Finite Group Databasebeta

← C2×#8329 #185002

Subgroups — C2×#8329 #185002

Order 512.

4486 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3C22×#328C214yesmaximal, complemented
4C2×C22⋊#95C211yesmaximal, characteristic
5#8329C218yesmaximal, direct product factor
6C24×Q8C411yescharacteristic, complemented
7C25×C4C2211yescharacteristic
8C22×C22⋊Q8—21no—
9C22×C22⋊Q8—22no—
10C22×C22⋊Q8—21no—
11C22×C22⋊Q8—22no—
12C22×C22⋊Q8—216no—
13C22×C22⋊Q8C412yescomplemented
14C22×C22⋊Q8C414yescomplemented
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2214yes—
17C23×C4⋊C4—21no—
18C23×C4⋊C4C411yescharacteristic, complemented
19C23×C22⋊C4—21no—
20C23×C22⋊C4—21no—
21C22×C22⋊C8—21no—
22C22×C22⋊C8—21no—
23C22×C22⋊C8C2212yes—
24C22×#95—21no—
25C22×#95C2212yes—
26#1779—22no—
27#1779C2214yes—
28C2×#328—216no—
29C2×#328C22116yescomplemented
30C22⋊#95—22no—
31C22⋊#95C2214yes—
32C26D411yescharacteristic
33C23×Q8—28no—
34C23×Q8—22no—
35C23×Q8—21no—
36C23×Q8—44no—
37C23×Q8—44no—
38C23×Q8C2×C412yes—
39C23×Q8C2×C414yescomplemented
40C24×C4—21no—
41C24×C4—21no—
42C24×C4—28no—
43C24×C4—22no—
44C24×C4—44no—
45C24×C4—44no—
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C412yes—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D412yes—
53C24×C4D412yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C2×C22⋊Q8—232no—
58C2×C22⋊Q8—232no—
59C2×C22⋊Q8—216no—
60C2×C22⋊Q8—216no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—232no—
71C2×C22⋊Q8—232no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8C2×C414yes—
74C2×C22⋊Q8C2×C418yescomplemented
75C2×C22⋊Q8D4116yes—
76C2×C22⋊Q8C2314yes—
77C22×C4⋊C4—216no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C414yescomplemented
87C22×C22⋊C4—216no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—44no—
95C22×C22⋊C4—44no—
96C2×C22⋊C8—28no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—44no—
101C2×C22⋊C8—44no—
102C2×C22⋊C8C2314yes—
103C2×#95—28no—
104C2×#95—22no—
105C2×#95—24no—
106C2×#95—44no—
107C2×#95C2314yes—
108#328—232no—
109C25—24no—
110C25—24no—
111C25—24no—
112C25—24no—
113C25—42no—
114C25—42no—
115C25C22⋊C411yescharacteristic
116C25C22⋊C412yes—
117C25C22⋊C411yescharacteristic
118C25SD1614yes—
119C25Q1614yes—
120C25C2×D412yes—
121C25C2×D411yescharacteristic
122C22×Q8—28no—
123C22×Q8—216no—
124C22×Q8—216no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—48no—
129C22×Q8—464no—
130C22×Q8—48no—
131C22×Q8—44no—
132C22×Q8—48no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—48no—
136C22×Q8—44no—
137C22×Q8C22⋊C418yescomplemented
138C22×Q8C22×C414yes—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—28no—
143C23×C4—28no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—28no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—464no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C22⋊C412yes—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C412yes—
190C23×C4C22⋊C412yes—
191C23×C4C22⋊C412yes—
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C412yes—
194C23×C4C22×C412yes—
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C412yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D418yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C22×C8—24no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8—44no—
210C22⋊Q8—232no—
211C22⋊Q8—232no—
212C22⋊Q8—232no—
213C22⋊Q8—232no—
214C22⋊Q8—232no—
215C22⋊Q8—216no—
216C22⋊Q8—28no—
217C22⋊Q8—216no—
218C22⋊Q8—216no—
219C22⋊Q8—28no—
220C22⋊Q8—216no—
221C22⋊Q8C22⋊C4116yescomplemented
222C22⋊Q8C2×D4116yes—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—232no—
226C2×C4⋊C4—232no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4C22⋊C418yescomplemented
237C2×C4⋊C4C22×C414yes—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—232no—
240C2×C22⋊C4—232no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—216no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—216no—
246C2×C22⋊C4—432no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C22⋊C8—28no—
250C22⋊C8—48no—
251C22⋊C8—48no—
252C22⋊C8—44no—
253#95—28no—
254#95—48no—
255#95—44no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—24no—
265C24—24no—
266C24—216no—
267C24—216no—
268C24—24no—
269C24—22no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—22no—
274C24—24no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24—216no—
281C24—216no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—48no—
286C24—48no—
287C24—42no—
288C24—416no—
289C24—416no—
290C24—416no—
291C24—416no—
292C24#9914yes—
293C24#10314yes—
294C24#10318yes—
295C24#10314yes—
296C24#10414yes—
297C24C2×C22⋊C412yes—
298C24C2×C22⋊C411yescharacteristic
