Finite Group Databasebeta

← C2×#24300 #6445359

Subgroups — C2×#24300 #6445359

Order 512.

3738 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×#533C211yesmaximal, characteristic, complemented
2C22×#476C211yesmaximal, characteristic, complemented
3C22×#474C211yesmaximal, characteristic, complemented
4C2×C4×#121C211yesmaximal, characteristic
5#24300C2116yesmaximal, direct product factor
6C2×#2059C212yesmaximal, complemented
7C2×#2025C212yesmaximal, complemented
8C2×#1988C211yesmaximal, characteristic, complemented
9C2×#1987C211yesmaximal, characteristic, complemented
10C2×#1875C212yesmaximal
11C2×#1854C212yesmaximal, complemented
12C2×#1797C211yesmaximal, characteristic, complemented
13C2×#533—24no—
14C2×#533—28no—
15C2×#533—28no—
16C2×#533C2218yescomplemented
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×#121C2211yescharacteristic
23C22×C4×Q8—21no—
24C22×C4×Q8—21no—
25C22×C4×D4—21no—
26C22×C4×D4C2211yescharacteristic
27C22×#117C2212yes—
28C23×C42C2211yescharacteristic
29C2×#476—28no—
30C2×#476C2218yescomplemented
31C2×#474—28no—
32C2×#474C2218yescomplemented
33C22×#107C2212yes—
34C22×#106C2212yes—
35C22×C4⋊C8C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C22×#102C2212yes—
38C22×#102C2212yes—
39C22×C22⋊C8C2212yes—
40C4×#121—24no—
41C4×#121C2218yes—
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#386—21no—
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49#2059C22116yescomplemented
50#2025C22116yescomplemented
51#1988C2218yescomplemented
52#1987C2218yescomplemented
53#1875C22116yes—
54#1854C22116yescomplemented
55#1797C2218yescomplemented
56C2×#340C2212yes—
57C2×#340C2212yes—
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#336C2212yes—
61C23×Q8—21no—
62C23×Q8—21no—
63C23×D4—21no—
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4D411yescharacteristic
67C24×C4D411yescharacteristic
68C24×C4C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2312yes—
71#533—264no—
72#533—232no—
73#533—232no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8D418yescomplemented
79C2×C4⋊Q8D418yescomplemented
80C2×C4⋊Q8C2314yes—
81C2×C4⋊Q8C2314yes—
82C2×#125—28no—
83C2×#125—28no—
84C2×#125—24no—
85C2×#125C2318yes—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×#121—22no—
91C2×#121—24no—
92C2×#121Q818yescomplemented
93C2×#121C2314yes—
94C2×C4×Q8—216no—
95C2×C4×Q8—216no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×D4—28no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—48no—
103C2×C4×D4C2314yes—
104C2×#117—24no—
105C2×#117—28no—
106C2×#117—28no—
107C2×#117C2318yes—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—22no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—22no—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42—21no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D414yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#476—232no—
135#474—232no—
136C2×#107—28no—
137C2×#107C2318yes—
138C2×#106—28no—
139C2×#106C2318yes—
140C2×C4⋊C8—24no—
141C2×C4⋊C8—24no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#102—28no—
145C2×#102—28no—
146C2×#102C2318yes—
147C2×#102C2318yes—
148C2×C22⋊C8—28no—
149C2×C22⋊C8C2318yes—
150#390—22no—
151#390C2314yes—
152#389—22no—
153#389C2314yes—
154#388—28no—
155#386—28no—
156#380—22no—
157#380C2314yes—
158C4×C4⋊C4—22no—
159C4×C4⋊C4C2314yes—
160C4×C22⋊C4—28no—
161C2×#95—21no—
162C2×#95—21no—
163C2×#95—21no—
164#340C2318yes—
165#340C2318yes—
166#340C2318yes—
167#340C2318yes—
168#336C2318yes—
169C25—21no—
170C25C2×D411yescharacteristic
171C22×Q8—28no—
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4C2×Q814yes—
190C22×D4C2412yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—24no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D414yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×Q814yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—24no—
227C22×C8—24no—
228C22×C8C2414yes—
229C22×C8C2414yes—
230C4⋊Q8—216no—
231C4⋊Q8—216no—
232C4⋊Q8—28no—
233C4⋊Q8—28no—
234C4⋊Q8C2×D4116yescomplemented
235C4⋊Q8C2×D4116yescomplemented
236#125—232no—
237#125—216no—
238#125—216no—
239C22⋊Q8—264no—
240C22⋊Q8—264no—
241#121—216no—
242#121—28no—
243#121C2×Q8116yescomplemented
244C4×Q8—264no—
245C4×Q8—264no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—432no—
249C4×D4—48no—
250#117—232no—
251#117—216no—
252#117—216no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
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—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×Q814yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C42—216no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—216no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2412yes—
