Finite Group Databasebeta

← C2×#30065 #6434733

Subgroups — C2×#30065 #6434733

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2861C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic, complemented
3#30065C2116yesmaximal, direct product factor
4C2×#2097C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2059C211yesmaximal, characteristic, complemented
7C2×#2059C211yesmaximal, characteristic, complemented
8C2×#2042C211yesmaximal, characteristic
9C2×#2042C211yesmaximal, characteristic
10C2×#2026C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic
12C2×#1928C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic
14C2×#1927C211yesmaximal, characteristic, complemented
15C2×#1925C211yesmaximal, characteristic
16C2×#1839C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32#2861—24no—
33#2861C2218yes—
34C4×#125—24no—
35C4×#125C2218yescomplemented
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#388—21no—
45C2×#388—21no—
46C2×#387—21no—
47C2×#387—21no—
48C2×#382C2211yescharacteristic
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C43C2211yescharacteristic
54#2097C2218yescomplemented
55#2092C2218yes—
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#2042C2218yes—
59#2042C2218yes—
60#2026C2218yescomplemented
61#2025C2218yes—
62#1928C2218yescomplemented
63#1927C2218yescomplemented
64#1926C2218yes—
65#1925C2218yes—
66#1839C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#336C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—42no—
88C2×#125—42no—
89C2×#125C2314yes—
90C2×#125C2314yes—
91C2×#124—22no—
92C2×#124—42no—
93C2×#124—42no—
94C2×#124C2314yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C2×#107—42no—
123C2×#107—42no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106—42no—
127C2×#106—42no—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—42no—
133C2×#103—42no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—42no—
141C2×#102—42no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146#391—22no—
147#391—22no—
148#391C2314yes—
149#391C2314yes—
150#390—22no—
151#390—22no—
152#390C2314yes—
153#390C2314yes—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#388—28no—
159#388—28no—
160#388—28no—
161#388—28no—
162#387—28no—
163#387—28no—
164#382—22no—
165#382C2314yes—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—28no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4C2314yes—
171C4×C4⋊C4C2314yes—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C43D414yes—
183C43C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2411yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
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×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241#125—216no—
242#125—28no—
243#125—28no—
244#125—44no—
245#125—44no—
246#125—48no—
247#124—44no—
248#124—44no—
249#124—48no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
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—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4#4114yes—
279C2×C4⋊C4#4114yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322#107—44no—
323#107—44no—
324#106—44no—
325#106—44no—
326#103—28no—
327#103—28no—
328#103—44no—
329#103—44no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344C24—22no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24C22×D411yescharacteristic
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4#13514yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8—44no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4C2×#4114yes—
478C4⋊C4C2×#4114yes—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C42—24no—
486C42—24no—
487C42—24no—
488C42—28no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—44no—
496C42—44no—
497C42—44no—
498C42—48no—
499C42—48no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23#52512yes—
515C23#53512yes—
516C23#53612yes—
517C23C2×#13512yes—
518C23#57512yes—
519C23#57512yes—
520C23#57512yes—
521C23C23×D411yescharacteristic
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—84no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—84no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4#52512yes—
584C2×C4#53512yes—
585C2×C4#53612yes—
586C2×C4#54214yes—
587C2×C4#54214yes—
588C2×C4#54214yes—
589C2×C4#54214yes—
590C2×C4C2×#13514yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+411yescharacteristic
596C2×C4C2×2−1+411yescharacteristic
597C8—42no—
598C8—42no—
599C8—42no—
600C8—42no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—168no—
609C22#263214yes—
610C22#310214yes—
611C22#311114yes—
612C22#321214yes—
613C22#321914yes—
614C22#323214yes—
615C22#324014yes—
616C22C2×#52511yescharacteristic
617C22C2×#53511yescharacteristic
618C22C2×#53611yescharacteristic
619C22C22×#13511yescharacteristic
620C22C2×#57511yescharacteristic
621C22C2×#57511yescharacteristic
622C22C2×#57511yescharacteristic
623C4—24no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#263214yes—
637C4C2×#54211yescharacteristic
638C4C2×#54211yescharacteristic
639C4C2×#54211yescharacteristic
640C4C2×#54211yescharacteristic
641C2—162no—
642C2#3006518yesdirect product factor
643C2C2×#263211yescharacteristic
644C2C2×#310211yescharacteristic
645C2C2×#311111yescharacteristic
646C2C2×#321211yescharacteristic
647C2C2×#321911yescharacteristic
648C2C2×#323211yescharacteristic
649C2C2×#324011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#125C2semidirect
3. #30065C2direct (N × H)
4. C2×#2097C2semidirect
6. C2×#2059C2semidirect
7. C2×#2059C2semidirect
10. C2×#2026C2semidirect
12. C2×#1928C2semidirect
14. C2×#1927C2semidirect
16. C2×#1839C2semidirect
35. C4×#125C22semidirect
54. #2097C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
60. #2026C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
66. #1839C22semidirect
642. C2#30065direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28612
C2×C4×#1252
#300652
C2×#20972
C2×#20922
C2×#20592
C2×#20422
C2×#20262
C2×#20252
C2×#19282
C2×#19262
C2×#19272
C2×#19252
C2×#18392