Finite Group Databasebeta

← C2×#27947 #6457323

Subgroups — C2×#27947 #6457323

Order 512.

2608 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#482C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3C22×#468C211yesmaximal, characteristic
4C22×#428C211yesmaximal, characteristic, complemented
5C2×#2499C211yesmaximal, characteristic, complemented
6C2×#2504C211yesmaximal, characteristic
7#27947C2116yesmaximal, direct product factor
8C2×#2097C211yesmaximal, characteristic, complemented
9C2×#2097C211yesmaximal, characteristic, complemented
10C2×#2094C211yesmaximal, characteristic, complemented
11C2×#2092C211yesmaximal, characteristic, complemented
12C2×#1973C211yesmaximal, characteristic, complemented
13C2×#1968C211yesmaximal, characteristic, complemented
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1846C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C2×#482—28no—
24C2×#482C2218yes—
25C2×#483—28no—
26C2×#483C2218yescomplemented
27C2×#468—28no—
28C2×#468C2218yes—
29C2×#428—28no—
30C2×#428C2218yescomplemented
31C22×#107C2211yescharacteristic
32C22×#107C2211yescharacteristic
33C22×#107C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C22×#103C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×C22⋊C8C2211yescharacteristic
47C22×#97C2211yescharacteristic
48C22×C4×C8C2211yescharacteristic
49#2504—24no—
50#2504C2218yes—
51#2499—24no—
52#2499C2218yescomplemented
53C2×#397—22no—
54C2×#397—22no—
55C2×#391C2211yescharacteristic
56C2×#391C2211yescharacteristic
57C2×#390C2211yescharacteristic
58C2×#390C2211yescharacteristic
59C2×#387—22no—
60C2×#386—22no—
61C2×#384C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×C4×C22⋊C4C2211yescharacteristic
66#2097C2218yescomplemented
67#2097C2218yescomplemented
68#2094C2218yescomplemented
69#2092C2218yescomplemented
70#1973C2218yescomplemented
71#1968C2218yescomplemented
72#1847C2218yescomplemented
73#1846C2218yescomplemented
74#1753C2218yescomplemented
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16C2314yes—
86C22×M16—24no—
87C22×M16—24no—
88C22×M16D418yes—
89C22×M16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8C2314yes—
98C2×#121—22no—
99C2×#121—24no—
100C2×#121—24no—
101C2×#121C2314yes—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117C2314yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4D412yes—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—232no—
129#482—232no—
130#468—232no—
131#428—232no—
132C2×#107—24no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106—24no—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106—24no—
144C2×#106—24no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2314yes—
150C2×#103—24no—
151C2×#103—24no—
152C2×#103—24no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102—24no—
157C2×#102—24no—
158C2×#102—24no—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8D418yes—
163C2×C22⋊C8C2314yes—
164C2×#97—24no—
165C2×#97C2314yes—
166C2×C4×C8—24no—
167C2×C4×C8C2314yes—
168#397—216no—
169#397—216no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#390—22no—
175#390—22no—
176#390C2314yes—
177#390C2314yes—
178#387—216no—
179#386—216no—
180#384—22no—
181#384C2314yes—
182#382—22no—
183#382—22no—
184#382—22no—
185#382C2314yes—
186#382C2314yes—
187#382C2314yes—
188C4×C22⋊C4D414yes—
189C4×C22⋊C4C2314yes—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95—22no—
193C2×#95—22no—
194C2×#95—22no—
195C2×#95D412yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C25C2×D411yescharacteristic
200C22×Q8—21no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8C2412yes—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—48no—
217C22×D4—44no—
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—84no—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C2×SD16—28no—
242C2×SD16—28no—
243C2×SD16—28no—
244C2×SD16—416no—
245C2×M16—232no—
246C2×M16—28no—
247C2×M16—28no—
248C2×M16C2×D4116yes—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8C2×D414yes—
256C22×C8C2×D414yes—
