Finite Group Databasebeta

← C2×#29374 #6461453

Subgroups — C2×#29374 #6461453

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic, complemented
3#29374C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2097C211yesmaximal, characteristic, complemented
6C2×#2042C211yesmaximal, characteristic
7C2×#2015C211yesmaximal, characteristic
8C2×#1927C211yesmaximal, characteristic, complemented
9C2×#1925C211yesmaximal, characteristic
10C2×#1880C211yesmaximal, characteristic
11C2×#1878C211yesmaximal, characteristic, complemented
12C2×#1878C211yesmaximal, characteristic, complemented
13C2×#1877C211yesmaximal, characteristic
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×#1839C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C4×#124—24no—
34C4×#124C2218yes—
35#2295—24no—
36#2295C2218yescomplemented
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—21no—
51C2×C4×C22⋊C4—21no—
52C2×#376—21no—
53C2×C43C2211yescharacteristic
54#2101C2218yescomplemented
55#2097C2218yescomplemented
56#2042C2218yes—
57#2015C2218yes—
58#1927C2218yescomplemented
59#1925C2218yes—
60#1880C2218yes—
61#1878C2218yescomplemented
62#1878C2218yescomplemented
63#1877C2218yes—
64#1864C2218yes—
65#1863C2218yescomplemented
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×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×Q16—24no—
82C22×Q16—24no—
83C22×Q16—42no—
84C22×Q16C2314yes—
85C22×SD16—24no—
86C22×SD16—24no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#124—24no—
94C2×#124—24no—
95C2×#124—22no—
96C2×#124C2314yes—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2314yes—
105C2×C4×D4—22no—
106C2×C4×D4—42no—
107C2×C4×D4—42no—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C2×#107—42no—
134C2×#107C2314yes—
135C2×#106—42no—
136C2×#106C2314yes—
137C2×#103—24no—
138C2×#103—24no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153#391—22no—
154#391—22no—
155#391—22no—
156#391C2314yes—
157#391C2314yes—
158#391C2314yes—
159#390—22no—
160#390C2314yes—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#388—28no—
166#388—28no—
167#387—28no—
168#387—28no—
169#382—22no—
170#382C2314yes—
171C4×C4⋊C4—28no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4—28no—
175C4×C22⋊C4—28no—
176#376—28no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C43D414yes—
189C43C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8#4114yes—
214C22×Q8#4114yes—
215C22×Q8C2412yes—
216C22×Q8C2412yes—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4#4114yes—
224C22×D4C2412yes—
225C23×C4—21no—
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—44no—
235C23×C4—44no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—416no—
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—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—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—216no—
268#124—28no—
269#124—28no—
270C4×Q8—216no—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—48no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4#4114yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—44no—
344#106—44no—
345#103—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#103—44no—
350#102—28no—
351#102—28no—
352#102—44no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364C24—24no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8C2×#4114yes—
398C2×Q8C2×#4114yes—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
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—48no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
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—48no—
459C22×C4—816no—
460C22×C4#13514yes—
461C22×C4#13714yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4C2511yescharacteristic
476Q16—416no—
477SD16—416no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—216no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4C2×#4114yes—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—216no—
519C42—24no—
520C42—24no—
521C42—24no—
522C42—28no—
523C42—28no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—44no—
531C42—44no—
532C42—48no—
533C42—48no—
534C23—24no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23#52512yes—
549C23#52512yes—
550C23#53512yes—
551C23C2×#13512yes—
552C23C2×#13712yes—
553C23#57512yes—
554C23#57512yes—
555C23C23×D411yescharacteristic
556Q8—24no—
557Q8—24no—
558Q8—24no—
559Q8—24no—
560Q8—48no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566D4—24no—
567D4—24no—
568D4—48no—
569D4—42no—
570D4—42no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—816no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4#52512yes—
623C2×C4#52512yes—
624C2×C4#53512yes—
625C2×C4#55114yes—
626C2×C4#55114yes—
627C2×C4#55314yes—
628C2×C4#55314yes—
629C2×C4C2×#13514yes—
630C2×C4C2×#13714yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—816no—
649C22#259814yes—
650C22#310214yes—
651C22#310814yes—
652C22#311114yes—
653C22#311614yes—
654C22#321214yes—
655C22#321514yes—
656C22C2×#52511yescharacteristic
657C22C2×#52511yescharacteristic
658C22C2×#53511yescharacteristic
659C22C22×#13511yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57511yescharacteristic
663C4—24no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#259814yes—
677C4C2×#55111yescharacteristic
678C4C2×#55111yescharacteristic
679C4C2×#55311yescharacteristic
680C4C2×#55311yescharacteristic
681C2—84no—
682C2#2937418yesdirect product factor
683C2C2×#259811yescharacteristic
684C2C2×#310211yescharacteristic
685C2C2×#310811yescharacteristic
686C2C2×#311611yescharacteristic
687C2C2×#311111yescharacteristic
688C2C2×#321211yescharacteristic
689C2C2×#321511yescharacteristic
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×#2295C2semidirect
3. #29374C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2097C2semidirect
8. C2×#1927C2semidirect
11. C2×#1878C2semidirect
12. C2×#1878C2semidirect
14. C2×#1863C2semidirect
16. C2×#1839C2semidirect
36. #2295C22semidirect
54. #2101C22semidirect
55. #2097C22semidirect
58. #1927C22semidirect
61. #1878C22semidirect
62. #1878C22semidirect
65. #1863C22semidirect
66. #1839C22semidirect
682. C2#29374direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#22952
#293742
C2×#21012
C2×#20972
C2×#20422
C2×#20152
C2×#19272
C2×#19252
C2×#18802
C2×#18782
C2×#18772
C2×#18632
C2×#18642
C2×#18392