Finite Group Databasebeta

← C2×#9180 #197220

Subgroups — C2×#9180 #197220

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3037C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5C2×#1479C211yesmaximal, characteristic, complemented
6#9180C218yesmaximal, direct product factor
7C2×#1303C211yesmaximal, characteristic, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9C22×#137—21no—
10C2×#532C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13#3037—22no—
14#3037—22no—
15#3037C414yescomplemented
16#3037C2214yes—
17C2×#464—22no—
18C2×#462C412yescomplemented
19C2×#460C412yescomplemented
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417—21no—
24#2351—22no—
25#2351—22no—
26#2351—22no—
27#2351C2214yescomplemented
28C2×#389—22no—
29C2×#389—21no—
30C2×#382—21no—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34#1517—22no—
35#1517C2214yes—
36#1502—22no—
37#1502C2214yes—
38C2×#327—21no—
39C2×#327—21no—
40C2×#327—22no—
41#1479—22no—
42#1479C2214yescomplemented
43C2×Q8⋊C8—22no—
44#1303C2214yescomplemented
45#1286C2214yescomplemented
46C22×#41—21no—
47C23×Q8D411yescharacteristic
48C2×#137—24no—
49C2×#137—24no—
50C2×#137—44no—
51C2×#137—44no—
52C22×Q16—22no—
53C22×SD16—22no—
54C22×M16—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#127C2×C411yescharacteristic
65C2×#124—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—44no—
77C2×C4×Q8—44no—
78C2×C4×Q8C2×C412yes—
79C2×C4×Q8C2×C412yes—
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×C4×Q8D412yes—
83C2×C4×Q8C2312yes—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4D411yescharacteristic
92C22×C42—21no—
93C22×C42—22no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#464—28no—
97#464—28no—
98#462—24no—
99#462C2×C418yescomplemented
100#460—24no—
101#460C2×C418yescomplemented
102#431—21no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2312yes—
109#431C2312yes—
110#423—21no—
111#423—21no—
112#423C2×C412yes—
113#423C2312yes—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2×C412yes—
118#417—24no—
119#417—24no—
120C2×#103—22no—
121C2×#102—22no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126#389—24no—
127#389—28no—
128#389—24no—
129#389—28no—
130#382—24no—
131#382—21no—
132#382—21no—
133#382—24no—
134#382D412yes—
135#382C2312yes—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4D412yes—
141C4×C4⋊C4C2312yes—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146C2×#95D412yes—
147#327—24no—
148#327—24no—
149#327—28no—
150Q8⋊C8—28no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—41no—
154C2×#41—41no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8C22⋊C412yes—
166C22×Q8C22⋊C412yes—
167C22×Q8D814yes—
168C22×Q8SD1614yes—
169C22×Q8C2×D412yes—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—41no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183#137—416no—
184#137—48no—
185#137—48no—
186C2×Q16—28no—
187C2×Q16—48no—
188C2×SD16—28no—
189C2×SD16—48no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—42no—
193C2×M16—42no—
194C22×C8—22no—
195C22×C8—41no—
196C22×C8—41no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22×C412yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C22×C412yes—
205#124—22no—
206#124—22no—
207#124—41no—
208#124—41no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—41no—
214#121—41no—
215#121—41no—
216#121—41no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—48no—
230C4×Q8—416no—
231C4×Q8—48no—
232C4×Q8C22⋊C414yes—
233C4×Q8C22⋊C414yes—
234C4×Q8C22×C414yes—
235C4×Q8C2×D414yes—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—41no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—41no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C22⋊C412yes—
267C2×C4⋊C4C22⋊C412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—48no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42C2411yescharacteristic
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8C22×C414yes—
303#103—28no—
