Finite Group Databasebeta

← C2×#33076 #6629549

Subgroups — C2×#33076 #6629549

Order 512.

2168 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×#2275C211yesmaximal, characteristic
3#33076C2116yesmaximal, direct product factor
4C2×#1949C211yesmaximal, characteristic, complemented
5C2×#1948C211yesmaximal, characteristic, complemented
6C2×#1919C211yesmaximal, characteristic, complemented
7C2×#1919C211yesmaximal, characteristic, complemented
8C2×#1920C211yesmaximal, characteristic, complemented
9C2×#1918C211yesmaximal, characteristic, complemented
10C2×#1864C211yesmaximal, characteristic
11C2×#1864C211yesmaximal, characteristic
12C2×#1864C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic
14C2×#1832C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×#1802C211yesmaximal, characteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30#2647—24no—
31#2647C2218yescomplemented
32C2×#400—21no—
33C2×#400—21no—
34C2×#398—21no—
35C2×#398—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42#2275—24no—
43#2275C2218yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4—21no—
50C2×#376C2211yescharacteristic
51C22×#95—21no—
52#1949C2218yescomplemented
53#1948C2218yescomplemented
54#1920C2218yescomplemented
55#1919C2218yescomplemented
56#1919C2218yescomplemented
57#1918C2218yescomplemented
58#1864C2218yes—
59#1864C2218yes—
60#1864C2218yes—
61#1862C2218yes—
62#1832C2218yescomplemented
63#1804C2218yes—
64#1802C2218yes—
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C24×C4—21no—
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—24no—
93C2×C4×D4Q818yes—
94C2×C4×D4C2314yes—
95C2×C4×D4C2314yes—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107C2314yes—
121C2×#106—24no—
122C2×#106—24no—
123C2×#106—24no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8—24no—
128C2×C4⋊C8—24no—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#102—24no—
132C2×#102—24no—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139#400—28no—
140#400—28no—
141#398—28no—
142#398—28no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389C2314yes—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#384—22no—
152#384C2314yes—
153#382—22no—
154#382C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4C2314yes—
159C4×C4⋊C4C2314yes—
160C4×C4⋊C4C2314yes—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—28no—
163C4×C22⋊C4—28no—
164#376D414yes—
165#376C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—28no—
176C2×#95D411yescharacteristic
177C2×#95D411yescharacteristic
178C2×#95D411yescharacteristic
179C2×#95D411yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4C2×Q814yes—
198C22×D4#4114yes—
199C22×D4#4114yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—82no—
215C23×C4—82no—
216C23×C4—82no—
217C23×C4—82no—
218C23×C4—82no—
219C23×C4—82no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4×D4—216no—
244C4×D4—216no—
245C4×D4—28no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—28no—
249C4×D4—216no—
250C4×D4—28no—
251C4×D4C2×Q8116yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C2×Q814yes—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—82no—
306C2×C42—82no—
307C2×C42—82no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×Q814yes—
312C2×C42#4114yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315C2×C42C2412yes—
316#107—28no—
317#106—28no—
318#106—28no—
319#106—28no—
320C4⋊C8—28no—
321C4⋊C8—28no—
322#102—28no—
323#102—28no—
324#102—28no—
325#102—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24C22×D411yescharacteristic
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4#125116yes—
358C2×D4C22×Q814yes—
359C2×D4C2×#4114yes—
360C2×D4C2×#4114yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—816no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—816no—
416C22×C4—816no—
417C22×C4C22⋊Q814yes—
418C22×C4C22⋊Q814yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q812yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2511yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8—44no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4C22×Q814yes—
474C4⋊C4C2×#4114yes—
475C4⋊C4C2×#4114yes—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C42—24no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—84no—
486C42—84no—
487C42—84no—
488C42C22⋊Q818yes—
489C42C22×Q814yes—
490C42C2×#4114yes—
491C23—216no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C2×C22⋊Q812yes—
506C23#52512yes—
507C23#52512yes—
508C23#57612yes—
509C23#57712yes—
510C23#57712yes—
511C23#57712yes—
512C23C23×D411yescharacteristic
513D4—216no—
514D4—24no—
515D4—24no—
516D4—24no—
517D4C2×#12518yes—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4C2×C22⋊Q814yes—
572C2×C4C2×C22⋊Q814yes—
573C2×C4C2×C22⋊Q812yes—
574C2×C4#52512yes—
575C2×C4#52512yes—
576C2×C4C2×#12514yes—
577C2×C4#52914yes—
578C2×C4#52914yes—
579C2×C4#53314yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C22—28no—
591C22—432no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22#285114yes—
599C22#308114yes—
600C22#308414yes—
601C22#311814yes—
602C22#311914yes—
603C22#312314yes—
604C22#312514yes—
605C22C22×C22⋊Q811yescharacteristic
606C22C2×#52511yescharacteristic
607C22C2×#52511yescharacteristic
608C22C2×#57611yescharacteristic
609C22C2×#57711yescharacteristic
610C22C2×#57711yescharacteristic
611C22C2×#57711yescharacteristic
612C4—24no—
613C4—24no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#285114yes—
626C4C22×#12511yescharacteristic
627C4C2×#52911yescharacteristic
628C4C2×#52911yescharacteristic
629C4C2×#53311yescharacteristic
630C2—48no—
631C2#3307618yesdirect product factor
632C2C2×#285111yescharacteristic
633C2C2×#308111yescharacteristic
634C2C2×#308411yescharacteristic
635C2C2×#311811yescharacteristic
636C2C2×#311911yescharacteristic
637C2C2×#312311yescharacteristic
638C2C2×#312511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2647C2semidirect
3. #33076C2direct (N × H)
4. C2×#1949C2semidirect
5. C2×#1948C2semidirect
6. C2×#1919C2semidirect
7. C2×#1919C2semidirect
8. C2×#1920C2semidirect
9. C2×#1918C2semidirect
14. C2×#1832C2semidirect
31. #2647C22semidirect
52. #1949C22semidirect
53. #1948C22semidirect
54. #1920C22semidirect
55. #1919C22semidirect
56. #1919C22semidirect
57. #1918C22semidirect
62. #1832C22semidirect
631. C2#33076direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#22752
#330762
C2×#19492
C2×#19482
C2×#19192
C2×#19202
C2×#19182
C2×#18642
C2×#18622
C2×#18322
C2×#18042
C2×#18022