Finite Group Databasebeta

← #30615516

Subgroups — #30615516

Order 15616.

1382 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840830C211yesmaximal, characteristic
2C2×#16840773C211yesmaximal, characteristic
3C22×#14519840C211yesmaximal, characteristic
4#21462300C211yesmaximal, characteristic
5#21461785C214yesmaximal, complemented
6#21461783C214yesmaximal, complemented
7#21461504C211yesmaximal, characteristic
8#21461487C211yesmaximal, characteristic
9#419×D61C211yesmaximal, characteristic
10C22×D61⋊C8C2211yescharacteristic
11#16840830—24no—
12#16840830—24no—
13C2×Q8×#6923—21no—
14C2×Q8×#6923—21no—
15C2×Q8×#6923C2212yes—
16C2×Q8×#6923C2212yes—
17C2×#14519856—22no—
18C2×#14519856C2211yescharacteristic
19C2×#14519856C2211yescharacteristic
20C22×C4⋊#6923C2211yescharacteristic
21#16840773—24no—
22C2×#14519846C2212yes—
23C2×#14519845C2212yes—
24C2×#14519840—24no—
25C2×#14519840C2214yes—
26#16840713C2214yes—
27#16840709—22no—
28#16840708—21no—
29#16840706—21no—
30#16840635C2212yes—
31#16840635C2212yes—
32(C2×#6923)⋊C8C2214yes—
33C22×Q8×D61C2211yescharacteristic
34C2×M16×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#16839772C411yescharacteristic, complemented
37#16839771C411yescharacteristic, complemented
38#16839770C411yescharacteristic, complemented
39#16839769C414yescomplemented
40#103×D61C2214yes—
41C61×#419C411yescharacteristic, complemented
42C23×#6923—21no—
43C2×D61⋊C8—22no—
44C2×D61⋊C8—22no—
45C2×D61⋊C8—21no—
46C2×D61⋊C8C2312yes—
47C22×#92420—21no—
48Q8×#6923—28no—
49Q8×#6923—28no—
50Q8×#6923—24no—
51Q8×#6923—24no—
52Q8×#6923—24no—
53Q8×#6923—24no—
54#14519856—216no—
55#14519856—22no—
56#14519856—22no—
57#14519856—22no—
58#14519856D414yes—
59C2×C4⋊#6923—22no—
60C2×C4⋊#6923—22no—
61C2×C4⋊#6923—22no—
62C2×C4⋊#6923—22no—
63C2×C4⋊#6923—21no—
64C2×C4⋊#6923—24no—
65C2×C4⋊#6923D411yescharacteristic
66C2×C4⋊#6923D411yescharacteristic
67C2×C4⋊#6923C2312yes—
68C2×C4⋊#6923C2311yescharacteristic
69C2×C4⋊#6923C2311yescharacteristic
70C2×C22⋊#6923—21no—
71C2×C22⋊#6923D412yes—
72C2×C4×#6923—22no—
73C2×C4×#6923—22no—
74C2×C4×#6923—21no—
75C2×C4×#6923C2312yes—
76#14519846—24no—
77#14519845—24no—
78#14519840—28no—
79(C2×#6923)⋊C4—24no—
80C22×Dic122—21no—
81C22×Dic122C2×C411yescharacteristic
82C2×Q8×D61—21no—
83C2×Q8×D61—22no—
84C2×Q8×D61—22no—
85C2×Q8×D61—21no—
86C2×Q8×D61C2×C412yes—
87C2×Q8×D61D414yes—
88C2×Q8×D61Q814yes—
89C2×Q8×D61C2312yes—
90C22×C4×D61—21no—
91C22×C4×D61C2311yescharacteristic
92C2×#11758194C2×C411yescharacteristic
93C2×#11758189C2×C412yes—
94M16×D61—22no—
95M16×D61—22no—
96C2×C8×D61C2312yes—
97#14519736C2×C411yescharacteristic
98#14519735C2×C412yes—
99#14519734—21no—
100#14519733—21no—
101C4⋊C4×D61C2312yes—
102C22⋊C4×D61—21no—
103C42×D61—21no—
104#14519702C2×C414yescomplemented
105#14519701C2×C414yescomplemented
106#14519700C2×C414yescomplemented
107C22×Q8×C61C2×C411yescharacteristic
108C2×M16×C61C2×C411yescharacteristic
109#117×C61C2×C411yescharacteristic
110#103×C61C2×C414yescomplemented
111C23×D61C2×D411yescharacteristic
112C22×#6923—21no—
