Finite Group Databasebeta

← C2×#9235 #198017

Subgroups — C2×#9235 #198017

Order 512.

1524 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic, complemented
4C2×#1503C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#9235C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1276C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C22×#117—21no—
12#3101—22no—
13#3101—22no—
14#3101C414yescomplemented
15#3101C2214yes—
16C2×#483—21no—
17C2×#476C412yescomplemented
18C2×#461C412yescomplemented
19C2×#450—21no—
20C2×C4×SD16—22no—
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2293—22no—
25#2293—22no—
26#2293—22no—
27#2293C2214yes—
28C2×#390—21no—
29C2×#389—21no—
30C2×#388—22no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33C2×C43C2211yescharacteristic
34#1994—22no—
35#1994C2214yescomplemented
36C2×#339—21no—
37C2×#339—21no—
38#1503—22no—
39#1503C2214yescomplemented
40C2×#328—22no—
41#1486—22no—
42#1486C2214yes—
43C2×D4⋊C8—22no—
44#1303C2214yescomplemented
45#1276C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×SD16—22no—
49C22×M16—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533C2312yes—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—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×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C2×C4×D4D412yes—
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4C2312yes—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117—21no—
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#483—24no—
95#483—24no—
96#476—24no—
97#476C2×C418yescomplemented
98#461—24no—
99#461C2×C418yescomplemented
100#450—24no—
101#450—24no—
102C4×SD16—28no—
103C4×SD16—28no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431C2×C412yes—
110#431C2312yes—
111#431C2312yes—
112C2×#106—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×#103—22no—
122C2×#102—22no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127#390—24no—
128#390—24no—
129#389—24no—
130#389—24no—
131#388—28no—
132#388—28no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4D412yes—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C2×#95—22no—
140C2×#95D412yes—
141C43—21no—
142C43—21no—
143C43D412yes—
144C43C2312yes—
145#339—24no—
146#339—24no—
147#328—28no—
148D4⋊C8—28no—
149C25—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—44no—
156C22×D4C22⋊C412yes—
157C22×D4C22⋊C412yes—
158C22×D4C2×D412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×SD16—28no—
177C2×SD16—48no—
178C2×M16—22no—
179C2×M16—41no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—22no—
183C22×C8—41no—
184C22×C8—41no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193#125—22no—
194#125—22no—
195#125—41no—
196#125—41no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C22⋊C414yes—
226C4×D4C22⋊C414yes—
227C4×D4C22×C414yes—
228C4×D4C2×D414yes—
229#117—22no—
230#117—22no—
231#117—48no—
232#117—416no—
233#117—41no—
234#117—41no—
235#117—48no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—41no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—41no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42SD1614yescomplemented
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42C2411yescharacteristic
295#106—28no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303C4⋊C8C22×C414yes—
304#103—28no—
305#102—28no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—41no—
314#95—28no—
315#95—22no—
316#95C2×D414yes—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—44no—
321C24—44no—
322C24—42no—
323C24—42no—
324C24#12011yescharacteristic
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—44no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4C2×C22⋊C414yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—44no—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12311yescharacteristic
395C22×C4#12311yescharacteristic
396C22×C4C2×SD1612yes—
397C22×C4#13512yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4#13712yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403SD16—416no—
404M16—42no—
405M16—42no—
406M16—82no—
407C2×C8—24no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—82no—
412C2×C8—82no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—416no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4C2×C22⋊C414yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C42—22no—
441C42—24no—
442C42—22no—
443C42—22no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42—42no—
454C42C2×C22⋊C411yescharacteristic
455C42C2×C22⋊C411yescharacteristic
456C42C4×D411yescharacteristic
457C42C4×D411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42#12111yescharacteristic
461C42#12111yescharacteristic
462C42#12111yescharacteristic
463C42#12111yescharacteristic
464C42#12311yescharacteristic
465C42#12311yescharacteristic
466C42C2×SD1614yescomplemented
467C42C23×C411yescharacteristic
468C42C22×D411yescharacteristic
469C42C2×#4111yescharacteristic
470C42C2×#4111yescharacteristic
471C23—22no—
472C23—22no—
473C23—24no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—88no—
485C23—88no—
486C23#38611yescharacteristic
487C23#38611yescharacteristic
488C23#45012yes—
489C23#45212yes—
490C23C2×#12011yescharacteristic
491Q8—44no—
492Q8—82no—
493Q8—82no—
494Q8—82no—
495D4—24no—
496D4—24no—
497D4—44no—
498D4—44no—
499D4—44no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—88no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4#42114yes—
545C2×C4C4×SD1612yes—
546C2×C4C4×SD1612yes—
547C2×C4#44012yes—
548C2×C4#44012yes—
549C2×C4#45012yes—
550C2×C4#45212yes—
551C2×C4#46112yes—
552C2×C4#46112yes—
553C2×C4#46312yes—
554C2×C4#46312yes—
555C2×C4C22×C22⋊C411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×#12111yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4C22×SD1611yescharacteristic
562C2×C4C2×#13511yescharacteristic
563C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
564C2×C4C2×#13711yescharacteristic
565C8—44no—
566C8—82no—
567C8—82no—
568C22—21no—
569C22—21no—
570C22—24no—
571C22—24no—
572C22—21no—
573C22—21no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—84no—
578C22—84no—
579C22—88no—
580C22—88no—
581C22#187712yes—
582C22#187712yes—
583C22#189712yes—
584C22#190612yes—
585C22C2×#38611yescharacteristic
586C22C2×#45011yescharacteristic
587C22C2×#45211yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4#187712yes—
601C4#187712yes—
602C4#189712yes—
603C4C2×#42112yes—
604C4C2×C4×SD1611yescharacteristic
605C4C2×#44011yescharacteristic
606C4C2×#46111yescharacteristic
607C4C2×#46311yescharacteristic
608C2—21no—
609C2—21no—
610C2—22no—
611C2—84no—
612C2#923514yesdirect product factor
613C2C2×#187711yescharacteristic
614C2C2×#189711yescharacteristic
615C2C2×#190611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23#386
prime radicalC23×D4D4
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
3. C2×#1994C2semidirect
4. C2×#1503C2semidirect
6. #9235C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1276C2semidirect
14. #3101C4semidirect
17. C2×#476C4semidirect
18. C2×#461C4semidirect
35. #1994C22semidirect
39. #1503C22semidirect
44. #1303C22semidirect
45. #1276C22semidirect
97. #476C2×C4semidirect
99. #461C2×C4semidirect
280. C2×C42SD16semidirect
466. C42C2×SD16semidirect
612. C2#9235direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#22932
C2×#19942
C2×#15032
C2×#14862
#92352
C2×#13032
C2×#12762