Finite Group Databasebeta

← C2×#13005 #494476

Subgroups — C2×#13005 #494476

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3067C211yesmaximal, characteristic
2C2×#2912C211yesmaximal, characteristic, complemented
3C2×#2145C211yesmaximal, characteristic, complemented
4C22×#360C211yesmaximal, characteristic, complemented
5C2×#2117C211yesmaximal, characteristic, complemented
6#13005C218yesmaximal, direct product factor
7C2×#1389C211yesmaximal, characteristic, complemented
8C2×#1383C211yesmaximal, characteristic, complemented
9C2×#575—21no—
10C22×#135C2211yescharacteristic
11C22×#131C2211yescharacteristic
12C2×#520—21no—
13C22×M32C2211yescharacteristic
14C22×M32C2211yescharacteristic
15#3067—22no—
16#3067—22no—
17#3067C414yescomplemented
18#3067C2214yes—
19C2×#473C412yescomplemented
20C2×#472C412yescomplemented
21C2×#471—22no—
22C22×#108C2211yescharacteristic
23#2912—22no—
24#2912—22no—
25#2912—22no—
26#2912C2214yescomplemented
27C2×#428—22no—
28C2×#428—21no—
29C2×#427—21no—
30C2×#427C2211yescharacteristic
31C22×#106—21no—
32C22×#106C2211yescharacteristic
33C2×#413—21no—
34#2145—22no—
35#2145C2214yescomplemented
36C2×#360—22no—
37C2×#360—22no—
38C2×#360—21no—
39C2×#360—21no—
40C2×#360C2214yescomplemented
41#2117—22no—
42#2117C2214yescomplemented
43C2×#350—22no—
44#1389C2214yescomplemented
45#1383C2214yescomplemented
46C22×#41—21no—
47C22×#41D411yescharacteristic
48#575—24no—
49#575—24no—
50#575—42no—
51#575—42no—
52C2×#137—22no—
53C2×#137—41no—
54C2×AGL(1,ℤ/8)—22no—
55C2×AGL(1,ℤ/8)—41no—
56C2×#135—21no—
57C2×#135—21no—
58C2×#135—21no—
59C2×#135—21no—
60C2×#135—21no—
61C2×#135—21no—
62C2×#135—21no—
63C2×#135—21no—
64C2×#135—21no—
65C2×#135—21no—
66C2×#135—42no—
67C2×#135C2×C412yescomplemented
68C2×#135C2312yes—
69C22×Q16C2×C411yescharacteristic, complemented
70C22×SD16—21no—
71C22×D8C2×C411yescharacteristic, complemented
72C2×#131—21no—
73C2×#131—21no—
74C2×#131—21no—
75C2×#131—21no—
76C2×#131—21no—
77C2×#131—21no—
78C2×#131C2×C412yes—
79C2×#131C2×C412yes—
80C2×#131D412yes—
81C2×#131D412yes—
82C2×#131D412yes—
83C2×#131C2312yes—
84C22×M16—21no—
85C22×M16—21no—
86C22×M16D411yescharacteristic
87C23×C8—21no—
88C23×C8C2311yescharacteristic
89#520—24no—
90#520—24no—
91#520—42no—
92#520—42no—
93C2×C4×Q8—41no—
94C2×C4×D4—42no—
95C2×C4×D4—41no—
96C2×#117—22no—
97C2×#117—21no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C2×M32—21no—
101C2×M32—21no—
102C2×M32—21no—
103C2×M32—21no—
104C2×M32—21no—
105C2×M32—21no—
106C2×M32—21no—
107C2×M32—21no—
108C2×M32—21no—
109C2×M32—21no—
110C2×M32C2312yes—
111C2×M32C2312yes—
112C22×C16—21no—
113C22×C16—21no—
114#473—24no—
115#473C2×C418yescomplemented
116#472—24no—
117#472C2×C418yescomplemented
118#471—28no—
119#471—28no—
120C2×#108—21no—
121C2×#108—21no—
122C2×#108—21no—
123C2×#108C2×C412yescomplemented
124C2×#108C2×C412yescomplemented
125C2×#108C2312yes—
126#428—24no—
127#428—28no—
128#428—24no—
129#428—28no—
130#427—24no—
131#427—21no—
132#427—21no—
133#427—24no—
134#427D412yes—
135#427C2312yes—
136C2×#107—22no—
137C2×#107—21no—
138C2×#106—22no—
139C2×#106—24no—
140C2×#106—21no—
141C2×#106—21no—
142C2×#106—21no—
143C2×#106—24no—
144C2×#106D411yescharacteristic
