Finite Group Databasebeta

← C2×#9383 #200388

Subgroups — C2×#9383 #200388

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2C2×#2564C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4C2×#1520C212yesmaximal, complemented
5#9383C218yesmaximal, direct product factor
6C2×#1294C212yesmaximal, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2972—24no—
10#2972—22no—
11#2972C2214yes—
12C2×C8×D4—24no—
13C2×#431—21no—
14C22×#99—21no—
15C22×C22⋊C8—21no—
16C22×C22⋊C8C2212yes—
17#2564—22no—
18#2564—22no—
19#2564C414yescomplemented
20#2564C2214yes—
21C2×#400C411yescharacteristic, complemented
22C2×#399C411yescharacteristic, complemented
23C2×#395C411yescharacteristic, complemented
24C2×#393C411yescharacteristic, complemented
25C2×#387—21no—
26C2×#386—21no—
27C2×#385—21no—
28C2×#380—21no—
29C2×#380C2212yes—
30C22×#95C2211yescharacteristic
31#1799—22no—
32#1799C2214yes—
33C2×#342—21no—
34#1520—24no—
35#1520C2218yescomplemented
36C2×#328—24no—
37#1294C2218yescomplemented
38C23×Q8C2×C411yescharacteristic
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—22no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8C2312yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4D412yes—
59C2×C4×D4D412yes—
60C2×C4×D4C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42D411yescharacteristic
74C8×D4—216no—
75C8×D4—216no—
76#431—24no—
77#431—24no—
78C2×C4⋊C8—22no—
79C2×#99—24no—
80C2×#99—44no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8D414yes—
88C2×C22⋊C8D414yes—
89C2×C22⋊C8C2314yes—
90C2×C4×C8—22no—
91#400—22no—
92#400C2×C414yescomplemented
93#399—22no—
94#399C2×C414yescomplemented
95#395—22no—
96#395C2×C414yescomplemented
97#393—22no—
98#393C2×C414yescomplemented
99#387—24no—
100#387—24no—
101#386—24no—
102#386—24no—
103#385—24no—
104#385—24no—
105#380—24no—
106#380—24no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380C2314yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2312yes—
123#342—24no—
124#328—216no—
125C25C22⋊C411yescharacteristic
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—82no—
131C22×Q8C22×C412yes—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4C22⋊C412yes—
136C22×D4SD1614yes—
137C22×D4Q1614yes—
138C22×D4C2×D412yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4#4112yes—
168C23×C4#4112yes—
169C23×C4C2411yescharacteristic
170C22×C8—24no—
171C22×C8—28no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—84no—
175C22×C8—84no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22×C412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22⋊C412yes—
238C2×C42C2×D412yes—
239C4⋊C8—28no—
240#99—48no—
241C22⋊C8—28no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C22⋊C8C2×D418yes—
246C4×C8—28no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—24no—
256#95—42no—
257#95C22×C412yes—
258#95C22×C412yes—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24#10314yes—
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C412yes—
269C24C2×C22⋊C411yescharacteristic
270C24#12112yes—
271C24#12112yes—
272C24C2×SD1612yes—
273C24C2×Q1612yes—
274C24#13514yes—
275C24C22×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—84no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—84no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4#10318yes—
291C2×D4C2×SD1614yes—
292C2×D4C2×Q1614yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—82no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—82no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4—81no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—816no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4#10314yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D412yes—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12012yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12311yescharacteristic
367C22×C4#12311yescharacteristic
368C22×C4#12312yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4112yes—
373C2×C8—28no—
374C2×C8—48no—
375C2×C8—84no—
376C2×C8—84no—
377C2×C8—88no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—48no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—48no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—48no—
425C23—42no—
426C23—42no—
427C23—81no—
428C23—84no—
429C23—81no—
430C23#38612yes—
431C23#38611yescharacteristic
432C23#38611yescharacteristic
433C23C2×#10312yes—
434C23C2×#10314yes—
435C23C2×#10312yes—
436C23#41612yes—
437C23#41612yes—
438C23#46514yes—
439C23#46514yes—
440C23#46714yes—
441C23#46714yes—
442C23C22×C22⋊C411yescharacteristic
443C23C2×#12112yes—
444C23C22×SD1611yescharacteristic
445C23C22×Q1611yescharacteristic
446C23C2×#13512yes—
447Q8—84no—
448Q8—84no—
449D4—216no—
450D4—28no—
451D4—28no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—88no—
500C2×C4—88no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—168no—
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4#38612yes—
513C2×C4C2×#10314yes—
514C2×C4C2×C4×D412yes—
515C2×C4C2×#12011yescharacteristic
516C2×C4C2×#12311yescharacteristic
517C8—48no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—24no—
523C22—28no—
524C22—24no—
525C22—28no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—48no—
541C22—42no—
542C22—82no—
543C22—84no—
544C22#188412yes—
545C22#189212yes—
546C22#189214yes—
547C22#189212yes—
548C22#191312yes—
549C22C2×#38611yescharacteristic
550C22C22×#10311yescharacteristic
551C22C2×#41611yescharacteristic
552C22C2×#46512yes—
553C22C2×#46712yes—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—22no—
559C4—22no—
560C4—24no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4—82no—
567C4—164no—
568C2—21no—
569C2—22no—
570C2—21no—
571C2—22no—
572C2—24no—
573C2—22no—
574C2—44no—
575C2#938314yesdirect product factor
576C2C2×#188411yescharacteristic
577C2C2×#189211yescharacteristic
578C2C2×#191311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1520C2semidirect
5. #9383C2direct (N × H)
6. C2×#1294C2semidirect
19. #2564C4semidirect
21. C2×#400C4semidirect
22. C2×#399C4semidirect
23. C2×#395C4semidirect
24. C2×#393C4semidirect
35. #1520C22semidirect
37. #1294C22semidirect
92. #400C2×C4semidirect
94. #399C2×C4semidirect
96. #395C2×C4semidirect
98. #393C2×C4semidirect
575. C2#9383direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29722
C2×#25642
C2×#17992
C2×#15202
#93832
C2×#12942