113C22×#6923—21no—
114C22×#6923—22no—
115C22×#6923—42no—
116C22×#6923—82no—
117C22×#6923C2×D412yes—
118C22×#6923C2×D412yes—
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61C4⋊C414yescomplemented
122C2×Q8×C61C22×C412yes—
123C22×C4×C61—21no—
124C22×C4×C61C22×C411yescharacteristic
125C22×Dic61—21no—
126C22×Dic61#4111yescharacteristic
127Q8×D61—24no—
128Q8×D61—24no—
129Q8×D61—24no—
130Q8×D61—24no—
131Q8×D61—24no—
132Q8×D61—22no—
133Q8×D61—22no—
134Q8×D61C4⋊C418yescomplemented
135Q8×D61C2×D414yes—
136Q8×D61C2×Q814yes—
137C2×C4×D61—22no—
138C2×C4×D61—24no—
139C2×C4×D61—24no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61C22×C411yescharacteristic
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×Q812yes—
147C2×C4×D61#4112yes—
148C2×C4×D61#4111yescharacteristic
149C2×C4×D61C2411yescharacteristic
150C2×Dic122—22no—
151C2×Dic122—21no—
152C2×Dic122—22no—
153C2×Dic122—24no—
154C2×Dic122—24no—
155C2×Dic122—22no—
156C2×Dic122C4⋊C414yescomplemented
157C2×Dic122C22×C412yes—
158C22⋊#6923—24no—
159C22⋊#6923—24no—
160C2×#92420—22no—
161C2×#92420—42no—
162C4⋊#6923—28no—
163C4⋊#6923—28no—
164C4⋊#6923—22no—
165C4⋊#6923—22no—
166C4⋊#6923—21no—
167C4⋊#6923—21no—
168C4⋊#6923—22no—
169C4⋊#6923—22no—
170C4⋊#6923—42no—
171C4⋊#6923—84no—
172C4⋊#6923C2×D412yes—
173C4⋊#6923C2×D412yes—
174C4×#6923—28no—
175C4×#6923—28no—
176C4×#6923—24no—
177C4×#6923—22no—
178C4×#6923—22no—
179D61⋊C8—22no—
180D61⋊C8—22no—
181D61⋊C8—42no—
182M16×C61—22no—
183C2×C8×C61C22×C412yes—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C22⋊Dic61—21no—
188#11758198—22no—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193#11758194—22no—
194C2×#92418C22×C412yes—
195#11758189—24no—
196C8×D61—22no—
197C8×D61—22no—
198C23×C61C4×D411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61C4×D411yescharacteristic
202C22×D61#12012yes—
203C22×D61C22⋊Q812yes—
204C22×D61C22×D411yescharacteristic
205C2×#6923—22no—
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—24no—
210C2×#6923—24no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×#6923—44no—
215C2×#6923—84no—
216Q8×C61—22no—
217Q8×C61—24no—
218Q8×C61C2×C4⋊C414yes—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61C2×C4⋊C412yes—
223C2×C4×C61C4×D412yes—
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C23×C411yescharacteristic
226C2×Dic61—22no—
227C2×Dic61—24no—
228C2×Dic61—22no—
229C2×Dic61C4×D411yescharacteristic
230C2×Dic61C22⋊Q812yes—
231C2×Dic61#12312yes—
232C2×Dic61C2×#4111yescharacteristic
233C4×D61—24no—
234C4×D61—24no—
235C4×D61—28no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61C2×C4⋊C412yes—
239C4×D61C4×D412yes—
240C4×D61#12012yes—
241C4×D61C22⋊Q812yes—
242C4×D61C22⋊Q812yes—
243C4×D61#12312yes—
244C4×D61C22×D411yescharacteristic
245C4×D61C22×D411yescharacteristic
246C4×D61C22×Q811yescharacteristic
247C4×D61C2×#4111yescharacteristic
248Dic122—22no—
249Dic122—24no—
250Dic122—24no—
251Dic122—24no—
252Dic122—28no—
253Dic122C2×C4⋊C414yes—
254#92420—42no—
255C8×C61—22no—
256#92418—22no—
