Finite Group Databasebeta

← C2×#10374 #238249

Subgroups — C2×#10374 #238249

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic, complemented
2C2×#1964C211yesmaximal, characteristic
3C2×#1519C211yesmaximal, characteristic
4C22×#327C211yesmaximal, characteristic
5#10374C218yesmaximal, direct product factor
6C2×#1294C211yesmaximal, characteristic, complemented
7C2×#1289C211yesmaximal, characteristic, complemented
8C2×#1282C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8—21no—
10C22×#121C2211yescharacteristic
11C22×#99—21no—
12C22×#99—21no—
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2549—22no—
16#2549—22no—
17#2549—22no—
18#2549C2214yescomplemented
19C2×#398—21no—
20C2×#395—21no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#385—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1964—22no—
29#1964C2214yes—
30#1519—22no—
31#1519C2214yes—
32C2×#327—21no—
33C2×#327—21no—
34C2×#327—22no—
35C2×#327—21no—
36C2×#327—21no—
37C2×#327C2214yes—
38#1294C2214yescomplemented
39#1289C2214yescomplemented
40#1282C2214yescomplemented
41C23×Q8—21no—
42C23×D4—21no—
43C23×D4Q811yescharacteristic, complemented
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8—21no—
47C23×C8—21no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—44no—
51C2×C22⋊Q8—44no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121Q812yescomplemented
63C2×#121C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4Q811yescharacteristic, complemented
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—21no—
78C2×#99—24no—
79C2×#99—24no—
80C2×#99—44no—
81C2×#99—44no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94#398—24no—
95#398—24no—
96#395—24no—
97#395—24no—
98#391—24no—
99#391—24no—
100#389—24no—
101#389—24no—
102#385—24no—
103#385—24no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380D412yes—
110#380C2312yes—
111#380C2312yes—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4D412yes—
115C4×C22⋊C4C2312yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125C2×#95D412yes—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128C2×#95C2312yes—
129#327—24no—
130#327—24no—
131#327—24no—
132#327—24no—
133C25—21no—
134C25C2×D411yescharacteristic
135C22×Q8—24no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8—82no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—41no—
147C22×D4—41no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—82no—
155C22×D4C2×Q812yescomplemented
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—41no—
169C23×C4—82no—
170C23×C4—82no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q811yescharacteristic
177C23×C4C2×Q811yescharacteristic
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4#4111yescharacteristic
183C23×C4#4111yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—41no—
188C22×C8—41no—
189C22×C8—41no—
190C22×C8—41no—
191C22×C8—82no—
192C22×C8—82no—
193C22⋊Q8—48no—
194C22⋊Q8—416no—
195C22⋊Q8—48no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—41no—
205#121—41no—
206#121—41no—
207#121—41no—
208#121—41no—
209#121—41no—
210#121—41no—
211#121—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×Q812yescomplemented
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4D814yes—
253C2×C22⋊C4SD1614yes—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—82no—
259#99—48no—
260#99—48no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—41no—
281#95C2×D412yes—
282#95C2×D412yes—
283#95C2×D412yes—
284#95C2×D412yes—
285C24—22no—
286C24—42no—
287C24—42no—
288C24—41no—
289C24—41no—
290C24—42no—
291C24—42no—
292C24—82no—
293C24—82no—
294C24#12011yescharacteristic
295C24#12011yescharacteristic
296C24C22⋊Q811yescharacteristic
297C24C22⋊Q811yescharacteristic
298C24#12411yescharacteristic
299C24#12411yescharacteristic
300C24C2×D812yes—
301C24C2×SD1612yes—
302C24AGL(1,ℤ/8)12yes—