145C2×#106D412yes—
146C2×#106D412yes—
147C2×#106D411yescharacteristic
148C2×#106C2312yes—
149#413—24no—
150#413—24no—
151C2×#101—22no—
152C2×#100—22no—
153#360—28no—
154#360—24no—
155#360—24no—
156#350—28no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—41no—
166C2×#41—41no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41—41no—
171C2×#41—41no—
172C2×#41—41no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41C22⋊C412yes—
176C2×#41C22⋊C412yes—
177C2×#41D812yes—
178C2×#41SD1612yes—
179C2×#41SD1612yes—
180C2×#41C2×D412yes—
181C22×Q8—21no—
182C22×Q8D811yescharacteristic
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—41no—
186C22×D4D811yescharacteristic
187C23×C4—21no—
188C23×C4—42no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4C2×D411yescharacteristic
192#137—28no—
193#137—44no—
194#137—44no—
195#137—44no—
196AGL(1,ℤ/8)—28no—
197AGL(1,ℤ/8)—44no—
198AGL(1,ℤ/8)—44no—
199AGL(1,ℤ/8)—44no—
200#135—22no—
201#135—22no—
202#135—22no—
203#135—22no—
204#135—22no—
205#135—22no—
206#135—22no—
207#135—22no—
208#135—48no—
209#135—48no—
210#135—41no—
211#135—41no—
212#135—41no—
213#135—41no—
214#135—41no—
215#135—41no—
216#135—41no—
217#135—41no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—42no—
224C2×Q16C22×C412yescomplemented
225C2×SD16—22no—
226C2×SD16—22no—
227C2×SD16—42no—
228C2×SD16—42no—
229C2×SD16—41no—
230C2×SD16—41no—
231C2×SD16—41no—
232C2×SD16—41no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—42no—
239C2×D8C22×C412yescomplemented
240#131—22no—
241#131—22no—
242#131—22no—
243#131—22no—
244#131—22no—
245#131—22no—
246#131—22no—
247#131—22no—
248#131—22no—
249#131—22no—
250#131—22no—
251#131—22no—
252#131C22⋊C414yes—
253#131C22⋊C414yes—
254#131C22×C414yes—
255#131C2×D414yes—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—41no—
269C2×M16—41no—
270C2×M16—41no—
271C2×M16C22⋊C412yes—
272C2×M16C22⋊C412yes—
273C2×M16C2×D412yes—
274C22×C8—21no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—41no—
282C22×C8C22×C411yescharacteristic
283C22×C8C22×C411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8#4111yescharacteristic
292C22×C8#4111yescharacteristic
293C22×C8#4111yescharacteristic
294C22×C8#4111yescharacteristic
295C22×C8C2411yescharacteristic
296C4×Q8—44no—
297C4×Q8—44no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302#117—28no—
303#117—22no—
304#117—22no—
305#117—44no—
306#117—41no—
307#117—41no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—41no—
317C2×C4⋊C4—41no—
318C2×C4⋊C4—41no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—41no—
321C2×C42—42no—
322C2×C42—41no—
323M32—22no—
324M32—22no—
325M32—22no—
326M32—22no—
327M32—41no—
328M32—41no—
329M32—41no—
330M32—41no—
331C2×C16—22no—
332C2×C16—22no—
333C2×C16—41no—
334C2×C16—41no—
335C2×C16—41no—
336C2×C16—41no—
337#108—22no—
338#108—22no—
339#108—22no—
340#108C22×C414yescomplemented
341#107—28no—
342#107—22no—
343#107—22no—
344#106—28no—
345#106—21no—
346#106—22no—
347#106—22no—
348#106—21no—
349#106—22no—
350#106C2×D412yes—
351#106C2×D414yes—
352#106C2×D412yes—
353#101—28no—
354#100—28no—
355C24—21no—
356C24—41no—
357C24AGL(1,ℤ/8)11yescharacteristic
358#41—22no—
