Finite Group Databasebeta

← C2×#10400 #239402

Subgroups — C2×#10400 #239402

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#1872C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5C22×#324C211yesmaximal, characteristic
6C2×#1441C211yesmaximal, characteristic
7#10400C218yesmaximal, direct product factor
8C2×#1294C211yesmaximal, characteristic
9C22×C22⋊Q8C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#107—21no—
13C22×#103—21no—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2627—22no—
17#2627—22no—
18#2627—22no—
19#2627C2214yes—
20C2×#400—21no—
21C2×#398—21no—
22C2×#398—21no—
23C2×#395—21no—
24C2×#385—21no—
25C2×#381—21no—
26C2×#380C2211yescharacteristic
27C22×#95—21no—
28C22×#95—21no—
29C22×#95C2211yescharacteristic
30#1872—22no—
31#1872C2214yes—
32#1852—22no—
33#1852C2214yes—
34C2×#340—21no—
35C2×#340—21no—
36C2×#328—24no—
37C2×#328C2214yes—
38C2×#324—24no—
39C2×#324C2214yes—
40#1441—22no—
41#1441C2214yes—
42#1294C2214yes—
43C23×Q8D411yescharacteristic
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8—21no—
50C23×C8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C2×#107—24no—
85C2×#107—44no—
86C2×#103—24no—
87C2×#103—44no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8D414yes—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96#400—24no—
97#400—24no—
98#398—24no—
99#398—24no—
100#398—24no—
101#398—24no—
102#395—24no—
103#395—24no—
104#385—24no—
105#385—24no—
106#381—24no—
107#381—24no—
108#380—21no—
109#380—21no—
110#380D412yes—
111#380C2312yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—44no—
125C2×#95—44no—
126C2×#95C2312yes—
127#340—24no—
128#340—24no—
129#328—28no—
130#324—28no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—41no—
168C23×C4—41no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4Q1614yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C22×C8—24no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261#107—48no—
262#103—48no—
263C22⋊C8—24no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—42no—
268C22⋊C8C2×D414yes—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—48no—
280#95—48no—
281#95—42no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24#12011yescharacteristic
287C24#12011yescharacteristic
288C24#12112yes—
289C24#12312yes—
290C24C2×Q1612yes—
291C24#13512yes—
292C24AGL(1,ℤ/8)12yes—
293C24#13712yes—
294C24C22×D411yescharacteristic
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—48no—
300C2×Q8—42no—
301C2×Q8—81no—
302C2×Q8—81no—
303C2×Q8—84no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
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—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4#12312yes—
404C22×C4#12312yes—
405C22×C4#12312yes—
406C22×C4#12412yes—
407C22×C4#12412yes—
408C22×C4C2×Q1614yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—48no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—84no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23#39412yes—
474C23#39412yes—
475C23#44912yes—
476C23#44912yes—
477C23#45112yes—
478C23#45112yes—
479C23#45412yes—
480C23#45612yes—
481C23B(4,𝔽2)12yes—
482C23#45812yes—
483C23#46714yes—
484C23#46814yes—
485C23#48414yes—
486C23#48514yes—
487C23C2×#12011yescharacteristic
488C23C2×#12111yescharacteristic
489C23C2×#12311yescharacteristic
490C23C22×Q1611yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494Q8—44no—
495Q8—82no—
496Q8—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—816no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—88no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—82no—
568C2×C4#39412yes—
569C2×C4#39412yes—
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12311yescharacteristic
573C2×C4C2×#12411yescharacteristic
574C8—44no—
575C8—82no—
576C22—28no—
577C22—28no—
578C22—28no—
579C22—28no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—21no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22#159714yes—
599C22#160514yes—
600C22#160914yes—
601C22#161714yes—
602C22#203614yes—
603C22#203614yes—
604C22#203814yes—
605C22#204414yes—
606C22C2×#39411yescharacteristic
607C22C2×#45111yescharacteristic
608C22C2×#44911yescharacteristic
609C22C2×#45411yescharacteristic
610C22C2×#45611yescharacteristic
611C22C2×B(4,𝔽2)11yescharacteristic
612C22C2×#45811yescharacteristic
613C22C2×#46711yescharacteristic
614C22C2×#46811yescharacteristic
615C22C2×#48511yescharacteristic
616C22C2×#48411yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
623C4—44no—
624C4—84no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—24no—
636C2#1040018yesdirect product factor
637C2C2×#159711yescharacteristic
638C2C2×#160911yescharacteristic
639C2C2×#160511yescharacteristic
640C2C2×#161711yescharacteristic
641C2C2×#203611yescharacteristic
642C2C2×#203811yescharacteristic
643C2C2×#204411yescharacteristic
subgroupG/H
center, socleC24#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10400C2direct (N × H)
636. C2#10400direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#18722
C2×#18522
C22×#3282
C22×#3242
C2×#14412
#104002
C2×#12942