Finite Group Databasebeta

← C2×#31659 #7505338

Subgroups — C2×#31659 #7505338

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic
3#31659C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic
5C2×#2074C211yesmaximal, characteristic
6C2×#2056C211yesmaximal, characteristic
7C2×#2024C211yesmaximal, characteristic
8C2×#2024C211yesmaximal, characteristic
9C2×#2016C211yesmaximal, characteristic
10C2×#1960C211yesmaximal, characteristic
11C2×#1935C211yesmaximal, characteristic
12C2×#1917C211yesmaximal, characteristic
13C2×#1878C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic
15C2×#1878C211yesmaximal, characteristic
16C2×#1805C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2568—24no—
34#2568C2218yes—
35#2446—24no—
36#2446C2218yes—
37C2×#400—21no—
38C2×#400—21no—
39C2×#400—21no—
40C2×#398—21no—
41C2×#391—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#387—21no—
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2218yes—
55#2074C2218yes—
56#2056C2218yes—
57#2024C2218yes—
58#2024C2218yes—
59#2016C2218yes—
60#1960C2218yes—
61#1935C2218yes—
62#1917C2218yes—
63#1878C2218yes—
64#1878C2218yes—
65#1878C2218yes—
66#1805C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8—21no—
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C22×Q16—24no—
82C22×Q16—24no—
83C22×Q16—24no—
84C22×Q16—42no—
85C22×Q16—42no—
86C22×Q16—42no—
87C22×Q16C2314yes—
88C22×Q16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2314yes—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8—42no—
106C2×C4×Q8—42no—
107C2×C4×Q8C2314yes—
108C2×C4×Q8C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C2×C4⋊C8—42no—
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—28no—
152#391—28no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#391C2314yes—
163#389—22no—
164#389C2314yes—
165#387—28no—
166#387—28no—
167#382—22no—
168#382—22no—
169#382C2314yes—
170#382C2314yes—
171C4×C4⋊C4D414yes—
172C4×C4⋊C4C2314yes—
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×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95D411yescharacteristic
189C2×#95D411yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—24no—
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—44no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8#4114yes—
224C22×Q8#4114yes—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—28no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16—416no—
250C2×Q16—416no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—28no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—48no—
272C4⋊Q8—48no—
273C4×Q8—28no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—48no—
278C4×Q8—48no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C4⋊C8—44no—
332#103—28no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#97—44no—
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#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24C22×D411yescharacteristic
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—28no—
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—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—816no—
400C2×Q8C2×#4114yes—
401C2×Q8C2×#4114yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
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—22no—
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—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4#13714yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466Q16—416no—
467Q16—416no—
468C2×C8—22no—
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×C8—44no—
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—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—48no—
517C42—48no—
518C23#53512yes—
519C23#54612yes—
520C23#54812yes—
521C23C2×#13712yes—
522C23#57512yes—
523C23#57812yes—
524C23#57812yes—
525C23C23×D411yescharacteristic
526Q8—24no—
527Q8—24no—
528Q8—24no—
529Q8—24no—
530Q8—42no—
531Q8—42no—
532Q8—48no—
533Q8—48no—
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—816no—
541Q8—84no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
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—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4#53512yes—
598C2×C4#53814yes—
599C2×C4#53914yes—
600C2×C4#54612yes—
601C2×C4#54812yes—
602C2×C4#55114yes—
603C2×C4#56214yes—
604C2×C4C2×#13714yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C8—42no—
612C8—42no—
613C8—42no—
614C8—42no—
615C22#269614yes—
616C22#321514yes—
617C22#322314yes—
618C22#331314yes—
619C22#332814yes—
620C22#337414yes—
621C22#337714yes—
622C22C2×#53511yescharacteristic
623C22C2×#54611yescharacteristic
624C22C2×#54811yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57811yescharacteristic
628C22C2×#57811yescharacteristic
629C4—24no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#269614yes—
644C4C2×#53811yescharacteristic
645C4C2×#53911yescharacteristic
646C4C2×#55111yescharacteristic
647C4C2×#56211yescharacteristic
648C2#3165918yesdirect product factor
649C2C2×#269611yescharacteristic
650C2C2×#321511yescharacteristic
651C2C2×#322311yescharacteristic
652C2C2×#331311yescharacteristic
653C2C2×#332811yescharacteristic
654C2C2×#337411yescharacteristic
655C2C2×#337711yescharacteristic
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. #31659C2direct (N × H)
648. C2#31659direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24462
#316592
C2×#21012
C2×#20742
C2×#20562
C2×#20242
C2×#20162
C2×#19602
C2×#19352
C2×#19172
C2×#18782
C2×#18052