Finite Group Databasebeta

← C2×#9281 #198781

Subgroups — C2×#9281 #198781

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2C2×#2981C211yesmaximal, characteristic, complemented
3C2×#1505C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic, complemented
5C2×#1496C211yesmaximal, characteristic, complemented
6#9281C218yesmaximal, direct product factor
7C2×#1287C211yesmaximal, characteristic, complemented
8C2×#1274C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3207—22no—
12#3207—22no—
13#3207C414yescomplemented
14#3207C2214yes—
15C2×#493C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#463—22no—
20C2×#449—22no—
21#2981—22no—
22#2981—22no—
23#2981—22no—
24#2981C2214yescomplemented
25C2×#436—21no—
26C2×#435—22no—
27C2×C8×D4—21no—
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#408—22no—
33C2×#376C2211yescharacteristic
34#1505—22no—
35#1505—22no—
36#1505C2214yescomplemented
37#1505C2214yescomplemented
38C2×#327—22no—
39C2×#327—22no—
40#1496—22no—
41#1496C2214yescomplemented
42C2×D4⋊C8—21no—
43C2×D4⋊C8—21no—
44#1287C2214yescomplemented
45#1274C2214yescomplemented
46C22×#41—21no—
47C22×#41—21no—
48C22×SD16—22no—
49C22×D8—22no—
50C22×M16—22no—
51C23×C8—22no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C412yes—
63#532C2×C412yes—
64#532D412yes—
65#532D412yes—
66#532C2312yes—
67#532C2312yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×#127C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#124—21no—
73C2×#124—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121D412yes—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4D412yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#493—22no—
85#493C2×C414yescomplemented
86#492—22no—
87#492C2×C414yescomplemented
88#487—22no—
89#487C2×C414yescomplemented
90#486—22no—
91#486C2×C414yescomplemented
92#463—28no—
93#463—28no—
94#449—28no—
95#449—28no—
96#436—24no—
97#436—24no—
98#435—28no—
99#435—28no—
100C8×D4—24no—
101C8×D4—24no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432D412yes—
108#432C2312yes—
109#432C2312yes—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431C2×C412yes—
115#431D412yes—
116#431C2312yes—
117#431C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×#103—22no—
123C2×#102—22no—
124#408—28no—
125#408—28no—
126C2×C22⋊C8—22no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8D412yes—
132C2×#97—21no—
133C2×#97—21no—
134C2×C4×C8C2×C411yescharacteristic
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8D411yescharacteristic
137C2×C4×C8D411yescharacteristic
138#376—21no—
139#376—21no—
140#376—21no—
141#376C2312yes—
142C2×#95—21no—
143C2×#95—21no—
144#327—28no—
145#327—28no—
146D4⋊C8—24no—
147D4⋊C8—24no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—84no—
157C2×#41—84no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—41no—
167C22×D4C22⋊C412yes—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4—41no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×SD16—28no—
175C2×SD16—84no—
176C2×D8—28no—
177C2×D8—84no—
178C2×M16—28no—
179C2×M16—84no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—28no—
186C22×C8—41no—
187C22×C8—84no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22⋊C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196#127—21no—
197#127—21no—
198#127—21no—
199#127—21no—
200#127—21no—
201#127C22⋊C412yes—
202#127C22×C412yes—
203#127C22×C412yes—
204#124—22no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124—22no—
209#124—22no—
210#124—41no—
211#124—41no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—22no—
221#121—22no—
222#121—41no—
223#121—41no—
224#121—41no—
225#121—41no—
226#121C22⋊C414yes—
227#121C2×D414yes—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4C22⋊C414yes—
243C4×D4C2×D414yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—41no—
276#103—28no—
277#102—28no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287C22⋊C8C2×D414yes—
288#97—22no—
289#97—22no—
290#97—22no—
291#97—41no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8C22×C412yes—
297C4×C8C22×C412yes—
298C4×C8C2×D412yes—
299C4×C8C2×D412yes—
300#95—22no—
301#95—22no—
302#95—41no—
303#95—41no—
304C24—22no—
305C24—41no—
306C24#12111yescharacteristic
307#41—84no—
308#41—84no—
309#41—84no—
310#41—84no—
311#41—84no—
312#41—84no—
313#41—84no—
314#41—84no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—44no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4C2×C22⋊C414yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—84no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12011yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12311yescharacteristic
378C22×C4#12311yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382SD16—88no—
383D8—88no—
384M16—88no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—42no—
391C2×C8—44no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—88no—
397C2×C8—82no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4C2×C22⋊C414yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C42—24no—
415C42—44no—
416C42—42no—
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42C4×D411yescharacteristic
422C42C4×D411yescharacteristic
423C42#12011yescharacteristic
424C42#12011yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42C23×C411yescharacteristic
430C42C22×D411yescharacteristic
431C42C22×D411yescharacteristic
432C42C2×#4111yescharacteristic
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23#38611yescharacteristic
445C23#38611yescharacteristic
446C23C2×#12111yescharacteristic
447Q8—44no—
448Q8—82no—
449Q8—82no—
450Q8—82no—
451D4—44no—
452D4—44no—
453D4—44no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—84no—
465D4—84no—
466D4—84no—
467D4—84no—
468D4—82no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—88no—
500C2×C4—82no—
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38611yescharacteristic
505C2×C4#38611yescharacteristic
506C2×C4#38611yescharacteristic
507C2×C4#44312yes—
508C2×C4#44412yes—
509C2×C4C22×C22⋊C411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×#12011yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12311yescharacteristic
515C8—42no—
516C8—44no—
517C8—42no—
518C8—42no—
519C8—42no—
520C8—82no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—88no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—164no—
536C22#187612yes—
537C22#190412yes—
538C22#191012yes—
539C22C2×#38611yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—44no—
551C4—84no—
552C4—82no—
553C4#187612yes—
554C4C2×#44311yescharacteristic
555C4C2×#44411yescharacteristic
556C2—41no—
557C2—41no—
558C2—84no—
559C2—162no—
560C2#928114yesdirect product factor
561C2C2×#187611yescharacteristic
562C2C2×#190411yescharacteristic
563C2C2×#191011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
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
2. C2×#2981C2semidirect
3. C2×#1505C2semidirect
4. C2×#1505C2semidirect
5. C2×#1496C2semidirect
6. #9281C2direct (N × H)
7. C2×#1287C2semidirect
8. C2×#1274C2semidirect
13. #3207C4semidirect
15. C2×#493C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
24. #2981C22semidirect
36. #1505C22semidirect
37. #1505C22semidirect
41. #1496C22semidirect
44. #1287C22semidirect
45. #1274C22semidirect
85. #493C2×C4semidirect
87. #492C2×C4semidirect
89. #487C2×C4semidirect
91. #486C2×C4semidirect
560. C2#9281direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32072
C2×#29812
C2×#15052
C2×#14962
#92812
C2×#12872
C2×#12742