8×D61—22no—
128Q8×D61—42no—
129Q8×D61—42no—
130Q8×D61—42no—
131Q8×D61—42no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61C22×C411yescharacteristic
138C2×C4×D61C2×D412yes—
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61#4112yes—
141C2×C4×D61#4112yes—
142C2×C4×D61#4112yes—
143C2×C4×D61#4111yescharacteristic
144C2×C4×D61C2411yescharacteristic
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—21no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152C2×Dic122C22×C412yes—
153C22⋊#6923—24no—
154#11758217—42no—
155C2×#92420—22no—
156C4⋊#6923—22no—
157C4⋊#6923—21no—
158C4⋊#6923—21no—
159C4⋊#6923—22no—
160C4⋊#6923—42no—
161C4⋊#6923C2×D412yes—
162C4⋊#6923C2×D412yes—
163C4×#6923—24no—
164C4×#6923—22no—
165C4×#6923—22no—
166C4×#6923—22no—
167C4×#6923—22no—
168C4×#6923—44no—
169C4×#6923—44no—
170#11758213—42no—
171D61⋊C8—22no—
172D61⋊C8—22no—
173M16×C61—22no—
174C2×C8×C61C22×C412yes—
175C4⋊C4×C61C22×C412yes—
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—21no—
179#11758198—22no—
180C4⋊Dic61C22×C412yes—
181#11758196—22no—
182C4×Dic61—21no—
183C4×Dic61—22no—
184#11758194—22no—
185C2×#92418C22×C412yes—
186#11758189—24no—
187C8×D61—22no—
188C8×D61—22no—
189C23×C61C4×D411yescharacteristic
190C22×D61—22no—
191C22×D61—22no—
192C22×D61C4×D411yescharacteristic
193C22×D61#12112yes—
194C22×D61#12312yes—
195C22×D61C22×D411yescharacteristic
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—22no—
201C2×#6923—22no—
202C2×#6923—22no—
203C2×#6923—48no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—44no—
208C2×#6923—84no—
209C2×#6923#13714yes—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—42no—
216C2×C4×C61#11712yes—
217C2×C4×C61C4×D412yes—
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C23×C411yescharacteristic
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—42no—
223C2×Dic61C4×D411yescharacteristic
224C2×Dic61#12412yes—
225C2×Dic61#12612yes—
226C2×Dic61C2×#4111yescharacteristic
227C4×D61—42no—
228C4×D61—42no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—44no—
232C4×D61#11712yes—
233C4×D61C4×D412yes—
234C4×D61#12112yes—
235C4×D61#12312yes—
236C4×D61#12412yes—
237C4×D61#12612yes—
238C4×D61C22×D411yescharacteristic
239C4×D61C2×#4111yescharacteristic
240C4×D61C2×#4111yescharacteristic
241C4×D61C2×#4111yescharacteristic
242Dic122—22no—
243Dic122—22no—
244Dic122—42no—
245Dic122—42no—
246Dic122—42no—
247Dic122—44no—
248#92420—42no—
249C8×C61—22no—
250#92418—22no—
251#21061—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61#38812yes—
254C22×C61C2×C4×D411yescharacteristic
255C2×D61—22no—
256C2×D61—24no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61#38812yes—
260C2×D61C2×#12111yescharacteristic
261C2×D61C2×#12311yescharacteristic
262C2×D61C2×#13712yes—
263C2×D61#57512yes—
264#6923—24no—
265#6923—42no—
266#6923—42no—
267#6923—42no—
268#6923—82no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61#38812yes—
272C4×C61C2×#11711yescharacteristic
273C4×C61C2×C4×D411yescharacteristic
274Dic61—42no—
275Dic61—42no—
276Dic61#38812yes—
277Dic61C2×#12411yescharacteristic
278Dic61C2×#12611yescharacteristic
279C2×#428—611no—
280C2×#419—611no—
281C2×#419—611no—
282C2×#391—611no—
283C4×#117—611no—
284#1928—614no—
285#1927—614no—
286#1736—611no—
287#1728—611no—
288C2×C61—22no—
289C2×C61C2×#38811yescharacteristic
290C2×C61C4×#13712yes—
291D61—22no—
292D61#303911yescharacteristic
293D61#304011yescharacteristic
294D61#309512yes—
295C23×Q8—611no—
296C22×M16—611no—
297C22×M16—611no—
298C2×#117—611no—
299C2×#117—611no—
300C22×C4⋊C4—611no—
301C22×C42—611no—
302#428—1224no—
303C2×#107—612no—
304C2×#106—612no—
305#419—611no—
306#419—611no—
307#419—611no—
308#419—614no—
309#419—1224no—
310#419#692311yescharacteristic, complemented
311C2×#103—614no—
312C2×#103—612no—
313C2×#103—612no—
314#391—614no—
315#391—1221no—
316#391—1221no—
317C4×C4⋊C4—612no—
318C4×C4⋊C4—612no—
319C4×C22⋊C4—1222no—
320#376—1221no—
321C2×#95—1222no—
322C43—1221no—
323#340—614no—
324#336—614no—
325C61#2106111yescharacteristic, complemented, Sylow
326C22×Q8—612no—
327C22×Q8—612no—
328C22×Q8—611no—
329C22×Q8—1221no—
330C22×Q8—1221no—
331C22×Q8—1221no—
332C22×Q8—1222no—
333C22×Q8—1222no—
334C22×Q8—1222no—
335C22×Q8—1222no—
336C22×Q8C2×#692311yescharacteristic
337C23×C4—611no—
338C23×C4—612no—
339C23×C4—1221no—
340C23×C4—1221no—
