Finite Group Databasebeta

← C2×#11222 #358010

Subgroups — C2×#11222 #358010

Order 512.

3806 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3C22×#342C212yesmaximal
4C22×#342C211yesmaximal, characteristic
5C2×C22⋊#95C212yesmaximal
6#11222C218yesmaximal, direct product factor
7C25×C4C2211yescharacteristic
8C23×C4⋊C4—21no—
9C23×C22⋊C4C2212yes—
10C23×C22⋊C4C2212yes—
11C23×C22⋊C4C2211yescharacteristic
12C23×C22⋊C4C2211yescharacteristic
13#2315—24no—
14#2315—22no—
15#2315C2214yes—
16C2×#385—24no—
17C2×#385—22no—
18C2×#385—24no—
19C2×#385C418yescomplemented
20C2×#385C2214yes—
21C2×#380—21no—
22C2×#380—21no—
23C22×#95—21no—
24C22×#95—24no—
25C2×#342—24no—
26C2×#342—28no—
27C2×#342C2218yes—
28C2×#342C2214yes—
29C22⋊#95—24no—
30C22⋊#95C2218yes—
31C26C2311yescharacteristic
32C24×C4—21no—
33C24×C4—22no—
34C24×C4—21no—
35C24×C4—22no—
36C24×C4—22no—
37C24×C4—21no—
38C24×C4—22no—
39C24×C4—24no—
40C24×C4—48no—
41C24×C4C2×C414yes—
42C24×C4D414yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2312yes—
46C22×C4⋊C4—28no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—48no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—24no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—24no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C22⋊C4C2×C414yes—
73C22×C22⋊C4C2×C414yes—
74C22×C22⋊C4D414yes—
75C22×C22⋊C4D412yes—
76C22×C22⋊C4D414yes—
77C22×C22⋊C4D414yes—
78C22×C22⋊C4Q818yes—
79C22×C22⋊C4Q814yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2312yes—
84#385—216no—
85#385—216no—
86#385—216no—
87#385—28no—
88#385C2×C4116yescomplemented
89#380—24no—
90#380—24no—
91#380—24no—
92#380—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—216no—
98C2×#95—416no—
99#342—28no—
100#342—216no—
101C25—44no—
102C25—44no—
103C25C22×C412yes—
104C25C2×D411yescharacteristic
105C25C2×D412yes—
106C25C2×D411yescharacteristic
107C25C2×D412yes—
108C25C2×D412yes—
109C25C2×D412yes—
110C25C2×Q812yes—
111C25C2×Q814yes—
112C25#4112yes—
113C25#4112yes—
114C25#4114yes—
115C25#4114yes—
116C25C2411yescharacteristic
117C23×C4—28no—
118C23×C4—216no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—28no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—28no—
135C23×C4—24no—
136C23×C4—28no—
137C23×C4—28no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—28no—
143C23×C4—416no—
144C23×C4—48no—
145C23×C4—416no—
146C23×C4—416no—
147C23×C4—48no—
148C23×C4—432no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—48no—
153C23×C4—41no—
154C23×C4—42no—
155C23×C4—41no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—48no—
163C23×C4—48no—
164C23×C4—48no—
165C23×C4—48no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4C22⋊C414yes—
169C23×C4C22⋊C414yes—
170C23×C4C22×C414yes—
171C23×C4C2×D418yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C2×C4⋊C4—216no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—432no—
178C2×C4⋊C4—416no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4—84no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—216no—
199C2×C22⋊C4—216no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4C22⋊C418yes—
229C2×C22⋊C4C4⋊C4116yescomplemented
230C2×C22⋊C4C22×C414yes—
231C2×C22⋊C4C22×C414yes—
232C2×C22⋊C4C2×D414yes—
233C2×C22⋊C4C2×D414yes—
234C2×C22⋊C4C2×Q818yes—
235C2×C22⋊C4C2×Q814yes—
236#95—28no—
237#95—28no—
238#95—216no—
239#95—432no—
240C24—24no—
241C24—28no—
242C24—24no—
243C24—28no—
244C24—44no—
245C24—44no—
246C24—42no—
247C24—44no—
248C24—48no—
249C24—42no—
250C24—44no—
251C24—44no—
252C24—42no—
253C24—44no—
254C24—48no—
255C24—48no—
256C24—416no—
257C24—48no—
258C24—48no—
259C24—48no—
260C24—416no—
261C24—42no—
262C24C2×C22⋊C412yes—
263C24C2×C22⋊C412yes—
264C24C2×C4⋊C414yes—
265C24#11714yes—
266C24C4×D414yes—
267C24C4×D414yes—
268C24C4×Q818yes—
269C24#12012yes—
270C24#12011yescharacteristic
271C24#12011yescharacteristic
272C24#12112yes—
273C24#12114yes—
274C24#12114yes—
275C24#12112yes—
276C24#12114yes—
277C24#12114yes—
278C24#12114yes—
279C24C22⋊Q812yes—
280C24C22⋊Q814yes—
281C24C22⋊Q812yes—
282C24C22⋊Q814yes—
283C24C22⋊Q814yes—
284C24C22⋊Q818yes—
285C24C22⋊Q818yes—
286C24#12312yes—
287C24#12314yes—
288C24#12312yes—
289C24#12314yes—
290C24#12412yes—
291C24#12412yes—
292C24#12518yes—
293C24#12518yes—
294C24#12618yes—
295C24C4⋊Q814yes—
296C24C4⋊Q812yes—
297C24C4⋊Q814yes—
298C24C4⋊Q812yes—
299C24C23×C411yescharacteristic
300C24C22×D411yescharacteristic
301C24C22×D412yes—
302C24C22×D411yescharacteristic
303C24C22×D411yescharacteristic
304C24C22×Q811yescharacteristic
305C24C22×Q812yes—
306C24C2×#4111yescharacteristic
307C24C2×#4111yescharacteristic
308C24C2×#4112yes—
309C24C2×#4112yes—
310C22×C4—216no—
311C22×C4—216no—
312C22×C4—28no—
