Finite Group Databasebeta

← C2×#31636 #7505316

Subgroups — C2×#31636 #7505316

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic, complemented
3#31636C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2072C211yesmaximal, characteristic, complemented
6C2×#2055C211yesmaximal, characteristic
7C2×#2024C211yesmaximal, characteristic, complemented
8C2×#2024C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10C2×#1956C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic
12C2×#1916C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1877C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8—21no—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2542—24no—
36#2542C2218yes—
37#2446—24no—
38#2446C2218yescomplemented
39C2×#400—21no—
40C2×#399—21no—
41C2×#399—21no—
42C2×#398—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#387—21no—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2218yescomplemented
55#2072C2218yescomplemented
56#2055C2218yes—
57#2024C2218yescomplemented
58#2024C2218yescomplemented
59#2013C2218yes—
60#1956C2218yescomplemented
61#1934C2218yes—
62#1916C2218yescomplemented
63#1880C2218yes—
64#1877C2218yes—
65#1877C2218yes—
66#1803C2218yescomplemented
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×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—48no—
105C2×C22⋊Q8—48no—
106C2×C4×Q8—22no—
107C2×C4×Q8—24no—
108C2×C4×Q8—42no—
109C2×C4×Q8C2314yes—
110C2×C4×D4—22no—
111C2×C4×D4—42no—
112C2×C4×D4—42no—
113C2×C4×D4C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
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⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C2×C4⋊C8—42no—
140C2×C4⋊C8C2314yes—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103—42no—
146C2×#103—42no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102—24no—
154C2×#102—42no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C4×C8—42no—
158C2×C4×C8C2314yes—
159#400—28no—
160#399—28no—
161#399—28no—
162#398—28no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2314yes—
168#391C2314yes—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2314yes—
173#389—22no—
174#389C2314yes—
175#387—28no—
176#387—28no—
177#384—22no—
178#384—22no—
179#384C2314yes—
180#384C2314yes—
181C4×C4⋊C4D414yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
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—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—21no—
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—22no—
216C22×Q8—22no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C22×Q8#4114yes—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4#4114yes—
230C22×D4C2412yes—
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—44no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—416no—
256C2×SD16—416no—
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×C8—22no—
268C22×C8—22no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278C4⋊Q8—48no—
279C22⋊Q8—432no—
280C22⋊Q8—432no—
281C4×Q8—28no—
282C4×Q8—44no—
283C4×Q8—48no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—48no—
287C2×C4⋊C4—24no—
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—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4SD1618yescomplemented
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C4⋊C8—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—28no—
361#102—44no—
362C4×C8—44no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D414yes—
371#95C2×D414yes—
372C24—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—816no—
410C2×Q8C2×#4114yes—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4C2×#4114yes—
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—24no—
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—48no—
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—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—816no—
470C22×C4C2×SD1614yes—
471C22×C4C2×SD1614yes—
472C22×C4C2×SD1614yes—
473C22×C4AGL(1,ℤ/8)14yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+412yes—
487C22×C4C2511yescharacteristic
488SD16—416no—
489SD16—416no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—44no—
503C2×C8—44no—
504C2×C8—44no—
505C2×C8—44no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4C2×SD16116yescomplemented
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—48no—
547C23—24no—
548C23—48no—
549C23—48no—
550C23—42no—
551C23—42no—
552C23—48no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#53512yes—
562C23#54612yes—
563C23#54812yes—
564C23C22×SD1612yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23#57812yes—
567C23#57812yes—
568C23C23×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—42no—
572Q8—48no—
573Q8—42no—
574Q8—42no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—816no—
579Q8—84no—
580D4—24no—
581D4—24no—
582D4—42no—
583D4—48no—
584D4—42no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
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—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—816no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4#53512yes—
635C2×C4#53814yes—
636C2×C4#53914yes—
637C2×C4#54612yes—
638C2×C4#54812yes—
639C2×C4#55114yes—
640C2×C4#56214yes—
641C2×C4C22×SD1614yes—
642C2×C4C22×SD1614yes—
643C2×C4C22×SD1614yes—
644C2×C4C2×AGL(1,ℤ/8)14yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×H(5,𝔽2)11yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C8—42no—
652C8—42no—
653C8—42no—
654C8—42no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22#269614yes—
664C22#320814yes—
665C22#322314yes—
666C22#329814yes—
667C22#330814yes—
668C22#335914yes—
669C22#336714yes—
670C22C2×#53511yescharacteristic
671C22C2×#54611yescharacteristic
672C22C2×#54811yescharacteristic
673C22C23×SD1611yescharacteristic
674C22C22×AGL(1,ℤ/8)11yescharacteristic
675C22C2×#57811yescharacteristic
676C22C2×#57811yescharacteristic
677C4—24no—
678C4—24no—
679C4—24no—
680C4—24no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#269614yes—
691C4C2×#53811yescharacteristic
692C4C2×#53911yescharacteristic
693C4C2×#55111yescharacteristic
694C4C2×#56211yescharacteristic
695C2—84no—
696C2#3163618yesdirect product factor
697C2C2×#269611yescharacteristic
698C2C2×#320811yescharacteristic
699C2C2×#322311yescharacteristic
700C2C2×#329811yescharacteristic
701C2C2×#330811yescharacteristic
702C2C2×#335911yescharacteristic
703C2C2×#336711yescharacteristic
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. #31636C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2072C2semidirect
7. C2×#2024C2semidirect
8. C2×#2024C2semidirect
10. C2×#1956C2semidirect
12. C2×#1916C2semidirect
16. C2×#1803C2semidirect
38. #2446C22semidirect
54. #2099C22semidirect
55. #2072C22semidirect
57. #2024C22semidirect
58. #2024C22semidirect
60. #1956C22semidirect
62. #1916C22semidirect
66. #1803C22semidirect
321. C2×C4⋊C4SD16semidirect
531. C4⋊C4C2×SD16semidirect
696. C2#31636direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24462
#316362
C2×#20992
C2×#20722
C2×#20552
C2×#20242
C2×#20132
C2×#19562
C2×#19342
C2×#19162
C2×#18802
C2×#18772
C2×#18032