Finite Group Databasebeta

← C2×#31663 #7505340

Subgroups — C2×#31663 #7505340

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic, complemented
3#31663C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2074C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10C2×#1960C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic
16C2×#1805C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2563—24no—
34#2563C2218yes—
35#2446—24no—
36#2446C2218yescomplemented
37C2×#400—21no—
38C2×#398—21no—
39C2×#393—21no—
40C2×#393—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#387—21no—
48C2×#387—21no—
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380—21no—
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2074C2218yescomplemented
56#2055C2218yes—
57#2024C2218yescomplemented
58#2024C2218yescomplemented
59#2013C2218yes—
60#1960C2218yescomplemented
61#1936C2218yes—
62#1917C2218yescomplemented
63#1880C2218yes—
64#1877C2218yes—
65#1877C2218yes—
66#1805C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8C2314yes—
101C2×C4⋊Q8C2314yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8C2314yes—
106C2×C4×D4—22no—
107C2×C4×D4—42no—
108C2×C4×D4—42no—
109C2×C4×D4C2314yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
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×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#103—42no—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#97—42no—
150C2×#97C2314yes—
151#400—28no—
152#398—28no—
153#393—28no—
154#393—28no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389C2314yes—
167#387—28no—
168#387—28no—
169#382—22no—
170#382—22no—
171#382C2314yes—
172#382C2314yes—
173#380—28no—
174#380—28no—
175C4×C4⋊C4D414yes—
176C4×C4⋊C4C2314yes—
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—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—21no—
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—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—44no—
216C22×Q8—82no—
217C22×Q8—82no—
218C22×Q8#4114yes—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4#4114yes—
228C22×D4C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×SD16—28no—
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—416no—
254C2×SD16—416no—
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×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—28no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—48no—
276C4⋊Q8—48no—
277C4×Q8—28no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
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—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
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—44no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2412yes—
341C4⋊C8—44no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—28no—
349#102—44no—
350#97—44no—
351#95—24no—
352#95—24no—
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#95C2×D414yes—
363#95C2×D414yes—
364C24—24no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—816no—
402C2×Q8C2×#4114yes—
403C2×D4—28no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4C2×#4114yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
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—48no—
442C22×C4—48no—
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—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—816no—
465C22×C4#13714yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480SD16—416no—
481SD16—416no—
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—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—44no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C4⋊C4—28no—
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—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—48no—
535C23—24no—
536C23—48no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—48no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23#53512yes—
550C23#54612yes—
551C23#54812yes—
552C23C2×#13712yes—
553C23#57512yes—
554C23#57712yes—
555C23#57712yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—42no—
560Q8—48no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—816no—
567Q8—84no—
568D4—24no—
569D4—24no—
570D4—42no—
571D4—48no—
572D4—42no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
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—44no—
594C2×C4—42no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
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—42no—
618C2×C4—816no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4#53512yes—
626C2×C4#53814yes—
627C2×C4#53914yes—
628C2×C4#54612yes—
629C2×C4#54812yes—
630C2×C4#55114yes—
631C2×C4#56214yes—
632C2×C4C2×#13714yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—42no—
640C8—42no—
641C8—42no—
642C8—42no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—82no—
647C22—82no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22#269614yes—
652C22#321514yes—
653C22#322214yes—
654C22#332114yes—
655C22#332914yes—
656C22#337214yes—
657C22#337614yes—
658C22C2×#53511yescharacteristic
659C22C2×#54611yescharacteristic
660C22C2×#54811yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57511yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57711yescharacteristic
665C4—24no—
666C4—42no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#269614yes—
679C4C2×#53811yescharacteristic
680C4C2×#53911yescharacteristic
681C4C2×#55111yescharacteristic
682C4C2×#56211yescharacteristic
683C2—84no—
684C2#3166318yesdirect product factor
685C2C2×#269611yescharacteristic
686C2C2×#321511yescharacteristic
687C2C2×#322211yescharacteristic
688C2C2×#332111yescharacteristic
689C2C2×#332911yescharacteristic
690C2C2×#337211yescharacteristic
691C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2446C2semidirect
3. #31663C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2074C2semidirect
7. C2×#2024C2semidirect
8. C2×#2024C2semidirect
10. C2×#1960C2semidirect
12. C2×#1917C2semidirect
16. C2×#1805C2semidirect
36. #2446C22semidirect
54. #2101C22semidirect
55. #2074C22semidirect
57. #2024C22semidirect
58. #2024C22semidirect
60. #1960C22semidirect
62. #1917C22semidirect
66. #1805C22semidirect
684. C2#31663direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#24462
#316632
C2×#21012
C2×#20742
C2×#20552
C2×#20242
C2×#20132
C2×#19602
C2×#19362
C2×#19172
C2×#18802
C2×#18772
C2×#18052