Finite Group Databasebeta

← C2×#25457 #6446064

Subgroups — C2×#25457 #6446064

Order 512.

2132 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
1C22×#499C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic
4C2×#2370C211yesmaximal, characteristic, complemented
5#25457C2116yesmaximal, direct product factor
6C2×#2084C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2067C211yesmaximal, characteristic
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13C2×#1939C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic
15C2×#1926C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#499—28no—
21C2×#499C2218yescomplemented
22C2×#478—28no—
23C2×#478C2218yes—
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2566—24no—
39#2566C2218yes—
40C2×#396—21no—
41C2×#396—21no—
42#2370—24no—
43#2370C2218yescomplemented
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#387—21no—
51C2×#385—21no—
52C2×#385—21no—
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×C4×C4⋊C4—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2085C2218yescomplemented
61#2084C2218yescomplemented
62#2067C2218yes—
63#2067C2218yes—
64#1989C2218yescomplemented
65#1949C2218yescomplemented
66#1947C2218yescomplemented
67#1939C2218yescomplemented
68#1926C2218yes—
69#1926C2218yes—
70C4×#102C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125D418yes—
91C2×#125C2314yes—
92C2×#125C2314yes—
93C2×C4×D4—22no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#499—232no—
126#478—232no—
127C2×#107—24no—
128C2×#107—24no—
129C2×#107—24no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106—24no—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—24no—
141C2×C4⋊C8—24no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102Q818yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C4×C8—24no—
154C2×C4×C8C2314yes—
155#396—28no—
156#396—28no—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#390C2314yes—
161#389—28no—
162#389—22no—
163#389—22no—
164#389C2314yes—
165#389C2314yes—
166#387—28no—
167#387—28no—
168#385—28no—
169#385—28no—
170#384—22no—
171#384C2314yes—
172#382—22no—
173#382C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—28no—
178C4×C4⋊C4D414yes—
179C4×C4⋊C4C2314yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95D411yescharacteristic
191C2×#95D411yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25—21no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2×Q814yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—82no—
219C23×C4—82no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8C2×Q814yes—
238C22×C8#4114yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243#125—216no—
244#125—28no—
245#125—28no—
246#125—28no—
247#125—48no—
248#125C2×D4116yes—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4—48no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×Q814yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—82no—
319C2×C42—82no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D414yes—
324C2×C42#4114yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2412yes—
328#107—28no—
329#107—28no—
330#107—28no—
331#106—28no—
332#106—28no—
333#106—28no—
334C4⋊C8—28no—
335C4⋊C8—28no—
336C4⋊C8—28no—
337#102—28no—
338#102—28no—
339#102—28no—
340#102C2×Q8116yes—
341C4×C8—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C24—24no—
352C24—22no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C22×D411yescharacteristic
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4C22×Q814yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
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—42no—
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—48no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—816no—
420C22×C4#12114yes—
421C22×C4#12114yes—
422C22×C4#13514yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×Q812yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C42−1+412yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437C2×C8—28no—
438C2×C8—28no—
439C2×C8—22no—
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—44no—
450C2×C8—44no—
451C2×C8C22×Q814yes—
452C2×C8C2×#4114yes—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—48no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4C22×D414yes—
482C4⋊C4C22×D414yes—
483C4⋊C4C22×Q814yes—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—84no—
497C42—84no—
498C42#12118yes—
499C42C22×D414yes—
500C42C2×#4114yes—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23C2×#12112yes—
516C23#54712yes—
517C23D4×Q812yes—
518C23C2×#13512yes—
519C23#57512yes—
520C23#57612yes—
521C23#57712yes—
522C23C23×D411yescharacteristic
523D4—24no—
524D4—24no—
525D4—48no—
526D4—42no—
527D4—42no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—48no—
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—42no—
572C2×C4—42no—
573C2×C4—816no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4C2×#12114yes—
579C2×C4C2×#12114yes—
580C2×C4C2×#12112yes—
581C2×C4#53314yes—
582C2×C4#54114yes—
583C2×C4#54712yes—
584C2×C4#54814yes—
585C2×C4D4×Q812yes—
586C2×C4D4×Q814yes—
587C2×C4C2×#13514yes—
588C2×C4C23×D411yescharacteristic
589C2×C4C23×Q811yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C8—24no—
595C8—24no—
596C8—42no—
597C8—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—816no—
605C22—82no—
606C22#242914yes—
607C22#303514yes—
608C22#304514yes—
609C22#333914yes—
610C22#335114yes—
611C22#338114yes—
612C22#338914yes—
613C22C22×#12111yescharacteristic
614C22C2×#54711yescharacteristic
615C22C2×D4×Q811yescharacteristic
616C22C22×#13511yescharacteristic
617C22C2×#57511yescharacteristic
618C22C2×#57611yescharacteristic
619C22C2×#57711yescharacteristic
620C4—24no—
621C4—24no—
622C4—24no—
623C4—44no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#242914yes—
634C4C2×#53311yescharacteristic
635C4C2×#54111yescharacteristic
636C4C2×#54811yescharacteristic
637C4C2×D4×Q811yescharacteristic
638C2—84no—
639C2#2545718yesdirect product factor
640C2C2×#242911yescharacteristic
641C2C2×#303511yescharacteristic
642C2C2×#304511yescharacteristic
643C2C2×#333911yescharacteristic
644C2C2×#335111yescharacteristic
645C2C2×#338111yescharacteristic
646C2C2×#338911yescharacteristic
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. C22×#499C2semidirect
4. C2×#2370C2semidirect
5. #25457C2direct (N × H)
6. C2×#2084C2semidirect
7. C2×#2085C2semidirect
10. C2×#1989C2semidirect
11. C2×#1949C2semidirect
12. C2×#1947C2semidirect
13. C2×#1939C2semidirect
21. C2×#499C22semidirect
43. #2370C22semidirect
60. #2085C22semidirect
61. #2084C22semidirect
64. #1989C22semidirect
65. #1949C22semidirect
66. #1947C22semidirect
67. #1939C22semidirect
639. C2#25457direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4992
C22×#4782
C2×#25662
C2×#23702
#254572
C2×#20842
C2×#20852
C2×#20672
C2×#19892
C2×#19492
C2×#19472
C2×#19392
C2×#19262
C2×C4×#1022