Finite Group Databasebeta

← C2×#10231 #233921

Subgroups — C2×#10231 #233921

Order 512.

2122 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic, complemented
2C22×#461C211yesmaximal, characteristic, complemented
3C2×#2537C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C22×D4⋊C8C211yesmaximal, characteristic, complemented
6C22×D4⋊C8C211yesmaximal, characteristic, complemented
7#10231C218yesmaximal, direct product factor
8C2×#1296C211yesmaximal, characteristic, complemented
9C23×SD16—21no—
10C22×C4⋊Q8C2211yescharacteristic
11C22×#121—21no—
12C22×#121—21no—
13C22×C4×D4C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#474—24no—
16C2×#474—24no—
17C2×#474—22no—
18C2×#474C2214yescomplemented
19C2×#461—24no—
20C2×#461—24no—
21C2×#461—22no—
22C2×#461C2214yescomplemented
23C22×#107—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103—21no—
27C22×#102—21no—
28C22×C4×C8C2211yescharacteristic
29#2537—22no—
30#2537—22no—
31#2537—22no—
32#2537C2214yes—
33C2×#394—22no—
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36C2×#380—21no—
37#1990—22no—
38#1990C2214yescomplemented
39C2×#340—22no—
40C2×D4⋊C8—24no—
41C2×D4⋊C8—24no—
42C2×D4⋊C8C2214yescomplemented
43C2×D4⋊C8C2214yescomplemented
44#1296C2214yescomplemented
45C23×Q8—21no—
46C23×D4—22no—
47C23×D4D411yescharacteristic
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C22×SD16—28no—
52C22×SD16—24no—
53C22×SD16—44no—
54C22×SD16—44no—
55C23×C8—21no—
56C23×C8—21no—
57C23×C8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8D414yescomplemented
65C2×C4⋊Q8C2312yes—
66C2×#121—24no—
67C2×#121—24no—
68C2×#121—24no—
69C2×#121—24no—
70C2×#121—44no—
71C2×#121—44no—
72C2×#121—44no—
73C2×#121—44no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4D412yes—
87C2×C4×D4D412yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D412yes—
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C42C2311yescharacteristic
102#474—216no—
103#474—28no—
104#474—28no—
105#461—216no—
106#461—28no—
107#461—28no—
108C2×#107—28no—
109C2×#107—24no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8D414yescomplemented
115C2×C4⋊C8D414yescomplemented
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×#103—28no—
119C2×#103—24no—
120C2×#102—28no—
121C2×#102—24no—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—21no—
125C2×C4×C8C2312yes—
126#394—28no—
127#394—28no—
128#384—21no—
129#384—21no—
130#384D412yes—
131#384C2312yes—
132#380—24no—
133#380—24no—
134#380—24no—
135#380—24no—
136C2×#95—22no—
137#340—28no—
138D4⋊C8—28no—
139D4⋊C8—28no—
140C25—21no—
141C25—21no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—41no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×D4—28no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—48no—
164C22×D4—48no—
165C22×D4—84no—
166C22×D4D814yes—
167C22×D4SD1614yescomplemented
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—41no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C2×SD16—216no—
194C2×SD16—48no—
195C2×SD16—432no—
196C2×SD16—48no—
197C22×C8—24no—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8—42no—
208C22×C8—42no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8C2×D414yescomplemented
219#121—48no—
220#121—48no—
221#121—416no—
222#121—416no—
223#121—48no—
224#121—48no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—42no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42#4112yes—
291C2×C42C2411yescharacteristic
292#107—216no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8C2×D414yescomplemented
298C4⋊C8C2×D414yescomplemented
299#103—216no—
300#102—216no—
301C4×C8—22no—
302C4×C8—22no—
303C4×C8—42no—
304#95—28no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24#12011yescharacteristic
322C2×Q8—24no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—28no—
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—88no—
362C2×D4—832no—
363C2×D4C2×D814yes—
364C2×D4C2×SD1614yescomplemented
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4#12011yescharacteristic
410C22×C4#12012yes—
411C22×C4#12012yes—
412C22×C4#12011yescharacteristic
413C22×C4#12011yescharacteristic
