Finite Group Databasebeta

← C2×#29489 #6464240

Subgroups — C2×#29489 #6464240

Order 512.

2144 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic, complemented
2C2×#2324C211yesmaximal, characteristic
3#29489C2116yesmaximal, direct product factor
4C2×#2075C211yesmaximal, characteristic, complemented
5C2×#2036C211yesmaximal, characteristic, complemented
6C2×#2034C211yesmaximal, characteristic
7C2×#2000C211yesmaximal, characteristic
8C2×#1973C211yesmaximal, characteristic, complemented
9C2×#1944C211yesmaximal, characteristic, complemented
10C2×#1934C211yesmaximal, characteristic, complemented
11C2×#1933C211yesmaximal, characteristic, complemented
12C2×#1927C211yesmaximal, characteristic, complemented
13C2×#1921C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1750C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2625—24no—
36#2625C2218yescomplemented
37C2×#400—21no—
38C2×#398—21no—
39C2×#397—21no—
40C2×#397—21no—
41#2324—24no—
42#2324C2218yes—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390—21no—
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#386—21no—
52C2×#385—21no—
53C2×#380C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2075C2218yescomplemented
57#2036C2218yescomplemented
58#2034C2218yes—
59#2000C2218yes—
60#1973C2218yescomplemented
61#1944C2218yescomplemented
62#1934C2218yescomplemented
63#1933C2218yescomplemented
64#1927C2218yescomplemented
65#1921C2218yescomplemented
66#1877C2218yescomplemented
67#1862C2218yescomplemented
68#1750C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—42no—
85C22×SD16C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8—42no—
93C2×C22⋊Q8C2314yes—
94C2×#121—24no—
95C2×#121—22no—
96C2×#121—24no—
97C2×#121C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4C2314yes—
102C2×#117—42no—
103C2×#117—42no—
104C2×#117D414yes—
105C2×#117C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C2×#107—24no—
131C2×#107—42no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106C2314yes—
136C2×#103—24no—
137C2×#103—24no—
138C2×#103—42no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102—24no—
143C2×#102—24no—
144C2×#102—42no—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8—24no—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97—42no—
153C2×#97C2314yes—
154C2×C4×C8—42no—
155C2×C4×C8C2314yes—
156#400—28no—
157#398—28no—
158#397—28no—
159#397—28no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#390—28no—
165#390—22no—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#388—28no—
174#386—28no—
175#385—28no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C25C2×D411yescharacteristic
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8C2412yes—
210C22×D4—28no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—82no—
219C22×D4—82no—
220C22×D4#4114yes—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4—82no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
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#4114yes—
262C22×C8#4114yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270#121—216no—
271#121—28no—
272#121—28no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276#117—48no—
277#117—44no—
278#117—44no—
279C2×C4⋊C4—21no—
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—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—82no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—28no—
344#107—44no—
345#106—28no—
346#103—28no—
347#103—28no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—44no—
352C22⋊C8—28no—
353C22⋊C8—28no—
354#97—44no—
355C4×C8—44no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
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—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×D4—216no—
397C2×D4—24no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—816no—
412C2×D4C2×#4114yes—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
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—24no—
431C22×C4—24no—
432C22×C4—24no—
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—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
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—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—816no—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4C2511yescharacteristic
486SD16—416no—
487C2×C8—28no—
488C2×C8—28no—
489C2×C8—22no—
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—44no—
500C2×C8—44no—
501C2×C8C2×#4114yes—
502C2×C8C2×#4114yes—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
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—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4C2×#4114yes—
527C22⋊C4—216no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—84no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—44no—
552C23—48no—
553C23—44no—
554C23—48no—
555C23—44no—
556C23—48no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23#52512yes—
572C23#53212yes—
573C23#53412yes—
574C23#57612yes—
575C23#57712yes—
576C23#57712yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—84no—
584D4—24no—
585D4—24no—
586D4—48no—
587D4—42no—
588D4—42no—
589D4—816no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
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—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—48no—
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—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—42no—
636C2×C4—42no—
637C2×C4—816no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4#52512yes—
645C2×C4#53212yes—
646C2×C4#53412yes—
647C2×C4#55114yes—
648C2×C4#55314yes—
649C2×C4#55314yes—
650C2×C4#55914yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C8—24no—
658C8—24no—
659C8—42no—
660C8—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—42no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—88no—
674C22—816no—
675C22—82no—
676C22—82no—
677C22#260214yes—
678C22#312314yes—
679C22#312414yes—
680C22#317814yes—
681C22#317914yes—
682C22#319814yes—
683C22#320114yes—
684C22C2×#52511yescharacteristic
685C22C2×#53211yescharacteristic
686C22C2×#53411yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C22C2×#57711yescharacteristic
690C22C2×#57811yescharacteristic
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#260214yes—
704C4C2×#55111yescharacteristic
705C4C2×#55311yescharacteristic
706C4C2×#55311yescharacteristic
707C4C2×#55911yescharacteristic
708C2—82no—
709C2—84no—
710C2#2948918yesdirect product factor
711C2C2×#260211yescharacteristic
712C2C2×#312311yescharacteristic
713C2C2×#312411yescharacteristic
714C2C2×#317811yescharacteristic
715C2C2×#317911yescharacteristic
716C2C2×#319811yescharacteristic
717C2C2×#320111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2625C2semidirect
3. #29489C2direct (N × H)
4. C2×#2075C2semidirect
5. C2×#2036C2semidirect
8. C2×#1973C2semidirect
9. C2×#1944C2semidirect
10. C2×#1934C2semidirect
11. C2×#1933C2semidirect
12. C2×#1927C2semidirect
13. C2×#1921C2semidirect
14. C2×#1877C2semidirect
15. C2×#1862C2semidirect
16. C2×#1750C2semidirect
36. #2625C22semidirect
56. #2075C22semidirect
57. #2036C22semidirect
60. #1973C22semidirect
61. #1944C22semidirect
62. #1934C22semidirect
63. #1933C22semidirect
64. #1927C22semidirect
65. #1921C22semidirect
66. #1877C22semidirect
67. #1862C22semidirect
68. #1750C22semidirect
710. C2#29489direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26252
C2×#23242
#294892
C2×#20752
C2×#20362
C2×#20342
C2×#20002
C2×#19732
C2×#19442
C2×#19342
C2×#19332
C2×#19272
C2×#19212
C2×#18772
C2×#18622
C2×#17502