Finite Group Databasebeta

← C2×#31920 #7506605

Subgroups — C2×#31920 #7506605

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2511C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic
3#31920C2116yesmaximal, direct product factor
4C2×#2096C211yesmaximal, characteristic
5C2×#2066C211yesmaximal, characteristic
6C2×#2066C211yesmaximal, characteristic
7C2×#2056C211yesmaximal, characteristic
8C2×#2056C211yesmaximal, characteristic
9C2×#2016C211yesmaximal, characteristic
10C2×#1981C211yesmaximal, characteristic
11C2×#1971C211yesmaximal, characteristic
12C2×#1940C211yesmaximal, characteristic
13C2×#1935C211yesmaximal, characteristic
14C2×#1919C211yesmaximal, characteristic
15C2×#1878C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2511—24no—
36#2511C2218yes—
37#2446—24no—
38#2446C2218yes—
39C2×#400—21no—
40C2×#400—21no—
41C2×#400—21no—
42C2×#398—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#387—21no—
52C2×#387—21no—
53C2×#387—21no—
54C2×#387—21no—
55C2×#384C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2096C2218yes—
59#2066C2218yes—
60#2066C2218yes—
61#2056C2218yes—
62#2056C2218yes—
63#2016C2218yes—
64#1981C2218yes—
65#1971C2218yes—
66#1940C2218yes—
67#1935C2218yes—
68#1919C2218yes—
69#1878C2218yes—
70#1803C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C22×Q16—24no—
81C22×Q16—24no—
82C22×Q16—42no—
83C22×Q16—42no—
84C22×Q16—42no—
85C22×Q16—42no—
86C22×Q16C2314yes—
87C22×Q16C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—42no—
95C2×C4⋊Q8C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—24no—
98C2×C4×Q8—42no—
99C2×C4×Q8C2314yes—
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×C42—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#106—24no—
120C2×#106—42no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8—42no—
124C2×C4⋊C8—42no—
125C2×C4⋊C8—42no—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103—24no—
130C2×#103—42no—
131C2×#103—42no—
132C2×#103—42no—
133C2×#103—42no—
134C2×#103—42no—
135C2×#103—42no—
136C2×#103—42no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#97—42no—
146C2×#97C2314yes—
147#400—28no—
148#400—28no—
149#400—28no—
150#398—28no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2314yes—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#391C2314yes—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#387—28no—
168#387—28no—
169#387—28no—
170#387—28no—
171#384—22no—
172#384C2314yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D414yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8#4114yes—
221C22×Q8C2412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×Q16—28no—
240C2×Q16—28no—
241C2×Q16—44no—
242C2×Q16—44no—
243C2×Q16—44no—
244C2×Q16—44no—
245C2×Q16—416no—
246C2×Q16—416no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8#4114yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—28no—
264C4⋊Q8—44no—
265C4⋊Q8—48no—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
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—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
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⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—82no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#106—28no—
324#106—44no—
325C4⋊C8—44no—
326C4⋊C8—44no—
327C4⋊C8—44no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335#103—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#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95C2×D414yes—
353#95C2×D414yes—
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—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8C2×#4114yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
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—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
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#13714yes—
443C22×C4#13714yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458Q16—416no—
459Q16—416no—
460C2×C8—28no—
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—44no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8C2×#4114yes—
476C4⋊C4—28no—
477C4⋊C4—28no—
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—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C42—24no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—48no—
510C42—84no—
511C23#53712yes—
512C23#54612yes—
513C23#55012yes—
514C23C2×#13712yes—
515C23C2×#13712yes—
516C23#57712yes—
517C23#57812yes—
518C23C23×D411yescharacteristic
519Q8—24no—
520Q8—24no—
521Q8—42no—
522Q8—48no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—84no—
533Q8—84no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
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—42no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#53712yes—
589C2×C4#53914yes—
590C2×C4#54214yes—
591C2×C4#54612yes—
592C2×C4#55012yes—
593C2×C4#55114yes—
594C2×C4#56014yes—
595C2×C4C2×#13714yes—
596C2×C4C2×#13714yes—
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C8—24no—
604C8—42no—
605C8—42no—
606C8—42no—
607C22#270514yes—
608C22#325514yes—
609C22#325814yes—
610C22#331214yes—
611C22#331614yes—
612C22#339814yes—
613C22#340014yes—
614C22C2×#53711yescharacteristic
615C22C2×#54611yescharacteristic
616C22C2×#55011yescharacteristic
617C22C22×#13711yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57711yescharacteristic
620C22C2×#57811yescharacteristic
621C4—24no—
622C4—24no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#270514yes—
636C4C2×#53911yescharacteristic
637C4C2×#54211yescharacteristic
638C4C2×#55111yescharacteristic
639C4C2×#56011yescharacteristic
640C2#3192018yesdirect product factor
641C2C2×#270511yescharacteristic
642C2C2×#325511yescharacteristic
643C2C2×#325811yescharacteristic
644C2C2×#331611yescharacteristic
645C2C2×#331211yescharacteristic
646C2C2×#339811yescharacteristic
647C2C2×#340011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
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
3. #31920C2direct (N × H)
640. C2#31920direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25112
C2×#24462
#319202
C2×#20962
C2×#20662
C2×#20562
C2×#20162
C2×#19812
C2×#19712
C2×#19402
C2×#19352
C2×#19192
C2×#18782
C2×#18032