Finite Group Databasebeta

← C2×#18623 #6348198

Subgroups — C2×#18623 #6348198

Order 512.

3170 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3503C211yesmaximal, characteristic
2C2×#3230C211yesmaximal, characteristic, complemented
3C22×#494C211yesmaximal, characteristic, complemented
4C2×#3079C211yesmaximal, characteristic
5#18623C2116yesmaximal, direct product factor
6C2×#1633C214yesmaximal, complemented
7C2×#1632C214yesmaximal, complemented
8C2×#1548C211yesmaximal, characteristic, complemented
9C2×#1493C211yesmaximal, characteristic, complemented
10C2×#1476C211yesmaximal, characteristic
11C23×Q16—22no—
12#3503—24no—
13#3503—24no—
14#3503—24no—
15#3503C2218yes—
16C2×Q82C2212yes—
17C2×#557—21no—
18C2×#554—22no—
19C2×#554C2212yes—
20C2×D4×Q8—22no—
21C2×D4×Q8—21no—
22C2×#544—22no—
23C2×#544—21no—
24C2×#533—21no—
25C2×#533C2211yescharacteristic
26C22×C4⋊Q8—21no—
27C22×C4⋊Q8C2211yescharacteristic
28C22×C4⋊Q8C2211yescharacteristic
29C2×#520—21no—
30C2×C4×#41C2211yescharacteristic
31#3230—24no—
32#3230C2218yescomplemented
33C2×#500—21no—
34C2×#494—22no—
35C2×#494—24no—
36C2×#494C2214yes—
37C2×#494C2218yescomplemented
38C2×#487—21no—
39C2×#484—22no—
40#3079—24no—
41#3079C2218yes—
42C2×#477C2214yes—
43C2×#477C2212yes—
44C2×#475—21no—
45C2×#476C2212yes—
46C2×#474—21no—
47C2×#431C2211yescharacteristic
48C2×#428—21no—
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51C22×C4×C8C2211yescharacteristic
52#1633C22132yescomplemented
53#1632C22132yescomplemented
54#1548C2218yescomplemented
55C2×#333C2214yes—
56#1493C2218yescomplemented
57#1476C2218yes—
58C2×Q8⋊C8C2212yes—
59C2×Q8⋊C8C2214yes—
60C2×D4⋊C8C2212yes—
61C22×#41D411yescharacteristic
62C23×Q8—22no—
63C23×Q8—21no—
64C23×Q8D412yes—
65C22×Q16—28no—
66C22×Q16—216no—
67C22×Q16—416no—
68C22×M16—21no—
69C23×C8—21no—
70Q82—24no—
71Q82—24no—
72Q82—24no—
73Q82C2318yes—
74#557—28no—
75#557—28no—
76#554—216no—
77#554—24no—
78#554—216no—
79#554—24no—
80#554—24no—
81#554C2318yes—
82D4×Q8—216no—
83D4×Q8—28no—
84D4×Q8—216no—
85D4×Q8—28no—
86#544—216no—
87#544—28no—
88#544—216no—
89#544—28no—
90#533—28no—
91#533—28no—
92#533—22no—
93#533—22no—
94#533D414yes—
95#533C2314yes—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—22no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8D411yescharacteristic
114C2×C4⋊Q8D412yes—
115C2×C4⋊Q8D414yes—
116C2×C4⋊Q8D411yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2314yes—
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2314yes—
121C2×#125—22no—
122C2×#125—24no—
123C2×#125—22no—
124C2×#125—21no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—24no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8D412yes—
132#520—28no—
133#520—28no—
134C4×#41—22no—
135C4×#41—22no—
136C4×#41D414yes—
137C4×#41C2314yes—
138C2×C4×Q8—22no—
139C2×C4×Q8—22no—
140C2×C4×Q8—22no—
141C2×C4×Q8—21no—
142C2×C4×Q8D411yescharacteristic
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2312yes—
145C2×C4×D4—24no—
146C2×C4×D4—22no—
147C2×C4×D4—21no—
148C2×C4×D4—21no—
149C2×C4×D4D411yescharacteristic
150C2×C4×D4C2311yescharacteristic
151C2×#117—22no—
152C2×#117—21no—
153C2×#117—21no—
154C2×#117D411yescharacteristic
155C22×C4⋊C4—22no—
156C22×C4⋊C4—21no—
157C22×C4⋊C4—21no—
158C22×C4⋊C4—22no—
159C22×C4⋊C4—21no—
160C22×C4⋊C4—21no—
161C22×C42—21no—
162C22×C42C2311yescharacteristic
163#500—28no—
164#494—28no—
165#494—216no—
166#494C23116yes—
167#487—28no—
168#484—216no—
169#477—24no—
170#477—28no—
171#477C2318yes—
172#477C23116yes—
