Finite Group Databasebeta

← C2×#36498 #7630184

Subgroups — C2×#36498 #7630184

Order 512.

7414 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×#2854C212yesmaximal, complemented
3#36498C2132yesmaximal, direct product factor
4C2×#2535C211yesmaximal, characteristic, complemented
5C2×#2535C211yesmaximal, characteristic, complemented
6C2×#2597C214yesmaximal, complemented
7C2×#2497C212yesmaximal, complemented
8C2×#2457C214yesmaximal, complemented
9C2×#2428C212yesmaximal, complemented
10C2×#2456C214yesmaximal, complemented
11C2×#2364C214yesmaximal, complemented
12C2×#2357C211yesmaximal, characteristic, complemented
13C2×#2349C211yesmaximal, characteristic, complemented
14C2×C4×#127C211yesmaximal, characteristic, complemented
15C2×C4×#121C212yesmaximal, complemented
16C2×C42×D4C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#123C2214yes—
19C22×#121C2212yes—
20C22×#121C2214yes—
21C22×#121C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×C4×D4C2212yes—
27C22×C4×D4C2214yes—
28C22×C4×D4C2214yes—
29C23×C42C2212yes—
30#2864C22116yescomplemented
31#2854C22132yescomplemented
32#2597C22164yescomplemented
33#2535C22116yescomplemented
34#2535C22116yescomplemented
35#2497C22132yescomplemented
36#2457C22164yescomplemented
37#2456C22164yescomplemented
38#2428C22132yescomplemented
39C2×#400C2214yes—
40C2×#397C2212yes—
41C2×#397C2212yes—
42C2×#396C2214yes—
43C2×#394C2214yes—
44C2×#392C2212yes—
45C2×#392C2212yes—
46C2×#392C2214yes—
47#2364C22164yescomplemented
48#2357C22116yescomplemented
49#2349C22116yescomplemented
50C4×#127C22116yescomplemented
51C4×#121C22132yescomplemented
52C2×#390C2211yescharacteristic
53C2×#390C2212yes—
54C2×#390C2214yes—
55C2×#390C2211yescharacteristic
56C2×#389C2212yes—
57C2×#389C2212yes—
58C2×#389C2212yes—
59C2×#388C2214yes—
60C2×#388C2214yes—
61C2×#388C2214yes—
62C2×#388C2214yes—
63C2×#388C2214yes—
64C2×#387C2214yes—
65C2×#387C2214yes—
66C2×#387C2214yes—
67C2×#386C2214yes—
68C2×#386C2214yes—
69C2×#386C2214yes—
70C2×#386C2214yes—
71C2×#386C2214yes—
72C2×#385C2214yes—
73C2×#384C2211yescharacteristic
74C2×#382C2211yescharacteristic
75C2×#382C2211yescharacteristic
76C2×#381C2212yes—
77C2×#380C2212yes—
78C2×#380C2212yes—
79C2×#380C2212yes—
80C42×D4C22116yescomplemented
81C2×C4×C4⋊C4C2212yes—
82C2×C4×C4⋊C4C2211yescharacteristic
83C2×C4×C4⋊C4C2212yes—
84C2×C4×C22⋊C4C2214yes—
85C2×C4×C22⋊C4C2212yes—
86C2×C4×C22⋊C4C2212yes—
87C2×C4×C22⋊C4C2212yes—
88C2×C4×C22⋊C4C2212yes—
89C22×#95C2212yes—
90C2×C43C2211yescharacteristic
91C23×D4C2311yescharacteristic
92C23×D4C2311yescharacteristic
93C23×D4C2314yes—
94C23×D4C2314yes—
95C23×D4C2311yescharacteristic
96C23×D4C2314yes—
97C24×C4C2312yes—
98C24×C4C2312yes—
99C24×C4C2312yes—
100C24×C4C2312yes—
101C24×C4C2312yes—
102C24×C4C2314yes—
103C2×#127—28no—
104C2×#127C2318yes—
105C2×#123—416no—
106C2×#123C23132yes—
107C2×#121—216no—
108C2×#121—232no—
109C2×#121—48no—
110C2×#121C23132yes—
111C2×#121C23116yes—
112C2×#121C23116yes—
113C2×C4×D4—216no—
114C2×C4×D4—28no—
115C2×C4×D4—28no—
116C2×C4×D4—28no—
117C2×C4×D4—416no—
118C2×C4×D4—48no—
119C2×C4×D4—416no—
120C2×C4×D4C2318yes—
121C2×C4×D4C2318yes—
122C2×C4×D4C23116yes—
123C2×C4×D4C2318yes—
124C2×C4×D4C23132yes—
125C2×C4×D4C23116yes—
126C2×C4×D4C23132yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C42—48no—
152C22×C42C2311yescharacteristic
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157C22×C42C23116yes—
158C22×C42C2312yes—
159C22×C42C2312yes—
160C22×C42C2314yes—
161C22×C42C2311yescharacteristic
162#400C23132yes—
163#397C23116yes—
164#397C23116yes—
165#396C23132yes—
166#394C23132yes—
167#392C23132yes—
168#392C23116yes—
169#392C23116yes—
170#390C23132yes—
171#390C2318yes—
172#390C2318yes—
173#390C23116yes—
174#389C23116yes—
175#389C23116yes—
176#389C23116yes—
177#388C23132yes—
178#388C23132yes—
179#388C23132yes—
180#388C23132yes—
181#388C23132yes—
182#387C23132yes—
183#387C23132yes—
184#387C23132yes—
185#386C23132yes—
186#386C23132yes—
187#386C23132yes—
188#386C23132yes—
189#386C23132yes—
