Finite Group Databasebeta

← C2×#30363 #6433926

Subgroups — C2×#30363 #6433926

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#30363C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2097C211yesmaximal, characteristic, complemented
6C2×#2076C211yesmaximal, characteristic
7C2×#2077C211yesmaximal, characteristic
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2036C211yesmaximal, characteristic
10C2×#2035C211yesmaximal, characteristic
11C2×#2026C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1943C211yesmaximal, characteristic, complemented
14C2×#1935C211yesmaximal, characteristic, complemented
15C2×#1935C211yesmaximal, characteristic, complemented
16C2×#1749C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31C22×#97C2211yescharacteristic
32#2855—24no—
33#2855C2218yes—
34#2851—24no—
35#2851C2218yes—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#387—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×#385—21no—
48C2×#385—21no—
49C2×#382C2211yescharacteristic
50C2×#381—21no—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54#2101C2218yescomplemented
55#2097C2218yescomplemented
56#2077C2218yes—
57#2076C2218yes—
58#2059C2218yescomplemented
59#2036C2218yes—
60#2035C2218yes—
61#2026C2218yescomplemented
62#1944C2218yescomplemented
63#1943C2218yescomplemented
64#1935C2218yescomplemented
65#1935C2218yescomplemented
66#1749C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4D411yescharacteristic
79C24×C4D411yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#125—22no—
86C2×#125—42no—
87C2×#125—42no—
88C2×#125C2314yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—42no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8C2314yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
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×C42—42no—
112C22×C42—42no—
113C22×C42D414yes—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C2×#107—24no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—24no—
124C2×#106—42no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—42no—
130C2×#103—42no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8—24no—
136C2×C22⋊C8—24no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97—42no—
140C2×#97—42no—
141C2×#97C2314yes—
142C2×#97C2314yes—
143#391—22no—
144#391—22no—
145#391C2314yes—
146#391C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#387—28no—
156#387—28no—
157#387—28no—
158#387—28no—
159#385—28no—
160#385—28no—
161#385—28no—
162#382—22no—
163#382C2314yes—
164#381—28no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2314yes—
170#380C2314yes—
171C2×#95—21no—
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—
182C2×#95—21no—
183C2×#95—21no—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8C2412yes—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D412yes—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8#4114yes—
244C22×C8#4114yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249#125—44no—
250#125—44no—
251#125—48no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254C22⋊Q8—48no—
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—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
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—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—48no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2412yes—
319C2×C42C2412yes—
320#107—28no—
321#107—44no—
322#106—28no—
323#106—44no—
324#103—28no—
325#103—28no—
326#103—44no—
327#103—44no—
328C22⋊C8—28no—
329C22⋊C8—28no—
330#97—44no—
331#97—44no—
332#95—24no—
333#95—24no—
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—
344#95—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24#13714yes—
352C24C22×D412yes—
353C24C22×D411yescharacteristic
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
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—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
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—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C42−1+412yes—
446C22×C42−1+412yes—
447C22×C4C2511yescharacteristic
448C2×C8—28no—
449C2×C8—28no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8C2×#4114yes—
463C2×C8C2×#4114yes—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—44no—
503C42—44no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—28no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23#53212yes—
518C23#53412yes—
519C23#53612yes—
520C23C2×#13714yes—
521C23C2×#13712yes—
522C23#57512yes—
523C23#57512yes—
524C23#57512yes—
525C23C23×D411yescharacteristic
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—84no—
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—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—416no—
547C2×C4—416no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
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—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—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#53212yes—
591C2×C4#53412yes—
592C2×C4#53612yes—
593C2×C4#53914yes—
594C2×C4#53914yes—
595C2×C4#54214yes—
596C2×C4#54214yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C8—24no—
604C8—24no—
605C8—42no—
606C8—42no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—416no—
614C22—42no—
615C22#266314yes—
616C22#318114yes—
617C22#318314yes—
618C22#319614yes—
619C22#319714yes—
620C22#323614yes—
621C22#324014yes—
622C22C2×#53211yescharacteristic
623C22C2×#53411yescharacteristic
624C22C2×#53611yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57511yescharacteristic
628C22C2×#57511yescharacteristic
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4#266314yes—
643C4C2×#53911yescharacteristic
644C4C2×#53911yescharacteristic
645C4C2×#54211yescharacteristic
646C4C2×#54211yescharacteristic
647C2—44no—
648C2#3036318yesdirect product factor
649C2C2×#266311yescharacteristic
650C2C2×#318111yescharacteristic
651C2C2×#318311yescharacteristic
652C2C2×#319611yescharacteristic
653C2C2×#319711yescharacteristic
654C2C2×#323611yescharacteristic
655C2C2×#324011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #30363C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2097C2semidirect
8. C2×#2059C2semidirect
11. C2×#2026C2semidirect
12. C2×#1944C2semidirect
13. C2×#1943C2semidirect
14. C2×#1935C2semidirect
15. C2×#1935C2semidirect
54. #2101C22semidirect
55. #2097C22semidirect
58. #2059C22semidirect
61. #2026C22semidirect
62. #1944C22semidirect
63. #1943C22semidirect
64. #1935C22semidirect
65. #1935C22semidirect
648. C2#30363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#28512
#303632
C2×#21012
C2×#20972
C2×#20762
C2×#20772
C2×#20592
C2×#20362
C2×#20352
C2×#20262
C2×#19442
C2×#19432
C2×#19352
C2×#17492