Finite Group Databasebeta

← C2×#32909 #6628910

Subgroups — C2×#32909 #6628910

Order 512.

5004 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C22×#464C212yesmaximal, complemented
3C22×#451C211yesmaximal, characteristic, complemented
4C22×#448C211yesmaximal, characteristic
5C2×#2850C211yesmaximal, characteristic
6#32909C2116yesmaximal, direct product factor
7C2×#1880C212yesmaximal, complemented
8C2×#1878C212yesmaximal, complemented
9C2×#1874C211yesmaximal, characteristic
10C2×#1872C211yesmaximal, characteristic, complemented
11C2×#1805C212yesmaximal, complemented
12C2×#1797C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C23×Q16C2212yes—
15C23×SD16C2212yes—
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×Q8—21no—
20C22×C4×Q8—21no—
21C22×C4×Q8—24no—
22C22×C4×Q8—216no—
23C22×C4×Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2218yescomplemented
26C23×C4⋊C4—21no—
27C23×C4⋊C4C2211yescharacteristic
28C23×C42—21no—
29C23×C42C2211yescharacteristic
30C2×#464—216no—
31C2×#464C22116yescomplemented
32C2×#451—28no—
33C2×#451C2218yescomplemented
34C2×#448—28no—
35C2×#448C2218yes—
36C22×C4⋊C8C2212yes—
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2850—24no—
44#2850C2218yes—
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#388—21no—
48C2×#388—21no—
49C2×#386—21no—
50C2×#386—21no—
51C2×#382C2211yescharacteristic
52#1880C22116yescomplemented
53#1878C22116yescomplemented
54#1874C2218yes—
55#1872C2218yescomplemented
56#1805C22116yescomplemented
57#1797C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—28no—
62C23×Q8—28no—
63C23×Q8—22no—
64C23×Q8—21no—
65C23×Q8—24no—
66C23×Q8D418yes—
67C23×Q8C2311yescharacteristic
68C23×Q8C2314yes—
69C23×Q8C2312yes—
70C23×D4—21no—
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4—21no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4C2311yescharacteristic
77C22×Q16—28no—
78C22×Q16—28no—
79C22×Q16—28no—
80C22×Q16C2318yes—
81C22×SD16—28no—
82C22×SD16—28no—
83C22×SD16—28no—
84C22×SD16C2318yes—
85C23×C8C2312yes—
86C23×C8C2312yes—
87C2×#124—22no—
88C2×#124—28no—
89C2×#124C2314yes—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8C2314yes—
94C2×#121—22no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121C2314yes—
98C2×C4×Q8—216no—
99C2×C4×Q8—264no—
100C2×C4×Q8—216no—
101C2×C4×Q8—28no—
102C2×C4×Q8—264no—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—28no—
108C2×C4×Q8—28no—
109C2×C4×Q8D4116yescomplemented
110C2×C4×Q8C2318yes—
111C2×C4×Q8C2318yes—
112C22×C4⋊C4—216no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—28no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C42—216no—
131C22×C42—22no—
132C22×C42—22no—
133C22×C42—28no—
134C22×C42—28no—
135C22×C42D414yes—
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2314yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#464—264no—
142#451—232no—
143#448—232no—
144C2×C4⋊C8—28no—
145C2×C4⋊C8C2318yes—
146C2×#103—28no—
147C2×#103—28no—
148C2×#103—28no—
149C2×#103C2318yes—
150C2×#103C2318yes—
151C2×#103C2318yes—
152C2×#102—28no—
153C2×#102C2318yes—
154C2×C22⋊C8—24no—
155C2×C22⋊C8—24no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#391—22no—
159#391C2314yes—
160#390—22no—
161#390C2314yes—
162#388—28no—
163#388—28no—
164#386—28no—
165#386—28no—
166#382—22no—
167#382C2314yes—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172#340C2318yes—
173#340C2318yes—
174#336C2318yes—
175C25—21no—
176C25C2×D411yescharacteristic
177C22×Q8—216no—
178C22×Q8—28no—
179C22×Q8—216no—
180C22×Q8—264no—
181C22×Q8—28no—
182C22×Q8—264no—
183C22×Q8—216no—
184C22×Q8—216no—
185C22×Q8—28no—
186C22×Q8—216no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—21no—
191C22×Q8—28no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8C2×D4116yes—
199C22×Q8C2×D418yes—
200C22×Q8C2×D418yes—
201C22×Q8#4118yes—
202C22×Q8C2412yes—
203C22×Q8C2414yes—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4C2412yes—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D414yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4C2×D411yescharacteristic
244C23×C4#4114yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C2×Q16—216no—
248C2×Q16—216no—
249C2×Q16—216no—
250C2×Q16—432no—
251C2×SD16—216no—
252C2×SD16—216no—
253C2×SD16—216no—
254C2×SD16—432no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—24no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263#124—216no—
264#124—48no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—48no—
268#121—28no—
269#121—28no—
270#121—48no—
271C4×Q8—232no—
272C4×Q8—264no—
273C4×Q8—264no—
274C4×Q8—216no—
275C4×Q8—216no—
276C4×Q8—232no—
277C4×Q8—216no—
278C4×Q8C2×D4132yescomplemented
