Finite Group Databasebeta

← C2×#29510 #6461541

Subgroups — C2×#29510 #6461541

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic, complemented
3#29510C2116yesmaximal, direct product factor
4C2×#2084C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic, complemented
6C2×#2042C211yesmaximal, characteristic, complemented
7C2×#2012C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic
9C2×#1968C211yesmaximal, characteristic, complemented
10C2×#1935C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1880C211yesmaximal, characteristic, complemented
13C2×#1879C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1746C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×D8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2644—24no—
35#2644C2218yescomplemented
36C2×#397—21no—
37C2×#397—21no—
38C2×#397—21no—
39C2×#396—21no—
40#2326—24no—
41#2326C2218yescomplemented
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52C2×#385—21no—
53C2×#385—21no—
54C2×#384C2211yescharacteristic
55C2×#376C2211yescharacteristic
56#2084C2218yescomplemented
57#2071C2218yescomplemented
58#2042C2218yescomplemented
59#2012C2218yes—
60#1972C2218yes—
61#1968C2218yescomplemented
62#1935C2218yescomplemented
63#1934C2218yescomplemented
64#1880C2218yescomplemented
65#1879C2218yes—
66#1865C2218yescomplemented
67#1864C2218yescomplemented
68#1746C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—24no—
82C22×SD16—24no—
83C22×SD16—42no—
84C22×SD16C2314yes—
85C22×D8—24no—
86C22×D8—24no—
87C22×D8—42no—
88C22×D8C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#124—24no—
94C2×#124—24no—
95C2×#124—22no—
96C2×#124C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#106—42no—
132C2×#106—42no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#103—24no—
136C2×#103—24no—
137C2×#103—42no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102—24no—
142C2×#102—24no—
143C2×#102—42no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#97—24no—
152C2×#97C2314yes—
153C2×C4×C8—24no—
154C2×C4×C8C2314yes—
155#397—28no—
156#397—28no—
157#397—28no—
158#396—28no—
159#391—22no—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2314yes—
175#388—28no—
176#388—28no—
177#385—28no—
178#385—28no—
179#384—22no—
180#384C2314yes—
181#376D414yes—
182#376C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8#4114yes—
209C22×Q8C2412yes—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4#4114yes—
224C22×D4C2412yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—28no—
244C2×SD16—28no—
245C2×SD16—44no—
246C2×SD16—416no—
247C2×D8—28no—
248C2×D8—28no—
249C2×D8—44no—
250C2×D8—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8#4114yes—
262C22×C8#4114yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—216no—
268#124—28no—
269#124—28no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#106—44no—
345#106—44no—
346#103—28no—
347#103—28no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—44no—
352#102—44no—
353#102—44no—
354#97—28no—
355C4×C8—28no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95C2×D414yes—
362#95C2×D414yes—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24C22×D411yescharacteristic
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8C2×#4114yes—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4C2×#4114yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
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—48no—
447C22×C4—42no—
448C22×C4—42no—
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—84no—
465C22×C4—84no—
466C22×C4—816no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4C2511yescharacteristic
481SD16—416no—
482D8—416no—
483C2×C8—28no—
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8C2×#4114yes—
498C2×C8C2×#4114yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—48no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C22⋊C4—216no—
521C22⋊C4—216no—
522C22⋊C4—48no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C42—24no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—48no—
541C23—24no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#52512yes—
570C23#53212yes—
571C23#53512yes—
572C23#57612yes—
573C23#57712yes—
574C23#57712yes—
575C23#57712yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582D4—24no—
583D4—24no—
584D4—48no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—84no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—816no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#52512yes—
642C2×C4#53212yes—
643C2×C4#53512yes—
644C2×C4#55114yes—
645C2×C4#55114yes—
646C2×C4#55314yes—
647C2×C4#55314yes—
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C8—24no—
655C8—24no—
656C8—42no—
657C8—42no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—816no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—168no—
674C22#260314yes—
675C22#311914yes—
676C22#312514yes—
677C22#317314yes—
678C22#317914yes—
679C22#322114yes—
680C22#322414yes—
681C22C2×#52511yescharacteristic
682C22C2×#53211yescharacteristic
683C22C2×#53511yescharacteristic
684C22C2×#57611yescharacteristic
685C22C2×#57711yescharacteristic
686C22C2×#57711yescharacteristic
687C22C2×#57711yescharacteristic
688C4—42no—
689C4—42no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#260314yes—
701C4C2×#55111yescharacteristic
702C4C2×#55111yescharacteristic
703C4C2×#55311yescharacteristic
704C4C2×#55311yescharacteristic
705C2—84no—
706C2—162no—
707C2#2951018yesdirect product factor
708C2C2×#260311yescharacteristic
709C2C2×#311911yescharacteristic
710C2C2×#312511yescharacteristic
711C2C2×#317311yescharacteristic
712C2C2×#317911yescharacteristic
713C2C2×#322411yescharacteristic
714C2C2×#322111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#2644C2semidirect
2. C2×#2326C2semidirect
3. #29510C2direct (N × H)
4. C2×#2084C2semidirect
5. C2×#2071C2semidirect
6. C2×#2042C2semidirect
9. C2×#1968C2semidirect
10. C2×#1935C2semidirect
11. C2×#1934C2semidirect
12. C2×#1880C2semidirect
14. C2×#1865C2semidirect
15. C2×#1864C2semidirect
16. C2×#1746C2semidirect
35. #2644C22semidirect
41. #2326C22semidirect
56. #2084C22semidirect
57. #2071C22semidirect
58. #2042C22semidirect
61. #1968C22semidirect
62. #1935C22semidirect
63. #1934C22semidirect
64. #1880C22semidirect
66. #1865C22semidirect
67. #1864C22semidirect
68. #1746C22semidirect
707. C2#29510direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26442
C2×#23262
#295102
C2×#20842
C2×#20712
C2×#20422
C2×#20122
C2×#19722
C2×#19682
C2×#19352
C2×#19342
C2×#18802
C2×#18792
C2×#18652
C2×#18642
C2×#17462