Finite Group Databasebeta

← C2×#31500 #7504883

Subgroups — C2×#31500 #7504883

Order 512.

2086 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2497C211yesmaximal, characteristic, complemented
2C2×#2426C211yesmaximal, characteristic
3#31500C2116yesmaximal, direct product factor
4C2×#2086C211yesmaximal, characteristic, complemented
5C2×#2071C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2042C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic, complemented
9C2×#2034C211yesmaximal, characteristic
10C2×#1983C211yesmaximal, characteristic, complemented
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1893C211yesmaximal, characteristic
13C2×#1892C211yesmaximal, characteristic, complemented
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1850C211yesmaximal, characteristic, complemented
16C2×#1803C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2497—24no—
36#2497C2218yescomplemented
37#2426—24no—
38#2426C2218yes—
39C2×#400—21no—
40C2×#396—21no—
41C2×#396—21no—
42C2×#394—21no—
43C2×#390—21no—
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#387—21no—
51C2×#384C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2086C2218yescomplemented
57#2071C2218yescomplemented
58#2067C2218yescomplemented
59#2042C2218yescomplemented
60#2036C2218yescomplemented
61#2034C2218yes—
62#1983C2218yescomplemented
63#1920C2218yescomplemented
64#1893C2218yes—
65#1892C2218yescomplemented
66#1865C2218yescomplemented
67#1850C2218yescomplemented
68#1803C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—42no—
82C23×C8—42no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2314yes—
89C2×#124—24no—
90C2×#124—22no—
91C2×#124—42no—
92C2×#124C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—42no—
96C2×C22⋊Q8C2314yes—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121—42no—
100C2×#121C2314yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—24no—
103C2×C4×Q8—42no—
104C2×C4×Q8C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
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⋊C4—21no—
126C22×C22⋊C4D411yescharacteristic
127C22×C22⋊C4D411yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C2×#107—42no—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106C2314yes—
136C2×C4⋊C8—42no—
137C2×C4⋊C8—42no—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—42no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8—42no—
157C2×C22⋊C8C2314yes—
158#400—28no—
159#396—28no—
160#396—28no—
161#394—28no—
162#390—28no—
163#390—22no—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#388—28no—
172#388—28no—
173#387—28no—
174#384—22no—
175#384C2314yes—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4D414yes—
183C4×C22⋊C4C2314yes—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
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#4114yes—
209C22×Q8C2412yes—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—82no—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8—416no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263#124—28no—
264#124—44no—
265#124—48no—
266C22⋊Q8—28no—
267C22⋊Q8—44no—
268C22⋊Q8—48no—
269#121—28no—
270#121—44no—
271#121—48no—
272C4×Q8—28no—
273C4×Q8—44no—
274C4×Q8—48no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
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—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
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—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
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—24no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#106—28no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349#103—28no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357C22⋊C8—44no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95C2×D414yes—
365#95C2×D414yes—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24#13714yes—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8C2×#4114yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—84no—
412C2×D4—816no—
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—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—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
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—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
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#13514yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
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—416no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8—44no—
498C2×C8—44no—
499C4⋊C4—24no—
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—22no—
510C4⋊C4—22no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—48no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C2×#4114yes—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—48no—
541C42—84no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#53712yes—
570C23#54612yes—
571C23#54612yes—
572C23C2×#13512yes—
573C23C2×#13714yes—
574C23C2×#13712yes—
575C23#57612yes—
576C23#57712yes—
577C23C23×D411yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—42no—
581Q8—48no—
582Q8—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—816no—
588D4—84no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
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—42no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—48no—
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—84no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#53712yes—
642C2×C4#53914yes—
643C2×C4#54214yes—
644C2×C4#54612yes—
645C2×C4#54612yes—
646C2×C4#55314yes—
647C2×C4#56014yes—
648C2×C4C2×#13514yes—
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C8—42no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—416no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—168no—
675C22#269114yes—
676C22#324814yes—
677C22#325714yes—
678C22#330114yes—
679C22#330514yes—
680C22#331414yes—
681C22#331614yes—
682C22C2×#53711yescharacteristic
683C22C2×#54611yescharacteristic
684C22C2×#54611yescharacteristic
685C22C22×#13511yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57611yescharacteristic
688C22C2×#57711yescharacteristic
689C4—24no—
690C4—42no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#269114yes—
702C4C2×#53911yescharacteristic
703C4C2×#54211yescharacteristic
704C4C2×#55311yescharacteristic
705C4C2×#56011yescharacteristic
706C2—44no—
707C2—162no—
708C2#3150018yesdirect product factor
709C2C2×#269111yescharacteristic
710C2C2×#324811yescharacteristic
711C2C2×#325711yescharacteristic
712C2C2×#330111yescharacteristic
713C2C2×#330511yescharacteristic
714C2C2×#331411yescharacteristic
715C2C2×#331611yescharacteristic
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×#2497C2semidirect
3. #31500C2direct (N × H)
4. C2×#2086C2semidirect
5. C2×#2071C2semidirect
6. C2×#2067C2semidirect
7. C2×#2042C2semidirect
8. C2×#2036C2semidirect
10. C2×#1983C2semidirect
11. C2×#1920C2semidirect
13. C2×#1892C2semidirect
14. C2×#1865C2semidirect
15. C2×#1850C2semidirect
16. C2×#1803C2semidirect
36. #2497C22semidirect
56. #2086C22semidirect
57. #2071C22semidirect
58. #2067C22semidirect
59. #2042C22semidirect
60. #2036C22semidirect
62. #1983C22semidirect
63. #1920C22semidirect
65. #1892C22semidirect
66. #1865C22semidirect
67. #1850C22semidirect
68. #1803C22semidirect
708. C2#31500direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24972
C2×#24262
#315002
C2×#20862
C2×#20712
C2×#20672
C2×#20422
C2×#20362
C2×#20342
C2×#19832
C2×#19202
C2×#18932
C2×#18922
C2×#18652
C2×#18502
C2×#18032