Finite Group Databasebeta

← #470×#6923 #30617061

Subgroups — #470×#6923 #30617061

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×#6923C211yesmaximal, characteristic, complemented
2C2×M16×#6923C211yesmaximal, characteristic
3C22⋊Q8×#6923C212yesmaximal
4#21461967C211yesmaximal, characteristic, complemented
5#21461925C211yesmaximal, characteristic, complemented
6#21461910C212yesmaximal, complemented
7#21461834C212yesmaximal
8#21461697C211yesmaximal, characteristic
9#106×#6923C211yesmaximal, characteristic, complemented
10#103×#6923C212yesmaximal, complemented
11#470×D61C211yesmaximal, characteristic
12C2×D61⋊M16C2211yescharacteristic
13C2×Q8×#6923C2212yes—
14C22×C4×#6923C2211yescharacteristic
15Q16×#6923—24no—
16C2×#14519846C2211yescharacteristic
17C2×#14519845C2211yescharacteristic
18C2×#14519840C2212yes—
19C2×#14519840C2212yes—
20M16×#6923—24no—
21C2×#14519831C2211yescharacteristic
22C2×#14519831C2211yescharacteristic
23C2×C8×#6923C2211yescharacteristic
24C2×C8×#6923C2211yescharacteristic
25(C2×#6923)⋊Q8C2212yes—
26#16840713C2212yes—
27#16840711—22no—
28#16840709—22no—
29#16840704C2212yes—
30C4⋊C4×#6923—22no—
31C4⋊C4×#6923C2212yes—
32C22⋊C4×#6923—22no—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36C2×Q16×D61C2211yescharacteristic
37C2×M16×D61C2211yescharacteristic
38C22⋊Q8×D61C2212yes—
39#16840021C411yescharacteristic, complemented
40#16840020C411yescharacteristic, complemented
41#16840019C412yescomplemented
42#16840018C412yescomplemented
43#16840017C411yescharacteristic, complemented
44#106×D61C2211yescharacteristic
45#103×D61C2212yes—
46C61×#470C411yescharacteristic, complemented
47C23×#6923D411yescharacteristic
48C23×#6923D411yescharacteristic
49C2×#11758213—21no—
50D61⋊M16—22no—
51D61⋊M16—22no—
52C2×#11758217—21no—
53C2×D61⋊C8C2311yescharacteristic
54C2×D61⋊C8C2311yescharacteristic
55Q8×#6923—24no—
56Q8×#6923—24no—
57C2×C4⋊#6923—22no—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—22no—
62C2×C4×#6923—22no—
63C2×C4×#6923—22no—
64C2×C4×#6923—22no—
65C2×C4×#6923—22no—
66C2×C4×#6923D411yescharacteristic
67C2×C4×#6923D411yescharacteristic
68C2×C4×#6923C2311yescharacteristic
69C2×C4×#6923C2311yescharacteristic
70#14519846—22no—
71#14519845—22no—
72#14519840—24no—
73#14519840—24no—
74#14519831—22no—
75#14519831—22no—
76C8×#6923—22no—
77C8×#6923D414yes—
78(C2×#6923)⋊C4—22no—
79(C2×#6923)⋊C4—22no—
80(C2×#6923)⋊C4—22no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83C2×Q8×D61C2312yes—
84C22×C4×D61C2311yescharacteristic
85C2×Dic244C2×C411yescharacteristic, complemented
86C2×#11758202C2×C412yescomplemented
87Q16×D61—22no—
88Q16×D61—22no—
89C2×#11758194C2×C411yescharacteristic
90C2×#11758189C2×C411yescharacteristic
91C2×#11758189C2×C411yescharacteristic
92M16×D61—22no—
93M16×D61—22no—
94C2×C8×D61C2311yescharacteristic
95C2×C8×D61C2311yescharacteristic
96#14519757C2×C412yes—
97#14519756C2×C412yes—
98#14519755—22no—
99#14519754—22no—
100#14519753C2×C412yes—
101C4⋊C4×D61—22no—
102C4⋊C4×D61C2312yes—
103C22⋊C4×D61—22no—
104#14519709C2×C411yescharacteristic, complemented
105#14519708C2×C412yescomplemented
106#14519702C2×C412yescomplemented
107#14519701C2×C412yescomplemented
108#14519700C2×C412yescomplemented
109C2×Q16×C61C2×C411yescharacteristic, complemented
110C2×M16×C61C2×C411yescharacteristic
111C22⋊Q8×C61C2×C412yes—
112#106×C61C2×C411yescharacteristic, complemented
113#103×C61C2×C412yescomplemented
114C23×D61C2×D411yescharacteristic
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923—24no—
118C22×#6923—22no—
119C22×#6923—22no—
120C22×#6923—42no—
121C22×#6923C2×D411yescharacteristic
122C22×#6923C2×D411yescharacteristic
123C22×#6923C2×D411yescharacteristic
124C22×#6923C2×D411yescharacteristic
125C2×Q8×C61C22×C412yes—
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61#4111yescharacteristic
128Q8×D61—22no—
129Q8×D61—22no—
130Q8×D61—22no—
131Q8×D61—22no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61C22×C411yescharacteristic
139C2×C4×D61C2×D411yescharacteristic
140C2×C4×D61#4111yescharacteristic
141C2×C4×D61C2411yescharacteristic
142C2×Dic122—22no—
143C2×Dic122C22×C412yes—
