Finite Group Databasebeta

← C2×#32621 #7510474

Subgroups — C2×#32621 #7510474

Order 512.

2106 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#32621C2116yesmaximal, direct product factor
4C2×#2095C211yesmaximal, characteristic, complemented
5C2×#2072C211yesmaximal, characteristic, complemented
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2055C211yesmaximal, characteristic
8C2×#2043C211yesmaximal, characteristic, complemented
9C2×#2013C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1973C211yesmaximal, characteristic
12C2×#1935C211yesmaximal, characteristic, complemented
13C2×#1925C211yesmaximal, characteristic
14C2×#1918C211yesmaximal, characteristic, complemented
15C2×#1877C211yesmaximal, characteristic
16C2×#1877C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2568—24no—
36#2568C2218yescomplemented
37#2538—24no—
38#2538C2218yes—
39C2×#400—21no—
40C2×#400—21no—
41C2×#396—21no—
42C2×#396—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#382—21no—
54C2×#380—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2095C2218yescomplemented
57#2072C2218yescomplemented
58#2068C2218yescomplemented
59#2055C2218yes—
60#2043C2218yescomplemented
61#2013C2218yes—
62#1985C2218yescomplemented
63#1973C2218yes—
64#1935C2218yescomplemented
65#1925C2218yes—
66#1918C2218yescomplemented
67#1877C2218yes—
68#1877C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×SD16—24no—
83C22×SD16—24no—
84C22×SD16—24no—
85C22×SD16—42no—
86C22×SD16—42no—
87C22×SD16—42no—
88C22×SD16C2314yes—
89C22×SD16C2314yes—
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—42no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8C2314yes—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—48no—
100C2×C4×Q8—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8D414yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—42no—
106C2×C4×D4—42no—
107C2×C4×D4C2314yes—
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⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C2×#107—24no—
132C2×#107—42no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×C4⋊C8—42no—
136C2×C4⋊C8C2314yes—
137C2×#103—24no—
138C2×#103—42no—
139C2×#103—42no—
140C2×#103—42no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102—24no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#97—42no—
154C2×#97C2314yes—
155#400—28no—
156#400—28no—
157#396—28no—
158#396—28no—
159#391—28no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391C2314yes—
165#391C2314yes—
166#391C2314yes—
167#391C2314yes—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#384—22no—
177#384C2314yes—
178#382—28no—
179#380—28no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
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×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—21no—
205C22×Q8—21no—
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—42no—
219C22×Q8C2×D414yes—
220C22×Q8C2412yes—
221C22×Q8C2412yes—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4#4114yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—416no—
255C2×SD16—416no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8#4114yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—48no—
275C22⋊Q8—432no—
276C4×Q8—48no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
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—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—28no—
345#107—44no—
346C4⋊C8—44no—
347#103—28no—
348#103—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353#102—28no—
354#102—44no—
355#97—44no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
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—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—44no—
391C2×Q8—416no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—28no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4C2×#4114yes—
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—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—22no—
441C22×C4—22no—
442C22×C4—48no—
443C22×C4—48no—
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—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—48no—
466C22×C4—816no—
467C22×C4AGL(1,ℤ/8)14yes—
468C22×C4#13714yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483SD16—416no—
484SD16—416no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8C2×#4114yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—24no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—48no—
540C23—24no—
541C23—48no—
542C23—48no—
543C23—42no—
544C23—42no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23#53412yes—
555C23#54812yes—
556C23#55012yes—
557C23C2×AGL(1,ℤ/8)12yes—
558C23C2×#13712yes—
559C23#57712yes—
560C23#57812yes—
561C23C23×D411yescharacteristic
562Q8—416no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572Q8—84no—
573D4—24no—
574D4—24no—
575D4—42no—
576D4—48no—
577D4—42no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
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—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—816no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4#53412yes—
630C2×C4#53814yes—
631C2×C4#54214yes—
632C2×C4#54812yes—
633C2×C4#55012yes—
634C2×C4#55114yes—
635C2×C4#56214yes—
636C2×C4C2×AGL(1,ℤ/8)14yes—
637C2×C4C2×#13714yes—
638C2×C4C22×#4111yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C8—24no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—82no—
652C22—82no—
653C22—816no—
654C22—82no—
655C22—82no—
656C22#273014yes—
657C22#319314yes—
658C22#320114yes—
659C22#336714yes—
660C22#337214yes—
661C22#339714yes—
662C22#339814yes—
663C22C2×#53411yescharacteristic
664C22C2×#54811yescharacteristic
665C22C2×#55011yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C22×#13711yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57811yescharacteristic
670C4—24no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#273014yes—
684C4C2×#53811yescharacteristic
685C4C2×#54211yescharacteristic
686C4C2×#55111yescharacteristic
687C4C2×#56211yescharacteristic
688C2—84no—
689C2#3262118yesdirect product factor
690C2C2×#273011yescharacteristic
691C2C2×#319311yescharacteristic
692C2C2×#320111yescharacteristic
693C2C2×#337211yescharacteristic
694C2C2×#336711yescharacteristic
695C2C2×#339711yescharacteristic
696C2C2×#339811yescharacteristic
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×#2568C2semidirect
3. #32621C2direct (N × H)
4. C2×#2095C2semidirect
5. C2×#2072C2semidirect
6. C2×#2068C2semidirect
8. C2×#2043C2semidirect
10. C2×#1985C2semidirect
12. C2×#1935C2semidirect
14. C2×#1918C2semidirect
36. #2568C22semidirect
56. #2095C22semidirect
57. #2072C22semidirect
58. #2068C22semidirect
60. #2043C22semidirect
62. #1985C22semidirect
64. #1935C22semidirect
66. #1918C22semidirect
689. C2#32621direct (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×#25382
#326212
C2×#20952
C2×#20722
C2×#20682
C2×#20552
C2×#20432
C2×#20132
C2×#19852
C2×#19732
C2×#19352
C2×#19252
C2×#19182
C2×#18772