190#385C23132yes—
191#384C2318yes—
192#382C2318yes—
193#382C2318yes—
194#381C23116yes—
195#380C23116yes—
196#380C23116yes—
197#380C23116yes—
198C4×C4⋊C4C23116yes—
199C4×C4⋊C4C2318yes—
200C4×C4⋊C4C23116yes—
201C4×C22⋊C4C23132yes—
202C4×C22⋊C4C23116yes—
203C4×C22⋊C4C23116yes—
204C4×C22⋊C4C23116yes—
205C4×C22⋊C4C23116yes—
206C2×#95C23116yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2312yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2311yescharacteristic
215C2×#95C2312yes—
216C2×#95C2314yes—
217C2×#95C2311yescharacteristic
218C2×#95C2312yes—
219C2×#95C2312yes—
220C2×#95C2314yes—
221C43C2318yes—
222C25C2412yes—
223C25C2412yes—
224C25C2412yes—
225C22×D4—28no—
226C22×D4—216no—
227C22×D4—216no—
228C22×D4—216no—
229C22×D4—216no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—24no—
233C22×D4—216no—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—48no—
238C22×D4—48no—
239C22×D4—48no—
240C22×D4—48no—
241C22×D4#4118yes—
242C22×D4C2414yes—
243C22×D4C2414yes—
244C22×D4C24116yes—
245C22×D4C24116yes—
246C22×D4C2414yes—
247C22×D4C24116yes—
248C23×C4—28no—
249C23×C4—28no—
250C23×C4—28no—
251C23×C4—28no—
252C23×C4—28no—
253C23×C4—28no—
254C23×C4—28no—
255C23×C4—28no—
256C23×C4—28no—
257C23×C4—216no—
258C23×C4—44no—
259C23×C4—44no—
260C23×C4—48no—
261C23×C4—48no—
262C23×C4—48no—
263C23×C4—48no—
264C23×C4C2418yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2414yes—
270C23×C4C2414yes—
271C23×C4C2418yes—
272C23×C4C2418yes—
273C23×C4C2411yescharacteristic
274C23×C4C2412yes—
275C23×C4C2412yes—
276C23×C4C2411yescharacteristic
277C23×C4C2412yes—
278C23×C4C2418yes—
279C23×C4C2418yes—
280C23×C4C24116yes—
281C23×C4C2411yescharacteristic
282#127—232no—
283#123—464no—
284#121—264no—
285#121—2128no—
286#121—432no—
287C4×D4—264no—
288C4×D4—232no—
289C4×D4—232no—
290C4×D4—232no—
291C4×D4—464no—
292C4×D4—432no—
293C4×D4—464no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—216no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2418yes—
305C2×C4⋊C4C24116yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2418yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2418yes—
311C2×C4⋊C4C24116yes—
312C2×C4⋊C4C24116yes—
313C2×C4⋊C4C24116yes—
314C2×C22⋊C4—216no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4#41116yes—
328C2×C22⋊C4C24116yes—
329C2×C22⋊C4C24116yes—
330C2×C22⋊C4C2418yes—
331C2×C22⋊C4C2418yes—
332C2×C22⋊C4C2418yes—
333C2×C22⋊C4C2418yes—
334C2×C22⋊C4C24116yes—
335C2×C22⋊C4C24116yes—
336C2×C22⋊C4C24116yes—
337C2×C22⋊C4C24116yes—
338C2×C22⋊C4C24116yes—
339C2×C22⋊C4C24116yes—
340C2×C22⋊C4C24116yes—
341C2×C22⋊C4C2418yes—
342C2×C42—28no—
343C2×C42—24no—
344C2×C42—28no—
345C2×C42—28no—
346C2×C42—24no—
347C2×C42—432no—
348C2×C42—48no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42#4118yes—
352C2×C42C2418yes—
353C2×C42C2418yes—
354C2×C42C2418yes—
355C2×C42C2414yes—
356C2×C42C2414yes—
357C2×C42C24116yes—
358C2×C42C2418yes—
359C2×C42C2418yes—
360C2×C42C2414yes—
361#95C2414yes—
362#95C2418yes—
363#95C24116yes—
364#95C2414yes—
365#95C2418yes—
366#95C2418yes—
367#95C24116yes—
368#95C2418yes—
369#95C24116yes—
370#95C24116yes—
371#95C24116yes—
372#95C2418yes—
373#95C2418yes—
374#95C24116yes—
375C24—24no—
376C24—24no—
377C24—24no—
378C24—28no—
379C24—24no—
380C24—28no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—28no—
387C24—24no—
388C24—28no—
389C24—24no—
390C24C2×#4118yes—
391C24C2×#4118yes—
392C24C2514yes—
393C24C2514yes—
394C24C2514yes—
395C24C2511yescharacteristic
396C2×D4—232no—
397C2×D4—216no—
398C2×D4—232no—
399C2×D4—232no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—232no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—232no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—464no—
413C2×D4—416no—
414C2×D4—416no—
415C2×D4—464no—
416C2×D4—416no—
417C2×D4—832no—
418C2×D4C2×#41116yes—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—216no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—216no—
449C22×C4—216no—
450C22×C4—216no—
451C22×C4—216no—