144C22⋊#6923—44no—
145#11758217—24no—
146C2×#92420—21no—
147C2×#92420—21no—
148C4⋊#6923—22no—
149C4⋊#6923—22no—
150C4⋊#6923—22no—
151C4⋊#6923—22no—
152C4⋊#6923—44no—
153C4⋊#6923—44no—
154C4×#6923—22no—
155C4×#6923—22no—
156C4×#6923—21no—
157C4×#6923—21no—
158C4×#6923—44no—
159C4×#6923—44no—
160C4×#6923C2×D412yes—
161C4×#6923#4112yes—
162#11758213—24no—
163D61⋊C8—21no—
164D61⋊C8—21no—
165D61⋊C8C2×D412yes—
166D61⋊C8#4112yes—
167Q16×C61—22no—
168M16×C61—22no—
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C22×C411yescharacteristic
171C4⋊C4×C61—22no—
172C4⋊C4×C61C22×C412yes—
173C22⋊C4×C61—22no—
174C22⋊Dic61—22no—
175#11758202—24no—
176#11758198—22no—
177#11758198—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61C22×C412yes—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183#11758194—22no—
184C2×#92418C22×C411yescharacteristic
185C2×#92418C22×C411yescharacteristic
186Dic244—22no—
187#11758189—22no—
188#11758189—22no—
189C8×D61—21no—
190C8×D61—21no—
191C8×D61C2×D412yes—
192C8×D61#4112yes—
193C23×C61C4×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61C4×D411yescharacteristic
199C22×D61C22×D411yescharacteristic
200C2×#6923—22no—
201C2×#6923—21no—
202C2×#6923—21no—
203C2×#6923—22no—
204C2×#6923—22no—
205C2×#6923—44no—
206C2×#6923—44no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923#12112yes—
214C2×#6923#12112yes—
215C2×#6923#13714yes—
216Q8×C61—22no—
217Q8×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61C4×D411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61C4×D411yescharacteristic
229C2×Dic61C2×#4111yescharacteristic
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—21no—
233C4×D61—21no—
234C4×D61—21no—
235C4×D61—21no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61C22×D411yescharacteristic
241C4×D61C2×#4111yescharacteristic
242C4×D61C2×#4111yescharacteristic
243C4×D61C2×#4111yescharacteristic
244Dic122—22no—
245Dic122—22no—
246Dic122—42no—
247Dic122—42no—
248#92420—22no—
249#92420—22no—
250C8×C61—21no—
251C8×C61C4×D412yes—
252#92418—21no—
253#92418C4×D412yes—
254C4×#470—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—21no—
257C22×C61C2×C4×D411yescharacteristic
258C2×D61—22no—
259C2×D61—21no—
260C2×D61—42no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61C2×#12111yescharacteristic
264C2×D61#53211yescharacteristic
265C2×D61C2×#13712yes—
266C2×D61#57512yes—
267#6923—22no—
268#6923—41no—
269#6923—41no—
270#6923—82no—
271#6923—82no—
272#6923#47014yesdirect product factor
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61C2×C4×D411yescharacteristic
277C4×C61C4×#4111yescharacteristic
278Dic61—21no—
279Dic61—42no—
280Dic61—42no—
281Dic61#52511yescharacteristic
282Dic61#52911yescharacteristic
283C2×#470—611no—
284C2×C4×Q16—611no—
285C2×C4×M16—611no—
286C4×C22⋊Q8—612no—
287#1971—611no—
288#1944—611no—
289#1935—612no—
290#1894—612no—
291#1809—611no—
292C4×#106—611no—
293C4×#103—612no—
294C2×C61—41no—
295C2×C61C4×#12111yescharacteristic
296C2×C61C4×#13712yes—
297D61—41no—
298D61C2×#47011yescharacteristic
299D61#305411yescharacteristic
300D61#317212yes—
301C22×Q16—611no—
302C22×M16—611no—
303C22×M16—611no—
304C2×C22⋊Q8—612no—
305C2×C4×Q8—612no—
306C22×C42—611no—
307#470—611no—
308#470—611no—
309#470—612no—
310#470—611no—
311#470—612no—
312#470#692311yescharacteristic, direct product factor
313C4×Q16—1224no—
314C2×#107—611no—
315C2×#106—611no—
316C2×#106—611no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C4×M16—1224no—
321C2×#97—611no—
322C2×#97—611no—
323C2×C4×C8—611no—
324C2×C4×C8—611no—
325#391—612no—
326#389—612no—
327#387—1222no—
328#385—1222no—
329#380—612no—
330C4×C4⋊C4—612no—
331C4×C4⋊C4—1222no—
332C4×C22⋊C4—1222no—
333#340—612no—
334#340—612no—
335#340—612no—
336C61C4×#47011yescharacteristic, complemented, Sylow
337C22×Q8—612no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C2×Q16—612no—
342C2×Q16—611no—
343C2×Q16—1222no—
344C2×Q16—1222no—
345C2×Q16C2×#692311yescharacteristic, complemented
346C2×M16—611no—
347C2×M16—611no—
348C2×M16—611no—
349C2×M16—1222no—
350C2×M16—1222no—
351C2×M16—1222no—
352C2×M16—1222no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16C2×#692311yescharacteristic
