Finite Group Databasebeta

← C2×#18776 #6348692

Subgroups — C2×#18776 #6348692

Order 512.

4142 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#3226C211yesmaximal, characteristic, complemented
3C2×#3100C211yesmaximal, characteristic
4C2×#3037C211yesmaximal, characteristic, complemented
5#18776C2116yesmaximal, direct product factor
6C2×#1649C214yesmaximal, complemented
7C2×#1627C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C24×Q8—21no—
12C22×#137—21no—
13C2×D4×Q8—22no—
14C2×D4×Q8—24no—
15C2×D4×Q8—22no—
16C2×D4×Q8—24no—
17C2×D4×Q8—22no—
18C2×D4×Q8—24no—
19C2×D4×Q8C2214yes—
20C2×D4×Q8C2218yescomplemented
21C2×#532C2211yescharacteristic
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2211yescharacteristic
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8—22no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4—21no—
28C22×C4×D4C2211yescharacteristic
29#3226—24no—
30#3226C2218yescomplemented
31C2×#500C2212yescomplemented
32C2×#486C2212yescomplemented
33#3100—24no—
34#3100C2218yes—
35C2×#480—21no—
36C2×#480—22no—
37C2×#474—22no—
38C2×#474C2212yes—
39#3037—24no—
40#3037C2218yescomplemented
41C2×#464—22no—
42C2×#462C2212yescomplemented
43C2×#460C2212yescomplemented
44C2×#459C2212yes—
45C2×#447—21no—
46C2×C4×D8—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1649C22132yescomplemented
52#1627C22132yescomplemented
53#1546C2218yescomplemented
54C2×#332C2214yescomplemented
55#1486C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—22no—
61C23×Q8—28no—
62C23×Q8—24no—
63C23×Q8—24no—
64C23×Q8—28no—
65C23×Q8—21no—
66C23×Q8—48no—
67C23×Q8—48no—
68C23×Q8D411yescharacteristic
69C23×D4D411yescharacteristic
70C24×C4—22no—
71C24×C4—21no—
72C2×#137—28no—
73C2×#137—48no—
74C22×Q16—22no—
75C22×SD16—22no—
76C22×D8—22no—
77C22×M16—21no—
78D4×Q8—216no—
79D4×Q8—216no—
80D4×Q8—216no—
81D4×Q8—28no—
82D4×Q8—216no—
83D4×Q8—28no—
84D4×Q8—216no—
85D4×Q8—28no—
86D4×Q8—216no—
87D4×Q8C23116yes—
88#532—22no—
89#532—22no—
90#532—22no—
91#532C2314yes—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8D414yes—
101C2×C4⋊Q8D412yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2314yes—
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—216no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—216no—
115C2×#121—21no—
116C2×#121—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8D412yes—
122C2×C4×Q8D414yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2314yes—
125C2×C4×D4—28no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×C4×D4—28no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4—22no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4D414yes—
136C2×C4×D4D412yes—
137C2×C4×D4C2314yes—
138C2×C4×D4C2312yes—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4D412yes—
144C22×C4⋊C4D411yescharacteristic
145C22×C22⋊C4—22no—
146C22×C22⋊C4—21no—
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149#500—24no—
150#500C2318yescomplemented
151#486—24no—
152#486C2318yescomplemented
153#480—28no—
154#480—216no—
155#474—216no—
156#474—24no—
157#474C2318yes—
158#464—216no—
159#462—24no—
160#462C2318yescomplemented
161#460—24no—
162#460C2318yescomplemented
163#459—24no—
164#459C2318yes—
165#447—28no—
166C4×D8—216no—
167#431—22no—
168#431—22no—
169#431C2314yes—
170#431C2314yes—
171C2×#107—24no—
172C2×#107—22no—
173#423—22no—
174#423C2314yes—
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179#417—28no—
180C2×#103—22no—
181C2×#102—22no—
182C2×#102—24no—
183C2×#102—22no—
184C2×#102—22no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—21no—
187C2×C4×C8—21no—
188C2×C4×C8C2312yes—
189#332C23116yescomplemented
190Q8⋊C8C23116yescomplemented
