Finite Group Databasebeta

← C2×#31603 #7505230

Subgroups — C2×#31603 #7505230

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C211yesmaximal, characteristic
2C2×#2421C211yesmaximal, characteristic
3#31603C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic
8C2×#2066C211yesmaximal, characteristic, complemented
9C2×#2036C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1928C211yesmaximal, characteristic, complemented
12C2×#1892C211yesmaximal, characteristic
13C2×#1865C211yesmaximal, characteristic, complemented
14C2×#1850C211yesmaximal, characteristic
15C2×#1807C211yesmaximal, characteristic
16C2×#1805C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2495—24no—
32#2495C2218yes—
33#2421—24no—
34#2421C2218yes—
35C2×#400—21no—
36C2×#400—21no—
37C2×#397—21no—
38C2×#397—21no—
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45C2×#387—21no—
46C2×#383—21no—
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#2101C2218yescomplemented
55#2087C2218yescomplemented
56#2085C2218yescomplemented
57#2075C2218yes—
58#2066C2218yescomplemented
59#2036C2218yes—
60#1949C2218yescomplemented
61#1928C2218yescomplemented
62#1892C2218yes—
63#1865C2218yescomplemented
64#1850C2218yes—
65#1807C2218yes—
66#1805C2218yescomplemented
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×C8—24no—
80C23×C8—42no—
81C23×C8—42no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2314yes—
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—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C2×#107—42no—
118C2×#107—42no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106—42no—
122C2×#106—42no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8—24no—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—42no—
132C2×#103—42no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8—42no—
138C2×C22⋊C8C2314yes—
139#400—28no—
140#400—28no—
141#397—28no—
142#397—28no—
143#391—22no—
144#391C2314yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#388—28no—
152#388—28no—
153#387—28no—
154#383—28no—
155#382—22no—
156#382—22no—
157#382C2314yes—
158#382C2314yes—
159#380—22no—
160#380—22no—
161#380—22no—
162#380C2314yes—
163#380C2314yes—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C22⋊C4D414yes—
168C4×C22⋊C4C2314yes—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
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—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—21no—
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—28no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8—416no—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22⋊Q8—28no—
246C22⋊Q8—44no—
247C22⋊Q8—48no—
248C4×Q8—28no—
249C4×Q8—44no—
250C4×Q8—48no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
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—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—82no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—82no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2412yes—
312C2×C42C2412yes—
313#107—44no—
314#107—44no—
315#106—44no—
316#106—44no—
317C4⋊C8—28no—
318C4⋊C8—44no—
319#103—28no—
320#103—28no—
321#103—44no—
322#103—44no—
323C22⋊C8—44no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24#13514yes—
346C24C22×D412yes—
347C24C22×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8C2×#4114yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
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×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C22×C42−1+412yes—
433C22×C4C2511yescharacteristic
434C2×C8—28no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—416no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—44no—
450C2×C8C2×#4114yes—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C22⋊C4—28no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—84no—
483C42—24no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—48no—
490C42—48no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23#53712yes—
505C23#54612yes—
506C23#54712yes—
507C23C2×#13514yes—
508C23C2×#13512yes—
509C23#57512yes—
510C23#57712yes—
511C23#57812yes—
512C23C23×D411yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—42no—
516Q8—48no—
517Q8—42no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
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—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#53712yes—
576C2×C4#53814yes—
577C2×C4#53914yes—
578C2×C4#54612yes—
579C2×C4#54712yes—
580C2×C4#55214yes—
581C2×C4#56314yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×H(5,𝔽2)11yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C8—24no—
589C8—42no—
590C8—42no—
591C8—42no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—42no—
596C22—416no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22#269514yes—
601C22#325014yes—
602C22#325814yes—
603C22#330014yes—
604C22#332114yes—
605C22#333714yes—
606C22#334914yes—
607C22C2×#53711yescharacteristic
608C22C2×#54611yescharacteristic
609C22C2×#54711yescharacteristic
610C22C22×#13511yescharacteristic
611C22C2×#57511yescharacteristic
612C22C2×#57711yescharacteristic
613C22C2×#57811yescharacteristic
614C4—42no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#269514yes—
628C4C2×#53811yescharacteristic
629C4C2×#53911yescharacteristic
630C4C2×#55211yescharacteristic
631C4C2×#56311yescharacteristic
632C2—44no—
633C2#3160318yesdirect product factor
634C2C2×#269511yescharacteristic
635C2C2×#325011yescharacteristic
636C2C2×#325811yescharacteristic
637C2C2×#330011yescharacteristic
638C2C2×#332111yescharacteristic
639C2C2×#333711yescharacteristic
640C2C2×#334911yescharacteristic
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. #31603C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2087C2semidirect
6. C2×#2085C2semidirect
8. C2×#2066C2semidirect
10. C2×#1949C2semidirect
11. C2×#1928C2semidirect
13. C2×#1865C2semidirect
16. C2×#1805C2semidirect
54. #2101C22semidirect
55. #2087C22semidirect
56. #2085C22semidirect
58. #2066C22semidirect
60. #1949C22semidirect
61. #1928C22semidirect
63. #1865C22semidirect
66. #1805C22semidirect
633. C2#31603direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24952
C2×#24212
#316032
C2×#21012
C2×#20872
C2×#20852
C2×#20752
C2×#20662
C2×#20362
C2×#19492
C2×#19282
C2×#18922
C2×#18652
C2×#18502
C2×#18072
C2×#18052