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4#12712yes—
418C22×C4C2×D812yes—
419C22×C4C2×SD1612yes—
420C22×C4#13512yes—
421C22×C4#13512yes—
422C22×C4AGL(1,ℤ/8)12yes—
423C22×C4AGL(1,ℤ/8)12yes—
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4#13712yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429SD16—432no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—42no—
434C2×C8—48no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—48no—
441C2×C8—48no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C42#12012yes—
471C42#12012yes—
472C42#12112yes—
473C42#12112yes—
474C42#12112yes—
475C42#12712yes—
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C42C22×D411yescharacteristic
479C42C2×#4111yescharacteristic
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—48no—
499C23—48no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23#39212yes—
509C23#44712yes—
510C23#44912yes—
511C23#45012yes—
512C23#45212yes—
513C23#45512yes—
514C23#45612yes—
515C23C2×#12011yescharacteristic
516Q8—44no—
517Q8—44no—
518Q8—88no—
519Q8—82no—
520Q8—82no—
521D4—24no—
522D4—24no—
523D4—24no—
524D4—24no—
525D4—42no—
526D4—42no—
527D4—48no—
528D4—48no—
529D4—42no—
530D4—42no—
531D4—832no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—88no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—84no—
552C2×C4—88no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#39212yes—
565C2×C4#39212yes—
566C2×C4#39212yes—
567C2×C4#44714yes—
568C2×C4#44712yes—
569C2×C4#44814yes—
570C2×C4#44914yes—
571C2×C4#44914yes—
572C2×C4#44912yes—
573C2×C4#45012yes—
574C2×C4#45212yes—
575C2×C4#46514yes—
576C2×C4#46614yes—
577C2×C4#46814yes—
578C2×C4#46914yes—
579C2×C4C2×#12011yescharacteristic
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12111yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4C2×#12711yescharacteristic
585C2×C4C22×D811yescharacteristic
586C2×C4C22×SD1611yescharacteristic
587C2×C4C2×#13511yescharacteristic
588C2×C4C2×#13511yescharacteristic
589C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
590C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
591C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
592C2×C4C2×#13711yescharacteristic
593C8—44no—
594C8—44no—
595C8—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—816no—
611C22—816no—
612C22#163914yes—
613C22#164614yes—
614C22#164714yes—
615C22#165414yes—
616C22#199914yes—
617C22#200014yes—
618C22#200714yes—
619C22C2×#39211yescharacteristic
620C22C2×#44711yescharacteristic
621C22C2×#45011yescharacteristic
622C22C2×#44911yescharacteristic
623C22C2×#45211yescharacteristic
624C22C2×#45511yescharacteristic
625C22C2×#45611yescharacteristic
626C4—21no—
627C4—21no—
628C4—22no—
629C4—84no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4#199914yes—
634C4#200014yes—
635C4C2×#44711yescharacteristic
636C4C2×#44811yescharacteristic
637C4C2×#44911yescharacteristic
638C4C2×#44911yescharacteristic
639C4C2×#46511yescharacteristic
640C4C2×#46611yescharacteristic
641C4C2×#46811yescharacteristic
642C4C2×#46911yescharacteristic
643C2—84no—
644C2—84no—
645C2#1023118yesdirect product factor
646C2C2×#163911yescharacteristic
647C2C2×#164711yescharacteristic
648C2C2×#165411yescharacteristic
649C2C2×#164611yescharacteristic
650C2C2×#199911yescharacteristic
651C2C2×#200011yescharacteristic
652C2C2×#200711yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2537C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#474C2semidirect
2. C22×#461C2semidirect
4. C2×#1990C2semidirect
5. C22×D4⋊C8C2semidirect
6. C22×D4⋊C8C2semidirect
7. #10231C2direct (N × H)
8. C2×#1296C2semidirect
18. C2×#474C22semidirect
22. C2×#461C22semidirect
38. #1990C22semidirect
42. C2×D4⋊C8C22semidirect
43. C2×D4⋊C8C22semidirect
44. #1296C22semidirect
64. C2×C4⋊Q8D4semidirect
114. C2×C4⋊C8D4semidirect
115. C2×C4⋊C8D4semidirect
167. C22×D4SD16semidirect
218. C4⋊Q8C2×D4semidirect
297. C4⋊C8C2×D4semidirect
298. C4⋊C8C2×D4semidirect
364. C2×D4C2×SD16semidirect
645. C2#10231direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4742
C22×#4612
C2×#25372
C2×#19902
C22×D4⋊C82
#102312
C2×#12962