Finite Group Databasebeta

← C2×#29787 #6433249

Subgroups — C2×#29787 #6433249

Order 512.

2300 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#468C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic, complemented
3C2×#2625C211yesmaximal, characteristic, complemented
4C2×#2624C211yesmaximal, characteristic
5#29787C2116yesmaximal, direct product factor
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2066C211yesmaximal, characteristic, complemented
10C2×#2067C211yesmaximal, characteristic, complemented
11C2×#2065C211yesmaximal, characteristic, complemented
12C2×#2055C211yesmaximal, characteristic, complemented
13C2×#2055C211yesmaximal, characteristic, complemented
14C2×#1893C211yesmaximal, characteristic
15C2×#1892C211yesmaximal, characteristic, complemented
16C2×#1809C211yesmaximal, characteristic, complemented
17C23×SD16C2211yescharacteristic
18C23×M16C2211yescharacteristic
19C24×C8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C2×#468—28no—
24C2×#468C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424—28no—
29C2×#424C2218yescomplemented
30C22×C4⋊C8C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40#2625—24no—
41#2625C2218yescomplemented
42#2624—24no—
43#2624C2218yes—
44C2×#400—21no—
45C2×#399—21no—
46C2×#398—21no—
47C2×#397—21no—
48C2×#397—21no—
49C2×#397—21no—
50C2×#394—21no—
51C2×#394—21no—
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#389C2211yescharacteristic
60C2×#380C2211yescharacteristic
61C2×#380C2211yescharacteristic
62#2087C2218yescomplemented
63#2086C2218yescomplemented
64#2068C2218yescomplemented
65#2067C2218yescomplemented
66#2066C2218yescomplemented
67#2065C2218yescomplemented
68#2055C2218yescomplemented
69#2055C2218yescomplemented
70#1893C2218yes—
71#1892C2218yescomplemented
72#1809C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16—24no—
84C22×SD16—42no—
85C22×SD16—42no—
86C22×SD16C2314yes—
87C22×M16—42no—
88C22×M16—42no—
89C22×M16D418yes—
90C22×M16C2314yes—
91C23×C8—24no—
92C23×C8—42no—
93C23×C8—42no—
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2314yes—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8C2314yes—
103C2×#121—22no—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121C2314yes—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117D414yes—
110C2×#117C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#468—232no—
137C2×#107—24no—
138C2×#107—42no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106—42no—
142C2×#106C2314yes—
143#424—232no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—42no—
154C2×#103—42no—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164#400—28no—
165#399—28no—
166#398—28no—
167#397—28no—
168#397—28no—
169#397—28no—
170#394—28no—
171#394—28no—
172#391—22no—
173#391—22no—
174#391C2314yes—
175#391C2314yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2314yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#380—22no—
189#380—22no—
190#380C2314yes—
191#380C2314yes—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25—21no—
209C25C2×D411yescharacteristic
210C22×Q8—21no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4—82no—
228C22×D4—82no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D414yes—
248C23×C4#4114yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—416no—
255C2×M16—416no—
256C2×M16—44no—
257C2×M16—44no—
258C2×M16C2×D4116yes—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—28no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—44no—
271C22×C8—44no—
272C22×C8—416no—
273C22×C8C2×D414yes—
274C22×C8C2×D414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—48no—
282#121—28no—
283#121—28no—
284#121—48no—
285#117—216no—
286#117—28no—
287#117—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4—82no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4—82no—
344C2×C22⋊C4C2×D414yes—
345C2×C22⋊C4C2×D414yes—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2412yes—
354#107—28no—
355#107—44no—
356#106—44no—
357C4⋊C8—28no—
358C4⋊C8—28no—
359C4⋊C8—28no—
360C4⋊C8—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#102—28no—
365#102—44no—
366#102—44no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24#12114yes—
392C24C22×D412yes—
393C24C22×D411yescharacteristic
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—84no—
422C2×D4—84no—
423C2×D4—816no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4#12114yes—
477C22×C4#12118yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C22×C4C2511yescharacteristic
493SD16—416no—
494M16—416no—
495C2×C8—28no—
496C2×C8—28no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—416no—
508C2×C8—44no—
509C2×C8—44no—
510C2×C8C22×D414yes—
511C2×C8C22×D414yes—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C22⋊C4—28no—
538C22⋊C4—28no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—24no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C23—28no—
554C23—22no—
555C23—22no—
556C23—28no—
557C23—28no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23C2×#12114yes—
581C23C2×#12112yes—
582C23#53712yes—
583C23#53712yes—
584C23#57612yes—
585C23#57712yes—
586C23#57712yes—
587C23#57812yes—
588C23C23×D411yescharacteristic
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—84no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—816no—
599D4—84no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—216no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4C2×#12114yes—
653C2×C4C2×#12112yes—
654C2×C4#53414yes—
655C2×C4#53614yes—
656C2×C4#53712yes—
657C2×C4#53712yes—
658C2×C4#53914yes—
659C2×C4#53914yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×H(5,𝔽2)11yescharacteristic
664C2×C4C2×2−1+411yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C8—24no—
667C8—24no—
668C8—42no—
669C8—42no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—416no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—168no—
686C22#261814yes—
687C22#306214yes—
688C22#306314yes—
689C22#325714yes—
690C22#325814yes—
691C22#326014yes—
692C22#326214yes—
693C22C22×#12111yescharacteristic
694C22C2×#53711yescharacteristic
695C22C2×#53711yescharacteristic
696C22C2×#57611yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57711yescharacteristic
699C22C2×#57811yescharacteristic
700C4—24no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#261814yes—
713C4C2×#53411yescharacteristic
714C4C2×#53611yescharacteristic
715C4C2×#53911yescharacteristic
716C4C2×#53911yescharacteristic
717C2—44no—
718C2—162no—
719C2#2978718yesdirect product factor
720C2C2×#261811yescharacteristic
721C2C2×#306211yescharacteristic
722C2C2×#306311yescharacteristic
723C2C2×#325711yescharacteristic
724C2C2×#325811yescharacteristic
725C2C2×#326011yescharacteristic
726C2C2×#326211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#424C2semidirect
3. C2×#2625C2semidirect
5. #29787C2direct (N × H)
6. C2×#2087C2semidirect
7. C2×#2086C2semidirect
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#2067C2semidirect
11. C2×#2065C2semidirect
12. C2×#2055C2semidirect
13. C2×#2055C2semidirect
15. C2×#1892C2semidirect
16. C2×#1809C2semidirect
29. C2×#424C22semidirect
41. #2625C22semidirect
62. #2087C22semidirect
63. #2086C22semidirect
64. #2068C22semidirect
65. #2067C22semidirect
66. #2066C22semidirect
67. #2065C22semidirect
68. #2055C22semidirect
69. #2055C22semidirect
71. #1892C22semidirect
72. #1809C22semidirect
719. C2#29787direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4682
C22×#4242
C2×#26252
C2×#26242
#297872
C2×#20872
C2×#20862
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#20552
C2×#18932
C2×#18922
C2×#18092