Finite Group Databasebeta

← C2×#32616 #7510467

Subgroups — C2×#32616 #7510467

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2592C211yesmaximal, characteristic
3#32616C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic
5C2×#2072C211yesmaximal, characteristic, complemented
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic
8C2×#2042C211yesmaximal, characteristic
9C2×#2015C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1971C211yesmaximal, characteristic, complemented
12C2×#1936C211yesmaximal, characteristic
13C2×#1927C211yesmaximal, characteristic, complemented
14C2×#1919C211yesmaximal, characteristic, complemented
15C2×#1880C211yesmaximal, characteristic
16C2×#1878C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2592—24no—
35#2592C2218yes—
36#2582—24no—
37#2582C2218yescomplemented
38C2×#398—21no—
39C2×#398—21no—
40C2×#397—21no—
41C2×#393—21no—
42C2×#391—21no—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#384C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2094C2218yes—
57#2072C2218yescomplemented
58#2066C2218yescomplemented
59#2055C2218yes—
60#2042C2218yes—
61#2015C2218yes—
62#1985C2218yescomplemented
63#1971C2218yescomplemented
64#1936C2218yes—
65#1927C2218yescomplemented
66#1919C2218yescomplemented
67#1880C2218yes—
68#1878C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C22×Q16—24no—
82C22×Q16—24no—
83C22×Q16—42no—
84C22×Q16C2314yes—
85C22×SD16—24no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C2×#124—22no—
94C2×#124—42no—
95C2×#124—42no—
96C2×#124C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8D414yes—
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2314yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C2×#106—24no—
130C2×#106—42no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#103—24no—
136C2×#103—42no—
137C2×#103—42no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102—24no—
146C2×#102—42no—
147C2×#102—42no—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#97—42no—
152C2×#97C2314yes—
153#398—28no—
154#398—28no—
155#397—28no—
156#393—28no—
157#391—28no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#391C2314yes—
165#391C2314yes—
166#390—22no—
167#390—22no—
168#390C2314yes—
169#390C2314yes—
170#389—28no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#388—28no—
176#388—28no—
177#384—22no—
178#384C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
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—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×Q8—24no—
202C22×Q8—21no—
203C22×Q8—21no—
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—42no—
216C22×Q8—42no—
217C22×Q8C2×D414yes—
218C22×Q8#4114yes—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4C2412yes—
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—44no—
246C2×Q16—416no—
247C2×SD16—28no—
248C2×SD16—44no—
249C2×SD16—44no—
250C2×SD16—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#4114yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—44no—
268#124—44no—
269#124—48no—
270C4×Q8—48no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—48no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
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—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—82no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#106—28no—
341#106—44no—
342C4⋊C8—44no—
343#103—28no—
344#103—44no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—28no—
349#102—44no—
350#102—44no—
351#97—44no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24C22×D411yescharacteristic
372C2×Q8—24no—
373C2×Q8—28no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—416no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8C2×#4114yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
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—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4#13714yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475Q16—416no—
476SD16—416no—
477C2×C8—28no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—44no—
490C2×C8—44no—
491C2×C8—44no—
492C2×C8C2×#4114yes—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C42—24no—
523C42—24no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—48no—
532C42—84no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23#53412yes—
548C23#54812yes—
549C23#55012yes—
550C23C2×#13712yes—
551C23#57512yes—
552C23#57712yes—
553C23#57712yes—
554C23C23×D411yescharacteristic
555Q8—24no—
556Q8—24no—
557Q8—416no—
558Q8—42no—
559Q8—48no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—84no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—84no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4#53412yes—
623C2×C4#53814yes—
624C2×C4#54214yes—
625C2×C4#54812yes—
626C2×C4#55012yes—
627C2×C4#55114yes—
628C2×C4#56214yes—
629C2×C4C2×#13714yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+411yescharacteristic
636C8—24no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—168no—
648C22#273014yes—
649C22#319614yes—
650C22#320414yes—
651C22#336914yes—
652C22#337114yes—
653C22#340014yes—
654C22#340114yes—
655C22C2×#53411yescharacteristic
656C22C2×#54811yescharacteristic
657C22C2×#55011yescharacteristic
658C22C22×#13711yescharacteristic
659C22C2×#57511yescharacteristic
660C22C2×#57711yescharacteristic
661C22C2×#57711yescharacteristic
662C4—24no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#273014yes—
676C4C2×#53811yescharacteristic
677C4C2×#54211yescharacteristic
678C4C2×#55111yescharacteristic
679C4C2×#56211yescharacteristic
680C2—162no—
681C2#3261618yesdirect product factor
682C2C2×#273011yescharacteristic
683C2C2×#319611yescharacteristic
684C2C2×#320411yescharacteristic
685C2C2×#336911yescharacteristic
686C2C2×#337111yescharacteristic
687C2C2×#340011yescharacteristic
688C2C2×#340111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2582C2semidirect
3. #32616C2direct (N × H)
5. C2×#2072C2semidirect
6. C2×#2066C2semidirect
10. C2×#1985C2semidirect
11. C2×#1971C2semidirect
13. C2×#1927C2semidirect
14. C2×#1919C2semidirect
16. C2×#1878C2semidirect
37. #2582C22semidirect
57. #2072C22semidirect
58. #2066C22semidirect
62. #1985C22semidirect
63. #1971C22semidirect
65. #1927C22semidirect
66. #1919C22semidirect
68. #1878C22semidirect
681. C2#32616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25922
#326162
C2×#20942
C2×#20722
C2×#20662
C2×#20552
C2×#20422
C2×#20152
C2×#19852
C2×#19712
C2×#19362
C2×#19272
C2×#19192
C2×#18802
C2×#18782