303C24#13712yes—
304C24C22×D411yescharacteristic
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—84no—
308C2×Q8—816no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4#12011yescharacteristic
394C22×C4#12011yescharacteristic
395C22×C4C22⋊Q811yescharacteristic
396C22×C4C22⋊Q811yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4C22⋊Q811yescharacteristic
400C22×C4C22⋊Q811yescharacteristic
401C22×C4C22⋊Q811yescharacteristic
402C22×C4C22⋊Q811yescharacteristic
403C22×C4C22⋊Q811yescharacteristic
404C22×C4C22⋊Q811yescharacteristic
405C22×C4#12411yescharacteristic
406C22×C4#12411yescharacteristic
407C22×C4#12411yescharacteristic
408C22×C4#12411yescharacteristic
409C22×C4#12411yescharacteristic
410C22×C4#12411yescharacteristic
411C22×C4#12411yescharacteristic
412C22×C4#12411yescharacteristic
413C22×C4#12411yescharacteristic
414C22×C4#12411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×Q811yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—82no—
424C2×C8—82no—
425C2×C8—82no—
426C2×C8—82no—
427C2×C8—82no—
428C2×C8—82no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—82no—
456C22⋊C4C2×D814yes—
457C22⋊C4C2×SD1614yes—
458C42—82no—
459C42—82no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—44no—
474C23—41no—
475C23—41no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—164no—
490C23—164no—
491C23#39311yescharacteristic
492C23#39311yescharacteristic
493C23#39311yescharacteristic
494C23#39311yescharacteristic
495C23#44712yes—
496C23#44712yes—
497C23#44812yes—
498C23#44812yes—
499C23#47412yes—
500C23#47412yes—
501C23#47612yes—
502C23#47612yes—
503C23#48612yes—
504C23#48612yes—
505C23#48912yes—
506C23#48912yes—
507C23C2×#12011yescharacteristic
508C23C2×C22⋊Q811yescharacteristic
509C23C2×#12411yescharacteristic
510C23C22×D811yescharacteristic
511C23C22×SD1611yescharacteristic
512C23C2×AGL(1,ℤ/8)11yescharacteristic
513C23C2×#13711yescharacteristic
514Q8—816no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519D4—82no—
520D4—82no—
521D4—82no—
522D4—82no—
523D4—84no—
524D4—84no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
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—84no—
578C2×C4—84no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—164no—
589C2×C4#39311yescharacteristic
590C2×C4#39311yescharacteristic
591C2×C4#39311yescharacteristic
592C2×C4#39311yescharacteristic
593C2×C4C2×C22⋊Q811yescharacteristic
594C2×C4C2×C22⋊Q811yescharacteristic
595C2×C4C2×#12411yescharacteristic
596C2×C4C2×#12411yescharacteristic
597C8—82no—
598C8—82no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—162no—
631C22—162no—
632C22—164no—
633C22—164no—
634C22#202312yes—
635C22#202312yes—
636C22#202312yes—
637C22#202312yes—
638C22#203112yes—
639C22#203212yes—
640C22C2×#39311yescharacteristic
641C22C2×#44711yescharacteristic
642C22C2×#44811yescharacteristic
643C22C2×#47411yescharacteristic
644C22C2×#47611yescharacteristic
645C22C2×#48611yescharacteristic
646C22C2×#48911yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—162no—
663C2—21no—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—22no—
671C2—22no—
672C2—162no—
673C2#1037414yesdirect product factor
674C2C2×#202311yescharacteristic
675C2C2×#203111yescharacteristic
676C2C2×#203211yescharacteristic
subgroupG/H
center, socleC23#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2549C2semidirect
5. #10374C2direct (N × H)
6. C2×#1294C2semidirect
7. C2×#1289C2semidirect
8. C2×#1282C2semidirect
18. #2549C22semidirect
38. #1294C22semidirect
39. #1289C22semidirect
40. #1282C22semidirect
43. C23×D4Q8semidirect
62. C2×#121Q8semidirect
71. C22×C4⋊C4Q8semidirect
155. C22×D4C2×Q8semidirect
233. C2×C4⋊C4C2×Q8semidirect
673. C2#10374direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25492
C2×#19642
C2×#15192
C22×#3272
#103742
C2×#12942
C2×#12892
C2×#12822