Finite Group Databasebeta

← C2×#24759 #6442971

Subgroups — C2×#24759 #6442971

Order 512.

3158 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#468C212yesmaximal
2C22×#461C211yesmaximal, characteristic, complemented
3C22×#423C211yesmaximal, characteristic, complemented
4C2×#2857C211yesmaximal, characteristic, complemented
5C2×C4×#121C211yesmaximal, characteristic
6#24759C2116yesmaximal, direct product factor
7C2×#2068C212yesmaximal, complemented
8C2×#2055C212yesmaximal, complemented
9C2×#1981C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1894C212yesmaximal, complemented
12C2×#1802C211yesmaximal, characteristic, complemented
13C23×SD16C2212yes—
14C23×M16C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×#121C2211yescharacteristic
18C22×C4×D4—21no—
19C22×C4×D4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21C2×#468—216no—
22C2×#468C22116yes—
23C2×#461—28no—
24C2×#461C2218yescomplemented
25C22×#107C2212yes—
26C2×#423—28no—
27C2×#423C2218yescomplemented
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2212yes—
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2212yes—
32C22×#103C2212yes—
33C22×#103C2212yes—
34C22×#102C2212yes—
35#2857—24no—
36#2857C2218yescomplemented
37C4×#121—24no—
38C4×#121C2218yes—
39C2×#391C2212yes—
40C2×#390C2211yescharacteristic
41C2×#389—21no—
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#386—21no—
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×#380—21no—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52#2068C22116yescomplemented
53#2055C22116yescomplemented
54#1981C2218yescomplemented
55#1947C2218yescomplemented
56#1894C22116yescomplemented
57#1802C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C23×Q8C2312yes—
61C23×D4—21no—
62C23×D4C2311yescharacteristic
63C24×C4—21no—
64C24×C4D411yescharacteristic
65C24×C4D411yescharacteristic
66C24×C4C2311yescharacteristic
67C22×SD16—28no—
68C22×SD16—28no—
69C22×SD16—44no—
70C22×SD16C2318yes—
71C22×M16—28no—
72C22×M16—28no—
73C22×M16—44no—
74C22×M16C2318yes—
75C23×C8C2312yes—
76C23×C8C2312yes—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8C2314yes—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8C2318yes—
85C2×#121—22no—
86C2×#121—24no—
87C2×#121D418yes—
88C2×#121C2314yes—
89C2×C4×D4—28no—
90C2×C4×D4—22no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—48no—
94C2×C4×D4C2314yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—24no—
112C22×C42—24no—
113C22×C42—21no—
114C22×C42D414yes—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#468—264no—
121#461—232no—
122C2×#107—28no—
123C2×#107C2318yes—
124#423—232no—
125C2×C4⋊C8—28no—
126C2×C4⋊C8—24no—
127C2×C4⋊C8D418yescomplemented
128C2×C4⋊C8C2318yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103—28no—
132C2×#103—28no—
133C2×#103—44no—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2318yes—
137C2×#102—28no—
138C2×#102C2318yes—
139#391—24no—
140#391C2318yes—
141#390—22no—
142#390C2314yes—
143#389—28no—
144#389—28no—
145#389—22no—
146#389C2314yes—
147#388—28no—
148#386—28no—
149#384—22no—
150#384C2314yes—
151#380—28no—
152#380—28no—
153#380—22no—
154#380C2314yes—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—28no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—22no—
162C2×#95—22no—
163#340C2318yes—
164#340C2318yes—
165C25—21no—
166C25C2×D411yescharacteristic
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C2414yes—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—22no—
178C22×D4—21no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4C2×D414yes—
186C22×D4C2412yes—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—21no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D414yes—
209C23×C4C2×D412yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D414yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4#4114yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C2×SD16—216no—
219C2×SD16—216no—
220C2×SD16—48no—
221C2×SD16—432no—
222C2×M16—216no—
223C2×M16—216no—
224C2×M16—48no—
225C2×M16—432no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C2×D418yes—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—48no—
237C22⋊Q8—216no—
238C22⋊Q8—216no—
239C22⋊Q8—416no—
240#121—216no—
241#121—28no—
242#121C2×D4116yes—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—432no—