359#41—22no—
360#41—22no—
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—22no—
369#41—22no—
370#41—42no—
371#41—42no—
372#41—42no—
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—81no—
379#41—81no—
380#41—81no—
381#41—81no—
382#41—81no—
383#41—81no—
384#41—81no—
385#41—81no—
386#41#10214yes—
387#41#10214yes—
388#41C2×C22⋊C414yes—
389#41C2×SD1614yes—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8#10212yes—
400C2×Q8#10212yes—
401C2×Q8C2×D812yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4#10212yes—
424C2×D4#10212yes—
425C2×D4C2×D812yes—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—162no—
447C22×C4—164no—
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4#12011yescharacteristic
451C22×C4#12011yescharacteristic
452C22×C4#12311yescharacteristic
453C22×C4#12311yescharacteristic
454C22×C4C2×D811yescharacteristic
455C22×C4C2×D811yescharacteristic
456C22×C4C2×SD1611yescharacteristic
457C22×C4C2×SD1611yescharacteristic
458C22×C4AGL(1,ℤ/8)11yescharacteristic
459C22×C4#13711yescharacteristic
460C22×C4#13711yescharacteristic
461C22×C4C22×D411yescharacteristic
462Q16—22no—
463Q16—22no—
464Q16—48no—
465Q16—41no—
466Q16—41no—
467SD16—48no—
468SD16—48no—
469SD16—42no—
470SD16—42no—
471SD16—42no—
472SD16—42no—
473D8—22no—
474D8—22no—
475D8—48no—
476D8—41no—
477D8—41no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—24no—
482M16—24no—
483M16—24no—
484M16—24no—
485M16—24no—
486M16—42no—
487M16—42no—
488M16—42no—
489M16C2×C22⋊C414yes—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8C2×C22⋊C411yescharacteristic
501C2×C8C2×C22⋊C411yescharacteristic
502C2×C8C4×D411yescharacteristic
503C2×C8C4×D411yescharacteristic
504C2×C8C4×D411yescharacteristic
505C2×C8C4×D411yescharacteristic
506C2×C8C4×D411yescharacteristic
507C2×C8C4×D411yescharacteristic
508C2×C8C4×D411yescharacteristic
509C2×C8C4×D411yescharacteristic
510C2×C8#12011yescharacteristic
511C2×C8#12011yescharacteristic
512C2×C8#12111yescharacteristic
513C2×C8#12111yescharacteristic
514C2×C8#12111yescharacteristic
515C2×C8#12111yescharacteristic
516C2×C8#12111yescharacteristic
517C2×C8#12111yescharacteristic
518C2×C8#12111yescharacteristic
519C2×C8#12111yescharacteristic
520C2×C8#12311yescharacteristic
521C2×C8#12311yescharacteristic
522C2×C8C23×C411yescharacteristic
523C2×C8C22×D411yescharacteristic
524C2×C8C22×D411yescharacteristic
525C2×C8C22×D411yescharacteristic
526C2×C8C2×#4111yescharacteristic
527C2×C8C2×#4111yescharacteristic
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—48no—
532C4⋊C4—42no—
533C4⋊C4—41no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—41no—
537C22⋊C4—48no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C42—48no—
541C42—42no—
542C42—42no—
543C16—42no—
544C16—42no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23—162no—
554C23#41711yescharacteristic
555C23#41711yescharacteristic
556C23#44711yescharacteristic
557C23#44711yescharacteristic
558C23#48111yescharacteristic
559C23#48111yescharacteristic
560C23C2×AGL(1,ℤ/8)11yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—42no—
565Q8—42no—
566Q8—81no—
567Q8—81no—
568Q8C2×#10214yes—
569D4—24no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4—42no—
