Finite Group Databasebeta

← C2×#9305 #199165

Subgroups — C2×#9305 #199165

Order 512.

1516 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×#3036C211yesmaximal, characteristic
2C2×#2972C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6#9305C218yesmaximal, direct product factor
7C2×#1301C211yesmaximal, characteristic, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9C22×AGL(1,ℤ/8)—21no—
10C2×#532C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3036—22no—
13#3036—22no—
14#3036C414yescomplemented
15#3036C2214yes—
16C2×#463—22no—
17C2×#461C412yescomplemented
18C2×#459C412yescomplemented
19#2972—22no—
20#2972—22no—
21#2972—22no—
22#2972C2214yescomplemented
23C2×C8×D4—22no—
24C2×C8×D4—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#417—21no—
30C22×#102—21no—
31C22×C22⋊C8—22no—
32C2×#382C2211yescharacteristic
33#1804—22no—
34#1804C2214yescomplemented
35C2×#340—21no—
36#1517—22no—
37#1517C2214yescomplemented
38C2×#327—22no—
39#1480—22no—
40#1480C2214yescomplemented
41C2×D4⋊C8—22no—
42#1301C2214yescomplemented
43#1286C2214yescomplemented
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×AGL(1,ℤ/8)—24no—
48C2×AGL(1,ℤ/8)—24no—
49C2×AGL(1,ℤ/8)—44no—
50C2×AGL(1,ℤ/8)—44no—
51C22×SD16—22no—
52C22×D8—22no—
53C22×M16—21no—
54C23×C8—24no—
55C23×C8—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C412yes—
63#532C2312yes—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#127C2×C411yescharacteristic
66C2×#124—21no—
67C2×#121—21no—
68C2×#121—21no—
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×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×C4×D4C2312yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87#463—28no—
88#463—28no—
89#461—24no—
90#461C2×C418yescomplemented
91#459—24no—
92#459C2×C418yescomplemented
93C8×D4—28no—
94C8×D4—28no—
95C8×D4—28no—
96C8×D4—28no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431D412yes—
102#431D412yes—
103#431C2312yes—
104#431C2312yes—
105#423—21no—
106#423—21no—
107#423C2×C412yes—
108#423C2312yes—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2×C412yes—
118C2×C4⋊C8C2312yes—
119#417—24no—
120#417—24no—
121C2×#103—22no—
122C2×#102—22no—
123C2×#102—24no—
124C2×#102—44no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8—28no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—28no—
129C2×C22⋊C8D412yes—
130C2×C4×C8—21no—
131C2×C4×C8D412yes—
132#382—21no—
133#382—21no—
134#382—21no—
135#382C2312yes—
136C2×#95—21no—
137C2×#95—21no—
138#340—24no—
139#327—28no—
140D4⋊C8—28no—
141C25—21no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—41no—
160C22×D4C22⋊C412yes—
161C22×D4C22⋊C412yes—
162C22×D4D814yes—
163C22×D4SD1614yes—
164C22×D4C2×D412yes—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—84no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178AGL(1,ℤ/8)—416no—
179AGL(1,ℤ/8)—48no—
180AGL(1,ℤ/8)—48no—
181C2×SD16—28no—
182C2×SD16—48no—
183C2×D8—28no—
184C2×D8—48no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—42no—
188C2×M16—42no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—216no—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8—41no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—88no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C22×C412yes—
203#127—21no—
204#127—21no—
205#127—21no—
206#127C22×C412yes—
207#124—22no—
208#124—22no—
209#124—41no—
210#124—41no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121—41no—
216#121—41no—
217#121—41no—
218#121—41no—
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×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4C22⋊C414yes—
240C4×D4C22⋊C414yes—
241C4×D4C22×C414yes—
242C4×D4C2×D414yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C22⋊C412yes—
253C2×C4⋊C4C22⋊C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42C2411yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286C4⋊C8C22×C414yes—
287#103—28no—
288#102—28no—
289#102—48no—
290C22⋊C8—216no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8C2×D414yes—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8C2×D414yes—
299#95—22no—
300#95—22no—
301#95—41no—
302#95—41no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—41no—
309C24—82no—