313C22×C4—232no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—216no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—48no—
333C22×C4—416no—
334C22×C4—48no—
335C22×C4—432no—
336C22×C4—416no—
337C22×C4—48no—
338C22×C4—44no—
339C22×C4—416no—
340C22×C4—48no—
341C22×C4—44no—
342C22×C4—416no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—432no—
346C22×C4—416no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—416no—
351C22×C4—432no—
352C22×C4—416no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—432no—
370C22×C4—48no—
371C22×C4—432no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—816no—
388C22×C4—816no—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C414yes—
391C4⋊C4—432no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C22⋊C4—24no—
395C22⋊C4—28no—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—216no—
400C22⋊C4—216no—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C22⋊C4C2×C4⋊C418yescomplemented
413C23—28no—
414C23—216no—
415C23—216no—
416C23—28no—
417C23—24no—
418C23—28no—
419C23—28no—
420C23—24no—
421C23—24no—
422C23—28no—
423C23—24no—
424C23—28no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—28no—
434C23—216no—
435C23—216no—
436C23—28no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—48no—
446C23—48no—
447C23—44no—
448C23—416no—
449C23—48no—
450C23—44no—
451C23—48no—
452C23—44no—
453C23—44no—
454C23—48no—
455C23—48no—
456C23—416no—
457C23—416no—
458C23—48no—
459C23—42no—
460C23—44no—
461C23—416no—
462C23—416no—
463C23—416no—
464C23—48no—
465C23—44no—
466C23—41no—
467C23—42no—
468C23—41no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23#38014yes—
476C23#38014yes—
477C23#38614yes—
478C23#38612yes—
479C23#38612yes—
480C23#38718yes—
481C23#38718yes—
482C23#38918yes—
483C23#38918yes—
484C23#39114yes—
485C23#39114yes—
486C23#39412yes—
487C23#39414yes—
488C23#39412yes—
489C23#39512yes—
490C23#39514yes—
491C23#39512yes—
492C23#39618yes—
493C23#39618yes—
494C23#39814yes—
495C23#39818yes—
496C23#39814yes—
497C23#39818yes—
498C23#40014yes—
499C23#40014yes—
500C23B(4,𝔽2)14yes—
501C23#45814yes—
502C23C22×C22⋊C411yescharacteristic
503C23C22×C4⋊C411yescharacteristic
504C23C2×#11711yescharacteristic
505C23C2×C4×D412yes—
506C23C2×C4×Q812yes—
507C23C2×#12011yescharacteristic
508C23C2×#12112yes—
509C23C2×#12111yescharacteristic
510C23C2×#12112yes—
511C23C2×#12111yescharacteristic
512C23C2×C22⋊Q812yes—
513C23C2×C22⋊Q811yescharacteristic
514C23C2×C22⋊Q812yes—
515C23C2×C22⋊Q812yes—
516C23C2×C22⋊Q811yescharacteristic
517C23C2×#12312yes—
518C23C2×#12311yescharacteristic
519C23C2×#12411yescharacteristic
520C23C2×#12512yes—
521C23C2×#12512yes—
522C23C2×#12612yes—
523C23C2×C4⋊Q812yes—
524C23C2×C4⋊Q811yescharacteristic
525C2×C4—28no—
526C2×C4—216no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—432no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—48no—
537C2×C4—416no—
538C2×C4—48no—
539C2×C4—416no—
540C2×C4—432no—
541C2×C4—416no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—82no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—84no—
560C2×C4—832no—
561C2×C4—88no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4—88no—
567C22—22no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—28no—
573C22—28no—
574C22—216no—
575C22—216no—
576C22—216no—
577C22—28no—
578C22—216no—
579C22—28no—
580C22—28no—
581C22—22no—
582C22—24no—
583C22—24no—
584C22—28no—
585C22—22no—
586C22—24no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—21no—
591C22—22no—
592C22—24no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—24no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22#144114yes—
618C22#144118yes—
619C22#144114yes—
620C22#188914yes—
621C22#189014yes—
622C22#204418yes—
623C22#206218yes—
624C22C2×#38011yescharacteristic
625C22C2×#38611yescharacteristic
626C22C2×#38712yes—
627C22C2×#38912yes—
628C22C2×#39111yescharacteristic
629C22C2×#39411yescharacteristic
630C22C2×#39511yescharacteristic
631C22C2×#39612yes—
632C22C2×#39811yescharacteristic
633C22C2×#39812yes—
634C22C2×#40011yescharacteristic
635C22C2×B(4,𝔽2)12yes—
636C22C2×#45812yes—
637C4—48no—
638C4—48no—
639C4—44no—
640C4—44no—
641C4—88no—
642C4—88no—
643C2—21no—
644C2—22no—
645C2—21no—
646C2—22no—
647C2—21no—
648C2—24no—
649C2—28no—
650C2—22no—
651C2—22no—
652C2—21no—
653C2#1122218yesdirect product factor
654C2C2×#144111yescharacteristic
655C2C2×#188911yescharacteristic
656C2C2×#189011yescharacteristic
657C2C2×#204412yes—
658C2C2×#206212yes—
subgroupG/H
center, socleC24#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11222C2direct (N × H)
19. C2×#385C4semidirect
88. #385C2×C4semidirect
229. C2×C22⋊C4C4⋊C4semidirect
412. C22⋊C4C2×C4⋊C4semidirect
653. C2#11222direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23152
C22×#3852
C22×#3422
C2×C22⋊#952
#112222