577D4—42no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—81no—
583D4—81no—
584D4C2×#10214yes—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—84no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—162no—
606C2×C4—162no—
607C2×C4—168no—
608C2×C4#38611yescharacteristic
609C2×C4#38611yescharacteristic
610C2×C4#38611yescharacteristic
611C2×C4#38611yescharacteristic
612C2×C4C2×#10211yescharacteristic
613C2×C4C2×#10211yescharacteristic
614C2×C4C2×#10211yescharacteristic
615C2×C4C2×#10211yescharacteristic
616C2×C4#41711yescharacteristic
617C2×C4#41711yescharacteristic
618C2×C4#44711yescharacteristic
619C2×C4#44711yescharacteristic
620C2×C4#44811yescharacteristic
621C2×C4#44811yescharacteristic
622C2×C4#44811yescharacteristic
623C2×C4#44811yescharacteristic
624C2×C4#48011yescharacteristic
625C2×C4#48011yescharacteristic
626C2×C4#48011yescharacteristic
627C2×C4#48011yescharacteristic
628C2×C4#48111yescharacteristic
629C2×C4#48111yescharacteristic
630C2×C4C22×C22⋊C411yescharacteristic
631C2×C4C2×#12011yescharacteristic
632C2×C4C2×#12311yescharacteristic
633C2×C4C22×D811yescharacteristic
634C2×C4C22×SD1611yescharacteristic
635C2×C4C2×#13711yescharacteristic
636C8—24no—
637C8—44no—
638C8#38611yescharacteristic
639C8#38611yescharacteristic
640C8#38611yescharacteristic
641C8#38611yescharacteristic
642C8C2×C4×D411yescharacteristic
643C8C2×C4×D411yescharacteristic
644C8C2×#12111yescharacteristic
645C8C2×#12111yescharacteristic
646C22—22no—
647C22—22no—
648C22—22no—
649C22—24no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—82no—
658C22—162no—
659C22—162no—
660C22—162no—
661C22#187411yescharacteristic
662C22#187411yescharacteristic
663C22#187411yescharacteristic
664C22#187411yescharacteristic
665C22C2×#41711yescharacteristic
666C22C2×#44711yescharacteristic
667C22C2×#48111yescharacteristic
668C4—24no—
669C4—82no—
670C4—84no—
671C4—82no—
672C4—82no—
673C4#187411yescharacteristic
674C4#187411yescharacteristic
675C4#187411yescharacteristic
676C4#187411yescharacteristic
677C4C2×#38611yescharacteristic
678C4C22×#10211yescharacteristic
679C4C2×#44811yescharacteristic
680C4C2×#48011yescharacteristic
681C2—21no—
682C2—21no—
683C2—21no—
684C2—21no—
685C2—21no—
686C2—21no—
687C2—44no—
688C2—162no—
689C2#1300512yesdirect product factor
690C2C2×#187411yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2912C2semidirect
3. C2×#2145C2semidirect
4. C22×#360C2semidirect
5. C2×#2117C2semidirect
6. #13005C2direct (N × H)
7. C2×#1389C2semidirect
8. C2×#1383C2semidirect
17. #3067C4semidirect
19. C2×#473C4semidirect
20. C2×#472C4semidirect
26. #2912C22semidirect
35. #2145C22semidirect
40. C2×#360C22semidirect
42. #2117C22semidirect
44. #1389C22semidirect
45. #1383C22semidirect
67. C2×#135C2×C4semidirect
69. C22×Q16C2×C4semidirect
71. C22×D8C2×C4semidirect
115. #473C2×C4semidirect
117. #472C2×C4semidirect
123. C2×#108C2×C4semidirect
124. C2×#108C2×C4semidirect
224. C2×Q16C22×C4semidirect
239. C2×D8C22×C4semidirect
340. #108C22×C4semidirect
689. C2#13005direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30672
C2×#29122
C2×#21452
C22×#3602
C2×#21172
#130052
C2×#13892
C2×#13832