310C24—82no—
311C24#12111yescharacteristic
312#41—42no—
313#41—42no—
314#41—42no—
315#41—42no—
316#41—44no—
317#41—44no—
318#41—82no—
319#41—82no—
320#41—82no—
321#41—82no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4#10218yes—
354C2×D4#10218yes—
355C2×D4C2×C22⋊C414yes—
356C2×D4C2×D814yes—
357C2×D4C2×SD1614yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4#12011yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12311yescharacteristic
397C22×C4#12311yescharacteristic
398C22×C4C2×D812yes—
399C22×C4C2×SD1612yes—
400C22×C4#13512yes—
401C22×C4#13512yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405SD16—416no—
406D8—416no—
407M16—42no—
408M16—42no—
409M16—44no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—44no—
414C2×C8—44no—
415C2×C8—44no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—816no—
420C2×C8—82no—
421C2×C8—82no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—82no—
432C4⋊C4C2×C22⋊C414yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C42C2×C22⋊C411yescharacteristic
442C42C2×C22⋊C411yescharacteristic
443C42C4×D411yescharacteristic
444C42C4×D411yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42#12011yescharacteristic
448C42#12011yescharacteristic
449C42#12111yescharacteristic
450C42#12111yescharacteristic
451C42#12311yescharacteristic
452C42#12311yescharacteristic
453C42C23×C411yescharacteristic
454C42C22×D411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C23—22no—
458C23—22no—
459C23—28no—
460C23—24no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—82no—
468C23—82no—
469C23—88no—
470C23—88no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23#38611yescharacteristic
477C23#38611yescharacteristic
478C23#46512yes—
479C23#46612yes—
480C23C2×#12111yescharacteristic
481Q8—42no—
482Q8—42no—
483Q8—82no—
484D4—24no—
485D4—28no—
486D4—28no—
487D4—24no—
488D4—24no—
489D4—44no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—82no—
494D4—82no—
495D4—84no—
496D4—82no—
497D4—82no—
498D4—82no—
499D4C2×#10218yes—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—164no—
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4#38611yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4#38611yescharacteristic
535C2×C4C2×#10212yes—
536C2×C4C2×#10212yes—
537C2×C4#41612yes—
538C2×C4#41612yes—
539C2×C4#46512yes—
540C2×C4#46512yes—
541C2×C4#46512yes—
542C2×C4#46612yes—
543C2×C4#46612yes—
544C2×C4#46612yes—
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×#12011yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4C22×D811yescharacteristic
552C2×C4C22×SD1611yescharacteristic
553C2×C4C2×#13511yescharacteristic
554C2×C4C2×#13511yescharacteristic
555C8—44no—
556C8—44no—
557C8—44no—
558C8—82no—
559C22—21no—
560C22—21no—
561C22—24no—
562C22—24no—
563C22—21no—
564C22—21no—
565C22—28no—
566C22—44no—
567C22—416no—
568C22—44no—
569C22—84no—
570C22—88no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—164no—
576C22#188112yes—
577C22#189312yes—
578C22#189312yes—
579C22#190412yes—
580C22C2×#38611yescharacteristic
581C22C2×#46511yescharacteristic
582C22C2×#46611yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—44no—
592C4—82no—
593C4—162no—
594C4#189312yes—
595C4#189312yes—
596C4C22×#10211yescharacteristic
597C4C2×#41611yescharacteristic
598C4C2×#46511yescharacteristic
599C4C2×#46611yescharacteristic
600C2—21no—
601C2—22no—
602C2—21no—
603C2—48no—
604C2—162no—
605C2#930514yesdirect product factor
606C2C2×#188111yescharacteristic
607C2C2×#189311yescharacteristic
608C2C2×#190411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
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×#2972C2semidirect
3. C2×#1804C2semidirect
4. C2×#1517C2semidirect
5. C2×#1480C2semidirect
6. #9305C2direct (N × H)
7. C2×#1301C2semidirect
8. C2×#1286C2semidirect
14. #3036C4semidirect
17. C2×#461C4semidirect
18. C2×#459C4semidirect
22. #2972C22semidirect
34. #1804C22semidirect
37. #1517C22semidirect
40. #1480C22semidirect
42. #1301C22semidirect
43. #1286C22semidirect
90. #461C2×C4semidirect
92. #459C2×C4semidirect
605. C2#9305direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30362
C2×#29722
C2×#18042
C2×#15172
C2×#14802
#93052
C2×#13012
C2×#12862