Finite Group Databasebeta

← C2×#8875 #192455

Subgroups — C2×#8875 #192455

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2976C211yesmaximal, characteristic
2C2×#1871C211yesmaximal, characteristic
3C2×#1855C211yesmaximal, characteristic, complemented
4C2×#1851C211yesmaximal, characteristic, complemented
5C2×#1501C211yesmaximal, characteristic
6#8875C218yesmaximal, direct product factor
7C2×#1297C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C23×M16C2211yescharacteristic
10C2×#525C2211yescharacteristic
11#2976—22no—
12#2976—22no—
13#2976—22no—
14#2976C2214yes—
15C2×#436—21no—
16C2×#435—21no—
17C2×#435—21no—
18C2×C8×D4—21no—
19C2×#433—21no—
20C22×#108—21no—
21C22×C4⋊C8—21no—
22C22×#104—21no—
23C2×#408—21no—
24C2×#407C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1871—22no—
30#1871C2214yes—
31#1855—22no—
32#1855C2214yescomplemented
33#1851—22no—
34#1851C2214yescomplemented
35C2×#341—21no—
36C2×#339—21no—
37C2×#336—21no—
38#1501C414yescomplemented
39#1501C2214yes—
40C2×#328C411yescharacteristic, complemented
41C2×#328C411yescharacteristic, complemented
42C2×#327C411yescharacteristic, complemented
43C2×#327C411yescharacteristic, complemented
44#1297C2214yescomplemented
45#1282C2214yescomplemented
46C22×#41D411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—21no—
52C22×M16—21no—
53C22×M16—21no—
54C22×M16—44no—
55C22×M16D411yescharacteristic
56C22×M16D411yescharacteristic
57C22×M16C2312yes—
58C23×C8—21no—
59C23×C8—21no—
60C23×C8—21no—
61C23×C8D411yescharacteristic
62C23×C8D411yescharacteristic
63#525—21no—
64#525—21no—
65#525—21no—
66#525—21no—
67#525C2×C412yes—
68#525D412yes—
69#525Q812yes—
70#525C2312yes—
71C2×#123—21no—
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×C22⋊Q8Q811yescharacteristic
74C2×#121C2×C411yescharacteristic
75C2×#121Q811yescharacteristic
76C2×#120—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×#117D411yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42—21no—
84#436—24no—
85#436—24no—
86#435—24no—
87#435—24no—
88#435—24no—
89#435—24no—
90C8×D4—24no—
91C8×D4—24no—
92#433—24no—
93#433—24no—
94C2×#108—24no—
95C2×#108—44no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—44no—
100C2×#104—24no—
101C2×#104—44no—
102#408—24no—
103#408—24no—
104#407—21no—
105#407—21no—
106#407D412yes—
107#407C2312yes—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8C2×C411yescharacteristic
120C2×C22⋊C8C2×C411yescharacteristic
121C2×C22⋊C8C2×C412yes—
122C2×C22⋊C8D412yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×#97—21no—
126C2×C4×C8—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4C2×C412yes—
133C4×C22⋊C4C2312yes—
134C4×C22⋊C4C2312yes—
135C2×#95—21no—
136C2×#95C2×C411yescharacteristic
137C2×#95C2×C411yescharacteristic
138#341—24no—
139#339—24no—
140#336—24no—
141#328C2×C414yescomplemented
142#328C2×C414yescomplemented
143#327C2×C414yescomplemented
144#327C2×C414yescomplemented
145C25#4111yescharacteristic
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41C4⋊C412yes—
151C2×#41C2×D412yes—
152C22×Q8C4⋊C411yescharacteristic
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C4⋊C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—41no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×Q811yescharacteristic
172C23×C4C2×Q811yescharacteristic
173C23×C4#4111yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4#4111yescharacteristic
176C23×C4C2411yescharacteristic
177C2×M16—24no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—41no—
191C2×M16—41no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—82no—
195C2×M16C2×D412yes—
196C2×M16C2×D412yes—
197C22×C8—24no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—24no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—41no—
214C22×C8—41no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—82no—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220#123—22no—
221#123—22no—
222#123—22no—
223#123—22no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8C4⋊C412yes—
229C22⋊Q8C4⋊C412yes—
230C22⋊Q8C22×C412yes—
231C22⋊Q8C2×Q812yes—
232#121—21no—
233#121—21no—
234#121—21no—
235#121—21no—
236#121C4⋊C412yes—
237#121C4⋊C412yes—
238#121C22×C412yes—
239#121C2×Q812yes—
240#120—22no—
241#120—22no—
242#120—22no—
243#120—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252#117—21no—
253#117—21no—
254#117C4⋊C412yes—
255#117C2×D412yes—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C4⋊C411yescharacteristic
