Finite Group Databasebeta

← C2×#18535 #6348081

Subgroups — C2×#18535 #6348081

Order 512.

2960 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#3141C211yesmaximal, characteristic
3C2×#3112C211yesmaximal, characteristic
4C2×#3112C211yesmaximal, characteristic
5#18535C2116yesmaximal, direct product factor
6C2×#1636C214yesmaximal, complemented
7C2×#1626C214yesmaximal, complemented
8C2×#1556C211yesmaximal, characteristic, complemented
9C2×#1491C211yesmaximal, characteristic, complemented
10C2×#1491C211yesmaximal, characteristic, complemented
11C2×#557—22no—
12C2×#557—24no—
13C2×#557—22no—
14C2×#557—24no—
15C2×#557—22no—
16C2×#557—24no—
17C2×#557C2214yes—
18C2×#557C2218yescomplemented
19C2×#532C2211yescharacteristic
20C22×C4⋊Q8—21no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125—21no—
23C22×#125—22no—
24C22×C4×Q8—21no—
25C22×C4×Q8—21no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28#3141—24no—
29#3141C2218yes—
30C2×#489—22no—
31C2×#487C2212yes—
32C2×#486C2212yes—
33#3112—24no—
34#3112—24no—
35#3112C2218yes—
36#3112C2218yes—
37C2×#480—21no—
38C2×#480—21no—
39C2×#477C2212yes—
40C2×#477C2212yes—
41C2×#460C2212yes—
42C2×#460C2212yes—
43C2×#448—21no—
44C2×#448—21no—
45C2×#440—22no—
46C2×#440—22no—
47C2×#432C2211yescharacteristic
48C2×#432C2211yescharacteristic
49C2×#417—22no—
50C2×C4×M16C2211yescharacteristic
51#1636C22132yescomplemented
52#1626C22132yescomplemented
53#1556C2218yescomplemented
54C2×#332C2214yes—
55#1491C2218yescomplemented
56#1491C2218yescomplemented
57C2×Q8⋊C8C2214yes—
58C2×Q8⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—21no—
61C23×Q8D411yescharacteristic
62C23×Q8D411yescharacteristic
63C22×SD16—22no—
64C22×SD16—22no—
65C22×M16—21no—
66#557—216no—
67#557—216no—
68#557—216no—
69#557—28no—
70#557—216no—
71#557—28no—
72#557—28no—
73#557—216no—
74#557—216no—
75#557C23116yes—
76#532—22no—
77#532—22no—
78#532—22no—
79#532C2314yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—28no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8D412yes—
89C2×C4⋊Q8D414yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2314yes—
93C2×#127C2311yescharacteristic
94C2×#125—216no—
95C2×#125—28no—
96C2×#125—28no—
97C2×#125—24no—
98C2×#125—28no—
99C2×#125—24no—
100C2×#125—216no—
101C2×#125—28no—
102C2×#124—21no—
103C2×#121—21no—
104C2×#121—21no—
105C2×C4×Q8—28no—
106C2×C4×Q8—24no—
107C2×C4×Q8—28no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—28no—
111C2×C4×Q8—24no—
112C2×C4×Q8—28no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8D414yes—
122C2×C4×Q8D414yes—
123C2×C4×Q8D412yes—
124C2×C4×Q8D412yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4D411yescharacteristic
139C22×C4⋊C4D412yes—
140C22×C42—22no—
141C22×C42—21no—
142C22×C42—21no—
143C22×C42C2311yescharacteristic
144#489—216no—
145#487—24no—
146#487C2318yes—
147#486—24no—
148#486C2318yes—
149#480—28no—
150#480—28no—
151#477—24no—
152#477—24no—
153#477C2318yes—
154#477C2318yes—
155#460—24no—
156#460—24no—
157#460C2318yes—
158#460C2318yes—
159#448—28no—
160#448—28no—
161#440—216no—
162#440—216no—
163#432—22no—
164#432—22no—
165#432C2314yes—
166#432C2314yes—
167C2×#107—22no—
168C2×#107—22no—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#417—216no—
172C2×#103—22no—
173C2×#103—22no—
174C2×#103—24no—
175C2×#102—22no—
176C2×#102—22no—
177C2×#102—24no—
178C2×C22⋊C8—21no—
179C2×C22⋊C8—21no—
180C4×M16—22no—
181C4×M16C2314yes—
182C2×#97—21no—
183C2×#97—21no—
184C2×#97—21no—
185C2×C4×C8C2312yes—
186#332C23116yes—
187Q8⋊C8C23116yes—
188Q8⋊C8C23116yes—
189C2×#41—24no—
190C2×#41—48no—
191C2×#41—42no—
192C22×Q8—28no—
193C22×Q8—22no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8—44no—
203C22×Q8—48no—
204C22×Q8C2×D414yes—
205C22×Q8C2×D412yes—
206C22×Q8C2×D412yes—
207C22×Q8C2×D414yes—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—41no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—21no—
217C23×C4—41no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C2×SD16—28no—
222C2×SD16—28no—
223C2×SD16—48no—
224C2×SD16—48no—
225C2×M16—24no—
226C2×M16—44no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C4⋊Q8—216no—
231C4⋊Q8—232no—
232C4⋊Q8—216no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—24no—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8C2×D414yes—
244C4⋊Q8C2×D418yes—
245C4⋊Q8C2412yes—
246C4⋊Q8C2414yes—
247#127—21no—
248#127—21no—
249#127—21no—
250#127C2412yes—
251#125—232no—
252#125—264no—