257#20877—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61#38512yes—
260C22×C61C2×C4×D411yescharacteristic
261C2×D61—22no—
262C2×D61—24no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61#38512yes—
266C2×D61C2×#12011yescharacteristic
267C2×D61C2×C22⋊Q811yescharacteristic
268C2×D61#57712yes—
269C2×D61#57812yes—
270#6923—42no—
271#6923—42no—
272#6923—42no—
273#6923—42no—
274#6923—82no—
275C4×C61—24no—
276C4×C61—42no—
277C4×C61#38512yes—
278C4×C61C22×C4⋊C411yescharacteristic
279C4×C61C2×C4×D411yescharacteristic
280Dic61—24no—
281Dic61—42no—
282Dic61#38512yes—
283Dic61C2×C22⋊Q811yescharacteristic
284Dic61C2×#12311yescharacteristic
285C2×#519—611no—
286C2×#428—611no—
287C2×#419—611no—
288C22×#103—611no—
289#2317—611no—
290#1865—614no—
291#1863—614no—
292#1735—611no—
293#1729—611no—
294C2×C61—22no—
295C2×C61C2×#38511yescharacteristic
296C2×C61#294712yes—
297D61—22no—
298D61#301011yescharacteristic
299D61#301111yescharacteristic
300D61#308212yes—
301C23×Q8—611no—
302C22×M16—611no—
303C23×C8—611no—
304#519—1224no—
305#519—1224no—
306C2×C4×Q8—612no—
307C2×C4×Q8—612no—
308C2×C4×Q8—1221no—
309C2×C4×Q8—1221no—
310C2×#117—611no—
311C2×#117—611no—
312C2×#117—611no—
313C2×#117—1222no—
314C22×C4⋊C4—611no—
315#428—1224no—
316C2×#107—612no—
317C2×#106—612no—
318#419—611no—
319#419—611no—
320#419—611no—
321#419—614no—
322#419#692311yescharacteristic, complemented
323C2×#103—614no—
324C2×#103—614no—
325C2×#103—1224no—
326#389—614no—
327#385—1222no—
328#384—1221no—
329#382—1221no—
330#340—612no—
331#340—612no—
332#336—614no—
333C61#2087711yescharacteristic, complemented, Sylow
334C22×Q8—614no—
335C22×Q8—614no—
336C22×Q8—612no—
337C22×Q8—612no—
338C22×Q8—611no—
339C22×Q8—1221no—
340C22×Q8—1221no—
341C22×Q8—1222no—
342C22×Q8—1221no—
343C22×Q8—1222no—
344C22×Q8C2×#692311yescharacteristic
345C23×C4—611no—
346C23×C4—1221no—
347C23×C4—1221no—
348C2×M16—611no—
349C2×M16—612no—
350C2×M16—1222no—
351C2×M16—1222no—
352C2×M16C2×#692311yescharacteristic
353C22×C8—612no—
354C22×C8—612no—
355C22×C8—1221no—
356C22×C8—1222no—
357C22×C8—1221no—
358C22×C8—1222no—
359C4×Q8—1224no—
360C4×Q8—1224no—
361C4×Q8—1224no—
362C4×Q8—1224no—
363C4×Q8—1228no—
364C4×Q8—1228no—
365#117—614no—
366#117—611no—
367#117—612no—
368#117—1221no—
369#117—1221no—
370#117—1222no—
371#117—1222no—
372#117—1222no—
373#117—12216no—
374#117C2×#692311yescharacteristic
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1221no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—612no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1222no—
394C2×C42—1222no—
395#107—1224no—
396#106—1224no—
397#103—614no—
398#103—614no—
399#103—614no—
400#103—1228no—
401#103C2×#692314yescomplemented
402#95—1224no—
403C24—611no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—614no—
407C2×Q8—618no—
408C2×Q8—614no—
409C2×Q8—614no—
410C2×Q8—612no—
411C2×Q8—1224no—
412C2×Q8—1224no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1224no—
416C2×Q8—1222no—
417C2×Q8—1224no—
418C2×Q8—1224no—
419C2×Q8—1224no—
420C2×Q8—1224no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1221no—
