Finite Group Databasebeta

← C2×#31371 #7504525

Subgroups — C2×#31371 #7504525

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2441C211yesmaximal, characteristic
2C2×#2423C211yesmaximal, characteristic
3#31371C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2035C211yesmaximal, characteristic
9C2×#2026C211yesmaximal, characteristic, complemented
10C2×#1992C211yesmaximal, characteristic, complemented
11C2×#1935C211yesmaximal, characteristic, complemented
12C2×#1924C211yesmaximal, characteristic
13C2×#1873C211yesmaximal, characteristic
14C2×#1865C211yesmaximal, characteristic, complemented
15C2×#1825C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2441—24no—
32#2441C2218yes—
33#2423—24no—
34#2423C2218yes—
35C2×#399—21no—
36C2×#399—21no—
37C2×#398—21no—
38C2×#396—21no—
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2100C2218yescomplemented
53#2088C2218yescomplemented
54#2084C2218yescomplemented
55#2077C2218yes—
56#2035C2218yes—
57#2026C2218yescomplemented
58#1992C2218yescomplemented
59#1935C2218yescomplemented
60#1924C2218yes—
61#1873C2218yes—
62#1865C2218yescomplemented
63#1825C2218yes—
64#1803C2218yescomplemented
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×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—24no—
84C2×#125—22no—
85C2×#125—42no—
86C2×#125C2314yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—42no—
90C2×C22⋊Q8C2314yes—
91C2×C4×Q8—22no—
92C2×C4×Q8—24no—
93C2×C4×Q8—42no—
94C2×C4×Q8C2314yes—
95C2×#117—22no—
96C2×#117—24no—
97C2×#117—42no—
98C2×#117C2314yes—
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⋊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⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×#106—42no—
128C2×#106—42no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#103—24no—
134C2×#103—24no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8—42no—
142C2×C22⋊C8—42no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97—42no—
146C2×#97C2314yes—
147#399—28no—
148#399—28no—
149#398—28no—
150#396—28no—
151#391—22no—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—28no—
157#389C2314yes—
158#389C2314yes—
159#389C2314yes—
160#388—28no—
161#387—28no—
162#387—28no—
163#384—22no—
164#384—22no—
165#384C2314yes—
166#384C2314yes—
167#380—22no—
168#380—22no—
169#380C2314yes—
170#380C2314yes—
171C4×C22⋊C4D414yes—
172C4×C22⋊C4C2314yes—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95D411yescharacteristic
181C2×#95D411yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8#4114yes—
202C22×Q8C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D412yes—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241#125—28no—
242#125—44no—
243#125—48no—
244C22⋊Q8—28no—
245C22⋊Q8—44no—
246C22⋊Q8—48no—
247C4×Q8—28no—
248C4×Q8—44no—
249C4×Q8—48no—
250#117—28no—
251#117—44no—
252#117—48no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4#4114yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#106—44no—
328#106—44no—
329C4⋊C8—44no—
330#103—28no—
331#103—28no—
332#103—44no—
333#103—44no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336#97—44no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—42no—
366C2×Q8—44no—
367C2×Q8C2×#4114yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
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—22no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4#13714yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)12yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C42−1+412yes—
439C22×C42−1+412yes—
440C22×C4C2511yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—44no—
454C2×C8—44no—
455C2×C8—44no—
456C2×C8—44no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4C2×#4114yes—
489C22⋊C4—28no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—48no—
502C42—48no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23#53612yes—
518C23#54612yes—
519C23#54612yes—
520C23C2×#13712yes—
521C23#57512yes—
522C23#57712yes—
523C23#57712yes—
524C23C23×D411yescharacteristic
525Q8—24no—
526Q8—24no—
527Q8—42no—
528Q8—48no—
529Q8—42no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#53612yes—
587C2×C4#53914yes—
588C2×C4#54214yes—
589C2×C4#54612yes—
590C2×C4#54612yes—
591C2×C4#55214yes—
592C2×C4#56214yes—
593C2×C4C2×#13714yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C8—42no—
601C8—42no—
602C8—42no—
603C8—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—88no—
612C22#268714yes—
613C22#323714yes—
614C22#324414yes—
615C22#331714yes—
616C22#331714yes—
617C22#332014yes—
618C22#333114yes—
619C22C2×#53611yescharacteristic
620C22C2×#54611yescharacteristic
621C22C2×#54611yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57711yescharacteristic
625C22C2×#57711yescharacteristic
626C4—24no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4#268714yes—
640C4C2×#53911yescharacteristic
641C4C2×#54211yescharacteristic
642C4C2×#55211yescharacteristic
643C4C2×#56211yescharacteristic
644C2—82no—
645C2#3137118yesdirect product factor
646C2C2×#268711yescharacteristic
647C2C2×#323711yescharacteristic
648C2C2×#324411yescharacteristic
649C2C2×#332011yescharacteristic
650C2C2×#331711yescharacteristic
651C2C2×#331711yescharacteristic
652C2C2×#333111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #31371C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2088C2semidirect
6. C2×#2084C2semidirect
9. C2×#2026C2semidirect
10. C2×#1992C2semidirect
11. C2×#1935C2semidirect
14. C2×#1865C2semidirect
16. C2×#1803C2semidirect
52. #2100C22semidirect
53. #2088C22semidirect
54. #2084C22semidirect
57. #2026C22semidirect
58. #1992C22semidirect
59. #1935C22semidirect
62. #1865C22semidirect
64. #1803C22semidirect
645. C2#31371direct (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×#24232
#313712
C2×#21002
C2×#20882
C2×#20842
C2×#20772
C2×#20352
C2×#20262
C2×#19922
C2×#19352
C2×#19242
C2×#18732
C2×#18652
C2×#18252
C2×#18032