Finite Group Databasebeta

← C2×#31367 #7504523

Subgroups — C2×#31367 #7504523

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic, complemented
2C2×#2422C211yesmaximal, characteristic
3#31367C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2076C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic
9C2×#2025C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic, complemented
13C2×#1875C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
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×#97C2211yescharacteristic
32#2441—24no—
33#2441C2218yescomplemented
34#2422—24no—
35#2422C2218yes—
36C2×#399—21no—
37C2×#397—21no—
38C2×#396—21no—
39C2×#394—21no—
40C2×#390C2211yescharacteristic
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#387—21no—
46C2×#385—21no—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2100C2218yescomplemented
53#2086C2218yescomplemented
54#2083C2218yescomplemented
55#2076C2218yescomplemented
56#2033C2218yes—
57#2025C2218yescomplemented
58#1992C2218yescomplemented
59#1936C2218yescomplemented
60#1924C2218yescomplemented
61#1875C2218yes—
62#1862C2218yescomplemented
63#1825C2218yescomplemented
64#1802C2218yescomplemented
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×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#125—24no—
86C2×#125—22no—
87C2×#125—42no—
88C2×#125C2314yes—
89C2×#123—28no—
90C2×#123—48no—
91C2×#121—22no—
92C2×#121—24no—
93C2×#121—42no—
94C2×#121C2314yes—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4C2314yes—
99C2×#117—22no—
100C2×#117—24no—
101C2×#117—42no—
102C2×#117C2314yes—
103C22×C4⋊C4—21no—
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⋊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×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107—42no—
132C2×#107—42no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8—42no—
146C2×C22⋊C8—42no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×#97—42no—
150C2×#97C2314yes—
151#399—28no—
152#397—28no—
153#396—28no—
154#394—28no—
155#390—22no—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—28no—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#387—28no—
165#385—28no—
166#384—22no—
167#384—22no—
168#384C2314yes—
169#384C2314yes—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174C4×C22⋊C4D414yes—
175C4×C22⋊C4C2314yes—
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×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—82no—
207C22×D4#4114yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—82no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
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×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#125—28no—
251#125—44no—
252#125—48no—
253#123—432no—
254#121—28no—
255#121—44no—
256#121—48no—
257C4×D4—28no—
258C4×D4—44no—
259C4×D4—48no—
260#117—28no—
261#117—44no—
262#117—48no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
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—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4#4114yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
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—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—44no—
338#107—44no—
339C4⋊C8—44no—
340#102—28no—
341#102—28no—
342#102—44no—
343#102—44no—
344C22⋊C8—44no—
345C22⋊C8—44no—
346#97—44no—
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×D4—28no—
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—84no—
386C2×D4—816no—
387C2×D4C2×#4114yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
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—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
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—48no—
445C22×C4—84no—
446C22×C4—816no—
447C22×C4AGL(1,ℤ/8)14yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
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—44no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
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—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4C2×#4114yes—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—48no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—48no—
541C23—48no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23#53612yes—
552C23#54612yes—
553C23#54612yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23#57512yes—
556C23#57612yes—
557C23#57812yes—
558C23C23×D411yescharacteristic
559D4—24no—
560D4—24no—
561D4—42no—
562D4—48no—
563D4—42no—
564D4—816no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
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—816no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4#53612yes—
619C2×C4#53914yes—
620C2×C4#54214yes—
621C2×C4#54612yes—
622C2×C4#54612yes—
623C2×C4#55214yes—
624C2×C4#56214yes—
625C2×C4C2×AGL(1,ℤ/8)14yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—88no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#268714yes—
653C22#323514yes—
654C22#324314yes—
655C22#330914yes—
656C22#331514yes—
657C22#331814yes—
658C22#333014yes—
659C22C2×#53611yescharacteristic
660C22C2×#54611yescharacteristic
661C22C2×#54611yescharacteristic
662C22C22×AGL(1,ℤ/8)11yescharacteristic
663C22C2×#57511yescharacteristic
664C22C2×#57611yescharacteristic
665C22C2×#57811yescharacteristic
666C4—24no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#268714yes—
679C4C2×#53911yescharacteristic
680C4C2×#54211yescharacteristic
681C4C2×#55211yescharacteristic
682C4C2×#56211yescharacteristic
683C2—82no—
684C2—84no—
685C2#3136718yesdirect product factor
686C2C2×#268711yescharacteristic
687C2C2×#323511yescharacteristic
688C2C2×#324311yescharacteristic
689C2C2×#331811yescharacteristic
690C2C2×#331511yescharacteristic
691C2C2×#330911yescharacteristic
692C2C2×#333011yescharacteristic
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×#2441C2semidirect
3. #31367C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2086C2semidirect
6. C2×#2083C2semidirect
7. C2×#2076C2semidirect
9. C2×#2025C2semidirect
10. C2×#1992C2semidirect
11. C2×#1936C2semidirect
12. C2×#1924C2semidirect
14. C2×#1862C2semidirect
15. C2×#1825C2semidirect
16. C2×#1802C2semidirect
33. #2441C22semidirect
52. #2100C22semidirect
53. #2086C22semidirect
54. #2083C22semidirect
55. #2076C22semidirect
57. #2025C22semidirect
58. #1992C22semidirect
59. #1936C22semidirect
60. #1924C22semidirect
62. #1862C22semidirect
63. #1825C22semidirect
64. #1802C22semidirect
685. C2#31367direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24412
C2×#24222
#313672
C2×#21002
C2×#20862
C2×#20832
C2×#20762
C2×#20332
C2×#20252
C2×#19922
C2×#19362
C2×#19242
C2×#18752
C2×#18622
C2×#18252
C2×#18022