304#102—28no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—41no—
312C4×C8—41no—
313#95—28no—
314#95—22no—
315#95—21no—
316#95—21no—
317#95—22no—
318#95—22no—
319#95C2×D412yes—
320#95C2×D412yes—
321#95C2×D414yes—
322C24—41no—
323C24#12011yescharacteristic
324#41—42no—
325#41—42no—
326#41—42no—
327#41—42no—
328#41—44no—
329#41—44no—
330#41—82no—
331#41—82no—
332#41—82no—
333#41—82no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8#10218yes—
352C2×Q8#10218yes—
353C2×Q8C2×C22⋊C414yes—
354C2×Q8C2×D814yes—
355C2×Q8C2×SD1614yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4#12011yescharacteristic
406C22×C4#12011yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12311yescharacteristic
412C22×C4#12311yescharacteristic
413C22×C4C2×D812yes—
414C22×C4C2×SD1612yes—
415C22×C4AGL(1,ℤ/8)12yes—
416C22×C4#13712yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420Q16—416no—
421SD16—416no—
422M16—42no—
423M16—42no—
424M16—44no—
425C2×C8—24no—
426C2×C8—44no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—82no—
430C2×C8—82no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—44no—
449C4⋊C4C2×C22⋊C414yes—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C42—24no—
455C42—24no—
456C42—24no—
457C42—24no—
458C42—416no—
459C42—42no—
460C42—42no—
461C42—44no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C2×#4111yescharacteristic
477C42C2×#4111yescharacteristic
478C23—22no—
479C23—22no—
480C23—42no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#38611yescharacteristic
485C23#38611yescharacteristic
486C23#44712yes—
487C23#44812yes—
488C23C2×#12011yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—24no—
492Q8—28no—
493Q8—28no—
494Q8—42no—
495Q8—42no—
496Q8—82no—
497Q8C2×#10218yes—
498D4—42no—
499D4—42no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504D4—84no—
505D4—82no—
506D4—82no—
507D4—82no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—88no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4#38611yescharacteristic
547C2×C4#38611yescharacteristic
548C2×C4#38611yescharacteristic
549C2×C4#38611yescharacteristic
550C2×C4#38611yescharacteristic
551C2×C4#38611yescharacteristic
552C2×C4C2×#10212yes—
553C2×C4C2×#10212yes—
554C2×C4#41712yes—
555C2×C4#41712yes—
556C2×C4#44712yes—
557C2×C4#44812yes—
558C2×C4#48012yes—
559C2×C4#48012yes—
560C2×C4#48112yes—
561C2×C4#48112yes—
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4C22×D811yescharacteristic
569C2×C4C22×SD1611yescharacteristic
570C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
571C2×C4C2×#13711yescharacteristic
572C8—44no—
573C8—82no—
574C8—82no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—24no—
580C22—24no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—164no—
585C22#187412yes—
586C22#187412yes—
587C22#190512yes—
588C22#190612yes—
589C22C2×#38611yescharacteristic
590C22C2×#44711yescharacteristic
591C22C2×#44811yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—28no—
601C4—44no—
602C4—44no—
603C4—84no—
604C4—82no—
605C4—82no—
606C4#187412yes—
607C4#187412yes—
608C4C22×#10211yescharacteristic
609C4C2×#41711yescharacteristic
610C4C2×#48011yescharacteristic
611C4C2×#48111yescharacteristic
612C2—21no—
613C2—22no—
614C2—21no—
615C2—162no—
616C2#918014yesdirect product factor
617C2C2×#187411yescharacteristic
618C2C2×#190511yescharacteristic
619C2C2×#190611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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×#2351C2semidirect
5. C2×#1479C2semidirect
6. #9180C2direct (N × H)
7. C2×#1303C2semidirect
8. C2×#1286C2semidirect
15. #3037C4semidirect
18. C2×#462C4semidirect
19. C2×#460C4semidirect
27. #2351C22semidirect
42. #1479C22semidirect
44. #1303C22semidirect
45. #1286C22semidirect
99. #462C2×C4semidirect
101. #460C2×C4semidirect
616. C2#9180direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#30372
C2×#23512
C2×#15172
C2×#15022
C2×#14792
#91802
C2×#13032
C2×#12862