279C2×C4⋊C4—216no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—264no—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—216no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—216no—
315C2×C42—264no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—216no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—216no—
323C2×C42—216no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D418yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C4⋊C8—216no—
332#103—216no—
333#103—216no—
334#103—216no—
335#102—216no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24#12114yes—
355C24#13514yes—
356C24#13714yes—
357C24C22×D412yes—
358C24C22×D411yescharacteristic
359C2×Q8—216no—
360C2×Q8—2128no—
361C2×Q8—264no—
362C2×Q8—264no—
363C2×Q8—216no—
364C2×Q8—264no—
365C2×Q8—28no—
366C2×Q8—216no—
367C2×Q8—216no—
368C2×Q8—28no—
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8#121132yes—
385C2×Q8C22×D418yes—
386C2×Q8C22×D418yes—
387C2×Q8C2×#4118yes—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—42no—
396C2×D4—48no—
397C2×D4—42no—
398C2×D4—48no—
399C2×D4—42no—
400C2×D4—816no—
401C22×C4—216no—
402C22×C4—232no—
403C22×C4—264no—
404C22×C4—216no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—216no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—432no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4#12014yes—
455C22×C4#12014yes—
456C22×C4#12114yes—
457C22×C4#12118yes—
458C22×C4#13714yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4112yes—
471C22×C4C2511yescharacteristic
472Q16—432no—
473SD16—432no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—48no—
481C2×C8—48no—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—232no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4C22×D418yes—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C42—232no—
502C42—232no—
503C42—28no—
504C42—24no—
505C42—28no—
506C42—24no—
507C42#12018yes—
508C42C22×D418yes—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23#464116yescomplemented
534C23C2×#12012yes—
535C23C2×#12114yes—
536C23C2×#12112yes—
537C23#52512yes—
538C23C2×#13514yes—
539C23C2×#13512yes—
540C23C2×#13714yes—
541C23C2×#13712yes—
542C23C2×#13712yes—
543C23#57512yes—
544C23C23×D411yescharacteristic
545Q8—264no—
546Q8—232no—
547Q8—28no—
548Q8—28no—
549Q8—28no—
550Q8—44no—
551Q8—44no—
552Q8—84no—
553Q8C2×#121116yes—
554D4—44no—
555D4—44no—
556D4—816no—
557D4—84no—
558C2×C4—232no—
559C2×C4—264no—
560C2×C4—232no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—216no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—416no—
584C2×C4—432no—
585C2×C4—44no—
586C2×C4—416no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4C2×#12014yes—
602C2×C4C2×#12014yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12114yes—
605C2×C4C2×#12112yes—
606C2×C4C2×#12118yes—
607C2×C4#52512yes—
608C2×C4#52518yes—
609C2×C4C2×#13714yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C23×D412yes—
612C2×C4C22×#4112yes—
613C2×C4C22×#4111yescharacteristic
614C8—44no—
615C8—44no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—232no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—168no—
631C22#284714yes—
632C22#300214yes—
633C22#300414yes—
634C22C2×#46414yes—
635C22C2×#46418yescomplemented
636C22#303714yes—
637C22#310814yes—
638C22#311714yes—
639C22C22×#12011yescharacteristic
640C22C22×#12111yescharacteristic
641C22C2×#52511yescharacteristic
642C22C22×#13511yescharacteristic
643C22C22×#13711yescharacteristic
644C22C22×#13711yescharacteristic
645C22C2×#57511yescharacteristic
646C4—216no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—42no—
652C4—48no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#284714yes—
657C4C22×#12112yes—
658C4C2×#52512yes—
659C2—28no—
660C2—162no—
661C2#3290918yesdirect product factor
662C2C2×#284711yescharacteristic
663C2C2×#300211yescharacteristic
664C2C2×#300411yescharacteristic
665C2C22×#46411yescharacteristic
666C2C2×#303711yescharacteristic
667C2C2×#310811yescharacteristic
668C2C2×#311711yescharacteristic
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
1. C23×C4×Q8C2semidirect
2. C22×#464C2semidirect
3. C22×#451C2semidirect
6. #32909C2direct (N × H)
7. C2×#1880C2semidirect
8. C2×#1878C2semidirect
10. C2×#1872C2semidirect
11. C2×#1805C2semidirect
12. C2×#1797C2semidirect
25. C22×C4×Q8C22semidirect
31. C2×#464C22semidirect
33. C2×#451C22semidirect
52. #1880C22semidirect
53. #1878C22semidirect
55. #1872C22semidirect
56. #1805C22semidirect
57. #1797C22semidirect
109. C2×C4×Q8D4semidirect
278. C4×Q8C2×D4semidirect
533. C23#464semidirect
635. C22C2×#464semidirect
661. C2#32909direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C22×#4642
C22×#4512
C22×#4482
C2×#28502
#329092
C2×#18802
C2×#18782
C2×#18742
C2×#18722
C2×#18052
C2×#17972