Finite Group Databasebeta

← C2×#28996 #6460295

Subgroups — C2×#28996 #6460295

Order 512.

2222 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic, complemented
2C2×#2420C211yesmaximal, characteristic
3#28996C2116yesmaximal, direct product factor
4C2×#2075C211yesmaximal, characteristic, complemented
5C2×#2074C211yesmaximal, characteristic, complemented
6C2×#2023C211yesmaximal, characteristic, complemented
7C2×#2000C211yesmaximal, characteristic
8C2×#1993C211yesmaximal, characteristic, complemented
9C2×#1973C211yesmaximal, characteristic, complemented
10C2×#1933C211yesmaximal, characteristic, complemented
11C2×#1921C211yesmaximal, characteristic, complemented
12C2×#1895C211yesmaximal, characteristic
13C2×#1894C211yesmaximal, characteristic, complemented
14C2×#1877C211yesmaximal, characteristic, complemented
15C2×C4×#102C211yesmaximal, characteristic, complemented
16C2×#1753C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2492—24no—
34#2492C2218yescomplemented
35#2420—24no—
36#2420C2218yes—
37C2×#400—21no—
38C2×#397—21no—
39C2×#394—21no—
40C2×#393—21no—
41C2×#391—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#386—21no—
48C2×#382C2211yescharacteristic
49C2×#381—21no—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2075C2218yescomplemented
55#2074C2218yescomplemented
56#2023C2218yescomplemented
57#2000C2218yes—
58#1993C2218yescomplemented
59#1973C2218yescomplemented
60#1933C2218yescomplemented
61#1921C2218yescomplemented
62#1895C2218yes—
63#1894C2218yescomplemented
64#1877C2218yescomplemented
65C4×#102C2218yescomplemented
66#1753C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×D4—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C22×M16—24no—
88C22×M16—42no—
89C22×M16—42no—
90C22×M16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8C2314yes—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—42no—
102C2×C22⋊Q8C2314yes—
103C2×#121—22no—
104C2×#121—24no—
105C2×#121—42no—
106C2×#121C2314yes—
107C2×C4×D4—24no—
108C2×C4×D4—22no—
109C2×C4×D4—42no—
110C2×C4×D4C2314yes—
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×C4⋊C4C2311yescharacteristic
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×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#107—24no—
133C2×#107C2314yes—
134C2×#103—24no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102—24no—
141C2×#102—24no—
142C2×#102—42no—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8—24no—
147C2×C22⋊C8C2314yes—
148C2×#97—42no—
149C2×#97C2314yes—
150C2×C4×C8—42no—
151C2×C4×C8C2314yes—
152#400—28no—
153#397—28no—
154#394—28no—
155#393—28no—
156#391—22no—
157#391—22no—
158#391—28no—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390C2314yes—
163#389—22no—
164#389C2314yes—
165#388—28no—
166#386—28no—
167#382—22no—
168#382C2314yes—
169#381—28no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4D414yes—
177C4×C22⋊C4C2314yes—
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—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
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—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—44no—
211C22×Q8—44no—
212C22×Q8—82no—
213C22×Q8C2412yes—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—82no—
224C22×D4#4114yes—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—416no—
255C2×M16—28no—
256C2×M16—44no—
257C2×M16—44no—
258C2×M16—416no—
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×C8—22no—
268C22×C8—22no—
269C22×C8#4114yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278C22⋊Q8—28no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281#121—28no—
282#121—44no—
283#121—48no—
284C4×D4—28no—
285C4×D4—44no—
286C4×D4—48no—
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—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4#4114yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
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—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4C2×D414yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—28no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—44no—
352C22⋊C8—28no—
353#97—44no—
354C4×C8—44no—
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—
364#95—24no—
365#95C2×D414yes—
366#95C2×D414yes—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24C22×D412yes—
383C24C22×D411yescharacteristic
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—84no—
400C2×Q8—816no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—84no—
416C2×D4—816no—
417C2×D4C2×#4114yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
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—24no—
435C22×C4—24no—
436C22×C4—24no—
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—22no—
448C22×C4—22no—
449C22×C4—48no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
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—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—816no—
477C22×C4#13714yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4C2511yescharacteristic
491SD16—416no—
492M16—416no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—44no—
506C2×C8—44no—
507C2×C8C2×#4114yes—
508C2×C8C2×#4114yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—84no—
529C4⋊C4C2×#4114yes—
530C22⋊C4—28no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—84no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—48no—
561C23—48no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—48no—
568C23—42no—
569C23—42no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23#53212yes—
575C23#54612yes—
576C23#54612yes—
577C23C2×#13712yes—
578C23#57512yes—
579C23#57612yes—
580C23#57712yes—
581C23C23×D411yescharacteristic
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—816no—
587Q8—84no—
588D4—24no—
589D4—24no—
590D4—42no—
591D4—42no—
592D4—48no—
593D4—816no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—48no—
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—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—816no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#52914yes—
648C2×C4#53212yes—
649C2×C4#53814yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#55114yes—
653C2×C4#55914yes—
654C2×C4C2×#13714yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C8—24no—
662C8—24no—
663C8—42no—
664C8—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—82no—
676C22—82no—
677C22—88no—
678C22—816no—
679C22—82no—
680C22—82no—
681C22#255714yes—
682C22#317214yes—
683C22#317914yes—
684C22#331114yes—
685C22#331414yes—
686C22#332214yes—
687C22#332914yes—
688C22C2×#53211yescharacteristic
689C22C2×#54611yescharacteristic
690C22C2×#54611yescharacteristic
691C22C22×#13711yescharacteristic
692C22C2×#57511yescharacteristic
693C22C2×#57611yescharacteristic
694C22C2×#57711yescharacteristic
695C4—24no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4#255714yes—
708C4C2×#52911yescharacteristic
709C4C2×#53811yescharacteristic
710C4C2×#55111yescharacteristic
711C4C2×#55911yescharacteristic
712C2—82no—
713C2—84no—
714C2#2899618yesdirect product factor
715C2C2×#255711yescharacteristic
716C2C2×#317211yescharacteristic
717C2C2×#317911yescharacteristic
718C2C2×#331411yescharacteristic
719C2C2×#331111yescharacteristic
720C2C2×#332211yescharacteristic
721C2C2×#332911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2492C2semidirect
3. #28996C2direct (N × H)
4. C2×#2075C2semidirect
5. C2×#2074C2semidirect
6. C2×#2023C2semidirect
8. C2×#1993C2semidirect
9. C2×#1973C2semidirect
10. C2×#1933C2semidirect
11. C2×#1921C2semidirect
13. C2×#1894C2semidirect
14. C2×#1877C2semidirect
15. C2×C4×#102C2semidirect
16. C2×#1753C2semidirect
34. #2492C22semidirect
54. #2075C22semidirect
55. #2074C22semidirect
56. #2023C22semidirect
58. #1993C22semidirect
59. #1973C22semidirect
60. #1933C22semidirect
61. #1921C22semidirect
63. #1894C22semidirect
64. #1877C22semidirect
65. C4×#102C22semidirect
66. #1753C22semidirect
714. C2#28996direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#24202
#289962
C2×#20752
C2×#20742
C2×#20232
C2×#20002
C2×#19932
C2×#19732
C2×#19332
C2×#19212
C2×#18952
C2×#18942
C2×#18772
C2×C4×#1022
C2×#17532