Finite Group Databasebeta

← C2×#31324 #7504414

Subgroups — C2×#31324 #7504414

Order 512.

2194 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2417C211yesmaximal, characteristic
3#31324C2116yesmaximal, direct product factor
4C2×#2076C211yesmaximal, characteristic, complemented
5C2×#2057C211yesmaximal, characteristic, complemented
6C2×#2036C211yesmaximal, characteristic, complemented
7C2×#2036C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic, complemented
9C2×#2024C211yesmaximal, characteristic, complemented
10C2×#1928C211yesmaximal, characteristic, complemented
11C2×#1921C211yesmaximal, characteristic, complemented
12C2×#1892C211yesmaximal, characteristic, complemented
13C2×#1869C211yesmaximal, characteristic
14C2×#1852C211yesmaximal, characteristic, complemented
15C2×#1824C211yesmaximal, characteristic, complemented
16C2×#1808C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2563—24no—
35#2563C2218yes—
36#2417—24no—
37#2417C2218yes—
38C2×#398—21no—
39C2×#396—21no—
40C2×#393—21no—
41C2×#393—21no—
42C2×#391C2211yescharacteristic
43C2×#390—21no—
44C2×#389C2211yescharacteristic
45C2×#386—21no—
46C2×#382—21no—
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2076C2218yescomplemented
55#2057C2218yescomplemented
56#2036C2218yescomplemented
57#2036C2218yescomplemented
58#2036C2218yescomplemented
59#2024C2218yescomplemented
60#1928C2218yescomplemented
61#1921C2218yescomplemented
62#1892C2218yescomplemented
63#1869C2218yes—
64#1852C2218yescomplemented
65#1824C2218yescomplemented
66#1808C2218yescomplemented
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×D4D411yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—42no—
85C23×C8—42no—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—42no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8C2314yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—42no—
99C2×C22⋊Q8—42no—
100C2×C22⋊Q8—42no—
101C2×C22⋊Q8C2314yes—
102C2×C22⋊Q8C2314yes—
103C2×C4×D4—42no—
104C2×C4×D4—42no—
105C2×C4×D4D414yes—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—42no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C2×#107—24no—
136C2×#107—42no—
137C2×#107—42no—
138C2×#107—42no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#103—24no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×C22⋊C8—24no—
152C2×C22⋊C8—42no—
153C2×C22⋊C8—42no—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157#398—28no—
158#396—28no—
159#393—28no—
160#393—28no—
161#391—22no—
162#391C2314yes—
163#390—28no—
164#389—22no—
165#389C2314yes—
166#386—28no—
167#382—28no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—28no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
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—82no—
212C22×Q8—82no—
213C22×Q8C2412yes—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4Q1618yes—
217C22×D4C2×D414yes—
218C23×C4—21no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—24no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—82no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
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—22no—
262C22×C8—28no—
263C22×C8—44no—
264C22×C8—44no—
265C22×C8—416no—
266C22×C8#4114yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—48no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C2×C4⋊C4—21no—
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—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4#4114yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350#107—28no—
351#107—44no—
352#107—44no—
353#107—44no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358C22⋊C8—28no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24C2×Q1614yes—
383C24C2×Q1614yes—
384C24C22×D412yes—
385C24C22×D412yes—
386C24C22×D411yescharacteristic
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8—816no—
404C2×D4—44no—
405C2×D4—416no—
406C2×D4—44no—
407C2×D4C2×Q16116yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
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—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
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—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—84no—
480C22×C4—816no—
481C22×C4C2×Q1614yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
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—22no—
506C2×C8—416no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8—44no—
510C2×C8C2×#4114yes—
511C4⋊C4—24no—
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—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—84no—
537C4⋊C4C2×#4114yes—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—48no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—28no—
562C23—28no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23#53412yes—
578C23#54612yes—
579C23#54712yes—
580C23C22×Q1614yes—
581C23C22×Q1614yes—
582C23C22×Q1612yes—
583C23#57512yes—
584C23#57612yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—816no—
592Q8—84no—
593D4—416no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—48no—
616C2×C4—48no—
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—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4#53412yes—
653C2×C4#53814yes—
654C2×C4#54214yes—
655C2×C4#54612yes—
656C2×C4#54712yes—
657C2×C4#55514yes—
658C2×C4#56214yes—
659C2×C4C22×Q1614yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C8—24no—
667C8—42no—
668C8—42no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—416no—
681C22—416no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22#268614yes—
687C22#318914yes—
688C22#319814yes—
689C22#329914yes—
690C22#331914yes—
691C22#333814yes—
692C22#334814yes—
693C22C2×#53411yescharacteristic
694C22C2×#54611yescharacteristic
695C22C2×#54711yescharacteristic
696C22C23×Q1611yescharacteristic
697C22C2×#57511yescharacteristic
698C22C2×#57611yescharacteristic
699C22C2×#57711yescharacteristic
700C4—24no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#268614yes—
713C4C2×#53811yescharacteristic
714C4C2×#54211yescharacteristic
715C4C2×#55511yescharacteristic
716C4C2×#56211yescharacteristic
717C2—44no—
718C2—44no—
719C2#3132418yesdirect product factor
720C2C2×#268611yescharacteristic
721C2C2×#318911yescharacteristic
722C2C2×#319811yescharacteristic
723C2C2×#329911yescharacteristic
724C2C2×#331911yescharacteristic
725C2C2×#333811yescharacteristic
726C2C2×#334811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #31324C2direct (N × H)
4. C2×#2076C2semidirect
5. C2×#2057C2semidirect
6. C2×#2036C2semidirect
7. C2×#2036C2semidirect
8. C2×#2036C2semidirect
9. C2×#2024C2semidirect
10. C2×#1928C2semidirect
11. C2×#1921C2semidirect
12. C2×#1892C2semidirect
14. C2×#1852C2semidirect
15. C2×#1824C2semidirect
16. C2×#1808C2semidirect
54. #2076C22semidirect
55. #2057C22semidirect
56. #2036C22semidirect
57. #2036C22semidirect
58. #2036C22semidirect
59. #2024C22semidirect
60. #1928C22semidirect
61. #1921C22semidirect
62. #1892C22semidirect
64. #1852C22semidirect
65. #1824C22semidirect
66. #1808C22semidirect
719. C2#31324direct (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×#24172
#313242
C2×#20762
C2×#20572
C2×#20362
C2×#20242
C2×#19282
C2×#19212
C2×#18922
C2×#18692
C2×#18522
C2×#18242
C2×#18082