Finite Group Databasebeta

← C2×#9315 #199320

Subgroups — C2×#9315 #199320

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#2988C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1806C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic, complemented
6#9315C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1292C211yesmaximal, characteristic, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#3079—22no—
12#3079—22no—
13#3079C414yescomplemented
14#3079C2214yes—
15C2×#477—21no—
16C2×#475C412yescomplemented
17C2×#476—21no—
18C2×#474C412yescomplemented
19#2988—22no—
20#2988—22no—
21#2988—22no—
22#2988C2214yes—
23C2×C8×Q8—21no—
24C2×C8×D4—22no—
25C2×C8×D4—21no—
26C2×#431—21no—
27C2×#431C2211yescharacteristic
28C2×#428—21no—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8—21no—
32C22×C4⋊C8C2211yescharacteristic
33C22×#104—21no—
34C2×#417—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1987—22no—
37#1987C2214yescomplemented
38#1806—22no—
39#1806C2214yes—
40C2×#340—22no—
41C2×#339—21no—
42#1548—22no—
43#1548C2214yescomplemented
44C2×#333—21no—
45C2×#333—21no—
46#1303C2214yescomplemented
47#1292C2214yescomplemented
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C22×M16—21no—
52C23×C8—22no—
53C23×C8D412yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2312yes—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41C2×C412yes—
71C4×#41D412yes—
72C4×#41D412yes—
73C4×#41C2312yes—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117D411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#477—24no—
89#477—24no—
90#476—24no—
91#476—24no—
92#475—24no—
93#475C2×C418yescomplemented
94#474—24no—
95#474C2×C418yescomplemented
96C8×Q8—24no—
97C8×Q8—24no—
98C8×D4—28no—
99C8×D4—24no—
100C8×D4—28no—
101C8×D4—24no—
102#431—24no—
103#431—21no—
104#431—21no—
105#431—24no—
106#431D412yes—
107#431C2312yes—
108#428—24no—
109#428—24no—
110C2×#107—22no—
111C2×#106—22no—
112#423—21no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423—21no—
117#423C2×C412yes—
118#423C2312yes—
119#423C2312yes—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—24no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—24no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8C2×C412yes—
131C2×C4⋊C8D412yes—
132C2×C4⋊C8D411yescharacteristic
133C2×C4⋊C8D411yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×#104—24no—
136C2×#104—44no—
137#417—24no—
138#417—24no—
139C2×#103—22no—
140C2×#102—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8—21no—
143C2×C4×C8—22no—
144C2×C4×C8—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2312yes—
149C2×#95—21no—
150#340—28no—
151#339—24no—
152#333—24no—
153#333—24no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41C22⋊C412yes—
159C2×#41SD1614yes—
160C2×#41Q1614yes—
161C2×#41C2×D412yes—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8—84no—
169C22×Q8C22⋊C411yescharacteristic
170C22×D4—21no—
171C22×D4C22⋊C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—41no—
176C23×C4—41no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4111yescharacteristic
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—41no—
186C2×M16—42no—
187C2×M16—42no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—28no—
193C22×C8—22no—
194C22×C8—44no—
195C22×C8—41no—
196C22×C8—84no—
197C22×C8C2×D414yes—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—41no—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213C4⋊Q8C22×C414yes—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8C22⋊C412yes—
219C4×Q8C22⋊C412yes—
220C4×Q8C22×C412yes—
221C4×Q8C2×D412yes—
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—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4C22⋊C412yes—
235C4×D4C22⋊C412yes—
236C4×D4C22×C412yes—
237C4×D4C2×D412yes—
238#117—21no—
239#117—21no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117C22⋊C412yes—
245#117C2×D412yes—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—41no—
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42#4111yescharacteristic
289C2×C42#4111yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42C2411yescharacteristic
292#107—28no—
293#106—28no—
294C4⋊C8—28no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—21no—
299C4⋊C8—22no—
300C4⋊C8—21no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—41no—
307C4⋊C8—41no—
308C4⋊C8C22×C414yes—
