Finite Group Databasebeta

← C2×#9237 #198018

Subgroups — C2×#9237 #198018

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic, complemented
3C2×#1962C211yesmaximal, characteristic, complemented
4C2×#1502C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#9237C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1276C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C22×#117—21no—
12#3100—22no—
13#3100—22no—
14#3100C414yescomplemented
15#3100C2214yes—
16C2×#480—21no—
17C2×#474C412yescomplemented
18C2×#459C412yescomplemented
19C2×#447—21no—
20C2×C4×D8—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#104—21no—
25C22×#104—21no—
26#2293—22no—
27#2293—22no—
28#2293—22no—
29#2293C2214yescomplemented
30C2×#390—21no—
31C2×#389—21no—
32C2×#388—22no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C43C2211yescharacteristic
36#1962—22no—
37#1962C2214yescomplemented
38#1502—22no—
39#1502C2214yescomplemented
40C2×#327—22no—
41#1486—22no—
42#1486C2214yescomplemented
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1276C2214yescomplemented
46C22×#41—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C22×D8—22no—
50C22×M16—21no—
51#532—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532C2×C412yes—
58#532C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#127C2×C411yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C412yes—
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—44no—
81C2×#117—44no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—22no—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94#480—24no—
95#480—24no—
96#474—24no—
97#474C2×C418yescomplemented
98#459—24no—
99#459C2×C418yescomplemented
100#447—24no—
101#447—24no—
102C4×D8—28no—
103C4×D8—28no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431C2×C412yes—
110#431C2312yes—
111#431C2312yes—
112C2×#107—22no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423C2312yes—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2×C412yes—
121C2×#104—24no—
122C2×#104—24no—
123C2×#104—44no—
124C2×#104—44no—
125C2×#102—22no—
126C2×#102—22no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#390—24no—
132#390—24no—
133#389—24no—
134#389—24no—
135#388—28no—
136#388—28no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C2×#95—22no—
144C2×#95D412yes—
145C43—21no—
146C43—21no—
147C43D412yes—
148C43C2312yes—
149#327—28no—
150D4⋊C8—28no—
151C25—21no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—44no—
168C22×D4—41no—
169C22×D4C22⋊C412yes—
170C22×D4C22⋊C412yes—
171C22×D4C2×D412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—41no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×D8—28no—
191C2×D8—48no—
192C2×M16—22no—
193C2×M16—41no—
194C2×M16—42no—
195C2×M16—42no—
196C22×C8—22no—
197C22×C8—41no—
198C22×C8—41no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C22×C412yes—
203#127—21no—
204#127—21no—
205#127—21no—
206#127C22×C412yes—
207#124—22no—
208#124—22no—
209#124—41no—
210#124—41no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—41no—
216#121—41no—
217#121—41no—
218#121—41no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4C22⋊C414yes—
240C4×D4C22⋊C414yes—
241C4×D4C22×C414yes—
242C4×D4C2×D414yes—
243#117—48no—
244#117—416no—
245#117—48no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42D814yescomplemented
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42C2411yescharacteristic
301#107—28no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8—41no—
309C4⋊C8C22×C414yes—
310#104—48no—
311#104—48no—
312#102—28no—
313#102—28no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—41no—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—41no—
322#95—28no—
323#95—22no—
324#95C2×D414yes—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—44no—
329C24—44no—
330C24—42no—
331C24—42no—
332C24—41no—
333C24#12011yescharacteristic
334#41—42no—
335#41—42no—
336#41—42no—
337#41—42no—
338#41—44no—
339#41—44no—
340#41—82no—
341#41—82no—
342#41—82no—
343#41—82no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—44no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4C2×C22⋊C414yes—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—41no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4#12011yescharacteristic
