Finite Group Databasebeta

← C2×#22156 #6375839

Subgroups — C2×#22156 #6375839

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C2×C4×#124C211yesmaximal, characteristic
3C2×#2295C211yesmaximal, characteristic, complemented
4#22156C2116yesmaximal, direct product factor
5C2×#1983C211yesmaximal, characteristic
6C2×#1973C211yesmaximal, characteristic
7C2×#1971C211yesmaximal, characteristic, complemented
8C2×#1949C211yesmaximal, characteristic, complemented
9C2×#1936C211yesmaximal, characteristic
10C2×#1935C211yesmaximal, characteristic, complemented
11C2×#1878C211yesmaximal, characteristic, complemented
12C2×#1877C211yesmaximal, characteristic
13C2×#1863C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1767C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×#464C4116yescomplemented
24C2×#464C2218yes—
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×#97C2211yescharacteristic
39C4×#124—24no—
40C4×#124C2218yes—
41#2295—24no—
42#2295C2218yescomplemented
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#388—21no—
50C2×#388—21no—
51C2×#387—21no—
52C2×#387—21no—
53C2×#382C2211yescharacteristic
54C2×C4×C4⋊C4—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4—21no—
57C2×C4×C22⋊C4—21no—
58C2×#376—21no—
59C2×C43C2211yescharacteristic
60#1983C2218yes—
61#1973C2218yes—
62#1971C2218yescomplemented
63#1949C2218yescomplemented
64#1936C2218yes—
65#1935C2218yescomplemented
66#1878C2218yescomplemented
67#1877C2218yes—
68#1864C2218yes—
69#1863C2218yescomplemented
70#1805C2218yescomplemented
71#1767C2218yescomplemented
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×Q16—42no—
83C22×Q16—42no—
84C22×Q16C2×C418yescomplemented
85C22×Q16C2314yes—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2×C418yes—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×#124—22no—
95C2×#124—24no—
96C2×#124C2×C418yes—
97C2×#124C2314yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—24no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8C2×C418yescomplemented
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—42no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#464C2×C4164yescomplemented
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#106—42no—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8C2×C418yescomplemented
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103C2×C418yescomplemented
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—42no—
154C2×#102—42no—
155C2×#102C2×C418yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#97—42no—
160C2×#97—42no—
161C2×#97C2314yes—
162C2×#97C2314yes—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#391C2314yes—
169#390—22no—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#388—28no—
176#388—28no—
177#387—28no—
178#387—28no—
179#382—22no—
180#382C2314yes—
181C4×C4⋊C4—28no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2314yes—
184C4×C22⋊C4—28no—
185C4×C22⋊C4—28no—
186#376—28no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C43D414yes—
199C43C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8C22×C414yes—
220C22×Q8C22×C414yes—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C22×C414yes—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×Q16—416no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16C22×C4116yescomplemented
253C2×SD16—416no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16C22×C4116yes—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C22×C414yes—
268C22×C8C22×C414yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273#124—216no—
274#124—28no—
275#124C22×C4116yes—
276C4×Q8—216no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×Q8C22×C4116yescomplemented
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C22×C414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C22×C414yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D412yes—
345C2×C42#4114yes—
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349#107—44no—
350#106—44no—
351C4⋊C8—28no—
352C4⋊C8C22×C4116yescomplemented
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103C22×C4116yescomplemented
358#102—44no—
359#102—44no—
360#102C22×C4116yes—
361#97—44no—
362#97—44no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8C23×C414yes—
408C2×Q8C23×C414yes—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4C23×C414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—816no—
469C22×C4C4×D414yes—
470C22×C4C4×D414yes—
471C22×C4#13714yes—
472C22×C4C23×C412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C42−1+412yes—
485C22×C4C2511yescharacteristic
486Q16—416no—
487SD16—416no—
488C2×C8—28no—
489C2×C8—28no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—44no—
501C2×C8—44no—
502C2×C8C23×C414yes—
503C2×C8C23×C414yes—
504C4⋊C4—216no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4C23×C414yes—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C42—216no—
529C42—24no—
530C42—24no—
531C42—28no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—48no—
540C42—48no—
541C42C4×D418yes—
542C42C23×C414yes—
543C42C2×#4114yes—
544C23—24no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—48no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23C2×C4×D412yes—
559C23#53512yes—
560C23#53712yes—
561C23C2×#13712yes—
562C23#57512yes—
563C23#57512yes—
564C23#57512yes—
565C23C23×D411yescharacteristic
566Q8—24no—
567Q8—24no—
568Q8—24no—
569Q8—24no—
570Q8—48no—
571Q8—48no—
572Q8—42no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576D4—24no—
577D4—24no—
578D4—48no—
579D4—42no—
580D4—42no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—816no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×C4×D414yes—
632C2×C4C2×C4×D414yes—
633C2×C4C2×C4×D412yes—
634C2×C4#51814yes—
635C2×C4#52014yes—
636C2×C4#53512yes—
637C2×C4#53712yes—
638C2×C4#54014yes—
639C2×C4#54214yes—
640C2×C4C2×#13714yes—
641C2×C4C24×C411yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C8—24no—
648C8—24no—
649C8—42no—
650C8—42no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—816no—
657C22—82no—
658C22—82no—
659C22#232414yes—
660C22#294314yes—
661C22C4×#13714yes—
662C22#321514yes—
663C22#321914yes—
664C22#325214yes—
665C22#325614yes—
666C22C22×C4×D411yescharacteristic
667C22C2×#53511yescharacteristic
668C22C2×#53711yescharacteristic
669C22C22×#13711yescharacteristic
670C22C2×#57511yescharacteristic
671C22C2×#57511yescharacteristic
672C22C2×#57511yescharacteristic
673C4—24no—
674C4—24no—
675C4—24no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#232414yes—
687C4C2×#51811yescharacteristic
688C4C2×#52011yescharacteristic
689C4C2×#54011yescharacteristic
690C4C2×#54211yescharacteristic
691C2—84no—
692C2#2215618yesdirect product factor
693C2C2×#232411yescharacteristic
694C2C2×#294311yescharacteristic
695C2C2×C4×#13711yescharacteristic
696C2C2×#321511yescharacteristic
697C2C2×#321911yescharacteristic
698C2C2×#325211yescharacteristic
699C2C2×#325611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×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
3. C2×#2295C2semidirect
4. #22156C2direct (N × H)
7. C2×#1971C2semidirect
8. C2×#1949C2semidirect
10. C2×#1935C2semidirect
11. C2×#1878C2semidirect
13. C2×#1863C2semidirect
15. C2×#1805C2semidirect
16. C2×#1767C2semidirect
23. C2×#464C4semidirect
42. #2295C22semidirect
62. #1971C22semidirect
63. #1949C22semidirect
65. #1935C22semidirect
66. #1878C22semidirect
69. #1863C22semidirect
70. #1805C22semidirect
71. #1767C22semidirect
84. C22×Q16C2×C4semidirect
103. C2×C4×Q8C2×C4semidirect
134. #464C2×C4semidirect
140. C2×C4⋊C8C2×C4semidirect
147. C2×#103C2×C4semidirect
252. C2×Q16C22×C4semidirect
281. C4×Q8C22×C4semidirect
352. C4⋊C8C22×C4semidirect
357. #103C22×C4semidirect
692. C2#22156direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4642
C2×C4×#1242
C2×#22952
#221562
C2×#19832
C2×#19732
C2×#19712
C2×#19492
C2×#19362
C2×#19352
C2×#18782
C2×#18772
C2×#18632
C2×#18642
C2×#18052
C2×#17672