Finite Group Databasebeta

← #30622080

Subgroups — #30622080

Order 15616.

886 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
1#21462352C211yesmaximal, characteristic, complemented
2D4×C4⋊#6923C211yesmaximal, characteristic
3#21462000C211yesmaximal, characteristic, complemented
4#21461930C211yesmaximal, characteristic, complemented
5#21461925C212yesmaximal, complemented
6#21461882C212yesmaximal, complemented
7#21461782C212yesmaximal
8#21461616C211yesmaximal, characteristic, complemented
9#21461590C212yesmaximal
10C4×#14519839C211yesmaximal, characteristic
11#476×D61C211yesmaximal, characteristic
12C2×D4×#6923C2212yes—
13C22×C4⋊#6923—21no—
14C2×#14519846C2212yes—
15C2×#14519839C2211yescharacteristic
16C2×#14519839C2211yescharacteristic
17C2×#14519831C2212yes—
18C4×D61⋊C8C2211yescharacteristic
19(C2×#6923)⋊Q8—22no—
20#16840713—21no—
21#16840713C2212yes—
22#16840713C2212yes—
23#16840709—22no—
24#16840708C2211yescharacteristic
25#16840704—21no—
26C4⋊C4×#6923—21no—
27C4⋊C4×#6923C2212yes—
28C4×C4⋊#6923C2211yescharacteristic
29C4×C4⋊#6923C2211yescharacteristic
30#16840636C2212yes—
31#16840635C2212yes—
32#16840635C2211yescharacteristic
33#16840635C2211yescharacteristic
34#16840635C2212yes—
35(C2×#6923)⋊C8C2212yes—
36C4⋊Q8×D61C2211yescharacteristic
37C4×D4×D61C2211yescharacteristic
38#16840053C411yescharacteristic, complemented
39#16840052C411yescharacteristic, complemented
40#16840051C412yescomplemented
41#16840050C412yescomplemented
42#16840049C411yescharacteristic, complemented
43#106×D61C2212yes—
44C4⋊C8×D61C2211yescharacteristic
45#102×D61C2212yes—
46C61×#476C411yescharacteristic, complemented
47C23×#6923—22no—
48C2×D61⋊C8C2311yescharacteristic
49C2×D61⋊C8C2311yescharacteristic
50D4×#6923—24no—
51D4×#6923—42no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—21no—
54C2×C4⋊#6923—22no—
55C2×C4⋊#6923—21no—
56C2×C4⋊#6923—44no—
57C2×C4⋊#6923D411yescharacteristic
58C2×C4⋊#6923D411yescharacteristic
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2311yescharacteristic
61C2×C4⋊#6923C2311yescharacteristic
62C2×C22⋊#6923—22no—
63C2×C4×#6923—21no—
64C2×C4×#6923—22no—
65C2×C4×#6923C2312yes—
66#14519846—42no—
67#14519839—22no—
68#14519839Q814yescomplemented
69#14519831—42no—
70C4×#92420—21no—
71C4×#92420—21no—
72(C2×#6923)⋊C4—22no—
73(C2×#6923)⋊C4—22no—
74(C2×#6923)⋊C4—22no—
75(C2×#6923)⋊C4—21no—
76(C2×#6923)⋊C4D412yes—
77C2×Q8×D61—21no—
78C2×D4×D61C2311yescharacteristic
79C22×C4×D61—21no—
80C2×C8×D61C2312yes—
81#14519777C2×C411yescharacteristic, complemented
82#14519776—21no—
83#14519776C2×C412yescomplemented
84#14519775—21no—
85C4×D244C2×C411yescharacteristic
86D4×Dic61C2×C411yescharacteristic
87#14519739C2×C411yescharacteristic
88C4×C22⋊D61—21no—
89#14519737—21no—
90C4⋊C4×D61—21no—
91C4⋊C4×D61C2311yescharacteristic
92C4⋊C4×D61C2312yes—
93C22⋊C4×D61—21no—
94C42×D61C2311yescharacteristic
95#14519709C2×C412yescomplemented
96#14519708C2×C412yescomplemented
97#14519708C2×C412yescomplemented
98C4⋊#92418C2×C411yescharacteristic, complemented
99#14519706C2×C412yescomplemented
100#14519699C2×C412yes—
101#14519698C2×C412yes—
102#14519697C2×C412yes—
103C4⋊Q8×C61C2×C411yescharacteristic, complemented