309C4⋊C8C2×D412yes—
310C4⋊C8C2×D412yes—
311#104—48no—
312#103—28no—
313#102—28no—
314C22⋊C8—28no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C4×C8—28no—
318C4×C8—22no—
319C4×C8—22no—
320#95—22no—
321#95—41no—
322C24—21no—
323C24#12111yescharacteristic
324#41—22no—
325#41—22no—
326#41—24no—
327#41—24no—
328#41—24no—
329#41—24no—
330#41—24no—
331#41#10318yes—
332#41C2×SD1614yes—
333#41C2×Q1614yes—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8#10314yes—
347C2×Q8C2×C22⋊C412yes—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4#10314yes—
355C2×D4C2×C22⋊C412yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—84no—
380C22×C4—82no—
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#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12311yescharacteristic
395C22×C4#12311yescharacteristic
396C22×C4C2×SD1612yes—
397C22×C4C2×Q1612yes—
398C22×C4#13512yes—
399C22×C4#13512yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403M16—42no—
404M16—42no—
405M16—44no—
406M16—42no—
407M16—42no—
408M16—82no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—44no—
418C2×C8—42no—
419C2×C8—82no—
420C2×C8—88no—
421C2×C8—82no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4C2×C22⋊C412yes—
440C4⋊C4C2×C22⋊C412yes—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—42no—
454C42—42no—
455C42—82no—
456C42C2×C22⋊C411yescharacteristic
457C42C2×C22⋊C411yescharacteristic
458C42C4×D411yescharacteristic
459C42C4×D411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42#12011yescharacteristic
463C42#12011yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42#12311yescharacteristic
467C42#12311yescharacteristic
468C42C23×C411yescharacteristic
469C42C22×D411yescharacteristic
470C42C22×D411yescharacteristic
471C42C2×#4111yescharacteristic
472C23—22no—
473C23—22no—
474C23—24no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23#38611yescharacteristic
479C23#38611yescharacteristic
480C23#46512yes—
481C23#46712yes—
482C23C2×#12111yescharacteristic
483Q8—22no—
484Q8—24no—
485Q8—84no—
486Q8—84no—
487Q8—84no—
488Q8—84no—
489Q8C2×#10314yes—
490D4—28no—
491D4—24no—
492D4—24no—
493D4—22no—
494D4—24no—
495D4C2×#10314yes—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—88no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4#38611yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4#38611yescharacteristic
535C2×C4#38611yescharacteristic
536C2×C4#38611yescharacteristic
537C2×C4C2×#10312yes—
538C2×C4C2×#10312yes—
539C2×C4#41612yes—
540C2×C4#41612yes—
541C2×C4#45512yes—
542C2×C4#45612yes—
543C2×C4#46512yes—
544C2×C4#46512yes—
545C2×C4#46512yes—
546C2×C4#46712yes—
547C2×C4#46712yes—
548C2×C4#46712yes—
549C2×C4C22×C22⋊C411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4C22×SD1611yescharacteristic
556C2×C4C22×Q1611yescharacteristic
557C2×C4C2×#13511yescharacteristic
558C2×C4C2×#13511yescharacteristic
559C8—44no—
560C8—44no—
561C8—42no—
562C8—42no—
563C8—82no—
564C22—21no—
565C22—21no—
566C22—24no—
567C22—24no—
568C22—21no—
569C22—21no—
570C22—24no—
571C22—42no—
572C22—48no—
573C22—42no—
574C22#188212yes—
575C22#189212yes—
576C22#189212yes—
577C22#190312yes—
578C22C2×#38611yescharacteristic
579C22C2×#46511yescharacteristic
580C22C2×#46711yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—24no—
588C4—42no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4#188212yes—
593C4#189212yes—
594C4#189212yes—
595C4C22×#10311yescharacteristic
596C4C2×#41611yescharacteristic
597C4C2×#45511yescharacteristic
598C4C2×#45611yescharacteristic
599C4C2×#46511yescharacteristic
600C4C2×#46711yescharacteristic
601C2—21no—
602C2—22no—
603C2—21no—
604C2—44no—
605C2#931514yesdirect product factor
606C2C2×#188211yescharacteristic
607C2C2×#189211yescharacteristic
608C2C2×#190311yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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×#1987C2semidirect
5. C2×#1548C2semidirect
6. #9315C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1292C2semidirect
13. #3079C4semidirect
16. C2×#475C4semidirect
18. C2×#474C4semidirect
37. #1987C22semidirect
43. #1548C22semidirect
46. #1303C22semidirect
47. #1292C22semidirect
93. #475C2×C4semidirect
95. #474C2×C4semidirect
605. C2#9315direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#29882
C2×#19872
C2×#18062
C2×#15482
#93152
C2×#13032
C2×#12922