Finite Group Databasebeta

← C2×#25462 #6446072

Subgroups — C2×#25462 #6446072

Order 512.

2170 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic
4C2×#2369C211yesmaximal, characteristic, complemented
5#25462C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2083C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#1989C211yesmaximal, characteristic, complemented
11C2×#1949C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13C2×#1940C211yesmaximal, characteristic, complemented
14C2×#1926C211yesmaximal, characteristic
15C2×#1925C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#501—28no—
21C2×#501C2218yescomplemented
22C2×#474—28no—
23C2×#474C2218yes—
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#97C2211yescharacteristic
38#2566—24no—
39#2566C2218yes—
40C2×#396—21no—
41C2×#396—21no—
42#2369—24no—
43#2369C2218yescomplemented
44C2×#391—21no—
45C2×#391—21no—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
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#2083C2218yescomplemented
62#2067C2218yes—
63#2065C2218yes—
64#1989C2218yescomplemented
65#1949C2218yescomplemented
66#1947C2218yescomplemented
67#1940C2218yescomplemented
68#1926C2218yes—
69#1925C2218yes—
70#1762C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8D418yes—
89C2×C4⋊Q8C2314yes—
90C2×#125—22no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125C2314yes—
94C2×C4×D4—22no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4C2314yes—
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#501—232no—
126#474—232no—
127C2×#107—24no—
128C2×#107—24no—
129C2×#107—24no—
130C2×#107—24no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106—24no—
136C2×#106—24no—
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×#102Q818yescomplemented
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#97—24no—
154C2×#97C2314yes—
155#396—28no—
156#396—28no—
157#391—28no—
158#391—28no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—28no—
164#389—22no—
165#389—22no—
166#389C2314yes—
167#389C2314yes—
168#385—28no—
169#385—28no—
170#384—22no—
171#384C2314yes—
172#382—22no—
173#382C2314yes—
174C4×C4⋊C4—28no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
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×Q8—28no—
202C22×Q8—24no—
203C22×Q8—82no—
204C22×Q8—82no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2×Q814yes—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—82no—
223C23×C4—82no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
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—22no—
241C22×C8C2×Q814yes—
242C22×C8#4114yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C4⋊Q8—216no—
248C4⋊Q8—28no—
249C4⋊Q8C2×D4116yes—
250#125—28no—
251#125—28no—
252#125—48no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—48no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
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—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×Q814yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4—82no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—82no—
321C2×C42—82no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D414yes—
326C2×C42#4114yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330#107—28no—
331#107—28no—
332#107—28no—
333#107—28no—
334#106—28no—
335#106—28no—
336C4⋊C8—28no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#102—28no—
340#102—28no—
341#102—28no—
342#102C2×Q8116yescomplemented
343#97—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—24no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24C22×D411yescharacteristic
362C2×Q8—216no—
363C2×Q8—816no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4C22×Q814yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
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—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—816no—
426C22×C4#12114yes—
427C22×C4#12114yes—
428C22×C4#13714yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C42−1+412yes—
441C22×C42−1+412yes—
442C22×C4C2511yescharacteristic
443C2×C8—28no—
444C2×C8—28no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—44no—
456C2×C8—44no—
457C2×C8C22×Q814yes—
458C2×C8C2×#4114yes—
459C4⋊C4—216no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C22×D414yes—
487C4⋊C4C22×D414yes—
488C4⋊C4C22×Q814yes—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—48no—
501C42—84no—
502C42—84no—
503C42#12118yes—
504C42C22×D414yes—
505C42C2×#4114yes—
506C23—24no—
507C23—48no—
508C23—48no—
509C23—42no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—84no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23C2×#12112yes—
521C23#54712yes—
522C23D4×Q812yes—
523C23C2×#13712yes—
524C23#57512yes—
525C23#57612yes—
526C23#57612yes—
527C23C23×D411yescharacteristic
528Q8—816no—
529D4—24no—
530D4—24no—
531D4—48no—
532D4—42no—
533D4—42no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—48no—
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—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—816no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4C2×#12114yes—
585C2×C4C2×#12114yes—
586C2×C4C2×#12112yes—
587C2×C4#53314yes—
588C2×C4#54114yes—
589C2×C4#54712yes—
590C2×C4#54814yes—
591C2×C4D4×Q812yes—
592C2×C4D4×Q814yes—
593C2×C4C2×#13714yes—
594C2×C4C23×D411yescharacteristic
595C2×C4C23×Q811yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+411yescharacteristic
600C8—24no—
601C8—24no—
602C8—42no—
603C8—42no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—816no—
611C22—82no—
612C22#242914yes—
613C22#303714yes—
614C22#304114yes—
615C22#334614yes—
616C22#334714yes—
617C22#338514yes—
618C22#338914yes—
619C22C22×#12111yescharacteristic
620C22C2×#54711yescharacteristic
621C22C2×D4×Q811yescharacteristic
622C22C22×#13711yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57611yescharacteristic
625C22C2×#57611yescharacteristic
626C4—24no—
627C4—24no—
628C4—24no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4#242914yes—
640C4C2×#53311yescharacteristic
641C4C2×#54111yescharacteristic
642C4C2×#54811yescharacteristic
643C4C2×D4×Q811yescharacteristic
644C2—84no—
645C2#2546218yesdirect product factor
646C2C2×#242911yescharacteristic
647C2C2×#303711yescharacteristic
648C2C2×#304111yescharacteristic
649C2C2×#334611yescharacteristic
650C2C2×#334711yescharacteristic
651C2C2×#338511yescharacteristic
652C2C2×#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×#501C2semidirect
4. C2×#2369C2semidirect
5. #25462C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2083C2semidirect
10. C2×#1989C2semidirect
11. C2×#1949C2semidirect
12. C2×#1947C2semidirect
13. C2×#1940C2semidirect
21. C2×#501C22semidirect
43. #2369C22semidirect
60. #2085C22semidirect
61. #2083C22semidirect
64. #1989C22semidirect
65. #1949C22semidirect
66. #1947C22semidirect
67. #1940C22semidirect
148. C2×#102Q8semidirect
342. #102C2×Q8semidirect
645. C2#25462direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4742
C2×#25662
C2×#23692
#254622
C2×#20852
C2×#20832
C2×#20672
C2×#20652
C2×#19892
C2×#19492
C2×#19472
C2×#19402
C2×#19262
C2×#19252
C2×#17622