313C2×C42C2412yes—
314#107—216no—
315#106—216no—
316C4⋊C8—28no—
317C4⋊C8—28no—
318#102—216no—
319#102—216no—
320C22⋊C8—216no—
321#95—24no—
322#95—24no—
323#95—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24C22⋊Q814yes—
339C24C22×D412yes—
340C24C22×D411yescharacteristic
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—432no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—432no—
349C2×D4—216no—
350C2×D4—24no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—48no—
359C2×D4—432no—
360C2×D4—48no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4C22×Q814yes—
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—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—416no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—416no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—816no—
418C22×C4#12014yes—
419C22×C4#12014yes—
420C22×C4C22⋊Q814yes—
421C22×C4C22⋊Q818yes—
422C22×C4#13514yes—
423C22×C4AGL(1,ℤ/8)14yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q812yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4112yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—48no—
446C2×C8—48no—
447C4⋊C4—232no—
448C4⋊C4—232no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4C22×D418yes—
465C4⋊C4C22×D418yes—
466C4⋊C4C22×Q814yes—
467C22⋊C4—232no—
468C22⋊C4—216no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C42—232no—
474C42—28no—
475C42—28no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—48no—
480C42#12018yes—
481C42C22×D414yes—
482C42C22×D414yes—
483C23—28no—
484C23—28no—
485C23—22no—
486C23—28no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23C2×#12012yes—
511C23C2×C22⋊Q814yes—
512C23C2×C22⋊Q812yes—
513C23#53612yes—
514C23C2×#13512yes—
515C23C2×AGL(1,ℤ/8)12yes—
516C23#57512yes—
517C23#57512yes—
518C23C23×D411yescharacteristic
519Q8—432no—
520Q8—432no—
521D4—24no—
522D4—24no—
523D4—432no—
524D4—48no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—216no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—416no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—416no—
551C2×C4—44no—
552C2×C4—416no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—816no—
569C2×C4#449116yes—
570C2×C4C2×#12014yes—
571C2×C4C2×#12012yes—
572C2×C4C2×#12014yes—
573C2×C4C2×C22⋊Q814yes—
574C2×C4C2×C22⋊Q812yes—
575C2×C4#53612yes—
576C2×C4#54814yes—
577C2×C4#54814yes—
578C2×C4D4×Q814yes—
579C2×C4D4×Q814yes—
580C2×C4C2×#13514yes—
581C2×C4C2×AGL(1,ℤ/8)14yes—
582C2×C4C23×D411yescharacteristic
583C2×C4C23×D411yescharacteristic
584C2×C4C23×Q811yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×2−1+411yescharacteristic
587C2×C4C2×2−1+411yescharacteristic
588C8—44no—
589C8—44no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—416no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—816no—
606C22#240914yes—
607C22C2×#44914yes—
608C22#300514yes—
609C22#307614yes—
610C22#307714yes—
611C22#323314yes—
612C22#323514yes—
613C22C22×#12011yescharacteristic
614C22C22×C22⋊Q811yescharacteristic
615C22C2×#53611yescharacteristic
616C22C22×#13511yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57511yescharacteristic
619C22C2×#57511yescharacteristic
620C4—24no—
621C4—24no—
622C4—44no—
623C4—42no—
624C4—48no—
625C4—42no—
626C4—48no—
627C4—82no—
628C4—82no—
629C4#240914yes—
630C4C2×#44918yes—
631C4C2×#54811yescharacteristic
632C4C2×#54811yescharacteristic
633C4C2×D4×Q811yescharacteristic
634C4C2×D4×Q811yescharacteristic
635C2—44no—
636C2—84no—
637C2#2430018yesdirect product factor
638C2C2×#240911yescharacteristic
639C2C22×#44911yescharacteristic
640C2C2×#300511yescharacteristic
641C2C2×#307611yescharacteristic
642C2C2×#307711yescharacteristic
643C2C2×#323311yescharacteristic
644C2C2×#323511yescharacteristic
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
1. C22×#533C2semidirect
2. C22×#476C2semidirect
3. C22×#474C2semidirect
5. #24300C2direct (N × H)
6. C2×#2059C2semidirect
7. C2×#2025C2semidirect
8. C2×#1988C2semidirect
9. C2×#1987C2semidirect
11. C2×#1854C2semidirect
12. C2×#1797C2semidirect
16. C2×#533C22semidirect
30. C2×#476C22semidirect
32. C2×#474C22semidirect
49. #2059C22semidirect
50. #2025C22semidirect
51. #1988C22semidirect
52. #1987C22semidirect
54. #1854C22semidirect
55. #1797C22semidirect
78. C2×C4⋊Q8D4semidirect
79. C2×C4⋊Q8D4semidirect
92. C2×#121Q8semidirect
234. C4⋊Q8C2×D4semidirect
235. C4⋊Q8C2×D4semidirect
243. #121C2×Q8semidirect
637. C2#24300direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5332
C22×#4762
C22×#4742
C2×C4×#1212
#243002
C2×#20592
C2×#20252
C2×#19882
C2×#19872
C2×#18752
C2×#18542
C2×#17972