Finite Group Databasebeta

← C2×#9323 #199420

Subgroups — C2×#9323 #199420

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic, complemented
4C2×#1506C211yesmaximal, characteristic, complemented
5C2×#1490C211yesmaximal, characteristic
6#9323C218yesmaximal, direct product factor
7C2×#1300C211yesmaximal, characteristic, complemented
8C2×#1285C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4×D4—21no—
11C22×C4×D4C2211yescharacteristic
12#3110—22no—
13#3110—22no—
14#3110C414yescomplemented
15#3110C2214yes—
16C2×#484—21no—
17C2×#474C412yescomplemented
18C2×#461C412yescomplemented
19C2×#450—21no—
20C2×#440—22no—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2275—22no—
25#2275—22no—
26#2275—22no—
27#2275C2214yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—22no—
31C2×C4×C22⋊C4—21no—
32C2×#376C2211yescharacteristic
33C22×#95—21no—
34#1986—22no—
35#1986C2214yescomplemented
36C2×#339—22no—
37#1506—22no—
38#1506C2214yescomplemented
39C2×#328—21no—
40C2×#328—21no—
41#1490—22no—
42#1490C2214yes—
43C2×D4⋊C8—22no—
44#1300C2214yescomplemented
45#1285C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4—21no—
49C22×SD16—22no—
50C22×M16—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533C2×C412yescomplemented
58#533C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic, complemented
60C2×C4⋊Q8C2×C411yescharacteristic, complemented
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—216no—
67C2×C4×D4—24no—
68C2×C4×D4—24no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×C4×D4C2×C412yes—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4C2312yes—
81C2×#117—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42D412yes—
89C22×C42C2311yescharacteristic
90#484—24no—
91#484—24no—
92#474—24no—
93#474C2×C418yescomplemented
94#461—24no—
95#461C2×C418yescomplemented
96#450—24no—
97#450—24no—
98#440—28no—
99#440—28no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432C2×C412yescomplemented
106#432C2312yes—
107#432C2312yes—
108C2×#107—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2×C412yescomplemented
111C2×#103—22no—
112C2×#102—22no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C4×M16—21no—
116C4×M16—21no—
117C4×M16—21no—
118C4×M16C2312yes—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—24no—
133#376—21no—
134#376—21no—
135#376D412yes—
136#376C2312yes—
137C2×#95—22no—
138C2×#95—24no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—24no—
142C2×#95D411yescharacteristic
143C2×#95D411yescharacteristic
144#339—28no—
145#328—24no—
146#328—24no—
147D4⋊C8—28no—
148C25—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×D4—28no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C22⋊C412yes—
156C22×D4C22⋊C412yes—
157C22×D4C2×D412yes—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—216no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—84no—
171C23×C4—84no—
172C23×C4—84no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×SD16—28no—
177C2×SD16—48no—
178C2×M16—24no—
179C2×M16—22no—
180C2×M16—42no—
181C2×M16—41no—
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×C412yescomplemented
192C4⋊Q8C22×C412yescomplemented
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—232no—
214C4×D4—232no—
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×D4—22no—
226C4×D4—22no—
227C4×D4C22⋊C414yes—
228C4×D4C22⋊C414yes—
229C4×D4C22×C414yes—
230C4×D4C2×D414yes—
231#117—22no—
232#117—22no—
233#117—41no—
234#117—41no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—41no—
259C2×C42—28no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—28no—
263C2×C42—22no—
264C2×C42—41no—
265C2×C42—44no—
266C2×C42—84no—
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D414yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—41no—
284C4⋊C8C22×C414yescomplemented
285#103—28no—
286#102—28no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—41no—
291#97—22no—
292#97—22no—
293#97—22no—
294#97—22no—
295#97—41no—
296#97—41no—
