Finite Group Databasebeta

← C2×#31856 #7506413

Subgroups — C2×#31856 #7506413

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2429C211yesmaximal, characteristic
3#31856C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2072C211yesmaximal, characteristic, complemented
6C2×#2056C211yesmaximal, characteristic, complemented
7C2×#2016C211yesmaximal, characteristic, complemented
8C2×#2001C211yesmaximal, characteristic
9C2×#2001C211yesmaximal, characteristic
10C2×#1960C211yesmaximal, characteristic, complemented
11C2×#1935C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic
13C2×#1878C211yesmaximal, characteristic, complemented
14C2×#1873C211yesmaximal, characteristic
15C2×#1873C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2499—24no—
38#2499C2218yes—
39#2429—24no—
40#2429C2218yes—
41C2×#398—21no—
42C2×#397—21no—
43C2×#397—21no—
44C2×#396—21no—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#387—21no—
52C2×#387—21no—
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2101C2218yescomplemented
57#2072C2218yescomplemented
58#2056C2218yescomplemented
59#2016C2218yescomplemented
60#2001C2218yes—
61#2001C2218yes—
62#1960C2218yescomplemented
63#1935C2218yescomplemented
64#1924C2218yes—
65#1878C2218yescomplemented
66#1873C2218yes—
67#1873C2218yes—
68#1803C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×Q16—24no—
82C22×Q16—42no—
83C22×Q16—42no—
84C22×Q16—42no—
85C22×Q16—42no—
86C22×Q16—42no—
87C22×Q16C2314yes—
88C22×Q16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8C2314yes—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—42no—
103C2×C22⋊Q8—42no—
104C2×C22⋊Q8—42no—
105C2×C22⋊Q8—42no—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—24no—
111C2×C4×Q8—42no—
112C2×C4×Q8C2314yes—
113C2×#117—22no—
114C2×#117—42no—
115C2×#117—42no—
116C2×#117C2314yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C2×C4⋊C8—42no—
142C2×C4⋊C8C2314yes—
143C2×#103—24no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103—42no—
148C2×#103—42no—
149C2×#103—42no—
150C2×#103—42no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×C22⋊C8—42no—
160C2×C22⋊C8—42no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×#97—42no—
164C2×#97C2314yes—
165#398—28no—
166#397—28no—
167#397—28no—
168#396—28no—
169#391—22no—
170#391—22no—
171#391—22no—
172#391C2314yes—
173#391C2314yes—
174#391C2314yes—
175#389—22no—
176#389—28no—
177#389C2314yes—
178#388—28no—
179#387—28no—
180#387—28no—
181#384—22no—
182#384C2314yes—
183#382—22no—
184#382C2314yes—
185C4×C22⋊C4D414yes—
186C4×C22⋊C4C2314yes—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206C25C2×D411yescharacteristic
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×Q8C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C2×Q16—28no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×Q16—44no—
257C2×Q16—44no—
258C2×Q16—44no—
259C2×Q16—416no—
260C2×Q16—416no—
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—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—44no—
279C4⋊Q8—48no—
280C22⋊Q8—28no—
281C22⋊Q8—28no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—48no—
287C22⋊Q8—48no—
288C22⋊Q8—48no—
289C4×Q8—28no—
290C4×Q8—44no—
291C4×Q8—48no—
292#117—44no—
293#117—44no—
294#117—48no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4#4114yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—44no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C4⋊C8—44no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366C22⋊C8—44no—
367C22⋊C8—44no—
368#97—44no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D414yes—
379#95C2×D414yes—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8C2×#4114yes—
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—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—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
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—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4#13714yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C4C2511yescharacteristic
501Q16—416no—
502Q16—416no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8—44no—
517C2×C8—44no—
518C2×C8—44no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C42—24no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—48no—
556C42—48no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23#53512yes—
572C23#54612yes—
573C23#54612yes—
574C23C2×#13712yes—
575C23#57512yes—
576C23#57812yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—42no—
582Q8—48no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—84no—
593Q8—84no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
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—22no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—48no—
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—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4#53512yes—
651C2×C4#53814yes—
652C2×C4#54014yes—
653C2×C4#54612yes—
654C2×C4#54612yes—
655C2×C4#55314yes—
656C2×C4#56014yes—
657C2×C4C2×#13714yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—88no—
676C22#270314yes—
677C22#321614yes—
678C22#322314yes—
679C22#331214yes—
680C22#331314yes—
681C22#331514yes—
682C22#332814yes—
683C22C2×#53511yescharacteristic
684C22C2×#54611yescharacteristic
685C22C2×#54611yescharacteristic
686C22C22×#13711yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57811yescharacteristic
689C22C2×#57811yescharacteristic
690C4—24no—
691C4—42no—
692C4—44no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#270314yes—
704C4C2×#53811yescharacteristic
705C4C2×#54011yescharacteristic
706C4C2×#55311yescharacteristic
707C4C2×#56011yescharacteristic
708C2—82no—
709C2#3185618yesdirect product factor
710C2C2×#270311yescharacteristic
711C2C2×#321611yescharacteristic
712C2C2×#322311yescharacteristic
713C2C2×#331211yescharacteristic
714C2C2×#331511yescharacteristic
715C2C2×#331311yescharacteristic
716C2C2×#332811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #31856C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2072C2semidirect
6. C2×#2056C2semidirect
7. C2×#2016C2semidirect
10. C2×#1960C2semidirect
11. C2×#1935C2semidirect
13. C2×#1878C2semidirect
16. C2×#1803C2semidirect
56. #2101C22semidirect
57. #2072C22semidirect
58. #2056C22semidirect
59. #2016C22semidirect
62. #1960C22semidirect
63. #1935C22semidirect
65. #1878C22semidirect
68. #1803C22semidirect
709. C2#31856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#24292
#318562
C2×#21012
C2×#20722
C2×#20562
C2×#20162
C2×#20012
C2×#19602
C2×#19352
C2×#19242
C2×#18782
C2×#18732
C2×#18032