Finite Group Databasebeta

← C2×#9207 #197572

Subgroups — C2×#9207 #197572

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2C2×C8×#41C211yesmaximal, characteristic
3C2×#1806C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic, complemented
5C2×#1493C211yesmaximal, characteristic, complemented
6#9207C218yesmaximal, direct product factor
7C2×#1301C211yesmaximal, characteristic, complemented
8C2×#1279C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#3076—22no—
12#3076—22no—
13#3076C414yescomplemented
14#3076C2214yes—
15C2×#479—21no—
16C2×#477C411yescharacteristic, complemented
17C2×#475C411yescharacteristic, complemented
18C2×#478—21no—
19C2×#476C411yescharacteristic, complemented
20C2×#474C411yescharacteristic, complemented
21C8×#41—22no—
22C8×#41—22no—
23C8×#41—22no—
24C8×#41C2214yes—
25C2×C8×Q8—21no—
26C2×C8×D4—22no—
27C2×C8×D4—21no—
28C2×#431—21no—
29C2×#431C2211yescharacteristic
30C2×#427—21no—
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#104—21no—
34C2×#416—21no—
35C22×C4×C8—21no—
36C22×C4×C8C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1806—22no—
39#1806C2214yes—
40C2×#339—21no—
41#1503—22no—
42#1503C2214yescomplemented
43C2×#328—22no—
44#1493—22no—
45#1493C2214yescomplemented
46C2×Q8⋊C8—21no—
47C2×Q8⋊C8—21no—
48#1301C2214yescomplemented
49#1279C2214yescomplemented
50C22×#41D411yescharacteristic
51C22×M16—21no—
52C23×C8—22no—
53C23×C8—21no—
54C23×C8—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533C2×C412yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41—21no—
71C4×#41—21no—
72C4×#41C2×C412yes—
73C4×#41D412yes—
74C4×#41D412yes—
75C4×#41C2312yes—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117D411yescharacteristic
87C22×C4⋊C4—21no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#479—24no—
92#479—24no—
93#478—24no—
94#478—24no—
95#477—22no—
96#477C2×C414yescomplemented
97#476—22no—
98#476C2×C414yescomplemented
99#475—22no—
100#475C2×C414yescomplemented
101#474—22no—
102#474C2×C414yescomplemented
103C8×Q8—24no—
104C8×Q8—24no—
105C8×D4—28no—
106C8×D4—24no—
107C8×D4—24no—
108C8×D4—28no—
109#431—24no—
110#431—21no—
111#431—21no—
112#431—24no—
113#431D412yes—
114#431C2312yes—
115#427—24no—
116#427—24no—
117C2×#107—21no—
118C2×#107—21no—
119C2×#106—21no—
120C2×#106—21no—
121#423—21no—
122#423—21no—
123#423—21no—
124#423C2312yes—
125C2×C4⋊C8—22no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2×C411yescharacteristic
133C2×C4⋊C8C2×C411yescharacteristic
134C2×C4⋊C8C2×C412yes—
135C2×C4⋊C8C2312yes—
136C2×#104—24no—
137C2×#104—44no—
138#416—24no—
139#416—24no—
140C2×#103—21no—
141C2×#103—21no—
142C2×#102—21no—
143C2×#102—21no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8D412yes—
146C2×C4×C8—22no—
147C2×C4×C8—24no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8—24no—
153C2×C4×C8D411yescharacteristic
154C2×C4×C8D411yescharacteristic
155C2×C4×C8D412yes—
156C2×C4×C8C2312yes—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2312yes—
161C2×#95—21no—
162#339—24no—
163#328—28no—
164Q8⋊C8—24no—
165Q8⋊C8—24no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41C22⋊C412yes—
171C2×#41SD1614yes—
172C2×#41Q1614yes—
173C2×#41C2×D412yes—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8C22⋊C411yescharacteristic
177C22×D4—21no—
178C22×D4C22⋊C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4111yescharacteristic
186C2×M16—22no—
187C2×M16—41no—
188C2×M16—42no—
189C2×M16—42no—
190C22×C8—21no—
191C22×C8—28no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—44no—
201C22×C8—48no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—41no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C412yes—
213#125—22no—
214#125—22no—
215#125—41no—
216#125—41no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8C22⋊C412yes—
238C4×Q8C22⋊C412yes—
239C4×Q8C22×C412yes—
240C4×Q8C2×D412yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C22⋊C412yes—
254C4×D4C22⋊C412yes—
255C4×D4C22×C412yes—
256C4×D4C2×D412yes—
257#117—21no—
258#117—21no—
259#117—22no—
260#117—22no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—22no—
265#117—41no—
266#117—41no—
267#117C22⋊C412yes—
268#117C2×D412yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—41no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—41no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—41no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42#4111yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42#4111yescharacteristic
310C2×C42C2411yescharacteristic
311#107—24no—
312#107—24no—
313#106—24no—
314#106—24no—
315C4⋊C8—28no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—41no—