173#476—24no—
174#476C2318yes—
175#475—28no—
176#474—28no—
177#431—22no—
178#431C2314yes—
179#428—28no—
180C2×#107—24no—
181C2×#107—22no—
182C2×#106—22no—
183#423—22no—
184#423C2314yes—
185C2×C4⋊C8—21no—
186C2×C4⋊C8C2312yes—
187C2×C4⋊C8C2312yes—
188#417—28no—
189C2×#103—22no—
190C2×#103—24no—
191C2×#102—22no—
192C2×C22⋊C8—21no—
193C2×C4×C8—21no—
194C2×C4×C8—22no—
195C2×C4×C8—21no—
196C2×C4×C8C2312yes—
197C2×C4×C8C2314yes—
198#333C23116yes—
199Q8⋊C8C23116yes—
200Q8⋊C8C2318yes—
201D4⋊C8C2318yes—
202C2×#41—22no—
203C2×#41—24no—
204C2×#41Q1618yescomplemented
205C2×#41C2×D414yes—
206C22×Q8—24no—
207C22×Q8—216no—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—48no—
214C22×Q8—48no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8—88no—
218C22×Q8C2×D412yes—
219C22×Q8C2×D411yescharacteristic
220C22×Q8C2×D414yes—
221C22×Q8C2×D418yes—
222C22×D4—21no—
223C22×D4C2×D411yescharacteristic
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—41no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C2×Q16—232no—
234C2×Q16—432no—
235C2×Q16—464no—
236C2×M16—24no—
237C2×M16—44no—
238C22×C8—28no—
239C22×C8—24no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C4⋊Q8—216no—
244C4⋊Q8—216no—
245C4⋊Q8—28no—
246C4⋊Q8—216no—
247C4⋊Q8—28no—
248C4⋊Q8—216no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—42no—
269C4⋊Q8C2×D418yes—
270C4⋊Q8C2×D414yes—
271C4⋊Q8C2×D414yes—
272C4⋊Q8C2×D414yes—
273C4⋊Q8C2414yes—
274C4⋊Q8C2414yes—
275#125—216no—
276#125—232no—
277#125—28no—
278#125—28no—
279#125—24no—
280#125—24no—
281C22⋊Q8—216no—
282C22⋊Q8—28no—
283C22⋊Q8—216no—
284C22⋊Q8—232no—
285C22⋊Q8—232no—
286C22⋊Q8—28no—
287C22⋊Q8—216no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8C2×D418yes—
292C4×Q8—28no—
293C4×Q8—216no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—22no—
299C4×Q8—24no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8C2×D414yes—
305C4×Q8C2×D412yes—
306C4×Q8C2×D414yes—
307C4×Q8C2412yes—
308C4×Q8C2414yes—
309C4×D4—232no—
310C4×D4—28no—
311C4×D4—28no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—22no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4C2×D414yes—
320C4×D4C2×D412yes—
321C4×D4C2412yes—
322#117—216no—
323#117—24no—
324#117—22no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117C2×D414yes—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—216no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—28no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—41no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2412yes—
382#107—28no—
383#107—216no—
384#106—28no—
385C4⋊C8—24no—
386C4⋊C8—22no—
387C4⋊C8—22no—
388C4⋊C8C2414yes—
389C4⋊C8C2414yes—
390#103—28no—
391#103—216no—
392#102—28no—
393C22⋊C8—24no—
394C4×C8—24no—
395C4×C8—22no—
396C4×C8—24no—
397C4×C8C2414yes—
398C24—21no—
399C24#12011yescharacteristic
400#41—28no—
401#41—216no—
402#41C2×Q16116yescomplemented
403C2×Q8—216no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—28no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—28no—
411C2×Q8—24no—
412C2×Q8—48no—
413C2×Q8—416no—
414C2×Q8—48no—
415C2×Q8—416no—
416C2×Q8—42no—
417C2×Q8—44no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—88no—
421C2×Q8—88no—
422C2×Q8—88no—
423C2×Q8—832no—
424C2×Q8—84no—
425C2×Q8C2×Q1614yescomplemented
426C2×Q8C22×D414yes—
427C2×Q8C22×D412yes—
428C2×Q8C22×D418yes—
429C2×D4—24no—
430C2×D4—28no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4C2×Q1614yescomplemented