452C22×C4—232no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—432no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—432no—
461C22×C4—432no—
462C22×C4—816no—
463C22×C4—816no—
464C22×C4—832no—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)18yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4C2514yes—
474C22×C4C2518yes—
475C22×C4C2518yes—
476C22×C4C2514yes—
477C22×C4C2514yes—
478C22×C4C2514yes—
479C22×C4C2512yes—
480C22×C4C2514yes—
481C22×C4C2512yes—
482C22×C4C2514yes—
483C22×C4C2514yes—
484C22×C4C2512yes—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—232no—
491C4⋊C4—216no—
492C4⋊C4—232no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C22⋊C4—232no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—232no—
501C22⋊C4—232no—
502C22⋊C4—232no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4C2×#41132yes—
509C42—216no—
510C42—28no—
511C42—28no—
512C42—216no—
513C42—216no—
514C42—48no—
515C42—48no—
516C42—416no—
517C42C2×#41116yes—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—24no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—216no—
529C23—216no—
530C23—232no—
531C23—216no—
532C23—216no—
533C23—24no—
534C23—28no—
535C23—24no—
536C23—28no—
537C23—24no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—416no—
545C23—48no—
546C23—48no—
547C23—416no—
548C23—48no—
549C23—416no—
550C23C22×#4118yes—
551C23C22×#4118yes—
552C23C22×#4112yes—
553C23C22×#4114yes—
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)14yes—
557C23C2611yescharacteristic
558D4—232no—
559D4—232no—
560D4—464no—
561D4—464no—
562D4—416no—
563D4—832no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—216no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—216no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—432no—
594C2×C4—432no—
595C2×C4—48no—
596C2×C4—416no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—416no—
601C2×C4—432no—
602C2×C4—832no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4#553116yes—
606C2×C4C22×#4118yes—
607C2×C4C22×#4114yes—
608C2×C4C22×#4114yes—
609C2×C4C22×#4118yes—
610C2×C4C22×#4114yes—
611C2×C4C22×#4114yes—
612C2×C4C2×H(5,𝔽2)18yes—
613C2×C4C2×H(5,𝔽2)14yes—
614C22—28no—
615C22—28no—
616C22—28no—
617C22—216no—
618C22—28no—
619C22—44no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—432no—
624C22—432no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—48no—
633C22—816no—
634C22#344414yes—
635C22C2×#55314yes—
636C22#348118yes—
637C22#350814yes—
638C22#351918yes—
639C22C23×#4112yes—
640C22C23×#4111yescharacteristic
641C22C23×#4111yescharacteristic
642C22C22×H(5,𝔽2)11yescharacteristic
643C22C22×H(5,𝔽2)12yes—
644C4—28no—
645C4—28no—
646C4—24no—
647C4—24no—
648C4—28no—
649C4—44no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—48no—
655C4C2×#55318yes—
656C2—48no—
657C2—48no—
658C2—84no—
659C2#3649818yesdirect product factor
660C2C2×#344411yescharacteristic
661C2C22×#55311yescharacteristic
662C2C2×#348112yes—
663C2C2×#351912yes—
664C2C2×#350811yescharacteristic
subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2864C2semidirect
2. C2×#2854C2semidirect
3. #36498C2direct (N × H)
4. C2×#2535C2semidirect
5. C2×#2535C2semidirect
6. C2×#2597C2semidirect
7. C2×#2497C2semidirect
8. C2×#2457C2semidirect
9. C2×#2428C2semidirect
10. C2×#2456C2semidirect
11. C2×#2364C2semidirect
12. C2×#2357C2semidirect
13. C2×#2349C2semidirect
14. C2×C4×#127C2semidirect
15. C2×C4×#121C2semidirect
16. C2×C42×D4C2semidirect
30. #2864C22semidirect
31. #2854C22semidirect
32. #2597C22semidirect
33. #2535C22semidirect
34. #2535C22semidirect
35. #2497C22semidirect
36. #2457C22semidirect
37. #2456C22semidirect
38. #2428C22semidirect
47. #2364C22semidirect
48. #2357C22semidirect
49. #2349C22semidirect
50. C4×#127C22semidirect
51. C4×#121C22semidirect
80. C42×D4C22semidirect
659. C2#36498direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#28542
#364982
C2×#25352
C2×#25972
C2×#24972
C2×#24572
C2×#24282
C2×#24562
C2×#23642
C2×#23572
C2×#23492
C2×C4×#1272
C2×C4×#1212
C2×C42×D42