104C4×D4×C61C2×C411yescharacteristic
105#106×C61C2×C412yescomplemented
106C4⋊C8×C61C2×C411yescharacteristic, complemented
107#102×C61C2×C412yes—
108C23×D61—21no—
109C22×#6923—21no—
110C22×#6923—21no—
111C22×#6923—22no—
112C22×#6923—44no—
113C22×#6923—44no—
114C22×#6923—82no—
115C22×#6923C2×D412yes—
116C22×#6923C2×D412yes—
117C2×Q8×C61—21no—
118C2×D4×C61C22×C411yescharacteristic
119C22×C4×C61—21no—
120C2×C22⋊D61—21no—
121C22×Dic61—21no—
122Q8×D61—81no—
123Q8×D61—81no—
124D4×D61—21no—
125D4×D61—21no—
126D4×D61C2×Q812yes—
127D4×D61#4112yes—
128C2×D244C22×C411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—22no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—42no—
134C2×C4×D61—42no—
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2411yescharacteristic
138C2×Dic122—21no—
139C2×Dic122—22no—
140C22⋊#6923—44no—
141C2×#92420—21no—
142C2×#92420—21no—
143C4⋊#6923—22no—
144C4⋊#6923—22no—
145C4⋊#6923—21no—
146C4⋊#6923—22no—
147C4⋊#6923—21no—
148C4⋊#6923—22no—
149C4⋊#6923—21no—
150C4⋊#6923—44no—
151C4⋊#6923—42no—
152C4⋊#6923—81no—
153C4⋊#6923—82no—
154C4⋊#6923C2×Q812yes—
155C4×#6923—22no—
156C4×#6923—22no—
157C4×#6923—42no—
158C4×#6923—82no—
159D61⋊C8—21no—
160D61⋊C8—21no—
161D61⋊C8—21no—
162D61⋊C8C2×Q812yes—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61C22×C411yescharacteristic
167C22⋊C4×C61—21no—
168C42×C61C22×C411yescharacteristic
169C22⋊Dic61—21no—
170#11758198—21no—
171#11758198—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C22×C412yes—
174C4⋊Dic61C22×C411yescharacteristic
175#11758196—22no—
176#11758196—21no—
177#11758196—22no—
178C4×Dic61C22×C411yescharacteristic
179C4×Dic61#4111yescharacteristic
180C4×Dic61#4111yescharacteristic
181C2×#92418C22×C412yes—
182C8×D61—22no—
183C8×D61—22no—
184C23×C61—21no—
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61C22×D411yescharacteristic
193C2×#6923—22no—
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—22no—
200C2×#6923—41no—
201C2×#6923—41no—
202C2×#6923—42no—
203C2×#6923—41no—
204C2×#6923—41no—
205C2×#6923—42no—
206C2×#6923—41no—
207C2×#6923—42no—
208C2×#6923—88no—
209Q8×C61—81no—
210D4×C61—21no—
211D4×C61C4×Q812yes—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—42no—
215C2×C4×C61C4×D411yescharacteristic
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C23×C411yescharacteristic
218C22⋊D61—22no—
219C22⋊D61—22no—
220C2×Dic61—22no—
221C2×Dic61—21no—
222C2×Dic61—42no—
223C2×Dic61C4×D411yescharacteristic
224C2×Dic61C4×D411yescharacteristic
225C2×Dic61C2×#4111yescharacteristic
226D244—21no—
227D244C4×Q812yes—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—41no—
231C4×D61—41no—
232C4×D61—42no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61AGL(1,ℤ/8)12yes—
237C4×D61#13712yes—
238C4×D61C22×Q811yescharacteristic
239C4×D61C2×#4111yescharacteristic
240C4×D61H(5,𝔽2)11yescharacteristic
241C4×D612−1+411yescharacteristic
242Dic122—82no—
243Dic122—81no—
244#92420—22no—
245#92420—41no—
246C8×C61—22no—
247#92418—22no—
248#22969—611nomaximal, Sylow
249C22×C61—22no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61C2×C4×D411yescharacteristic
253C2×D61—22no—
254C2×D61—41no—
255C2×D61—41no—
256C2×D61—42no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—42no—