434C2×D4C22×D412yes—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—216no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4—84no—
457C22×C4#12012yes—
458C22×C4#12012yes—
459C22×C4#12011yescharacteristic
460C22×C4#12012yes—
461C22×C4#12011yescharacteristic
462C22×C4#12011yescharacteristic
463C22×C4C2×Q1614yes—
464C22×C4#13714yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471Q16—464no—
472M16—48no—
473C2×C8—28no—
474C2×C8—216no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—44no—
479C2×C8—44no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—28no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4C22×D412yes—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D412yes—
511C22⋊C4—216no—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C42—216no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—24no—
522C42—22no—
523C42—24no—
524C42#12012yes—
525C42#12012yes—
526C42#12012yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C42C22×D411yescharacteristic
530C42C22×D411yescharacteristic
531C42C22×D412yes—
532C42C22×D411yescharacteristic
533C42C2511yescharacteristic
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23#45114yes—
541C23C2×#12012yes—
542C23C2×#12011yescharacteristic
543Q8—24no—
544Q8—22no—
545Q8—28no—
546Q8—48no—
547Q8—48no—
548Q8—44no—
549Q8—416no—
550Q8—48no—
551Q8—84no—
552Q8—84no—
553Q8—816no—
554Q8—88no—
555Q8C22×Q1614yescomplemented
556D4—216no—
557D4—24no—
558D4—22no—
559D4C22×Q1614yescomplemented
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—216no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—88no—
584C2×C4—816no—
585C2×C4—88no—
586C2×C4—84no—
587C2×C4#45114yes—
588C2×C4#45118yes—
589C2×C4#45314yes—
590C2×C4C2×#12012yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12011yescharacteristic
593C2×C4C2×#12012yes—
594C2×C4C2×#12011yescharacteristic
595C2×C4C2×#12012yes—
596C2×C4C22×Q1612yes—
597C2×C4C22×Q1614yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13714yes—
600C2×C4#57814yes—
601C2×C4C23×D411yescharacteristic
602C2×C4C23×D411yescharacteristic
603C2×C4C23×D411yescharacteristic
604C8—28no—
605C8—44no—
606C8—44no—
607C22—28no—
608C22—22no—
609C22—22no—
610C22—24no—
611C22—48no—
612C22—42no—
613C22—42no—
614C22C2×#45114yes—
615C22C2×#45112yes—
616C22#301112yes—
617C22C2×#45312yes—
618C22C22×#12011yescharacteristic
619C4—24no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—42no—
624C4—44no—
625C4—42no—
626C4—48no—
627C4—44no—
628C4—84no—
629C4C2×#45114yes—
630C4C2×#45112yes—
631C4#301112yes—
632C4C2×#45314yes—
633C4C23×Q1611yescharacteristic
634C4C22×#13711yescharacteristic
635C4C2×#57812yes—
636C2—24no—
637C2—44no—
638C2#1862314yesdirect product factor
639C2C22×#45111yescharacteristic
640C2C2×#301111yescharacteristic
641C2C22×#45311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC22×#41D4
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
2. C2×#3230C2semidirect
3. C22×#494C2semidirect
5. #18623C2direct (N × H)
6. C2×#1633C2semidirect
7. C2×#1632C2semidirect
8. C2×#1548C2semidirect
9. C2×#1493C2semidirect
32. #3230C22semidirect
37. C2×#494C22semidirect
52. #1633C22semidirect
53. #1632C22semidirect
54. #1548C22semidirect
56. #1493C22semidirect
204. C2×#41Q16semidirect
402. #41C2×Q16semidirect
425. C2×Q8C2×Q16semidirect
433. C2×D4C2×Q16semidirect
555. Q8C22×Q16semidirect
559. D4C22×Q16semidirect
638. C2#18623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#35032
C2×#32302
C22×#4942
C2×#30792
#186232
C2×#16332
C2×#16322
C2×#15482
C2×#14932
C2×#14762