Finite Group Databasebeta

← C2×#31357 #7504512

Subgroups — C2×#31357 #7504512

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2439C211yesmaximal, characteristic, complemented
2C2×#2422C211yesmaximal, characteristic
3#31357C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2076C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic
9C2×#2023C211yesmaximal, characteristic, complemented
10C2×#1990C211yesmaximal, characteristic, complemented
11C2×#1934C211yesmaximal, characteristic, complemented
12C2×#1923C211yesmaximal, characteristic, complemented
13C2×#1875C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic, complemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×#123—21no—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32#2439—24no—
33#2439C2218yescomplemented
34#2422—24no—
35#2422C2218yes—
36C2×#397—21no—
37C2×#396—21no—
38C2×#394—21no—
39C2×#393—21no—
40C2×#391—21no—
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2087C2218yescomplemented
54#2083C2218yescomplemented
55#2076C2218yescomplemented
56#2033C2218yes—
57#2023C2218yescomplemented
58#1990C2218yescomplemented
59#1934C2218yescomplemented
60#1923C2218yescomplemented
61#1875C2218yes—
62#1862C2218yescomplemented
63#1825C2218yescomplemented
64#1804C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8C2314yes—
90C2×#123—28no—
91C2×#123—48no—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121—42no—
95C2×#121C2314yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—42no—
99C2×C4×D4C2314yes—
100C2×#117—22no—
101C2×#117—24no—
102C2×#117—42no—
103C2×#117C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
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⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#107—42no—
130C2×#107—42no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8—42no—
144C2×C22⋊C8—42no—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C4×C8—42no—
148C2×C4×C8C2314yes—
149#397—28no—
150#396—28no—
151#394—28no—
152#393—28no—
153#391—28no—
154#390—22no—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#387—28no—
163#385—28no—
164#384—22no—
165#384C2314yes—
166#382—22no—
167#382C2314yes—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172C4×C22⋊C4D414yes—
173C4×C22⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95D411yescharacteristic
182C2×#95D411yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—82no—
201C22×D4—24no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—82no—
211C22×D4#4114yes—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257#123—432no—
258#121—28no—
259#121—44no—
260#121—48no—
261C4×D4—28no—
262C4×D4—44no—
263C4×D4—48no—
264#117—28no—
265#117—44no—
266#117—48no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#107—44no—
336#107—44no—
337C4⋊C8—44no—
338#102—28no—
339#102—28no—
340#102—44no—
341#102—44no—
342C22⋊C8—44no—
343C22⋊C8—44no—
344C4×C8—44no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—84no—
374C2×Q8—816no—
375C2×D4—28no—
376C2×D4—24no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—84no—
390C2×D4—816no—
391C2×D4C2×#4114yes—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—84no—
449C22×C4—816no—
450C22×C4#13514yes—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—44no—
480C2×C8—44no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4C2×#4114yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—84no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—42no—
533C23—44no—
534C23—48no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—48no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23#53612yes—
553C23#54612yes—
554C23#54612yes—
555C23C2×#13512yes—
556C23C2×AGL(1,ℤ/8)12yes—
557C23#57612yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—816no—
561D4—24no—
562D4—24no—
563D4—42no—
564D4—48no—
565D4—42no—
566D4—816no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
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—816no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4#53612yes—
620C2×C4#53914yes—
621C2×C4#54214yes—
622C2×C4#54612yes—
623C2×C4#54612yes—
624C2×C4#55214yes—
625C2×C4#56214yes—
626C2×C4C2×#13514yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—88no—
649C22—816no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22#268714yes—
655C22#322814yes—
656C22#324314yes—
657C22#329714yes—
658C22#330214yes—
659C22#330814yes—
660C22#332514yes—
661C22C2×#53611yescharacteristic
662C22C2×#54611yescharacteristic
663C22C2×#54611yescharacteristic
664C22C22×#13511yescharacteristic
665C22C22×AGL(1,ℤ/8)11yescharacteristic
666C22C2×#57611yescharacteristic
667C22C2×#57811yescharacteristic
668C4—24no—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#268714yes—
681C4C2×#53911yescharacteristic
682C4C2×#54211yescharacteristic
683C4C2×#55211yescharacteristic
684C4C2×#56211yescharacteristic
685C2—82no—
686C2—84no—
687C2#3135718yesdirect product factor
688C2C2×#268711yescharacteristic
689C2C2×#322811yescharacteristic
690C2C2×#324311yescharacteristic
691C2C2×#329711yescharacteristic
692C2C2×#330211yescharacteristic
693C2C2×#330811yescharacteristic
694C2C2×#332511yescharacteristic
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×#2439C2semidirect
3. #31357C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2087C2semidirect
6. C2×#2083C2semidirect
7. C2×#2076C2semidirect
9. C2×#2023C2semidirect
10. C2×#1990C2semidirect
11. C2×#1934C2semidirect
12. C2×#1923C2semidirect
14. C2×#1862C2semidirect
15. C2×#1825C2semidirect
16. C2×#1804C2semidirect
33. #2439C22semidirect
52. #2101C22semidirect
53. #2087C22semidirect
54. #2083C22semidirect
55. #2076C22semidirect
57. #2023C22semidirect
58. #1990C22semidirect
59. #1934C22semidirect
60. #1923C22semidirect
62. #1862C22semidirect
63. #1825C22semidirect
64. #1804C22semidirect
687. C2#31357direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24392
C2×#24222
#313572
C2×#21012
C2×#20872
C2×#20832
C2×#20762
C2×#20332
C2×#20232
C2×#19902
C2×#19342
C2×#19232
C2×#18752
C2×#18622
C2×#18252
C2×#18042