325C4⋊C8—41no—
326C4⋊C8C22×C412yes—
327C4⋊C8C22×C412yes—
328#104—48no—
329#103—24no—
330#103—24no—
331#102—24no—
332#102—24no—
333C22⋊C8—28no—
334C22⋊C8—22no—
335C22⋊C8C2×D414yes—
336C4×C8—28no—
337C4×C8—21no—
338C4×C8—22no—
339C4×C8—21no—
340C4×C8—22no—
341C4×C8—22no—
342C4×C8—22no—
343C4×C8C2×D412yes—
344C4×C8C2×D412yes—
345#95—22no—
346#95—41no—
347C24—21no—
348C24#12111yescharacteristic
349#41—22no—
350#41—22no—
351#41—24no—
352#41—24no—
353#41—24no—
354#41—24no—
355#41—24no—
356#41#10318yes—
357#41C2×SD1614yes—
358#41C2×Q1614yes—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—82no—
368C2×Q8#10314yes—
369C2×Q8C2×C22⋊C412yes—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4#10314yes—
377C2×D4C2×C22⋊C412yes—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4#12011yescharacteristic
408C22×C4#12011yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4#12311yescharacteristic
416C22×C4C2×SD1612yes—
417C22×C4C2×Q1612yes—
418C22×C4#13512yes—
419C22×C4#13512yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423M16—42no—
424M16—42no—
425M16—82no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—416no—
435C2×C8—44no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—44no—
442C2×C8—42no—
443C2×C8—82no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4C2×C22⋊C412yes—
466C4⋊C4C2×C22⋊C412yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—82no—
486C42C2×C22⋊C411yescharacteristic
487C42C2×C22⋊C411yescharacteristic
488C42C4×D411yescharacteristic
489C42C4×D411yescharacteristic
490C42C4×D411yescharacteristic
491C42C4×D411yescharacteristic
492C42#12011yescharacteristic
493C42#12011yescharacteristic
494C42#12111yescharacteristic
495C42#12111yescharacteristic
496C42#12311yescharacteristic
497C42#12311yescharacteristic
498C42C23×C411yescharacteristic
499C42C22×D411yescharacteristic
500C42C22×D411yescharacteristic
501C42C2×#4111yescharacteristic
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—42no—
507C23—42no—
508C23#38611yescharacteristic
509C23#38611yescharacteristic
510C23#46512yes—
511C23#46712yes—
512C23C2×#12111yescharacteristic
513Q8—22no—
514Q8—24no—
515Q8—42no—
516Q8—42no—
517Q8—82no—
518Q8—82no—
519Q8—82no—
520Q8C2×#10314yes—
521D4—28no—
522D4—24no—
523D4—24no—
524D4—22no—
525D4—24no—
526D4C2×#10314yes—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—88no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4#38611yescharacteristic
563C2×C4#38611yescharacteristic
564C2×C4#38611yescharacteristic
565C2×C4#38611yescharacteristic
566C2×C4#38611yescharacteristic
567C2×C4#38611yescharacteristic
568C2×C4C2×#10312yes—
569C2×C4C2×#10312yes—
570C2×C4#41612yes—
571C2×C4#41612yes—
572C2×C4#44312yes—
573C2×C4#44412yes—
574C2×C4#46512yes—
575C2×C4#46512yes—
576C2×C4#46512yes—
577C2×C4#46712yes—
578C2×C4#46712yes—
579C2×C4#46712yes—
580C2×C4C22×C22⋊C411yescharacteristic
581C2×C4C2×C4×D411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12111yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4C22×SD1611yescharacteristic
587C2×C4C22×Q1611yescharacteristic
588C2×C4C2×#13511yescharacteristic
589C2×C4C2×#13511yescharacteristic
590C8—24no—
591C8—44no—
592C8—42no—
593C8—42no—
594C8—42no—
595C8—82no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—48no—
606C22#187612yes—
607C22#189212yes—
608C22#189212yes—
609C22#190612yes—
610C22C2×#38611yescharacteristic
611C22C2×#46511yescharacteristic
612C22C2×#46711yescharacteristic
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—24no—
620C4—42no—
621C4—42no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4#187612yes—
626C4#189212yes—
627C4#189212yes—
628C4C22×#10311yescharacteristic
629C4C2×#41611yescharacteristic
630C4C2×#44311yescharacteristic
631C4C2×#44411yescharacteristic
632C4C2×#46511yescharacteristic
633C4C2×#46711yescharacteristic
634C2—21no—
635C2—22no—
636C2—21no—
637C2—44no—
638C2#920714yesdirect product factor
639C2C2×#187611yescharacteristic
640C2C2×#189211yescharacteristic
641C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1503C2semidirect
5. C2×#1493C2semidirect
6. #9207C2direct (N × H)
7. C2×#1301C2semidirect
8. C2×#1279C2semidirect
13. #3076C4semidirect
16. C2×#477C4semidirect
17. C2×#475C4semidirect
19. C2×#476C4semidirect
20. C2×#474C4semidirect
42. #1503C22semidirect
45. #1493C22semidirect
48. #1301C22semidirect
49. #1279C22semidirect
96. #477C2×C4semidirect
98. #476C2×C4semidirect
100. #475C2×C4semidirect
102. #474C2×C4semidirect
638. C2#9207direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30762
C2×C8×#412
C2×#18062
C2×#15032
C2×#14932
#92072
C2×#13012
C2×#12792