422C22×C4#12011yescharacteristic
423C22×C4#12011yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12311yescharacteristic
429C22×C4#12311yescharacteristic
430C22×C4C2×D812yes—
431C22×C4#13512yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4AGL(1,ℤ/8)12yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437D8—416no—
438M16—42no—
439M16—42no—
440M16—82no—
441C2×C8—24no—
442C2×C8—44no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—82no—
446C2×C8—82no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—416no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4C2×C22⋊C414yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—416no—
463C22⋊C4—416no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C42—22no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42—44no—
474C42—44no—
475C42—44no—
476C42—42no—
477C42—42no—
478C42—44no—
479C42—42no—
480C42C2×C22⋊C411yescharacteristic
481C42C2×C22⋊C411yescharacteristic
482C42C4×D411yescharacteristic
483C42C4×D411yescharacteristic
484C42C4×D411yescharacteristic
485C42C4×D411yescharacteristic
486C42#12111yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12111yescharacteristic
490C42#12311yescharacteristic
491C42#12311yescharacteristic
492C42C2×D814yescomplemented
493C42C23×C411yescharacteristic
494C42C22×D411yescharacteristic
495C42C2×#4111yescharacteristic
496C42C2×#4111yescharacteristic
497C23—22no—
498C23—22no—
499C23—24no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—88no—
512C23—88no—
513C23—82no—
514C23—82no—
515C23—82no—
516C23#38611yescharacteristic
517C23#38611yescharacteristic
518C23#44712yes—
519C23#44912yes—
520C23C2×#12011yescharacteristic
521Q8—42no—
522Q8—42no—
523Q8—82no—
524D4—24no—
525D4—24no—
526D4—44no—
527D4—44no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—82no—
533D4—82no—
534D4—84no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—88no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4#38611yescharacteristic
579C2×C4#38611yescharacteristic
580C2×C4#38611yescharacteristic
581C2×C4#38611yescharacteristic
582C2×C4#38611yescharacteristic
583C2×C4#38611yescharacteristic
584C2×C4#42114yes—
585C2×C4C4×D812yes—
586C2×C4C4×D812yes—
587C2×C4#44212yes—
588C2×C4#44212yes—
589C2×C4#44712yes—
590C2×C4#44912yes—
591C2×C4#45912yes—
592C2×C4#45912yes—
593C2×C4#46312yes—
594C2×C4#46312yes—
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C2×C4C22×D811yescharacteristic
602C2×C4C2×#13511yescharacteristic
603C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
604C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
605C8—44no—
606C8—82no—
607C8—82no—
608C22—21no—
609C22—21no—
610C22—24no—
611C22—24no—
612C22—21no—
613C22—21no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—84no—
618C22—84no—
619C22—88no—
620C22—88no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—164no—
625C22#187912yes—
626C22#187912yes—
627C22#189712yes—
628C22#190612yes—
629C22C2×#38611yescharacteristic
630C22C2×#44711yescharacteristic
631C22C2×#44911yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—84no—
642C4—82no—
643C4#187912yes—
644C4#187912yes—
645C4#189712yes—
646C4C2×#42112yes—
647C4C2×C4×D811yescharacteristic
648C4C2×#44211yescharacteristic
649C4C2×#45911yescharacteristic
650C4C2×#46311yescharacteristic
651C2—21no—
652C2—21no—
653C2—22no—
654C2—84no—
655C2—162no—
656C2#923714yesdirect product factor
657C2C2×#187911yescharacteristic
658C2C2×#189711yescharacteristic
659C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC2×#3100C2
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
2. C2×#2293C2semidirect
3. C2×#1962C2semidirect
4. C2×#1502C2semidirect
5. C2×#1486C2semidirect
6. #9237C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1276C2semidirect
14. #3100C4semidirect
17. C2×#474C4semidirect
18. C2×#459C4semidirect
29. #2293C22semidirect
37. #1962C22semidirect
39. #1502C22semidirect
42. #1486C22semidirect
44. #1303C22semidirect
45. #1276C22semidirect
97. #474C2×C4semidirect
99. #459C2×C4semidirect
286. C2×C42D8semidirect
492. C42C2×D8semidirect
656. C2#9237direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31002
C2×#22932
C2×#19622
C2×#15022
C2×#14862
#92372
C2×#13032
C2×#12762