Finite Group Databasebeta

← #484×#6923 #30617190

Subgroups — #484×#6923 #30617190

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×#6923C211yesmaximal, characteristic
2C2×C4⋊C4×#6923C211yesmaximal, characteristic
3#21461899C212yesmaximal, complemented
4#21461883C212yesmaximal, complemented
5#21461804C211yesmaximal, characteristic
6#21461768C211yesmaximal, characteristic
7#21461723C211yesmaximal, characteristic
8#107×#6923C212yesmaximal, complemented
9#103×#6923C212yesmaximal, complemented
10C22⋊C8×#6923C211yesmaximal, characteristic
11#484×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2211yescharacteristic
13C22×C4⋊#6923C2211yescharacteristic
14C22×C4×#6923—21no—
15C22×C4×#6923C2211yescharacteristic
16C2×#14519846C2212yes—
17C2×#14519840C2212yes—
18D61⋊(C22⋊C8)C2211yescharacteristic
19C2×C8×#6923C2212yes—
20(C2×#6923)⋊Q8C2211yescharacteristic
21#16840713C2211yescharacteristic
22#16840711—21no—
23#16840709—21no—
24#16840704C2211yescharacteristic
25C4⋊C4×#6923—21no—
26C4⋊C4×#6923—21no—
27C4⋊C4×#6923—21no—
28C4⋊C4×#6923C2211yescharacteristic
29C4⋊C4×#6923C2212yes—
30C4⋊C4×#6923C2212yes—
31C22⋊C4×#6923—21no—
32C2×(C2×#6923)⋊C4—21no—
33#16840636C2212yes—
34#16840635C2212yes—
35#16840635C2212yes—
36#16840635C2212yes—
37(C2×#6923)⋊C8C2212yes—
38C22⋊Q8×D61C2211yescharacteristic
39C2×C4⋊C4×D61C2211yescharacteristic
40#16840099C412yescomplemented
41#16840098C411yescharacteristic, complemented
42#16840097C411yescharacteristic, complemented
43#16840096C412yescomplemented
44#16840095C411yescharacteristic, complemented
45#107×D61C2212yes—
46#103×D61C2212yes—
47C22⋊C8×D61C2211yescharacteristic
48C61×#484C411yescharacteristic, complemented
49C23×#6923—21no—
50C23×#6923D411yescharacteristic
51C23×#6923D411yescharacteristic
52C2×D61⋊C8C2312yes—
53Q8×#6923—24no—
54C2×C4⋊#6923—21no—
55C2×C4⋊#6923—24no—
56C2×C4⋊#6923—21no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923C2311yescharacteristic
59C2×C4⋊#6923C2311yescharacteristic
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—21no—
62C2×C4×#6923—24no—
63C2×C4×#6923—22no—
64C2×C4×#6923—22no—
65C2×C4×#6923—21no—
66C2×C4×#6923—21no—
67C2×C4×#6923—44no—
68C2×C4×#6923D411yescharacteristic
69C2×C4×#6923D411yescharacteristic
70C2×C4×#6923C2311yescharacteristic
71C2×C4×#6923C2311yescharacteristic
72#14519846—24no—
73#14519840—24no—
74#14519834—21no—
75C22⋊#92420—21no—
76C8×#6923—24no—
77(C2×#6923)⋊C4—21no—
78(C2×#6923)⋊C4—21no—
79(C2×#6923)⋊C4—21no—
80(C2×#6923)⋊C4—21no—
81(C2×#6923)⋊C4—22no—
82(C2×#6923)⋊C4—22no—
83(C2×#6923)⋊C4—21no—
84C2×Q8×D61C2311yescharacteristic
85C22×C4×D61—21no—
86C22×C4×D61C2311yescharacteristic
87C2×C8×D61C2312yes—
88#14519757C2×C411yescharacteristic
89#14519756C2×C411yescharacteristic
90#14519755—21no—
91#14519754—21no—
92#14519753C2×C411yescharacteristic
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×#11758196—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61C2×C412yes—
99C4⋊C4×D61C2311yescharacteristic
100C4⋊C4×D61C2312yes—
101C22⋊C4×D61—21no—
102#14519711C2×C412yescomplemented
103#14519710C2×C412yescomplemented
104#14519710C2×C412yescomplemented
105#14519702C2×C412yescomplemented
106#14519701C2×C412yescomplemented
107#14519700C2×C412yescomplemented
108C22⋊#92418C2×C411yescharacteristic
109#14519689C2×C412yes—
110C22⋊Q8×C61C2×C411yescharacteristic
111C2×C4⋊C4×C61C2×C411yescharacteristic
112#107×C61C2×C412yescomplemented
113#103×C61C2×C412yescomplemented
114C22⋊C8×C61C2×C411yescharacteristic
115C23×D61C2×D411yescharacteristic
116C22×#6923—22no—
117C22×#6923—22no—
118C22×#6923—24no—
119C22×#6923—22no—
120C22×#6923—22no—
121C22×#6923—22no—
122C22×#6923—21no—
123C22×#6923—21no—
124C22×#6923—41no—
125C22×#6923—42no—
126C22×#6923—42no—
127C22×#6923Q1614yes—
128C22×#6923C2×D411yescharacteristic
129C22×#6923C2×D411yescharacteristic
130C22×#6923C2×D411yescharacteristic
131C22×#6923C2×D411yescharacteristic
132C2×Q8×C61C22×C411yescharacteristic
133C22×C4×C61—21no—
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61#4111yescharacteristic
137Q8×D61—22no—