253#125—216no—
254#125—232no—
255#125—232no—
256#125—216no—
257#124—24no—
258#124—42no—
259#121—24no—
260#121—24no—
261#121—42no—
262#121—42no—
263C4×Q8—216no—
264C4×Q8—232no—
265C4×Q8—216no—
266C4×Q8—232no—
267C4×Q8—216no—
268C4×Q8—216no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—22no—
283C4×Q8C2×D418yes—
284C4×Q8C2×D418yes—
285C4×Q8C2×D414yes—
286C4×Q8C2×D414yes—
287C4×Q8C2414yes—
288C4×Q8C2414yes—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—42no—
293C2×C4⋊C4—216no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—216no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—216no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D414yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D418yes—
325C2×C22⋊C4—41no—
326C2×C22⋊C4—41no—
327C2×C42—216no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345#107—28no—
346#107—28no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8C2414yes—
350C4⋊C8C2414yes—
351#103—28no—
352#103—28no—
353#103—216no—
354#102—216no—
355#102—28no—
356#102—28no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359#97—24no—
360#97—24no—
361#97—24no—
362C4×C8—22no—
363C4×C8C2414yes—
364C24—41no—
365C24#12011yescharacteristic
366#41—416no—
367#41—88no—
368C2×Q8—216no—
369C2×Q8—28no—
370C2×Q8—216no—
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—416no—
387C2×Q8—416no—
388C2×Q8—44no—
389C2×Q8—41no—
390C2×Q8—48no—
391C2×Q8#13718yes—
392C2×Q8C22×D414yes—
393C2×Q8C22×D414yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—84no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—41no—
424C22×C4—44no—
425C22×C4—84no—
426C22×C4#12012yes—
427C22×C4#12012yes—
428C22×C4#12012yes—
429C22×C4#12011yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12011yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438SD16—416no—
439SD16—416no—
440M16—48no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C4⋊C4—232no—
448C4⋊C4—232no—
449C4⋊C4—232no—
450C4⋊C4—28no—
451C4⋊C4—24no—
452C4⋊C4—28no—
453C4⋊C4—24no—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4C22×D418yes—
470C4⋊C4C22×D414yes—
471C4⋊C4C22×D414yes—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C42—232no—
475C42—28no—
476C42—24no—
477C42—28no—
478C42—24no—
479C42—24no—
480C42—24no—
481C42#12012yes—
482C42#12012yes—
483C42#12012yes—
484C42C22×D411yescharacteristic
485C42C22×D412yes—
486C42C22×D412yes—
487C42C22×D412yes—
488C42C22×D411yescharacteristic
489C42C22×D411yescharacteristic
490C42C2511yescharacteristic
491C23—22no—
492C23—22no—
493C23—42no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23#45414yes—
498C23C2×#12012yes—
499C23C2×#12011yescharacteristic
500Q8—216no—
501Q8—28no—
502Q8—24no—
503Q8—24no—
504Q8—24no—
505Q8—48no—
506Q8—416no—
507Q8—44no—
508Q8—44no—
509Q8—48no—
510Q8—48no—
511Q8—82no—
512Q8C2×#13718yes—
513D4—44no—
514D4—44no—
515D4—44no—
516D4—82no—
517D4—88no—
518D4—82no—
519D4—82no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—48no—
538C2×C4—416no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—88no—
547C2×C4—82no—
548C2×C4C2×#12012yes—
549C2×C4C2×#12012yes—
550C2×C4C2×#12012yes—
551C2×C4C2×#12011yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12011yescharacteristic
554C2×C4C2×#13714yes—
555C2×C4#57514yes—
556C2×C4C23×D411yescharacteristic
557C2×C4C23×D411yescharacteristic
558C2×C4C23×D411yescharacteristic
559C8—44no—
560C8—44no—
561C8—44no—
562C22—28no—
563C22—22no—
564C22—22no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—164no—
569C22#300412yes—
570C22#300512yes—
571C22C2×#45414yes—
572C22C2×#45412yes—
573C22C22×#12011yescharacteristic
574C4—28no—
575C4—22no—
576C4—22no—
577C4—21no—
578C4—22no—
579C4—21no—
580C4—44no—
581C4—48no—
582C4—44no—
583C4—44no—
584C4—82no—
585C4#300412yes—
586C4#300512yes—
587C4C22×#13712yes—
588C4C2×#57512yes—
589C2—24no—
590C2—162no—
591C2#1853514yesdirect product factor
592C2C2×#300411yescharacteristic
593C2C2×#300511yescharacteristic
594C2C22×#45411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
5. #18535C2direct (N × H)
6. C2×#1636C2semidirect
7. C2×#1626C2semidirect
8. C2×#1556C2semidirect
9. C2×#1491C2semidirect
10. C2×#1491C2semidirect
18. C2×#557C22semidirect
51. #1636C22semidirect
52. #1626C22semidirect
53. #1556C22semidirect
55. #1491C22semidirect
56. #1491C22semidirect
591. C2#18535direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#31412
C2×#31122
#185352
C2×#16362
C2×#16262
C2×#15562
C2×#14912