297C4×C8—22no—
298C4×C8—41no—
299#95—28no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—21no—
304#95—22no—
305#95—22no—
306#95C2×D412yes—
307#95C2×D412yes—
308C24—28no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—44no—
313C24—42no—
314C24—42no—
315C24#12011yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—44no—
321C2×Q8—82no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4#10418yes—
332C2×D4C2×C22⋊C414yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—232no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4—832no—
368C22×C4—88no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12011yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12311yescharacteristic
383C22×C4#12311yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387SD16—416no—
388M16—24no—
389M16—44no—
390M16—42no—
391C2×C8—24no—
392C2×C8—44no—
393C2×C8—42no—
394C2×C8—44no—
395C2×C8—42no—
396C2×C8—82no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4C2×C22⋊C414yes—
415C22⋊C4—232no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C42—28no—
423C42—28no—
424C42—44no—
425C42—42no—
426C42—42no—
427C42—82no—
428C42—88no—
429C42C2×C22⋊C411yescharacteristic
430C42C2×C22⋊C411yescharacteristic
431C42C4×D411yescharacteristic
432C42C4×D411yescharacteristic
433C42C4×D411yescharacteristic
434C42C4×D411yescharacteristic
435C42#12111yescharacteristic
436C42#12111yescharacteristic
437C42#12111yescharacteristic
438C42#12111yescharacteristic
439C42#12311yescharacteristic
440C42#12311yescharacteristic
441C42C23×C411yescharacteristic
442C42C22×D411yescharacteristic
443C42C2×#4111yescharacteristic
444C42C2×#4111yescharacteristic
445C23—22no—
446C23—22no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—24no—
451C23—44no—
452C23—44no—
453C23—416no—
454C23—416no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23#38611yescharacteristic
461C23#38611yescharacteristic
462C23#45512yes—
463C23#45612yes—
464C23C2×#12011yescharacteristic
465Q8—44no—
466Q8—82no—
467Q8—82no—
468Q8—82no—
469D4—28no—
470D4—28no—
471D4—24no—
472D4—28no—
473D4C2×#10418yes—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—416no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—832no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#38611yescharacteristic
519C2×C4#38611yescharacteristic
520C2×C4#38611yescharacteristic
521C2×C4#38611yescharacteristic
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4C2×#10414yes—
525C2×C4C22×C22⋊C411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12311yescharacteristic
531C8—44no—
532C8—44no—
533C8—82no—
534C22—21no—
535C22—21no—
536C22—24no—
537C22—24no—
538C22—21no—
539C22—21no—
540C22—28no—
541C22—48no—
542C22—48no—
543C22—416no—
544C22—44no—
545C22—416no—
546C22—44no—
547C22#188212yes—
548C22#188212yes—
549C22#189612yes—
550C22#190412yes—
551C22C2×#38611yescharacteristic
552C22C2×#45511yescharacteristic
553C22C2×#45611yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—44no—
565C4—48no—
566C4—44no—
567C4—44no—
568C4—82no—
569C4—82no—
570C4#189612yes—
571C4C22×#10412yes—
572C2—21no—
573C2—21no—
574C2—22no—
575C2—48no—
576C2#932314yesdirect product factor
577C2C2×#188211yescharacteristic
578C2C2×#189611yescharacteristic
579C2C2×#190411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1986C2semidirect
4. C2×#1506C2semidirect
6. #9323C2direct (N × H)
7. C2×#1300C2semidirect
8. C2×#1285C2semidirect
14. #3110C4semidirect
17. C2×#474C4semidirect
18. C2×#461C4semidirect
35. #1986C22semidirect
38. #1506C22semidirect
44. #1300C22semidirect
45. #1285C22semidirect
57. #533C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
93. #474C2×C4semidirect
95. #461C2×C4semidirect
105. #432C2×C4semidirect
110. C2×C4⋊C8C2×C4semidirect
191. C4⋊Q8C22×C4semidirect
192. C4⋊Q8C22×C4semidirect
284. C4⋊C8C22×C4semidirect
576. C2#9323direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#22752
C2×#19862
C2×#15062
C2×#14902
#93232
C2×#13002
C2×#12852