138Q8×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61#4111yescharacteristic
150C2×C4×D61C2411yescharacteristic
151C2×Dic122—21no—
152C2×Dic122C22×C411yescharacteristic
153C22⋊#6923—42no—
154C2×#92420—22no—
155C4⋊#6923—24no—
156C4⋊#6923—22no—
157C4⋊#6923—22no—
158C4⋊#6923—22no—
159C4⋊#6923—42no—
160C4⋊#6923—42no—
161C4⋊#6923#4114yes—
162C4×#6923—22no—
163C4×#6923—22no—
164C4×#6923—22no—
165C4×#6923—44no—
166C4×#6923—42no—
167C4×#6923—42no—
168C4×#6923—44no—
169C4×#6923#4114yes—
170D61⋊C8—22no—
171D61⋊C8—22no—
172C2×C8×C61C22×C412yes—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C412yes—
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊Dic61—21no—
179#11758198—21no—
180#11758198—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61C22×C412yes—
184C4⋊Dic61C22×C411yescharacteristic
185#11758196—21no—
186#11758196—22no—
187#11758196—21no—
188#11758196—22no—
189#11758196—21no—
190C2×#92418C22×C412yes—
191C8×D61—22no—
192C8×D61—22no—
193C23×C61C4×D411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61C4×D411yescharacteristic
199C22×D61C2×Q1612yes—
200C22×D61#13512yes—
201C22×D61C22×D411yescharacteristic
202C2×#6923—21no—
203C2×#6923—21no—
204C2×#6923—21no—
205C2×#6923—21no—
206C2×#6923—28no—
207C2×#6923—21no—
208C2×#6923—24no—
209C2×#6923—21no—
210C2×#6923—21no—
211C2×#6923—44no—
212C2×#6923—48no—
213C2×#6923—42no—
214C2×#6923—42no—
215C2×#6923—41no—
216C2×#6923—41no—
217C2×#6923—41no—
218C2×#6923—41no—
219C2×#6923—41no—
220C2×#6923—41no—
221C2×#6923#12312yes—
222C2×#6923#12312yes—
223C2×#6923C2×Q1612yes—
224C2×#6923C2×Q1612yes—
225C2×#6923AGL(1,ℤ/8)12yes—
226Q8×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—42no—
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—22no—
237C2×Dic61—21no—
238C2×Dic61—42no—
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C2×#4111yescharacteristic
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—44no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—42no—
252C4×D61C2×#4112yes—
253C4×D61C2×#4112yes—
254Dic122—22no—
255Dic122—42no—
256#92420—42no—
257C8×C61—22no—
258#92418—22no—
259C4×#484—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61C4×Q1612yes—
263C22×C61C2×C4×D411yescharacteristic
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61C4×Q1612yes—
271C2×D61C2×#12311yescharacteristic
272C2×D61#52511yescharacteristic
273C2×D61C22×Q1611yescharacteristic
274C2×D61C2×#13511yescharacteristic
275C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
276C2×D61#57511yescharacteristic
277#6923—22no—
278#6923—22no—
279#6923—41no—
280#6923—44no—
281#6923—81no—
282#6923—81no—
283#6923#48414yesdirect product factor
284C4×C61—21no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61C4×#4112yes—
289Dic61—21no—
290Dic61—42no—
291Dic61—41no—
292Dic61—41no—
293Dic61#52912yes—
294C2×#484—611no—
295C4×C22⋊Q8—611no—
296C2×C4×C4⋊C4—611no—
297#1928—612no—
298#1921—612no—
299#1872—611no—
300#1852—611no—
301#1824—611no—
302C4×#107—612no—
303C4×#103—612no—
304C4×C22⋊C8—611no—
305C2×C61—22no—
306C2×C61C4×#12311yescharacteristic
307C2×C61C2×C4×Q1611yescharacteristic
308C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
309D61—22no—
310D61C2×#48411yescharacteristic
311D61#309111yescharacteristic
312D61#310411yescharacteristic
313D61#310511yescharacteristic
314C2×C22⋊Q8—611no—
315C2×C4×Q8—611no—
316C22×C4⋊C4—611no—
317C22×C4⋊C4—611no—
318C22×C42—611no—
319C22×C42—1221no—
320#484—611no—
321#484—611no—
322#484—612no—
323#484—612no—
324#484—611no—
325#484#692311yescharacteristic, direct product factor
326C2×#107—612no—
327C2×#107—612no—
328C2×#103—612no—
329C2×#103—612no—
330C2×C22⋊C8—611no—
331C2×C22⋊C8—611no—
332C2×C4×C8—612no—
333#391—611no—
334#389—611no—
335#387—1221no—
336#385—1221no—
337#380—611no—
338C4×C4⋊C4—612no—
339C4×C4⋊C4—612no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—1221no—
342C4×C4⋊C4—1221no—
343C4×C4⋊C4—1221no—
344C4×C22⋊C4—1221no—
345C2×#95—1221no—
346#340—612no—
347#340—612no—
348#340—612no—
349#340—612no—
350#336—612no—
351C61C4×#48411yescharacteristic, complemented, Sylow
352C22×Q8—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—1221no—
