Finite Group Databasebeta

← C2×#31536 #7504987

Subgroups — C2×#31536 #7504987

Order 512.

1884 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×#2567C211yesmaximal, characteristic
2C2×#2453C211yesmaximal, characteristic
3#31536C2116yesmaximal, direct product factor
4C2×#2096C211yesmaximal, characteristic
5C2×#2084C211yesmaximal, characteristic
6C2×#2066C211yesmaximal, characteristic
7C2×#2058C211yesmaximal, characteristic
8C2×#2026C211yesmaximal, characteristic
9C2×#2024C211yesmaximal, characteristic
10C2×#1992C211yesmaximal, characteristic
11C2×#1971C211yesmaximal, characteristic
12C2×#1917C211yesmaximal, characteristic
13C2×#1878C211yesmaximal, characteristic
14C2×#1878C211yesmaximal, characteristic
15C2×#1865C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic
17C23×Q16C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2567—24no—
34#2567C2218yes—
35#2453—24no—
36#2453C2218yes—
37C2×#400—21no—
38C2×#398—21no—
39C2×#395—21no—
40C2×#395—21no—
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#389—21no—
46C2×#389—21no—
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#387—21no—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2096C2218yes—
55#2084C2218yes—
56#2066C2218yes—
57#2058C2218yes—
58#2026C2218yes—
59#2024C2218yes—
60#1992C2218yes—
61#1971C2218yes—
62#1917C2218yes—
63#1878C2218yes—
64#1878C2218yes—
65#1865C2218yes—
66#1803C2218yes—
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
80C22×Q16—42no—
81C22×Q16—42no—
82C22×Q16Q818yescomplemented
83C22×Q16C2314yes—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8C2314yes—
92C2×#125—24no—
93C2×#125—22no—
94C2×#125—42no—
95C2×#125C2314yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8—24no—
99C2×C4×Q8—42no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
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⋊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×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C2×#106—42no—
129C2×#106—42no—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8—42no—
135C2×C4⋊C8C2314yes—
136C2×#103—24no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#97—42no—
149C2×#97C2314yes—
150#400—28no—
151#398—28no—
152#395—28no—
153#395—28no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2314yes—
159#391C2314yes—
160#391C2314yes—
161#391C2314yes—
162#389—28no—
163#389—28no—
164#389—28no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#387—28no—
170#384—22no—
171#384—22no—
172#384C2314yes—
173#384C2314yes—
174C4×C4⋊C4D414yes—
175C4×C4⋊C4C2314yes—
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×#95D411yescharacteristic
187C2×#95D411yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
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—44no—
212C22×Q8—82no—
213C22×Q8—82no—
214C22×Q8C2×Q814yes—
215C22×Q8C2×Q814yes—
216C22×Q8C2412yes—
217C22×Q8C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C2×Q16—416no—
234C2×Q16—44no—
235C2×Q16—44no—
236C2×Q16C2×Q8116yescomplemented
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8C2×Q814yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256#125—28no—
257#125—44no—
258#125—48no—
259C4×Q8—28no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C4×Q8—48no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
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—24no—
280C2×C4⋊C4—24no—
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—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4#4114yes—
309C2×C4⋊C4C2412yes—
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—
331#106—44no—
332#106—44no—
333#106—44no—
334C4⋊C8—44no—
335#103—28no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341#97—44no—
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—24no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
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—48no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×Q8—816no—
383C2×Q8C22×Q814yes—
384C2×Q8C22×Q814yes—
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—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
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—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—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4#13714yes—
434C22×C4#13714yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×Q812yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+412yes—
448C22×C4C2511yescharacteristic
449Q16—416no—
450C2×C8—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8—44no—
465C2×C8C22×Q814yes—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4C2×#4114yes—
497C42—24no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—48no—
507C42—48no—
508C23#53512yes—
509C23#54612yes—
510C23D4×Q812yes—
511C23C2×#13712yes—
512C23C2×#13712yes—
513C23#57712yes—
514C23#57712yes—
515C23C23×D411yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—24no—
519Q8—24no—
520Q8—42no—
521Q8—42no—
522Q8—48no—
523Q8—48no—
524Q8—42no—
525Q8—42no—
526Q8—816no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
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—22no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4#53512yes—
582C2×C4#54214yes—
583C2×C4#54314yes—
584C2×C4#54612yes—
585C2×C4D4×Q812yes—
586C2×C4#55414yes—
587C2×C4#56214yes—
588C2×C4C2×#13714yes—
589C2×C4C2×#13714yes—
590C2×C4C23×Q811yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×2−1+411yescharacteristic
596C8—24no—
597C8—42no—
598C8—42no—
599C8—42no—
600C22#269214yes—
601C22#321814yes—
602C22#322214yes—
603C22#331614yes—
604C22#333114yes—
605C22#338714yes—
606C22#338714yes—
607C22C2×#53511yescharacteristic
608C22C2×#54611yescharacteristic
609C22C2×D4×Q811yescharacteristic
610C22C22×#13711yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57711yescharacteristic
613C22C2×#57711yescharacteristic
614C4—24no—
615C4—24no—
616C4—42no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4#269214yes—
629C4C2×#54211yescharacteristic
630C4C2×#54311yescharacteristic
631C4C2×#55411yescharacteristic
632C4C2×#56211yescharacteristic
633C2#3153618yesdirect product factor
634C2C2×#269211yescharacteristic
635C2C2×#321811yescharacteristic
636C2C2×#322211yescharacteristic
637C2C2×#331611yescharacteristic
638C2C2×#333111yescharacteristic
639C2C2×#338711yescharacteristic
640C2C2×#338711yescharacteristic
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. #31536C2direct (N × H)
82. C22×Q16Q8semidirect
236. C2×Q16C2×Q8semidirect
633. C2#31536direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#24532
#315362
C2×#20962
C2×#20842
C2×#20662
C2×#20582
C2×#20262
C2×#20242
C2×#19922
C2×#19712
C2×#19172
C2×#18782
C2×#18652
C2×#18032