424C2×Q8C4⋊#692314yescomplemented
425C2×Q8C22×#692312yes—
426C22×C4—21no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—612no—
438C22×C4—1222no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1222no—
447C22×C4—4882no—
448C22×C4—2442no—
449C22×C4C22×#692311yescharacteristic
450M16—22no—
451M16—1224no—
452M16—1222no—
453C2×C8—612no—
454C2×C8—1222no—
455C2×C8—1222no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—2442no—
460C2×C8—2442no—
461C2×C8C22×#692312yes—
462C4⋊C4—612no—
463C4⋊C4—612no—
464C4⋊C4—612no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1222no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1228no—
473C4⋊C4—1228no—
474C4⋊C4—4884no—
475C4⋊C4—2442no—
476C4⋊C4C22×#692312yes—
477C22⋊C4—21no—
478C22⋊C4—1222no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—1221no—
482C42—21no—
483C42—1222no—
484C42—1222no—
485C42—1222no—
486C42—1224no—
487C42—1228no—
488C42—1228no—
489C42—1221no—
490C23—611no—
491C23—612no—
492C23—612no—
493C23—611no—
494C23—1222no—
495C23—1222no—
496C23D4×#692311yescharacteristic
497Q8—24no—
498Q8—22no—
499Q8—614no—
500Q8—1224no—
501Q8—1224no—
502Q8—1228no—
503Q8—1224no—
504Q8—1222no—
505Q8C2×C4⋊#692314yes—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—611no—
512C2×C4—611no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1228no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1222no—
535C2×C4—4884no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2444no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4C2×C4⋊#692312yes—
543C2×C4D4×#692311yescharacteristic
544C2×C4D4×#692312yes—
545C2×C4C23×#692311yescharacteristic
546C8—22no—
547C8—1222no—
548C8—2442no—
549C22—22no—
550C22—612no—
551C22—612no—
552C22—611no—
553C22—611no—
554C22—612no—
555C22—1222no—
556C22—1222no—
557C22—1224no—
558C22—1222no—
559C22#1684070912yes—
560C22C2×D4×#692311yescharacteristic
561C4—24no—
562C4—42no—
563C4—611no—
564C4—611no—
565C4—612no—
566C4—1224no—
567C4—4882no—
568C4—2442no—
569C4—2442no—
570C4—2442no—
571C4—2442no—
572C4—2442no—
573C4#1684070912yes—
574C4C22×C4⋊#692311yescharacteristic
575C4C2×D4×#692311yescharacteristic
576C2—22no—
577C2—611no—
578C2—611no—
579C2—612no—
580C2—1222no—
581C2C2×#1684070911yescharacteristic
582C2#2146257712yes—
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#20877
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. #21461785C2semidirect
6. #21461783C2semidirect
36. #16839772C4semidirect
37. #16839771C4semidirect
38. #16839770C4semidirect
39. #16839769C4semidirect
41. C61×#419C4semidirect
104. #14519702C2×C4semidirect
105. #14519701C2×C4semidirect
106. #14519700C2×C4semidirect
110. #103×C61C2×C4semidirect
121. C2×Q8×C61C4⋊C4semidirect
134. Q8×D61C4⋊C4semidirect
156. C2×Dic122C4⋊C4semidirect
322. #419#6923semidirect
333. C61#20877semidirect
401. #103C2×#6923semidirect
424. C2×Q8C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408302
C2×#168407732
C22×#145198402
#214623002
#214617852
#214617832
#214615042
#214614872
#419×D612
#2087761