357C23×C4—1221no—
358C22×C8—612no—
359C22×C8—612no—
360C22⋊Q8—611no—
361C22⋊Q8—611no—
362C22⋊Q8—611no—
363C22⋊Q8—1221no—
364C22⋊Q8—1221no—
365C22⋊Q8C2×#692311yescharacteristic
366C4×Q8—1224no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4C2×#692311yescharacteristic
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1224no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42—2444no—
395#107—612no—
396#107—612no—
397#107—612no—
398#107—1224no—
399#107C2×#692312yescomplemented
400#103—612no—
401#103—612no—
402#103—612no—
403#103—1224no—
404#103C2×#692312yescomplemented
405C22⋊C8—611no—
406C22⋊C8—612no—
407C22⋊C8—1221no—
408C22⋊C8—1221no—
409C22⋊C8C2×#692311yescharacteristic
410C4×C8—1224no—
411#95—1222no—
412#95—1222no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418C24—611no—
419C2×Q8—611no—
420C2×Q8—1221no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8C22×#692311yescharacteristic
424C22×C4—21no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—614no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2441no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4C22×#692311yescharacteristic
456C2×C8—612no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8C22×#692312yes—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—614no—
466C4⋊C4—612no—
467C4⋊C4—611no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1224no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C4⋊C4C22×#692312yes—
482C4⋊C4C22×#692311yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1221no—
485C22⋊C4—1221no—
486C22⋊C4—1221no—
487C22⋊C4—2442no—
488C42—614no—
489C42—1222no—
490C42—1222no—
491C42—1222no—
492C42—2444no—
493C42—2442no—
494C42—2442no—
495C42—2444no—
496C23—611no—
497C23—612no—
498C23—612no—
499C23—611no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23D4×#692311yescharacteristic
505Q8—22no—
506Q8—1222no—
507Q8—2442no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—1222no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1222no—
526C2×C4—1228no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1222no—
532C2×C4—1221no—
533C2×C4—1224no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2442no—
544C2×C4—2441no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2448no—
557C2×C4D4×#692311yescharacteristic
558C2×C4C23×#692311yescharacteristic
559C8—22no—
560C8—1222no—
561C8—2442no—
562C22—21no—
563C22—21no—
564C22—611no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22—611no—
569C22—611no—
570C22—612no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1221no—
576C22—1221no—
577C22Q16×#692312yes—
578C22C2×D4×#692311yescharacteristic
579C4—21no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—614no—
584C4—612no—
585C4—1222no—
586C4—1222no—
587C4—1221no—
588C4—4881no—
589C4—4881no—
590C4—2441no—
591C4—2444no—
592C4—2441no—
593C4—2441no—
594C4—2442no—
595C4#41×#692312yes—
596C2—22no—
597C2—611no—
598C2—611no—
599C2—611no—
600C2—611no—
601C2—1222no—
602C2#123×#692311yescharacteristic
603C2C2×Q16×#692311yescharacteristic
604C2AGL(1,ℤ/8)×#692311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#484C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#484#6923
hypocenterC61C4×#484
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #21461899C2semidirect
4. #21461883C2semidirect
8. #107×#6923C2semidirect
9. #103×#6923C2semidirect
40. #16840099C4semidirect
41. #16840098C4semidirect
42. #16840097C4semidirect
43. #16840096C4semidirect
44. #16840095C4semidirect
48. C61×#484C4semidirect
102. #14519711C2×C4semidirect
103. #14519710C2×C4semidirect
104. #14519710C2×C4semidirect
105. #14519702C2×C4semidirect
106. #14519701C2×C4semidirect
107. #14519700C2×C4semidirect
112. #107×C61C2×C4semidirect
113. #103×C61C2×C4semidirect
283. #6923#484direct (N × H)
325. #484#6923direct (N × H)
351. C61C4×#484semidirect
399. #107C2×#6923semidirect
404. #103C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×#69232
C2×C4⋊C4×#69232
#214618992
#214618832
#214618042
#214617682
#214617232
#107×#69232
#103×#69232
C22⋊C8×#69232
#484×D612
C4×#48461