Finite Group Databasebeta

← C2×#25604 #6449121

Subgroups — C2×#25604 #6449121

Order 512.

3814 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C22×#497C211yesmaximal, characteristic, complemented
3C22×#491C211yesmaximal, characteristic, complemented
4C2×#2857C211yesmaximal, characteristic, complemented
5#25604C2116yesmaximal, direct product factor
6C2×#2001C212yesmaximal, complemented
7C2×#2000C212yesmaximal
8C2×#1992C211yesmaximal, characteristic, complemented
9C2×#1959C211yesmaximal, characteristic, complemented
10C2×#1878C212yesmaximal, complemented
11C2×#1877C212yesmaximal, complemented
12C2×#1749C211yesmaximal, characteristic, complemented
13C23×Q16C2212yes—
14C23×SD16C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×#529—24no—
17C2×#529—28no—
18C2×#529—28no—
19C2×#529C2218yes—
20C22×#126—21no—
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#123—21no—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×#121C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4—21no—
29C22×C4×D4C2211yescharacteristic
30C22×#117—21no—
31C23×C42C2211yescharacteristic
32C2×#497—28no—
33C2×#497C2218yescomplemented
34C2×#491—28no—
35C2×#491C2218yescomplemented
36C22×#103C2212yes—
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#102C2212yes—
40C22×C22⋊C8C2212yes—
41C22×#97C2211yescharacteristic
42C22×#97C2211yescharacteristic
43#2857—24no—
44#2857C2218yescomplemented
45C2×#391C2212yes—
46C2×#389—21no—
47C2×#389—21no—
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380—21no—
51#2001C22116yescomplemented
52#2000C22116yes—
53#1992C2218yescomplemented
54#1959C2218yescomplemented
55#1878C22116yescomplemented
56#1877C22116yescomplemented
57#1749C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8C2312yes—
62C23×Q8C2311yescharacteristic
63C23×D4—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4D411yescharacteristic
67C24×C4D411yescharacteristic
68C24×C4C2311yescharacteristic
69C22×Q16—28no—
70C22×Q16—28no—
71C22×Q16—44no—
72C22×Q16C2318yes—
73C22×SD16—28no—
74C22×SD16—28no—
75C22×SD16—44no—
76C22×SD16C2318yes—
77C23×C8C2312yes—
78C23×C8C2312yes—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2314yes—
83#529—264no—
84#529—232no—
85#529—232no—
86C2×#126—216no—
87C2×#126—28no—
88C2×#125—22no—
89C2×#125—24no—
90C2×#125D418yes—
91C2×#125C2314yes—
92C2×#124—22no—
93C2×#124—24no—
94C2×#124D418yes—
95C2×#124C2314yes—
96C2×#123—216no—
97C2×#123—28no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2318yes—
106C2×#121—24no—
107C2×#121—22no—
108C2×#121—24no—
109C2×#121C2314yes—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—22no—
113C2×C4×Q8C2314yes—
114C2×C4×D4—216no—
115C2×C4×D4—24no—
116C2×C4×D4—28no—
117C2×C4×D4—24no—
118C2×C4×D4—22no—
119C2×C4×D4C2314yes—
120C2×#117—216no—
121C2×#117—28no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C22⋊C4—22no—
140C22×C42—21no—
141C22×C42—24no—
142C22×C42—24no—
143C22×C42D414yes—
144C22×C42D411yescharacteristic
145C22×C42D411yescharacteristic
146C22×C42C2314yes—
147C22×C42C2311yescharacteristic
148C22×C42C2311yescharacteristic
149#497—232no—
150#491—232no—
151C2×#103—28no—
152C2×#103—28no—
153C2×#103—44no—
154C2×#103C2318yes—
155C2×#103C2318yes—
156C2×#103C2318yes—
157C2×#102—44no—
158C2×#102C2318yes—
159C2×C22⋊C8—28no—
160C2×C22⋊C8C2318yes—
161C2×#97—24no—
162C2×#97—24no—
163C2×#97C2314yes—
164C2×#97C2314yes—
165#391—24no—
166#391C2318yes—
167#389—28no—
168#389—28no—
169#384—22no—
170#384C2314yes—
171#380—28no—
172#380—28no—
173C2×#95—22no—
174C2×#95—22no—
175#340C2318yes—
176#340C2318yes—
177#336C2318yes—
178C25—21no—
179C25C2×D411yescharacteristic
180C22×Q8—22no—
181C22×Q8—21no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8C2×D414yes—
192C22×Q8C2414yes—
193C22×Q8C2412yes—
194C22×D4—28no—
195C22×D4—24no—
196C22×D4—21no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4C2×D414yes—
203C22×D4C2412yes—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C2×Q16—216no—
234C2×Q16—216no—
235C2×Q16—48no—
236C2×Q16—432no—
237C2×SD16—216no—
238C2×SD16—216no—
239C2×SD16—48no—
240C2×SD16—432no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8#4118yes—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—48no—
252#126—264no—
253#125—216no—
254#125—28no—