257C22×C8C2×D414yes—
258C22×C8C2×D414yes—
259C22×C8#4114yes—
260C22×C8#4114yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268#121—28no—
269#121—28no—
270#121—48no—
271#117—28no—
272#117—28no—
273#117—48no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4—84no—
318C2×C22⋊C4C2×D418yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329#107—28no—
330#107—28no—
331#107—28no—
332#107—28no—
333#106—28no—
334#106—28no—
335#106—28no—
336#106—28no—
337#106—28no—
338#103—28no—
339#103—28no—
340#103—28no—
341#102—28no—
342#102—28no—
343#102—28no—
344C22⋊C8C2×D4116yes—
345#97—28no—
346C4×C8—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95C2×D418yes—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—44no—
362C24—44no—
363C24#12714yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—416no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—816no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—88no—
437C22×C4—816no—
438C22×C4#12714yes—
439C22×C4#12718yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C22×C4C2511yescharacteristic
453SD16—416no—
454M16—232no—
455C2×C8—28no—
456C2×C8—28no—
457C2×C8—28no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8#123116yes—
465C2×C8C22×D414yes—
466C2×C8C22×D414yes—
467C2×C8C22×D414yes—
468C2×C8C22×D414yes—
469C2×C8C2×#4114yes—
470C2×C8C2×#4114yes—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—48no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—48no—
513C23—48no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—88no—
520C23—88no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23C2×#12714yes—
525C23C2×#12712yes—
526C23#54712yes—
527C23#54712yes—
528C23#57514yes—
529C23#57714yes—
530C23C23×D411yescharacteristic
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—84no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—816no—
541D4—84no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—216no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—816no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4C2×#12314yes—
582C2×C4C2×#12714yes—
583C2×C4C2×#12712yes—
584C2×C4#53614yes—
585C2×C4#54714yes—
586C2×C4#54714yes—
587C2×C4#54712yes—
588C2×C4#54712yes—
589C2×C4C23×D411yescharacteristic
590C2×C4C23×D411yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C8—24no—
596C8—24no—
597C8—24no—
598C8C2×#12318yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—416no—
603C22—44no—
604C22—44no—
605C22—84no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—168no—
611C22#250114yes—
612C22#315314yes—
613C22#315414yes—
614C22#335018yes—
615C22#335118yes—
616C22C22×#12711yescharacteristic
617C22C2×#54711yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×#57512yes—
620C22C2×#57712yes—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4#250114yes—
632C4C22×#12311yescharacteristic
633C4C2×#53611yescharacteristic
634C4C2×#54711yescharacteristic
635C4C2×#54711yescharacteristic
636C2—44no—
637C2—162no—
638C2#2794718yesdirect product factor
639C2C2×#250111yescharacteristic
640C2C2×#315311yescharacteristic
641C2C2×#315411yescharacteristic
642C2C2×#335012yes—
643C2C2×#335112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#483C2semidirect
4. C22×#428C2semidirect
5. C2×#2499C2semidirect
7. #27947C2direct (N × H)
8. C2×#2097C2semidirect
9. C2×#2097C2semidirect
10. C2×#2094C2semidirect
11. C2×#2092C2semidirect
12. C2×#1973C2semidirect
13. C2×#1968C2semidirect
14. C2×#1847C2semidirect
15. C2×#1846C2semidirect
16. C2×#1753C2semidirect
26. C2×#483C22semidirect
30. C2×#428C22semidirect
52. #2499C22semidirect
66. #2097C22semidirect
67. #2097C22semidirect
68. #2094C22semidirect
69. #2092C22semidirect
70. #1973C22semidirect
71. #1968C22semidirect
72. #1847C22semidirect
73. #1846C22semidirect
74. #1753C22semidirect
638. C2#27947direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4822
C22×#4832
C22×#4682
C22×#4282
C2×#24992
C2×#25042
#279472
C2×#20972
C2×#20942
C2×#20922
C2×#19732
C2×#19682
C2×#18472
C2×#18462
C2×#17532