191D4⋊C8C23116yes—
192C25—21no—
193C2×#41—24no—
194C2×#41—42no—
195C2×#41—44no—
196C2×#41—44no—
197C22×Q8—216no—
198C22×Q8—28no—
199C22×Q8—216no—
200C22×Q8—232no—
201C22×Q8—216no—
202C22×Q8—22no—
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—416no—
211C22×Q8—416no—
212C22×Q8—432no—
213C22×Q8—432no—
214C22×Q8—464no—
215C22×Q8—416no—
216C22×Q8—432no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8D818yescomplemented
220C22×Q8C2×D412yes—
221C22×Q8C2×D414yes—
222C22×D4—28no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4—41no—
229C22×D4C2×D414yes—
230C22×D4C2×D412yes—
231C23×C4—21no—
232C23×C4—216no—
233C23×C4—28no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—416no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—41no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248#137—432no—
249C2×Q16—28no—
250C2×Q16—48no—
251C2×SD16—28no—
252C2×SD16—48no—
253C2×D8—28no—
254C2×D8—48no—
255C2×M16—24no—
256C2×M16—44no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C4⋊Q8—216no—
261C4⋊Q8—232no—
262C4⋊Q8—216no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8C2×D418yes—
274C4⋊Q8C2×D414yes—
275C4⋊Q8C2412yes—
276C4⋊Q8C2414yes—
277#127—21no—
278#127—21no—
279#127—21no—
280#127C2412yes—
281#124—24no—
282#124—42no—
283C22⋊Q8—216no—
284C22⋊Q8—232no—
285C22⋊Q8—232no—
286C22⋊Q8—264no—
287C22⋊Q8—216no—
288C22⋊Q8—232no—
289#121—24no—
290#121—24no—
291#121—42no—
292#121—42no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—22no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—22no—
300C4×Q8C2×D414yes—
301C4×Q8C2×D418yes—
302C4×Q8C2414yes—
303C4×D4—216no—
304C4×D4—232no—
305C4×D4—216no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—22no—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—42no—
316C4×D4—42no—
317C4×D4C2×D418yes—
318C4×D4C2×D414yes—
319C4×D4C2414yes—
320C2×C4⋊C4—216no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D414yes—
342C2×C4⋊C4C2×D414yes—
343C2×C4⋊C4C2×D418yes—
344C2×C22⋊C4—216no—
345C2×C22⋊C4—28no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—41no—
351C2×C22⋊C4—41no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2412yes—
364#107—28no—
365#107—216no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—22no—
369C4⋊C8—22no—
370C4⋊C8C2414yes—
371C4⋊C8C2414yes—
372#103—28no—
373#102—28no—
374#102—28no—
375#102—28no—
376#102—216no—
377C22⋊C8—24no—
378C22⋊C8—24no—
379C4×C8—24no—
380C4×C8—22no—
381C4×C8C2414yes—
382C24—28no—
383C24—24no—
384C24—22no—
385C24—44no—
386C24—44no—
387C24—41no—
388C24#12011yescharacteristic
389#41—48no—
390#41—48no—
391#41—88no—
392C2×Q8—232no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—28no—
399C2×Q8—24no—
400C2×Q8—28no—
401C2×Q8—216no—
402C2×Q8—22no—
403C2×Q8—432no—
404C2×Q8—464no—
405C2×Q8—464no—
406C2×Q8—4128no—
407C2×Q8—432no—
408C2×Q8—432no—
409C2×Q8—41no—
410C2×Q8—44no—
411C2×Q8—48no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—416no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8C2×D818yes—
419C2×Q8C2×D8116yescomplemented
420C2×Q8C22×D414yes—
421C2×D4—28no—
422C2×D4—216no—
423C2×D4—216no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—84no—
439C2×D4#13718yescomplemented
440C2×D4C22×D414yes—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—232no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—432no—
457C22×C4—464no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—41no—
468C22×C4—44no—
469C22×C4—84no—
470C22×C4#12012yes—
471C22×C4#12011yescharacteristic
472C22×C4#12011yescharacteristic
473C22×C4#12012yes—
474C22×C4#12012yes—
475C22×C4#12011yescharacteristic
476C22×C4C2×D814yes—