341C2×M16—611no—
342C2×M16—612no—
343C2×M16—1222no—
344C2×M16—1222no—
345C2×M16—1221no—
346C2×M16—1222no—
347C2×M16—1222no—
348C2×M16—1221no—
349C2×M16C2×#692311yescharacteristic
350C22×C8—612no—
351C22×C8—612no—
352#117—614no—
353#117—611no—
354#117—612no—
355#117—1221no—
356#117—1221no—
357#117—1222no—
358#117C2×#692311yescharacteristic
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1221no—
366C2×C22⋊C4—1221no—
367C2×C22⋊C4—1221no—
368C2×C42—611no—
369C2×C42—611no—
370C2×C42—612no—
371C2×C42—1221no—
372C2×C42—1221no—
373C2×C42—1222no—
374C2×C42—1222no—
375C2×C42—1222no—
376C2×C42—1222no—
377#107—1224no—
378#106—1224no—
379#103—614no—
380#103—614no—
381#103—614no—
382#103—1224no—
383#103—1224no—
384#103C2×#692314yescomplemented
385#95—1224no—
386#95—1224no—
387#95—1222no—
388#95—1222no—
389C24—611no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—612no—
394C2×Q8—1222no—
395C2×Q8—1222no—
396C2×Q8—1222no—
397C2×Q8—1224no—
398C2×Q8—1222no—
399C2×Q8—1222no—
400C2×Q8—1224no—
401C2×Q8—1222no—
402C2×Q8—1221no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—2442no—
406C2×Q8—2442no—
407C2×Q8—2442no—
408C2×Q8—2442no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×Q8C22×#692312yes—
412C22×C4—21no—
413C22×C4—611no—
414C22×C4—612no—
415C22×C4—612no—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1224no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1224no—
431C22×C4—1221no—
432C22×C4—1222no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2444no—
438C22×C4C22×#692311yescharacteristic
439M16—22no—
440M16—1224no—
441M16—1222no—
442M16—2442no—
443M16—2442no—
444C2×C8—612no—
445C2×C8—1222no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—1222no—
450C2×C8C22×#692312yes—
451C4⋊C4—612no—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—1222no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—2442no—
460C4⋊C4C22×#692312yes—
461C22⋊C4—21no—
462C22⋊C4—1222no—
463C22⋊C4—1224no—
464C22⋊C4—1221no—
465C42—21no—
466C42—1222no—
467C42—1222no—
468C42—1222no—
469C42—1222no—
470C42—1222no—
471C42—1224no—
472C42—1221no—
473C42—2444no—
474C42—2444no—
475C23—611no—
476C23—612no—
477C23—612no—
478C23—611no—
479C23—1222no—
480C23—1222no—
481C23D4×#692311yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—1222no—
486Q8—1222no—
487Q8—2442no—
488Q8—2442no—
489Q8—2444no—
490Q8—2442no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—612no—
495C2×C4—612no—
496C2×C4—611no—
497C2×C4—611no—
498C2×C4—612no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—612no—
502C2×C4—612no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—612no—
506C2×C4—611no—
507C2×C4—611no—
508C2×C4—614no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—4884no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2444no—
524C2×C4—2448no—
525C2×C4—2442no—
526C2×C4—2444no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4#1451985612yes—
531C2×C4D4×#692311yescharacteristic
532C2×C4D4×#692312yes—
533C2×C4C23×#692311yescharacteristic
534C8—22no—
535C8—1222no—
536C8—2442no—
537C22—22no—
538C22—611no—
539C22—611no—
540C22—612no—
541C22—612no—
542C22—612no—
543C22—1222no—
544C22—1224no—
545C22—1222no—
546C22—1222no—
547C22#1684071212yes—
548C22C2×D4×#692311yescharacteristic
549C4—42no—
550C4—42no—
551C4—611no—
552C4—611no—
553C4—612no—
554C4—1224no—
555C4—4882no—
556C4—2442no—
557C4—2442no—
558C4—2442no—
559C4—2442no—
560C4—2442no—
561C4#1684071212yes—
562C4C2×#1451985611yescharacteristic
563C4C2×D4×#692311yescharacteristic
564C2—22no—
565C2—612no—
566C2—611no—
567C2—611no—
568C2—1222no—
569C2C2×#1684071211yescharacteristic
570C2#137×#692312yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#419C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#419#6923
hypocenterC61#21061
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
5. #21461899C2semidirect
6. #21461898C2semidirect
34. #16839772C4semidirect
35. #16839771C4semidirect
36. #16839770C4semidirect
37. #16839769C4semidirect
39. C61×#419C4semidirect
96. #14519702C2×C4semidirect
97. #14519701C2×C4semidirect
98. #14519700C2×C4semidirect
102. #103×C61C2×C4semidirect
310. #419#6923semidirect
325. C61#21061semidirect
384. #103C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407732
C2×#168407552
C2×(C2×#6923)⋊Q82
#117×#69232
#214618992
#214618982
#214615082
#214614812
#419×D612
#2106161