299C24C2×C22⋊C412yes—
300C24C2×C22⋊C411yescharacteristic
301C24#12014yes—
302C24C2×SD1614yes—
303C24C2×SD1612yes—
304C24C2×Q1614yes—
305C24C2×Q1612yes—
306C24#13714yes—
307C24C22×D411yescharacteristic
308C2×Q8—216no—
309C2×Q8—24no—
310C2×Q8—28no—
311C2×Q8—464no—
312C2×Q8—464no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—48no—
316C2×Q8—432no—
317C2×Q8—416no—
318C2×Q8—416no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—48no—
322C2×Q8C2×C22⋊C418yescomplemented
323C22×C4—232no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—416no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—416no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—432no—
383C22×C4—432no—
384C22×C4—432no—
385C22×C4—432no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—432no—
389C22×C4—48no—
390C22×C4—432no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—416no—
400C22×C4—44no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4#9914yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C414yes—
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C414yes—
417C22×C4#12014yes—
418C22×C4#12014yes—
419C22×C4#12014yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D414yes—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—88no—
427C4⋊C4—216no—
428C4⋊C4—232no—
429C4⋊C4—24no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4C2×C22⋊C418yescomplemented
434C22⋊C4—232no—
435C22⋊C4—216no—
436C22⋊C4—216no—
437C22⋊C4—432no—
438C23—22no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—22no—
454C23—28no—
455C23—28no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—216no—
474C23—28no—
475C23—22no—
476C23—28no—
477C23—216no—
478C23—28no—
479C23—48no—
480C23—44no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—48no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—464no—
489C23—464no—
490C23—48no—
491C23—48no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23#328116yescomplemented
497C23#37912yes—
498C23#37912yes—
499C23C2×#9914yes—
500C23C2×#9912yes—
501C23C2×#9912yes—
502C23C2×#10314yes—
503C23C2×#10314yes—
504C23C2×#10312yes—
505C23C2×#10312yes—
506C23#41912yes—
507C23#41912yes—
508C23C2×#10414yes—
509C23C2×#10412yes—
510C23#42112yes—
511C23#44818yes—
512C23#45118yes—
513C23C22×C22⋊C411yescharacteristic
514C23C2×#12012yes—
515C23C22×SD1611yescharacteristic
516C23C22×Q1611yescharacteristic
517C23C2×#13712yes—
518Q8—48no—
519Q8—48no—
520Q8—416no—
521Q8—432no—
522C2×C4—216no—
523C2×C4—216no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—432no—
555C2×C4—432no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—84no—
567C2×C4—832no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4#37912yes—
572C2×C4#37912yes—
573C2×C4C2×#9914yes—
574C2×C4C22×C22⋊C412yes—
575C2×C4C2×#12012yes—
576C8—84no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—216no—
591C22—232no—
592C22—216no—
593C22—22no—
594C22—24no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—24no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—416no—
604C22—416no—
605C22—432no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—416no—
610C22—42no—
611C22C2×#32814yes—
612C22C2×#32818yescomplemented
613C22#150814yes—
614C22#151114yes—
615C22#152014yes—
616C22#178114yes—
617C22#178714yes—
618C22#178714yes—
619C22#178914yes—
620C22C2×#37911yescharacteristic
621C22C22×#9911yescharacteristic
622C22C22×#9911yescharacteristic
623C22C22×#10311yescharacteristic
624C22C2×#41911yescharacteristic
625C22C22×#10411yescharacteristic
626C22C2×#42111yescharacteristic
627C22C2×#44812yes—
628C22C2×#45112yes—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—22no—
634C4—42no—
635C4—48no—
636C4—48no—
637C4—88no—
638C2—22no—
639C2—22no—
640C2—28no—
641C2—24no—
642C2—44no—
643C2#832918yesdirect product factor
644C2C22×#32811yescharacteristic
645C2C2×#150811yescharacteristic
646C2C2×#151111yescharacteristic
647C2C2×#152011yescharacteristic
648C2C2×#178111yescharacteristic
649C2C2×#178711yescharacteristic
650C2C2×#178911yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#328C2semidirect
5. #8329C2direct (N × H)
6. C24×Q8C4semidirect
13. C22×C22⋊Q8C4semidirect
14. C22×C22⋊Q8C4semidirect
18. C23×C4⋊C4C4semidirect
29. C2×#328C22semidirect
39. C23×Q8C2×C4semidirect
74. C2×C22⋊Q8C2×C4semidirect
86. C22×C4⋊C4C2×C4semidirect
137. C22×Q8C22⋊C4semidirect
221. C22⋊Q8C22⋊C4semidirect
236. C2×C4⋊C4C22⋊C4semidirect
322. C2×Q8C2×C22⋊C4semidirect
433. C4⋊C4C2×C22⋊C4semidirect
496. C23#328semidirect
612. C22C2×#328semidirect
643. C2#8329direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#17792
C22×#3282
C2×C22⋊#952
#83292