477C22×C4AGL(1,ℤ/8)14yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484Q16—416no—
485SD16—416no—
486D8—416no—
487M16—48no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—44no—
492C2×C8—44no—
493C2×C8—44no—
494C4⋊C4—232no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4C22×D414yes—
512C4⋊C4C22×D414yes—
513C4⋊C4C22×D418yes—
514C22⋊C4—232no—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C42—28no—
521C42—24no—
522C42—24no—
523C42#12012yes—
524C42#12012yes—
525C42#12012yes—
526C42C22×D412yes—
527C42C22×D411yescharacteristic
528C42C22×D412yes—
529C42C22×D412yes—
530C42C22×D411yescharacteristic
531C42C22×D411yescharacteristic
532C42C2511yescharacteristic
533C23—22no—
534C23—22no—
535C23—216no—
536C23—24no—
537C23—28no—
538C23—432no—
539C23—48no—
540C23—48no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23#44714yes—
548C23C2×#12012yes—
549C23C2×#12011yescharacteristic
550Q8—28no—
551Q8—216no—
552Q8—24no—
553Q8—464no—
554Q8—416no—
555Q8—48no—
556Q8—44no—
557Q8—44no—
558Q8—42no—
559Q8—42no—
560Q8—82no—
561Q8C22×D818yes—
562D4—216no—
563D4—28no—
564D4—24no—
565D4—44no—
566D4—42no—
567D4—42no—
568D4—44no—
569D4—82no—
570D4—88no—
571D4—82no—
572D4—82no—
573D4C2×#13718yescomplemented
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—216no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—464no—
588C2×C4—48no—
589C2×C4—416no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—416no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—88no—
601C2×C4#44714yes—
602C2×C4#44718yes—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12011yescharacteristic
606C2×C4C2×#12012yes—
607C2×C4C2×#12011yescharacteristic
608C2×C4C2×#12011yescharacteristic
609C2×C4C22×D812yes—
610C2×C4C22×D814yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)14yes—
613C2×C4C2×#13714yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C23×D411yescharacteristic
616C2×C4C23×D411yescharacteristic
617C8—44no—
618C8—44no—
619C8—44no—
620C22—28no—
621C22—22no—
622C22—22no—
623C22—28no—
624C22—432no—
625C22—416no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—164no—
632C22C2×#44714yes—
633C22C2×#44712yes—
634C22#300412yes—
635C22#302012yes—
636C22C22×#12011yescharacteristic
637C4—28no—
638C4—22no—
639C4—21no—
640C4—22no—
641C4—21no—
642C4—48no—
643C4—44no—
644C4—44no—
645C4—82no—
646C4C2×#44714yes—
647C4C2×#44712yes—
648C4#300412yes—
649C4C23×D811yescharacteristic
650C4C22×AGL(1,ℤ/8)11yescharacteristic
651C4C22×#13712yes—
652C2—24no—
653C2—48no—
654C2—162no—
655C2#1877614yesdirect product factor
656C2C22×#44711yescharacteristic
657C2C2×#300411yescharacteristic
658C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3100C2
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×D4×Q8C2semidirect
2. C2×#3226C2semidirect
4. C2×#3037C2semidirect
5. #18776C2direct (N × H)
6. C2×#1649C2semidirect
7. C2×#1627C2semidirect
8. C2×#1546C2semidirect
9. C2×#1486C2semidirect
10. C2×#1479C2semidirect
20. C2×D4×Q8C22semidirect
30. #3226C22semidirect
31. C2×#500C22semidirect
32. C2×#486C22semidirect
40. #3037C22semidirect
42. C2×#462C22semidirect
43. C2×#460C22semidirect
51. #1649C22semidirect
52. #1627C22semidirect
53. #1546C22semidirect
54. C2×#332C22semidirect
55. #1486C22semidirect
56. #1479C22semidirect
57. C2×Q8⋊C8C22semidirect
150. #500C23semidirect
152. #486C23semidirect
160. #462C23semidirect
162. #460C23semidirect
189. #332C23semidirect
190. Q8⋊C8C23semidirect
219. C22×Q8D8semidirect
419. C2×Q8C2×D8semidirect
439. C2×D4#137semidirect
573. D4C2×#137semidirect
655. C2#18776direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#32262
C2×#31002
C2×#30372
#187762
C2×#16492
C2×#16272
C2×#15462
C2×#14862
C2×#14792