Finite Group Databasebeta

← C2×#29762 #6433185

Subgroups — C2×#29762 #6433185

Order 512.

2440 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C22×#482C211yesmaximal, characteristic
3C2×#2850C211yesmaximal, characteristic
4C2×#2296C211yesmaximal, characteristic, complemented
5#29762C2116yesmaximal, direct product factor
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2085C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2066C211yesmaximal, characteristic, complemented
10C2×#2067C211yesmaximal, characteristic, complemented
11C2×#2065C211yesmaximal, characteristic, complemented
12C2×#2042C211yesmaximal, characteristic, complemented
13C2×#2042C211yesmaximal, characteristic, complemented
14C2×#1875C211yesmaximal, characteristic
15C2×#1873C211yesmaximal, characteristic, complemented
16C2×#1797C211yesmaximal, characteristic, complemented
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#485—28no—
24C2×#485C2218yescomplemented
25C2×#482—28no—
26C2×#482C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×C22⋊C8C2211yescharacteristic
42C22×C22⋊C8C2211yescharacteristic
43#2850—24no—
44#2850C2218yes—
45#2296—24no—
46#2296C2218yescomplemented
47C2×#391C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#388—21no—
55C2×#386—21no—
56C2×#386—21no—
57C2×#385—21no—
58C2×#385—21no—
59C2×#382C2211yescharacteristic
60C2×C4×C4⋊C4—21no—
61C2×C4×C4⋊C4—21no—
62#2085C2218yescomplemented
63#2085C2218yescomplemented
64#2068C2218yescomplemented
65#2067C2218yescomplemented
66#2066C2218yescomplemented
67#2065C2218yescomplemented
68#2042C2218yescomplemented
69#2042C2218yescomplemented
70#1875C2218yes—
71#1873C2218yescomplemented
72#1797C2218yescomplemented
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4D411yescharacteristic
84C24×C4C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#124—22no—
90C2×#124—42no—
91C2×#124—42no—
92C2×#124C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2314yes—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121C2314yes—
101C2×#117—22no—
102C2×#117—24no—
103C2×#117—22no—
104C2×#117—42no—
105C2×#117—42no—
106C2×#117D418yes—
107C2×#117C2314yes—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42—42no—
131C22×C42—42no—
132C22×C42D414yes—
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2314yes—
138#485—232no—
139#482—232no—
140C2×#107—24no—
141C2×#107—24no—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106—24no—
145C2×#106—24no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8—42no—
149C2×C4⋊C8—42no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103—42no—
155C2×#103—42no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102—24no—
161C2×#102—24no—
162C2×#102—42no—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8—24no—
169C2×C22⋊C8—24no—
170C2×C22⋊C8C2314yes—
171C2×C22⋊C8C2314yes—
172#391—22no—
173#391C2314yes—
174#390—22no—
175#390C2314yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#388—28no—
185#388—28no—
186#386—28no—
187#386—28no—
188#385—28no—
189#385—28no—
190#382—22no—
191#382C2314yes—
192C4×C4⋊C4—28no—
193C4×C4⋊C4—28no—
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#336C2314yes—
207#336C2314yes—
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—44no—
228C22×D4—44no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D414yes—
255C23×C4C2×D414yes—
256C23×C4#4114yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#124—44no—
276#124—44no—
277#124—48no—
278C22⋊Q8—28no—
279C22⋊Q8—28no—
280C22⋊Q8—48no—
281#121—28no—
282#121—28no—
283#121—48no—
284#117—216no—
285#117—28no—
286#117—44no—
287#117—44no—
288#117—48no—
289#117C2×D4116yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
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—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—28no—
325C2×C22⋊C4—24no—
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—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C42—28no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—24no—
347C2×C42—48no—
348C2×C42—44no—
349C2×C42—44no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42—44no—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357C2×C42C2412yes—
358C2×C42C2412yes—
359#107—28no—
360#107—28no—
361#106—28no—
362#106—28no—
363C4⋊C8—44no—
364C4⋊C8—44no—
365#103—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24#12114yes—
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—816no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
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—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4#12114yes—
490C22×C4#12118yes—
491C22×C4#13514yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C42−1+412yes—
505C22×C42−1+412yes—
506C22×C4C2511yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—44no—
520C2×C8—44no—
521C2×C8—44no—
522C2×C8—44no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—48no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—48no—
544C4⋊C4C22×D414yes—
545C4⋊C4C22×D414yes—
546C22⋊C4—216no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C42—216no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—44no—
562C42—44no—
563C42—48no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—28no—
568C23—28no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23—84no—
589C23—84no—
590C23—84no—
591C23C2×#12114yes—
592C23C2×#12112yes—
593C23#53412yes—
594C23#53612yes—
595C23C2×#13512yes—
596C23#57512yes—
597C23#57512yes—
598C23#57512yes—
599C23C23×D411yescharacteristic
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—84no—
605D4—42no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—816no—
610D4—84no—
611C2×C4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—28no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—216no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4C2×#12114yes—
665C2×C4C2×#12112yes—
666C2×C4#53412yes—
667C2×C4#53612yes—
668C2×C4#53714yes—
669C2×C4#53714yes—
670C2×C4#53914yes—
671C2×C4#53914yes—
672C2×C4C2×#13514yes—
673C2×C4C23×D411yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×H(5,𝔽2)11yescharacteristic
676C2×C4C2×H(5,𝔽2)11yescharacteristic
677C2×C4C2×2−1+411yescharacteristic
678C2×C4C2×2−1+411yescharacteristic
679C8—42no—
680C8—42no—
681C8—42no—
682C8—42no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—42no—
687C22—416no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—168no—
699C22#261814yes—
700C22#303514yes—
701C22#304014yes—
702C22#319114yes—
703C22#319714yes—
704C22#323314yes—
705C22#324014yes—
706C22C22×#12111yescharacteristic
707C22C2×#53411yescharacteristic
708C22C2×#53611yescharacteristic
709C22C22×#13511yescharacteristic
710C22C2×#57511yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57511yescharacteristic
713C4—24no—
714C4—24no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4#261814yes—
726C4C2×#53711yescharacteristic
727C4C2×#53711yescharacteristic
728C4C2×#53911yescharacteristic
729C4C2×#53911yescharacteristic
730C2—44no—
731C2—162no—
732C2#2976218yesdirect product factor
733C2C2×#261811yescharacteristic
734C2C2×#303511yescharacteristic
735C2C2×#304011yescharacteristic
736C2C2×#319111yescharacteristic
737C2C2×#319711yescharacteristic
738C2C2×#323311yescharacteristic
739C2C2×#324011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#485C2semidirect
4. C2×#2296C2semidirect
5. #29762C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2085C2semidirect
8. C2×#2068C2semidirect
9. C2×#2066C2semidirect
10. C2×#2067C2semidirect
11. C2×#2065C2semidirect
12. C2×#2042C2semidirect
13. C2×#2042C2semidirect
15. C2×#1873C2semidirect
16. C2×#1797C2semidirect
24. C2×#485C22semidirect
46. #2296C22semidirect
62. #2085C22semidirect
63. #2085C22semidirect
64. #2068C22semidirect
65. #2067C22semidirect
66. #2066C22semidirect
67. #2065C22semidirect
68. #2042C22semidirect
69. #2042C22semidirect
71. #1873C22semidirect
72. #1797C22semidirect
732. C2#29762direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4822
C2×#28502
C2×#22962
#297622
C2×#20852
C2×#20682
C2×#20662
C2×#20672
C2×#20652
C2×#20422
C2×#18752
C2×#18732
C2×#17972