Finite Group Databasebeta

← C2×#27657 #6457201

Subgroups — C2×#27657 #6457201

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3C2×#2323C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#27657C2116yesmaximal, direct product factor
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2074C211yesmaximal, characteristic, complemented
9C2×#2072C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1824C211yesmaximal, characteristic
13C2×#1825C211yesmaximal, characteristic
14C2×#1805C211yesmaximal, characteristic, complemented
15C2×#1803C211yesmaximal, characteristic, complemented
16C2×#1798C211yesmaximal, characteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#485—28no—
23C2×#485C2218yes—
24C2×#483—28no—
25C2×#483C2218yes—
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2323—24no—
39#2323C2218yes—
40#2317—24no—
41#2317C2218yes—
42C2×#391C2211yescharacteristic
43C2×#389—21no—
44C2×#389—21no—
45C2×#389—21no—
46C2×#389C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#387—21no—
50C2×#385—21no—
51C2×#385—21no—
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2218yescomplemented
59#2084C2218yescomplemented
60#2074C2218yescomplemented
61#2072C2218yescomplemented
62#1949C2218yescomplemented
63#1948C2218yescomplemented
64#1825C2218yes—
65#1824C2218yes—
66#1805C2218yescomplemented
67#1803C2218yescomplemented
68#1798C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8D418yes—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8C2314yes—
94C2×#117—22no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D414yes—
100C2×#117C2314yes—
101C2×#117C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#485—232no—
134#483—232no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—24no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103—24no—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103—24no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#391—22no—
160#391C2314yes—
161#389—28no—
162#389—28no—
163#389—28no—
164#389—22no—
165#389C2314yes—
166#388—28no—
167#388—28no—
168#387—28no—
169#385—28no—
170#385—28no—
171#384—22no—
172#384—22no—
173#384C2314yes—
174#384C2314yes—
175#382—22no—
176#382—22no—
177#382C2314yes—
178#382C2314yes—
179#380—22no—
180#380C2314yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8C2×D414yes—
210C22×Q8C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—82no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D414yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22⋊Q8—216no—
254C22⋊Q8—28no—
255C22⋊Q8C2×D4116yes—
256C4×Q8—28no—
257C4×Q8—28no—
258C4×Q8—48no—
259#117—216no—
260#117—28no—
261#117—28no—
262#117—28no—
263#117—28no—
264#117—48no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4C2×D414yes—
325C2×C22⋊C4C2×D414yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42#4114yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#106—28no—
342#106—28no—
343#106—28no—
344C4⋊C8—28no—
345C4⋊C8—28no—
346#103—28no—
347#103—28no—
348#103—28no—
349#103—28no—
350C22⋊C8—28no—
351C22⋊C8—28no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24#12114yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8C22×D414yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—816no—
441C22×C4#12114yes—
442C22×C4#12118yes—
443C22×C4#12314yes—
444C22×C4#12314yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C42−1+412yes—
458C22×C42−1+412yes—
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8—44no—
476C4⋊C4—216no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—48no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4C22×D414yes—
507C22⋊C4—216no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—84no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—48no—
516C42—48no—
517C42—84no—
518C42#12318yes—
519C42C2×#4114yes—
520C42C2×#4114yes—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23C2×#12114yes—
535C23C2×#12112yes—
536C23C2×#12312yes—
537C23#53612yes—
538C23#57712yes—
539C23#57712yes—
540C23#57712yes—
541C23#57812yes—
542C23C23×D411yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—42no—
546Q8—42no—
547Q8—48no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—216no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—816no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4C2×#12114yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12314yes—
606C2×C4C2×#12314yes—
607C2×C4C2×#12312yes—
608C2×C4#53612yes—
609C2×C4#54714yes—
610C2×C4#54714yes—
611C2×C4#55214yes—
612C2×C4#55214yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C8—42no—
620C8—42no—
621C8—42no—
622C8—42no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22#248614yes—
632C22#304314yes—
633C22#304614yes—
634C22#309714yes—
635C22#309914yes—
636C22#324214yes—
637C22#324414yes—
638C22C22×#12111yescharacteristic
639C22C22×#12311yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#57711yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57711yescharacteristic
644C22C2×#57811yescharacteristic
645C4—24no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#248614yes—
659C4C2×#54711yescharacteristic
660C4C2×#54711yescharacteristic
661C4C2×#55211yescharacteristic
662C4C2×#55211yescharacteristic
663C2—44no—
664C2#2765718yesdirect product factor
665C2C2×#248611yescharacteristic
666C2C2×#304311yescharacteristic
667C2C2×#304611yescharacteristic
668C2C2×#309711yescharacteristic
669C2C2×#309911yescharacteristic
670C2C2×#324211yescharacteristic
671C2C2×#324411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #27657C2direct (N × H)
6. C2×#2084C2semidirect
7. C2×#2085C2semidirect
8. C2×#2074C2semidirect
9. C2×#2072C2semidirect
10. C2×#1949C2semidirect
11. C2×#1948C2semidirect
14. C2×#1805C2semidirect
15. C2×#1803C2semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2074C22semidirect
61. #2072C22semidirect
62. #1949C22semidirect
63. #1948C22semidirect
66. #1805C22semidirect
67. #1803C22semidirect
664. C2#27657direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4832
C2×#23232
C2×#23172
#276572
C2×#20842
C2×#20852
C2×#20742
C2×#20722
C2×#19492
C2×#19482
C2×#18242
C2×#18252
C2×#18052
C2×#18032
C2×#17982