Finite Group Databasebeta

← C2×#31574 #7505122

Subgroups — C2×#31574 #7505122

Order 512.

2188 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2445C211yesmaximal, characteristic
3#31574C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic, complemented
5C2×#2094C211yesmaximal, characteristic, complemented
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2071C211yesmaximal, characteristic, complemented
8C2×#2065C211yesmaximal, characteristic, complemented
9C2×#2041C211yesmaximal, characteristic
10C2×#1957C211yesmaximal, characteristic
11C2×#1941C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic, complemented
14C2×#1916C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C22×#127C2211yescharacteristic
18C22×#125—21no—
19C22×#123—21no—
20C22×C4×D4C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#2552—24no—
36#2552C2218yescomplemented
37#2445—24no—
38#2445C2218yes—
39C2×#398—21no—
40C2×#398—21no—
41C2×#394—21no—
42C2×#394—21no—
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#386—21no—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2094C2218yescomplemented
57#2094C2218yescomplemented
58#2084C2218yescomplemented
59#2071C2218yescomplemented
60#2065C2218yescomplemented
61#2041C2218yes—
62#1957C2218yes—
63#1941C2218yescomplemented
64#1926C2218yescomplemented
65#1926C2218yescomplemented
66#1916C2218yescomplemented
67#1864C2218yescomplemented
68#1802C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#127—24no—
87C2×#127—22no—
88C2×#127—42no—
89C2×#127C2314yes—
90C2×#125—28no—
91C2×#125—48no—
92C2×#123—28no—
93C2×#123—48no—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4C2314yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#106—24no—
127C2×#106—42no—
128C2×#106—42no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—42no—
135C2×C4⋊C8C2314yes—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102—42no—
142C2×#102—42no—
143C2×#102—42no—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C4×C8—42no—
153C2×C4×C8C2314yes—
154#398—28no—
155#398—28no—
156#394—28no—
157#394—28no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#390C2314yes—
164#389—28no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#386—28no—
172#384—22no—
173#384—22no—
174#384—22no—
175#384C2314yes—
176#384C2314yes—
177#384C2314yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4D414yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95D411yescharacteristic
187C2×#95D411yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—82no—
218C22×D4—82no—
219C22×D4#4114yes—
220C22×D4C2412yes—
221C22×D4C2412yes—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—82no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
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×C8#4114yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255#127—28no—
256#127—44no—
257#127—48no—
258#125—432no—
259#123—432no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—48no—
263C2×C4⋊C4—24no—
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—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
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—82no—
300C2×C4⋊C4SD1618yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D414yes—
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—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C22⋊C4—82no—
322C2×C22⋊C4—82no—
323C2×C42—22no—
324C2×C42—22no—
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×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#106—28no—
338#106—44no—
339#106—44no—
340#106—44no—
341C4⋊C8—44no—
342#102—28no—
343#102—28no—
344#102—28no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350C4×C8—44no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95C2×D414yes—
356#95C2×D414yes—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24C22×D411yescharacteristic
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—84no—
404C2×D4—84no—
405C2×D4—816no—
406C2×D4—816no—
407C2×D4C2×#4114yes—
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—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—48no—
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—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—84no—
449C22×C4—816no—
450C22×C4C2×SD1614yes—
451C22×C4C2×SD1614yes—
452C22×C4C2×SD1614yes—
453C22×C4AGL(1,ℤ/8)14yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C4C2511yescharacteristic
468C2×C8—28no—
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—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—44no—
483C2×C8C2×#4114yes—
484C4⋊C4—28no—
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—22no—
496C4⋊C4—22no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4C2×SD16116yes—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—48no—
530C23—24no—
531C23—42no—
532C23—48no—
533C23—42no—
534C23—48no—
535C23—48no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23#53612yes—
559C23#54612yes—
560C23#55012yes—
561C23C22×SD1612yes—
562C23C2×AGL(1,ℤ/8)12yes—
563C23#57612yes—
564C23#57712yes—
565C23C23×D411yescharacteristic
566D4—24no—
567D4—24no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—48no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—816no—
576D4—816no—
577D4—84no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—44no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—816no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4#53612yes—
624C2×C4#53914yes—
625C2×C4#53914yes—
626C2×C4#54612yes—
627C2×C4#55012yes—
628C2×C4#55214yes—
629C2×C4#56214yes—
630C2×C4C22×SD1614yes—
631C2×C4C22×SD1614yes—
632C2×C4C22×SD1614yes—
633C2×C4C2×AGL(1,ℤ/8)14yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—24no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—816no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—168no—
660C22#269414yes—
661C22#322714yes—
662C22#324114yes—
663C22#329514yes—
664C22#330614yes—
665C22#339214yes—
666C22#339914yes—
667C22C2×#53611yescharacteristic
668C22C2×#54611yescharacteristic
669C22C2×#55011yescharacteristic
670C22C23×SD1611yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57611yescharacteristic
673C22C2×#57711yescharacteristic
674C4—24no—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—42no—
679C4—42no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#269414yes—
687C4C2×#53911yescharacteristic
688C4C2×#53911yescharacteristic
689C4C2×#55211yescharacteristic
690C4C2×#56211yescharacteristic
691C2—84no—
692C2—162no—
693C2#3157418yesdirect product factor
694C2C2×#269411yescharacteristic
695C2C2×#322711yescharacteristic
696C2C2×#324111yescharacteristic
697C2C2×#329511yescharacteristic
698C2C2×#330611yescharacteristic
699C2C2×#339211yescharacteristic
700C2C2×#339911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. C2×#2552C2semidirect
3. #31574C2direct (N × H)
4. C2×#2094C2semidirect
5. C2×#2094C2semidirect
6. C2×#2084C2semidirect
7. C2×#2071C2semidirect
8. C2×#2065C2semidirect
11. C2×#1941C2semidirect
12. C2×#1926C2semidirect
13. C2×#1926C2semidirect
14. C2×#1916C2semidirect
15. C2×#1864C2semidirect
16. C2×#1802C2semidirect
36. #2552C22semidirect
56. #2094C22semidirect
57. #2094C22semidirect
58. #2084C22semidirect
59. #2071C22semidirect
60. #2065C22semidirect
63. #1941C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1916C22semidirect
67. #1864C22semidirect
68. #1802C22semidirect
693. C2#31574direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24452
#315742
C2×#20942
C2×#20842
C2×#20712
C2×#20652
C2×#20412
C2×#19572
C2×#19412
C2×#19262
C2×#19162
C2×#18642
C2×#18022