Finite Group Databasebeta

← C2×#32525 #7509976

Subgroups — C2×#32525 #7509976

Order 512.

1820 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2511C211yesmaximal, characteristic, complemented
2C2×#2512C211yesmaximal, characteristic
3#32525C2116yesmaximal, direct product factor
4C2×#2096C211yesmaximal, characteristic, complemented
5C2×#2094C211yesmaximal, characteristic
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2055C211yesmaximal, characteristic
9C2×#2055C211yesmaximal, characteristic
10C2×#1968C211yesmaximal, characteristic
11C2×#1948C211yesmaximal, characteristic, complemented
12C2×#1948C211yesmaximal, characteristic, complemented
13C2×#1941C211yesmaximal, characteristic, complemented
14C2×#1939C211yesmaximal, characteristic, complemented
15C2×#1926C211yesmaximal, characteristic
16C2×#1927C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C22×#126—21no—
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2512—24no—
35#2512C2218yes—
36#2511—24no—
37#2511C2218yescomplemented
38C2×#400—21no—
39C2×#400—21no—
40C2×#397—21no—
41C2×#396—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#387—21no—
52C2×#387—21no—
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2096C2218yescomplemented
59#2094C2218yes—
60#2066C2218yescomplemented
61#2065C2218yes—
62#2055C2218yes—
63#2055C2218yes—
64#1968C2218yes—
65#1948C2218yescomplemented
66#1948C2218yescomplemented
67#1941C2218yescomplemented
68#1939C2218yescomplemented
69#1927C2218yescomplemented
70#1926C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C22×SD16—24no—
80C22×SD16—42no—
81C22×SD16—42no—
82C22×SD16C2314yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#126—28no—
88C2×#126—48no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C42—21no—
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C2×#106—24no—
115C2×#106—42no—
116C2×#106—42no—
117C2×#106—42no—
118C2×#106—42no—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8—24no—
125C2×C4⋊C8—24no—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#103—24no—
131C2×#103—42no—
132C2×#103—42no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102—24no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C4×C8—42no—
143C2×C4×C8C2314yes—
144#400—28no—
145#400—28no—
146#397—28no—
147#396—28no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391C2314yes—
152#391C2314yes—
153#391C2314yes—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#389C2314yes—
164#388—28no—
165#387—28no—
166#387—28no—
167#384—22no—
168#384—22no—
169#384C2314yes—
170#384C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D414yes—
174C4×C4⋊C4C2314yes—
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×#95D411yescharacteristic
188C2×#95D411yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8C2412yes—
204C22×D4—21no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2411yescharacteristic
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—416no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8#4114yes—
241C22×C8#4114yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246#126—432no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
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—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4—82no—
277C2×C4⋊C4—82no—
278C2×C4⋊C4SD1618yescomplemented
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—82no—
298C2×C22⋊C4—82no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42—82no—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315#106—28no—
316#106—44no—
317#106—44no—
318#106—44no—
319#106—44no—
320C4⋊C8—28no—
321C4⋊C8—28no—
322C4⋊C8—44no—
323#103—28no—
324#103—44no—
325#103—44no—
326#102—28no—
327#102—44no—
328#102—44no—
329C4×C8—44no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95C2×D414yes—
341#95C2×D414yes—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24C22×D411yescharacteristic
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—44no—
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—22no—
387C22×C4—22no—
388C22×C4—22no—
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—42no—
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×C4C2×SD1614yes—
421C22×C4C2×SD1614yes—
422C22×C4C2×SD1614yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437SD16—416no—
438C2×C8—28no—
439C2×C8—28no—
440C2×C8—22no—
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—44no—
451C2×C8—44no—
452C2×C8C2×#4114yes—
453C2×C8C2×#4114yes—
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—22no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C2×SD16116yescomplemented
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—84no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#53712yes—
512C23#54712yes—
513C23#55012yes—
514C23C22×SD1612yes—
515C23#57512yes—
516C23#57712yes—
517C23#57712yes—
518C23C23×D411yescharacteristic
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—84no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—84no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
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—44no—
549C2×C4—44no—
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4#53712yes—
579C2×C4#53814yes—
580C2×C4#53914yes—
581C2×C4#54712yes—
582C2×C4#55012yes—
583C2×C4#55614yes—
584C2×C4#56314yes—
585C2×C4C22×SD1614yes—
586C2×C4C22×SD1614yes—
587C2×C4C22×SD1614yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C8—24no—
595C8—24no—
596C8—42no—
597C8—42no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—168no—
606C22#272714yes—
607C22#324614yes—
608C22#326314yes—
609C22#333514yes—
610C22#335114yes—
611C22#339214yes—
612C22#340114yes—
613C22C2×#53711yescharacteristic
614C22C2×#54711yescharacteristic
615C22C2×#55011yescharacteristic
616C22C23×SD1611yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57711yescharacteristic
619C22C2×#57711yescharacteristic
620C4—24no—
621C4—24no—
622C4—24no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#272714yes—
634C4C2×#53811yescharacteristic
635C4C2×#53911yescharacteristic
636C4C2×#55611yescharacteristic
637C4C2×#56311yescharacteristic
638C2—162no—
639C2#3252518yesdirect product factor
640C2C2×#272711yescharacteristic
641C2C2×#324611yescharacteristic
642C2C2×#326311yescharacteristic
643C2C2×#333511yescharacteristic
644C2C2×#335111yescharacteristic
645C2C2×#339211yescharacteristic
646C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. C2×#2511C2semidirect
3. #32525C2direct (N × H)
4. C2×#2096C2semidirect
6. C2×#2066C2semidirect
11. C2×#1948C2semidirect
12. C2×#1948C2semidirect
13. C2×#1941C2semidirect
14. C2×#1939C2semidirect
16. C2×#1927C2semidirect
37. #2511C22semidirect
58. #2096C22semidirect
60. #2066C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1941C22semidirect
68. #1939C22semidirect
69. #1927C22semidirect
278. C2×C4⋊C4SD16semidirect
479. C4⋊C4C2×SD16semidirect
639. C2#32525direct (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×#25122
#325252
C2×#20962
C2×#20942
C2×#20662
C2×#20652
C2×#20552
C2×#19682
C2×#19482
C2×#19412
C2×#19392
C2×#19262
C2×#19272