356C22×C8—611no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22⋊Q8—612no—
361C22⋊Q8—612no—
362C22⋊Q8—612no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8C2×#692312yes—
366C4×Q8—1224no—
367C4×Q8—1224no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1222no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—1222no—
381C2×C42—1222no—
382C2×C42—1222no—
383C2×C42—1222no—
384#107—1222no—
385#106—611no—
386#106—612no—
387#106—1222no—
388#106C2×#692311yescharacteristic, complemented
389#103—612no—
390#103—612no—
391#103—612no—
392#103—1224no—
393#103—1224no—
394#103C2×#692312yescomplemented
395#97—1222no—
396#97—1222no—
397C4×C8—614no—
398C4×C8—1222no—
399#95—1222no—
400#95—1222no—
401#95—1222no—
402#95—1222no—
403#95—1222no—
404C24—611no—
405C2×Q8—612no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8C22×#692312yes—
412C22×C4—611no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—2442no—
433C22×C4C22×#692311yescharacteristic
434Q16—22no—
435Q16—1224no—
436Q16—1222no—
437M16—22no—
438M16—1222no—
439M16—1222no—
440M16—1224no—
441M16—1224no—
442M16—1222no—
443C2×C8—611no—
444C2×C8—612no—
445C2×C8—612no—
446C2×C8—612no—
447C2×C8—612no—
448C2×C8—611no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8C22×#692311yescharacteristic
456C2×C8C22×#692311yescharacteristic
457C4⋊C4—22no—
458C4⋊C4—612no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—2444no—
468C4⋊C4—2444no—
469C4⋊C4C22×#692312yes—
470C22⋊C4—22no—
471C22⋊C4—1222no—
472C22⋊C4—1222no—
473C22⋊C4—1222no—
474C22⋊C4—2444no—
475C42—612no—
476C42—612no—
477C42—1221no—
478C42—1222no—
479C42—1221no—
480C42—1222no—
481C42—2444no—
482C42—2444no—
483C23—611no—
484C23—611no—
485C23—1222no—
486C23—1221no—
487C23—1221no—
488C23—1222no—
489C23D4×#692311yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—1222no—
493Q8—1222no—
494Q8—2442no—
495Q8—2442no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—612no—
501C2×C4—614no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—612no—
508C2×C4—611no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1222no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2444no—
529C2×C4—2444no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4D4×#692311yescharacteristic
537C2×C4C23×#692311yescharacteristic
538C8—21no—
539C8—612no—
540C8—1222no—
541C8—1222no—
542C8—1221no—
543C8D4×#692312yes—
544C22—21no—
545C22—22no—
546C22—611no—
547C22—611no—
548C22—612no—
549C22—612no—
550C22—1221no—
551C22—1222no—
552C22—2441no—
553C22—2441no—
554C22—2442no—
555C22C2×D4×#692311yescharacteristic
556C4—21no—
557C4—42no—
558C4—42no—
559C4—614no—
560C4—611no—
561C4—611no—
562C4—1222no—
563C4—1221no—
564C4—4882no—
565C4—4882no—
566C4—2441no—
567C4—2441no—
568C4—2442no—
569C4—2442no—
570C4C2×D4×#692311yescharacteristic
571C4#41×#692311yescharacteristic
572C2—41no—
573C2—611no—
574C2—611no—
575C2—612no—
576C2—2441no—
577C2#121×#692311yescharacteristic
578C2#137×#692312yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#470C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#470#6923
hypocenterC61C4×#470
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×#6923C2semidirect
4. #21461967C2semidirect
5. #21461925C2semidirect
6. #21461910C2semidirect
9. #106×#6923C2semidirect
10. #103×#6923C2semidirect
39. #16840021C4semidirect
40. #16840020C4semidirect
41. #16840019C4semidirect
42. #16840018C4semidirect
43. #16840017C4semidirect
46. C61×#470C4semidirect
85. C2×Dic244C2×C4semidirect
86. C2×#11758202C2×C4semidirect
104. #14519709C2×C4semidirect
105. #14519708C2×C4semidirect
106. #14519702C2×C4semidirect
107. #14519701C2×C4semidirect
108. #14519700C2×C4semidirect
109. C2×Q16×C61C2×C4semidirect
112. #106×C61C2×C4semidirect
113. #103×C61C2×C4semidirect
272. #6923#470direct (N × H)
312. #470#6923direct (N × H)
336. C61C4×#470semidirect
345. C2×Q16C2×#6923semidirect
388. #106C2×#6923semidirect
394. #103C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×#69232
C2×M16×#69232
C22⋊Q8×#69232
#214619672
#214619252
#214619102
#214618342
#214616972
#106×#69232
#103×#69232
#470×D612
C4×#47061