246C4×D4—48no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—216no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—44no—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302C2×C42C2412yes—
303C2×C42C2412yes—
304#107—216no—
305C4⋊C8—216no—
306C4⋊C8—28no—
307C4⋊C8C2×D4116yescomplemented
308#103—216no—
309#103—216no—
310#103—48no—
311#102—216no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24#12114yes—
332C24C22×D412yes—
333C24C22×D411yescharacteristic
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×D4—216no—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—48no—
357C2×D4—432no—
358C2×D4—48no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4C22×D414yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—44no—
395C22×C4—416no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—816no—
415C22×C4#12114yes—
416C22×C4#12114yes—
417C22×C4#12118yes—
418C22×C4#12114yes—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4#13714yes—
421C22×C4#13714yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437SD16—432no—
438M16—432no—
439C2×C8—216no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—48no—
446C2×C8C22×D418yes—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4C22×D414yes—
465C22⋊C4—216no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C42—28no—
472C42—28no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—48no—
477C42#12118yes—
478C42C22×D414yes—
479C42C2×#4114yes—
480C23—22no—
481C23—28no—
482C23—22no—
483C23—28no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—48no—
496C23—42no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23C2×#12114yes—
508C23C2×#12112yes—
509C23C2×#12112yes—
510C23#53712yes—
511C23C2×AGL(1,ℤ/8)12yes—
512C23C2×#13712yes—
513C23C2×#13712yes—
514C23#57512yes—
515C23C23×D411yescharacteristic
516Q8—44no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520Q8—88no—
521D4—24no—
522D4—24no—
523D4—432no—
524D4—48no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—216no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—416no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—88no—
564C2×C4—84no—
565C2×C4—88no—
566C2×C4—84no—
567C2×C4—816no—
568C2×C4#468116yescomplemented
569C2×C4C2×#12114yes—
570C2×C4C2×#12114yes—
571C2×C4C2×#12112yes—
572C2×C4C2×#12112yes—
573C2×C4C2×#12114yes—
574C2×C4#53712yes—
575C2×C4D4214yes—
576C2×C4#54714yes—
577C2×C4#54814yes—
578C2×C4#55314yes—
579C2×C4C2×AGL(1,ℤ/8)14yes—
580C2×C4C2×#13714yes—
581C2×C4C2×#13714yes—
582C2×C4C23×D411yescharacteristic
583C2×C4C23×D411yescharacteristic
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C8—28no—
589C8—44no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—816no—
604C22—82no—
605C22—82no—
606C22#241614yes—
607C22#303614yes—
608C22#303914yes—
609C22C2×#46814yes—
610C22#305614yes—
611C22#325214yes—
612C22#325414yes—
613C22C22×#12111yescharacteristic
614C22C22×#12111yescharacteristic
615C22C2×#53711yescharacteristic
616C22C22×AGL(1,ℤ/8)11yescharacteristic
617C22C22×#13711yescharacteristic
618C22C22×#13711yescharacteristic
619C22C2×#57511yescharacteristic
620C4—24no—
621C4—24no—
622C4—24no—
623C4—42no—
624C4—44no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4#241614yes—
630C4C2×#46818yescomplemented
631C4C2×D4211yescharacteristic
632C4C2×#54711yescharacteristic
633C4C2×#54811yescharacteristic
634C4C2×#55311yescharacteristic
635C2—44no—
636C2—84no—
637C2#2475918yesdirect product factor
638C2C2×#241611yescharacteristic
639C2C2×#303611yescharacteristic
640C2C2×#303911yescharacteristic
641C2C22×#46811yescharacteristic
642C2C2×#305611yescharacteristic
643C2C2×#325211yescharacteristic
644C2C2×#325411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#461C2semidirect
3. C22×#423C2semidirect
4. C2×#2857C2semidirect
6. #24759C2direct (N × H)
7. C2×#2068C2semidirect
8. C2×#2055C2semidirect
9. C2×#1981C2semidirect
10. C2×#1947C2semidirect
11. C2×#1894C2semidirect
12. C2×#1802C2semidirect
24. C2×#461C22semidirect
27. C2×#423C22semidirect
36. #2857C22semidirect
52. #2068C22semidirect
53. #2055C22semidirect
54. #1981C22semidirect
55. #1947C22semidirect
56. #1894C22semidirect
57. #1802C22semidirect
127. C2×C4⋊C8D4semidirect
307. C4⋊C8C2×D4semidirect
568. C2×C4#468semidirect
630. C4C2×#468semidirect
637. C2#24759direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4682
C22×#4612
C22×#4232
C2×#28572
C2×C4×#1212
#247592
C2×#20682
C2×#20552
C2×#19812
C2×#19472
C2×#18942
C2×#18022