259C2×C4⋊C4C4⋊C411yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—41no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—41no—
281#108—48no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—48no—
285#104—48no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8C22×C412yes—
301C22⋊C8C22×C412yes—
302#97—24no—
303C4×C8—24no—
304#95—21no—
305#95—21no—
306#95—22no—
307#95—41no—
308#95C22×C412yes—
309#95C22×C412yes—
310C24—22no—
311C24—22no—
312C24—21no—
313C24—42no—
314C24—42no—
315C24C4×D411yescharacteristic
316C24C4×D411yescharacteristic
317C24C22⋊Q811yescharacteristic
318C24C22⋊Q811yescharacteristic
319C24#12311yescharacteristic
320C24#12311yescharacteristic
321C24C2×#4111yescharacteristic
322#41—42no—
323#41—44no—
324#41—42no—
325#41—82no—
326#41—81no—
327#41—81no—
328C2×Q8—21no—
329C2×Q8—42no—
330C2×Q8C2×C4⋊C412yes—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4C2×C4⋊C412yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4C2×C4⋊C411yescharacteristic
385C22×C4C2×C4⋊C411yescharacteristic
386C22×C4C2×C4⋊C411yescharacteristic
387C22×C4C2×C4⋊C411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4C22⋊Q811yescharacteristic
400C22×C4C22⋊Q811yescharacteristic
401C22×C4C22⋊Q811yescharacteristic
402C22×C4C22⋊Q811yescharacteristic
403C22×C4C22⋊Q811yescharacteristic
404C22×C4#12311yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×Q811yescharacteristic
411C22×C4C2×#4111yescharacteristic
412M16—24no—
413M16—24no—
414M16—42no—
415M16—42no—
416M16—44no—
417M16—44no—
418M16—42no—
419M16—42no—
420M16—81no—
421M16—81no—
422M16—84no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—44no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—44no—
437C2×C8—42no—
438C2×C8—44no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—81no—
446C2×C8—81no—
447C2×C8—84no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4C2×C4⋊C412yes—
455C4⋊C4C2×C4⋊C412yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—42no—
471C42—22no—
472C42—22no—
473C42—42no—
474C42—44no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—82no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—84no—
507C23#38511yescharacteristic
508C23#38511yescharacteristic
509C23#38511yescharacteristic
510C23#38511yescharacteristic
511C23#44312yes—
512C23#44412yes—
513C23C2×C4×D411yescharacteristic
514C23C2×C22⋊Q811yescharacteristic
515C23C2×#12311yescharacteristic
516Q8—42no—
517Q8—81no—
518D4—42no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—81no—
523D4—82no—
524D4—82no—
525D4—82no—
526D4—82no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—82no—
578C2×C4—84no—
579C2×C4—82no—
580C2×C4—88no—
581C2×C4#38511yescharacteristic
582C2×C4#38511yescharacteristic
583C2×C4#38511yescharacteristic
584C2×C4#38511yescharacteristic
585C2×C4B(4,𝔽2)12yes—
586C2×C4#45812yes—
587C2×C4C22×C4⋊C411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C8—42no—
592C8—42no—
593C8—44no—
594C8—42no—
595C8—42no—
596C8—42no—
597C8—42no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—24no—
603C22—24no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—44no—
617C22—44no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22#186112yes—
629C22#186612yes—
630C22#186612yes—
631C22#186712yes—
632C22C2×#38511yescharacteristic
633C22C2×#44311yescharacteristic
634C22C2×#44411yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—82no—
646C4—84no—
647C4#186712yes—
648C4C2×B(4,𝔽2)11yescharacteristic
649C4C2×#45811yescharacteristic
650C2—21no—
651C2—21no—
652C2—22no—
653C2—41no—
654C2—41no—
655C2—41no—
656C2—41no—
657C2—82no—
658C2#887514yesdirect product factor
659C2C2×#186111yescharacteristic
660C2C2×#186611yescharacteristic
661C2C2×#186711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23#385
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1855C2semidirect
4. C2×#1851C2semidirect
6. #8875C2direct (N × H)
7. C2×#1297C2semidirect
8. C2×#1282C2semidirect
32. #1855C22semidirect
34. #1851C22semidirect
38. #1501C4semidirect
40. C2×#328C4semidirect
41. C2×#328C4semidirect
42. C2×#327C4semidirect
43. C2×#327C4semidirect
44. #1297C22semidirect
45. #1282C22semidirect
141. #328C2×C4semidirect
142. #328C2×C4semidirect
143. #327C2×C4semidirect
144. #327C2×C4semidirect
658. C2#8875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29762
C2×#18712
C2×#18552
C2×#18512
C2×#15012
#88752
C2×#12972
C2×#12822