Finite Group Databasebeta

← C2×#31535 #7504991

Subgroups — C2×#31535 #7504991

Order 512.

2112 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic, complemented
3#31535C2116yesmaximal, direct product factor
4C2×#2093C211yesmaximal, characteristic, complemented
5C2×#2085C211yesmaximal, characteristic, complemented
6C2×#2067C211yesmaximal, characteristic, complemented
7C2×#2059C211yesmaximal, characteristic, complemented
8C2×#2025C211yesmaximal, characteristic, complemented
9C2×#2025C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1972C211yesmaximal, characteristic
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1879C211yesmaximal, characteristic
14C2×#1879C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1804C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2566—24no—
34#2566C2218yescomplemented
35#2454—24no—
36#2454C2218yescomplemented
37C2×#399—21no—
38C2×#396—21no—
39C2×#396—21no—
40C2×#396—21no—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#385—21no—
48C2×#385—21no—
49C2×#385—21no—
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2093C2218yescomplemented
55#2085C2218yescomplemented
56#2067C2218yescomplemented
57#2059C2218yescomplemented
58#2025C2218yescomplemented
59#2025C2218yescomplemented
60#1992C2218yescomplemented
61#1972C2218yes—
62#1917C2218yescomplemented
63#1879C2218yes—
64#1879C2218yes—
65#1862C2218yescomplemented
66#1804C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C22×D8—42no—
82C22×D8—42no—
83C22×D8Q818yes—
84C22×D8C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#125—24no—
90C2×#125—22no—
91C2×#125—22no—
92C2×#125—42no—
93C2×#125—42no—
94C2×#125—42no—
95C2×#125C2314yes—
96C2×#125C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×C4×D4C2314yes—
104C2×C4×D4C2314yes—
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⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C2×#107—42no—
135C2×#107—42no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106—42no—
139C2×#106C2314yes—
140C2×C4⋊C8—42no—
141C2×C4⋊C8C2314yes—
142C2×#102—24no—
143C2×#102—42no—
144C2×#102—42no—
145C2×#102—42no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—42no—
155C2×#97C2314yes—
156#399—28no—
157#396—28no—
158#396—28no—
159#396—28no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#390C2314yes—
166#390C2314yes—
167#390C2314yes—
168#389—22no—
169#389—22no—
170#389C2314yes—
171#389C2314yes—
172#385—28no—
173#385—28no—
174#385—28no—
175#384—22no—
176#384C2314yes—
177#382—22no—
178#382C2314yes—
179#380—28no—
180C4×C4⋊C4D414yes—
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#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198C25—21no—
199C25—21no—
200C22×D4—21no—
201C22×D4—21no—
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×D4C2×Q814yes—
213C22×D4C2×Q814yes—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4—82no—
229C23×C4—82no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×D8—416no—
238C2×D8—44no—
239C2×D8—44no—
240C2×D8C2×Q8116yes—
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—22no—
251C22×C8—22no—
252C22×C8C2×Q814yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257#125—28no—
258#125—44no—
259#125—44no—
260#125—44no—
261#125—48no—
262#125—48no—
263C4×D4—28no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—48no—
268C4×D4—48no—
269C2×C4⋊C4—21no—
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—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4—82no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345#107—44no—
346#107—44no—
347#106—44no—
348C4⋊C8—44no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355#97—44no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24C22×D411yescharacteristic
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4C22×Q814yes—
404C2×D4C22×Q814yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
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—22no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
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—42no—
449C22×C4—42no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—816no—
454C22×C4—816no—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q812yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469D8—416no—
470C2×C8—28no—
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—22no—
481C2×C8—22no—
482C2×C8—44no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8C22×Q814yes—
486C4⋊C4—24no—
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—22no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4C2×#4114yes—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—24no—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C23—24no—
534C23—24no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—42no—
540C23—42no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23—84no—
559C23—84no—
560C23—84no—
561C23#53512yes—
562C23#54612yes—
563C23D4×Q812yes—
564C23#57512yes—
565C23#57512yes—
566C23#57612yes—
567C23#57612yes—
568C23C23×D411yescharacteristic
569D4—24no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—42no—
574D4—42no—
575D4—48no—
576D4—48no—
577D4—42no—
578D4—42no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
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—22no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—816no—
623C2×C4—816no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4#53512yes—
630C2×C4#54214yes—
631C2×C4#54314yes—
632C2×C4#54612yes—
633C2×C4D4×Q812yes—
634C2×C4#55414yes—
635C2×C4#56214yes—
636C2×C4C23×Q811yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+411yescharacteristic
642C8—24no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—816no—
653C22—816no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#269214yes—
663C22#321914yes—
664C22#322014yes—
665C22#330914yes—
666C22#332214yes—
667C22#338914yes—
668C22#338914yes—
669C22C2×#53511yescharacteristic
670C22C2×#54611yescharacteristic
671C22C2×D4×Q811yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57611yescharacteristic
675C22C2×#57611yescharacteristic
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4#269214yes—
689C4C2×#54211yescharacteristic
690C4C2×#54311yescharacteristic
691C4C2×#55411yescharacteristic
692C4C2×#56211yescharacteristic
693C2—84no—
694C2—84no—
695C2#3153518yesdirect product factor
696C2C2×#269211yescharacteristic
697C2C2×#321911yescharacteristic
698C2C2×#322011yescharacteristic
699C2C2×#330911yescharacteristic
700C2C2×#332211yescharacteristic
701C2C2×#338911yescharacteristic
702C2C2×#338911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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×#2566C2semidirect
2. C2×#2454C2semidirect
3. #31535C2direct (N × H)
4. C2×#2093C2semidirect
5. C2×#2085C2semidirect
6. C2×#2067C2semidirect
7. C2×#2059C2semidirect
8. C2×#2025C2semidirect
9. C2×#2025C2semidirect
10. C2×#1992C2semidirect
12. C2×#1917C2semidirect
15. C2×#1862C2semidirect
16. C2×#1804C2semidirect
34. #2566C22semidirect
36. #2454C22semidirect
54. #2093C22semidirect
55. #2085C22semidirect
56. #2067C22semidirect
57. #2059C22semidirect
58. #2025C22semidirect
59. #2025C22semidirect
60. #1992C22semidirect
62. #1917C22semidirect
65. #1862C22semidirect
66. #1804C22semidirect
695. C2#31535direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25662
C2×#24542
#315352
C2×#20932
C2×#20852
C2×#20672
C2×#20592
C2×#20252
C2×#19922
C2×#19722
C2×#19172
C2×#18792
C2×#18622
C2×#18042