260C2×D61#54611yescharacteristic
261C2×D61D4×Q811yescharacteristic
262C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
263C2×D61C2×#13711yescharacteristic
264C2×D61#57611yescharacteristic
265C2×D61#57711yescharacteristic
266#6923—42no—
267#6923—42no—
268#6923—42no—
269#6923—81no—
270#6923—82no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61#44012yes—
275C4×C61C2×C4×Q811yescharacteristic
276C4×C61#52011yescharacteristic
277Dic61—21no—
278Dic61—41no—
279Dic61—42no—
280Dic61#44012yes—
281Dic61#55211yescharacteristic
282Dic61#55411yescharacteristic
283C2×#476—611no—
284#2369—611no—
285D4×C4⋊C4—611no—
286#1990—611no—
287#1947—611no—
288#1944—612no—
289#1920—612no—
290#1862—612no—
291#1770—611no—
292#1762—612no—
293C4×#102—611no—
294C2×C61—42no—
295C2×C61#236011yescharacteristic
296C2×C61C2×#44011yescharacteristic
297C2×C61#294611yescharacteristic
298D61—42no—
299D61#332511yescharacteristic
300D61#333111yescharacteristic
301D61#338411yescharacteristic
302D61#338511yescharacteristic
303C2×C4⋊Q8—611no—
304C2×C4×D4—611no—
305C2×C4×D4—612no—
306C22×C4⋊C4—1221no—
307#476—611no—
308#476—611no—
309#476—612no—
310#476—611no—
311#476—612no—
312#476#692311yescharacteristic, complemented
313C2×#107—612no—
314C2×#106—612no—
315C2×C4⋊C8—611no—
316C2×#102—611no—
317C2×#102—612no—
318C2×#102—611no—
319C2×#97—612no—
320C2×C4×C8—611no—
321#391—1222no—
322#389—612no—
323#389—612no—
324#389—1221no—
325#385—1222no—
326#384—611no—
327#380—1221no—
328C4×C4⋊C4—611no—
329C4×C4⋊C4—612no—
330C4×C4⋊C4—611no—
331C4×C4⋊C4—1221no—
332#340—612no—
333#340—611no—
334#340—612no—
335#340—611no—
336#340—612no—
337#336—612no—
338C61#2296911yescharacteristic, complemented, Sylow
339C22×Q8—1221no—
340C22×D4—611no—
341C23×C4—1221no—
342C23×C4—1222no—
343C22×C8—611no—
344C22×C8—612no—
345C22×C8—611no—
346C4⋊Q8—612no—
347C4⋊Q8—611no—
348C4⋊Q8—1221no—
349C4⋊Q8—1221no—
350C4⋊Q8C2×#692311yescharacteristic, complemented
351C4×D4—611no—
352C4×D4—611no—
353C4×D4—611no—
354C4×D4—1221no—
355C4×D4—1221no—
356C4×D4—1224no—
357C4×D4—2442no—
358C4×D4C2×#692311yescharacteristic
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—2444no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1222no—
374C2×C42—611no—
375C2×C42—612no—
376C2×C42—1222no—
377C2×C42—1221no—
378#107—2442no—
379#106—612no—
380#106—612no—
381#106—612no—
382#106C2×#692312yescomplemented
383C4⋊C8—612no—
384C4⋊C8—611no—
385C4⋊C8C2×#692311yescharacteristic, complemented
386#102—614no—
387#102—612no—
388#102—612no—
389#102—612no—
390#102—1222no—
391#102C2×#692312yes—
392#97—2442no—
393C4×C8—1221no—
394C4×C8—1221no—
395#95—612no—
396#95—1222no—
397#95—1221no—
398#95—1222no—
399#95—1222no—
400C24—1221no—
401C2×Q8—21no—
402C2×Q8—1222no—
403C2×Q8—1221no—
404C2×Q8—4881no—
405C2×Q8—4881no—
406C2×D4—612no—
407C2×D4—612no—
408C2×D4—611no—
409C2×D4—1221no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4C22×#692311yescharacteristic
413C22×C4—21no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—4882no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4—2444no—
431C22×C4—2444no—
432C2×C8—612no—
433C2×C8—612no—
434C2×C8—1222no—
435C2×C8—1222no—
