Finite Group Databasebeta

← C2×#9107 #196308

Subgroups — C2×#9107 #196308

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic, complemented
4C22×#323C212yesmaximal, complemented
5#9107C218yesmaximal, direct product factor
6C2×#1282C212yesmaximal, complemented
7C22×#120C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10#2972—24no—
11#2972—22no—
12#2972C2214yescomplemented
13C2×C8×D4—24no—
14C2×#431—21no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2321—22no—
19#2321—22no—
20#2321C414yes—
21#2321C2214yes—
22C2×#390—21no—
23C2×#388—21no—
24C2×#388—21no—
25C2×#386—21no—
26C2×#386C411yescharacteristic
27C2×#385C411yescharacteristic
28C2×#379C411yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1459—22no—
31#1459C2214yescomplemented
32C2×#324—21no—
33C2×#323—24no—
34C2×#323—24no—
35C2×#323—21no—
36C2×#323C2218yescomplemented
37#1282C2218yescomplemented
38C26C2×C411yescharacteristic
39C23×D4—21no—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—22no—
46C23×C8—22no—
47C2×#120—22no—
48C2×#120—24no—
49C2×#120C818yescomplemented
50C2×#120C2×C414yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4D412yes—
57C2×C4×D4D412yes—
58C2×C4×D4C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4C2312yes—
79C22×C42—21no—
80C22×C42D411yescharacteristic
81C8×D4—216no—
82C8×D4—216no—
83#431—24no—
84#431—24no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8D414yescomplemented
98C2×C22⋊C8D414yescomplemented
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×C4×C8—22no—
102#390—24no—
103#390—24no—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#386—22no—
109#386—24no—
110#386—24no—
111#386C2×C414yes—
112#385—22no—
113#385C2×C414yes—
114#379—22no—
115#379C2×C414yes—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2312yes—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123#324—24no—
124#323—24no—
125#323—216no—
126C25—21no—
127C25—21no—
128C25—22no—
129C25—22no—
130C25—22no—
131C25—24no—
132C25—82no—
133C25—82no—
134C25C22⋊C411yescharacteristic
135C25C2×C814yescomplemented
136C25C22×C412yes—
137C25C22×C411yescharacteristic
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—28no—
142C22×D4—24no—
143C22×D4—82no—
144C22×D4—82no—
145C22×D4C22⋊C412yes—
146C22×D4C2×C814yes—
147C22×D4M1614yes—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C22×C8—24no—
179C22×C8—28no—
180C22×C8—24no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—84no—
186C22×C8—84no—
187C22×C8—84no—
188#120—216no—
189#120—28no—
190#120C2×C8116yescomplemented
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C2×C814yescomplemented
242C2×C22⋊C4C22×C412yes—
243C2×C22⋊C4C22×C412yes—
244C2×C22⋊C4C22×C412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—82no—
253C2×C42C22⋊C412yes—
254C2×C42C2×D412yes—
255C4⋊C8—28no—
256C22⋊C8—28no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—24no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—42no—
263C22⋊C8—42no—
264C22⋊C8C2×D418yescomplemented
265C4×C8—28no—
266#95—22no—
267#95—24no—
268#95—24no—
269#95—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—24no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—28no—
280C24—28no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—41no—
290C24—42no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24—44no—
295C24—44no—
296C24—84no—
297C24—84no—
298C24—88no—
299C24—82no—
300C24—82no—
301C24—82no—
302C24—82no—
303C24—88no—
304C24—84no—
305C24—84no—
306C24—816no—
307C24C22⋊C814yes—
308C24C2×C22⋊C411yescharacteristic
309C24C2×C22⋊C412yes—
310C24C2×C22⋊C411yescharacteristic
311C24C4×D412yes—
312C24C4×D412yes—
313C24C22×C812yes—
314C24C22×C814yescomplemented
315C24C2×M1612yes—
316C24#13114yes—
317C24C23×C411yescharacteristic
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—216no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—816no—
330C2×D4C22⋊C818yes—
331C2×D4C22×C814yes—