255#125C2×D4116yes—
256#124—216no—
257#124—28no—
258#124C2×D4116yes—
259#123—264no—
260C22⋊Q8—216no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—48no—
264C22⋊Q8—48no—
265C22⋊Q8—416no—
266#121—216no—
267#121—28no—
268#121—28no—
269C4×Q8—216no—
270C4×Q8—28no—
271C4×Q8—28no—
272C4×D4—264no—
273C4×D4—216no—
274C4×D4—28no—
275C4×D4—28no—
276#117—264no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—48no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—216no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42C2×D414yes—
329C2×C42C2×D412yes—
330C2×C42C2×D414yes—
331C2×C42C2×D412yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#103—216no—
337#103—216no—
338#103—48no—
339#102—48no—
340C22⋊C8—216no—
341#97—28no—
342#97—28no—
343#95—28no—
344#95—28no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24#13714yes—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—216no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—44no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8C22×D414yes—
384C2×D4—216no—
385C2×D4—24no—
386C2×D4—216no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—432no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—48no—
396C2×D4C22×D414yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—432no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4#12014yes—
451C22×C4#12014yes—
452C22×C4#13714yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4C2511yescharacteristic
464Q16—432no—
465SD16—432no—
466C2×C8—216no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—48no—
473C2×C8C2×#4118yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4C22×D414yes—
491C4⋊C4C22×D414yes—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—416no—
498C42—216no—
499C42—28no—
500C42—24no—
501C42—28no—
502C42—22no—
503C42—22no—
504C42#12018yes—
505C42C22×D414yes—
506C42C22×D414yes—
507C23—22no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23C2×#12012yes—
535C23#53212yes—
536C23#53512yes—
537C23C2×#13714yes—
538C23C2×#13712yes—
539C23C2×#13712yes—
540C23#57512yes—
541C23#57512yes—
542C23C23×D411yescharacteristic
543Q8—28no—
544Q8—48no—
545Q8—44no—
546Q8—44no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—88no—
551D4—28no—
552D4—432no—
553D4—48no—
554D4—44no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—432no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4C2×#12014yes—
599C2×C4C2×#12014yes—
600C2×C4C2×#12012yes—
601C2×C4#53212yes—
602C2×C4#53512yes—
603C2×C4#54618yes—
604C2×C4#55018yes—
605C2×C4C2×#13714yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4112yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C8—28no—
611C8—44no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—82no—
624C22—816no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#243114yes—
629C22#300714yes—
630C22#300714yes—
631C22#317214yes—
632C22#318114yes—
633C22#321814yes—
634C22#321914yes—
635C22C22×#12011yescharacteristic
636C22C2×#53211yescharacteristic
637C22C2×#53511yescharacteristic
638C22C22×#13711yescharacteristic
639C22C22×#13711yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57511yescharacteristic
642C4—24no—
643C4—24no—
644C4—24no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—84no—
652C4#243114yes—
653C4C2×#54612yes—
654C4C2×#55012yes—
655C2—44no—
656C2—84no—
657C2#2560418yesdirect product factor
658C2C2×#243111yescharacteristic
659C2C2×#300711yescharacteristic
660C2C2×#300711yescharacteristic
661C2C2×#317211yescharacteristic
662C2C2×#318111yescharacteristic
663C2C2×#321811yescharacteristic
664C2C2×#321911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#497C2semidirect
3. C22×#491C2semidirect
4. C2×#2857C2semidirect
5. #25604C2direct (N × H)
6. C2×#2001C2semidirect
8. C2×#1992C2semidirect
9. C2×#1959C2semidirect
10. C2×#1878C2semidirect
11. C2×#1877C2semidirect
12. C2×#1749C2semidirect
33. C2×#497C22semidirect
35. C2×#491C22semidirect
44. #2857C22semidirect
51. #2001C22semidirect
53. #1992C22semidirect
54. #1959C22semidirect
55. #1878C22semidirect
56. #1877C22semidirect
57. #1749C22semidirect
657. C2#25604direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5292
C22×#4972
C22×#4912
C2×#28572
#256042
C2×#20012
C2×#20002
C2×#19922
C2×#19592
C2×#18782
C2×#18772
C2×#17492