436C2×C8—1221no—
437C2×C8—1221no—
438C2×C8—1221no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8C22×#692312yes—
442C4⋊C4—21no—
443C4⋊C4—612no—
444C4⋊C4—612no—
445C4⋊C4—611no—
446C4⋊C4—1221no—
447C4⋊C4—1222no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1221no—
457C4⋊C4—4881no—
458C4⋊C4—4882no—
459C4⋊C4—2442no—
460C4⋊C4—2444no—
461C4⋊C4C22×#692312yes—
462C4⋊C4C22×#692311yescharacteristic
463C22⋊C4—21no—
464C22⋊C4—1221no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—2444no—
468C42—611no—
469C42—611no—
470C42—611no—
471C42—1222no—
472C42—1222no—
473C42—4882no—
474C42—2442no—
475C42C22×#692311yescharacteristic
476C23—21no—
477C23—611no—
478C23—1222no—
479C23—1222no—
480C23—1221no—
481C23—2441no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485Q8—81no—
486Q8—4881no—
487Q8—4882no—
488D4—21no—
489D4—612no—
490D4—1222no—
491D4—1222no—
492D4—1221no—
493D4Q8×#692312yes—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—612no—
500C2×C4—612no—
501C2×C4—611no—
502C2×C4—611no—
503C2×C4—611no—
504C2×C4—611no—
505C2×C4—611no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—4888no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2442no—
533C2×C4D4×#692311yescharacteristic
534C2×C4D4×#692311yescharacteristic
535C2×C4C23×#692311yescharacteristic
536C8—22no—
537C8—1222no—
538C8—1222no—
539C8—2441no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—611no—
544C22—611no—
545C22—611no—
546C22—611no—
547C22—611no—
548C22—611no—
549C22—1222no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22—2442no—
554C22—2441no—
555C22—2441no—
556C22C2×D4×#692311yescharacteristic
557C4—21no—
558C4—41no—
559C4—42no—
560C4—612no—
561C4—611no—
562C4—611no—
563C4—1221no—
564C4—4882no—
565C4—4881no—
566C4—2442no—
567C4—2442no—
568C4—2442no—
569C4—2441no—
570C4—2442no—
571C4#1684080112yes—
572C4C2×Q8×#692311yescharacteristic
573C4#1684083111yescharacteristic
574C2—42no—
575C2—611no—
576C2—611no—
577C2—611no—
578C2—611no—
579C2—2442no—
580C2#2146234311yescharacteristic
581C2C2×#1684080111yescharacteristic
582C2#2146257611yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#476C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#476#6923
hypocenterC61#22969
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #21462352C2semidirect
3. #21462000C2semidirect
4. #21461930C2semidirect
5. #21461925C2semidirect
6. #21461882C2semidirect
8. #21461616C2semidirect
38. #16840053C4semidirect
39. #16840052C4semidirect
40. #16840051C4semidirect
41. #16840050C4semidirect
42. #16840049C4semidirect
46. C61×#476C4semidirect
68. #14519839Q8semidirect
81. #14519777C2×C4semidirect
83. #14519776C2×C4semidirect
95. #14519709C2×C4semidirect
96. #14519708C2×C4semidirect
97. #14519708C2×C4semidirect
98. C4⋊#92418C2×C4semidirect
99. #14519706C2×C4semidirect
103. C4⋊Q8×C61C2×C4semidirect
105. #106×C61C2×C4semidirect
106. C4⋊C8×C61C2×C4semidirect
312. #476#6923semidirect
338. C61#22969semidirect
350. C4⋊Q8C2×#6923semidirect
382. #106C2×#6923semidirect
385. C4⋊C8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623522
D4×C4⋊#69232
#214620002
#214619302
#214619252
#214618822
#214617822
#214616162
#214615902
C4×#145198392
#476×D612
#2296961