332C2×D4C2×M1614yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—82no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—81no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4C22⋊C814yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4#12011yescharacteristic
390C22×C4#12011yescharacteristic
391C22×C4#12012yes—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4#12311yescharacteristic
397C22×C4#12311yescharacteristic
398C22×C4#12312yes—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4112yes—
403C2×C8—28no—
404C2×C8—48no—
405C2×C8—44no—
406C2×C8—84no—
407C2×C8—84no—
408C2×C8—88no—
409C2×C8—84no—
410C2×C8—84no—
411C2×C8—84no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—84no—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—216no—
422C22⋊C4—24no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4C22×C814yescomplemented
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—82no—
439C42—82no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—44no—
457C23—44no—
458C23—48no—
459C23—42no—
460C23—42no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—48no—
467C23—48no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—48no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—81no—
477C23—81no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—816no—
482C23—82no—
483C23—82no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23—816no—
489C23—84no—
490C23—84no—
491C23—88no—
492C23—88no—
493C23—816no—
494C23—88no—
495C23—816no—
496C23—82no—
497C23—82no—
498C23#38612yes—
499C23#38611yescharacteristic
500C23#38611yescharacteristic
501C23C2×C22⋊C812yes—
502C23C2×C22⋊C814yes—
503C23C2×C22⋊C812yes—
504C23#40812yes—
505C23#40812yes—
506C23C8×D414yes—
507C23C8×D414yes—
508C23#43514yes—
509C23#43514yes—
510C23C22×C22⋊C411yescharacteristic
511C23C2×C4×D412yes—
512C23C23×C811yescharacteristic
513C23C22×M1611yescharacteristic
514C23C2×#13112yes—
515D4—216no—
516D4—28no—
517D4—28no—
518D4—816no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—84no—
564C2×C4—82no—
565C2×C4—84no—
566C2×C4—82no—
567C2×C4—84no—
568C2×C4#38611yescharacteristic
569C2×C4#38611yescharacteristic
570C2×C4#38612yes—
571C2×C4C2×C22⋊C814yes—
572C2×C4C2×#12011yescharacteristic
573C2×C4C2×#12112yes—
574C2×C4C2×#12311yescharacteristic
575C8—48no—
576C8—84no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—22no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—24no—
586C22—24no—
587C22—28no—
588C22—28no—
589C22—24no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—48no—
603C22—44no—
604C22—82no—
605C22—84no—
606C22—88no—
607C22—82no—
608C22—82no—
609C22—88no—
610C22—816no—
611C22—88no—
612C22—82no—
613C22—82no—
614C22—88no—
615C22#186912yes—
616C22#186914yes—
617C22#186912yes—
618C22#191312yes—
619C22#191412yes—
620C22C2×#38611yescharacteristic
621C22C22×C22⋊C811yescharacteristic
622C22C2×#40811yescharacteristic
623C22C2×C8×D412yes—
624C22C2×#43512yes—
625C4—21no—
626C4—22no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—22no—
631C4—24no—
632C4—44no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C2—21no—
638C2—21no—
639C2—22no—
640C2—24no—
641C2—22no—
642C2—22no—
643C2—44no—
644C2—84no—
645C2#910714yesdirect product factor
646C2C2×#186911yescharacteristic
647C2C2×#191311yescharacteristic
648C2C2×#191411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC22×#120C4
LCSG ▷ C23 ▷ C2 ▷ 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×#2972C2semidirect
3. C2×#1459C2semidirect
4. C22×#323C2semidirect
5. #9107C2direct (N × H)
6. C2×#1282C2semidirect
12. #2972C22semidirect
31. #1459C22semidirect
36. C2×#323C22semidirect
37. #1282C22semidirect
49. C2×#120C8semidirect
97. C2×C22⋊C8D4semidirect
98. C2×C22⋊C8D4semidirect
135. C25C2×C8semidirect
190. #120C2×C8semidirect
241. C2×C22⋊C4C2×C8semidirect
264. C22⋊C8C2×D4semidirect
314. C24C22×C8semidirect
433. C22⋊C4C22×C8semidirect
645. C2#9107direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29722
C2×#23212
C2×#14592
C22×#3232
#91072
C2×#12822