Finite Group Databasebeta

← C2×#31226 #7503794

Subgroups — C2×#31226 #7503794

Order 512.

2302 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C22×#462C211yesmaximal, characteristic, complemented
3C2×#2448C211yesmaximal, characteristic
4C2×#2446C211yesmaximal, characteristic, complemented
5#31226C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2056C211yesmaximal, characteristic, complemented
9C2×#2055C211yesmaximal, characteristic
10C2×#2013C211yesmaximal, characteristic
11C2×#2015C211yesmaximal, characteristic
12C2×#1973C211yesmaximal, characteristic
13C2×#1971C211yesmaximal, characteristic, complemented
14C2×#1940C211yesmaximal, characteristic, complemented
15C2×#1863C211yesmaximal, characteristic, complemented
16C2×#1863C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×Q16C2211yescharacteristic
19C23×SD16C2211yescharacteristic
20C23×SD16C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C2×#464—28no—
26C2×#464C2218yes—
27C2×#462—28no—
28C2×#462C2218yescomplemented
29C22×#107C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×#103C2211yescharacteristic
38C22×#103C2211yescharacteristic
39C22×#103C2211yescharacteristic
40C22×#102C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#97C2211yescharacteristic
43#2448—24no—
44#2448C2218yes—
45#2446—24no—
46#2446C2218yescomplemented
47C2×#400—21no—
48C2×#399—21no—
49C2×#398—21no—
50C2×#397—21no—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#390C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#389C2211yescharacteristic
59C2×#388—21no—
60C2×#388—21no—
61C2×#387—21no—
62C2×#387—21no—
63C2×C4×C4⋊C4C2211yescharacteristic
64#2066C2218yescomplemented
65#2065C2218yes—
66#2056C2218yescomplemented
67#2055C2218yes—
68#2015C2218yes—
69#2013C2218yes—
70#1973C2218yes—
71#1971C2218yescomplemented
72#1940C2218yescomplemented
73#1863C2218yescomplemented
74#1863C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—42no—
85C22×Q16—42no—
86C22×Q16—42no—
87C22×Q16—42no—
88C22×Q16D418yes—
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91C22×SD16—24no—
92C22×SD16—24no—
93C22×SD16—42no—
94C22×SD16—42no—
95C22×SD16—42no—
96C22×SD16—42no—
97C22×SD16C2314yes—
98C22×SD16C2314yes—
99C23×C8C2311yescharacteristic
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—42no—
106C2×C4⋊Q8C2314yes—
107C2×#124—24no—
108C2×#124—22no—
109C2×#124—42no—
110C2×#124C2314yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—42no—
116C2×C4×Q8—42no—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2314yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#464—232no—
143#462—232no—
144C2×#107—42no—
145C2×#107C2314yes—
146C2×#106—42no—
147C2×#106C2314yes—
148C2×C4⋊C8—24no—
149C2×C4⋊C8—24no—
150C2×C4⋊C8—42no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—24no—
156C2×#103—24no—
157C2×#103—24no—
158C2×#103—24no—
159C2×#103—42no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102—24no—
167C2×#102—42no—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#97—42no—
171C2×#97C2314yes—
172#400—28no—
173#399—28no—
174#398—28no—
175#397—28no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391C2314yes—
181#391C2314yes—
182#391C2314yes—
183#391C2314yes—
184#390—22no—
185#390—22no—
186#390C2314yes—
187#390C2314yes—
188#389—22no—
189#389—22no—
190#389C2314yes—
191#389C2314yes—
192#388—28no—
193#388—28no—
194#387—28no—
195#387—28no—
196C4×C4⋊C4D414yes—
197C4×C4⋊C4C2314yes—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25—21no—
215C22×Q8—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8C2×D414yes—
235C22×Q8C2×D414yes—
236C22×Q8C2412yes—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×Q16—28no—
262C2×Q16—416no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×Q16—416no—
268C2×Q16C2×D4116yes—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—44no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×SD16—44no—
275C2×SD16—416no—
276C2×SD16—416no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2×D414yes—
288C22×C8#4114yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C4⋊Q8—28no—
294C4⋊Q8—44no—
295C4⋊Q8—48no—
296#124—28no—
297#124—44no—
298#124—48no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×Q8—48no—
304C4×Q8—48no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—82no—
348C2×C22⋊C4—82no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42—44no—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364C2×C42C2412yes—
365#107—44no—
366#106—44no—
367C4⋊C8—28no—
368C4⋊C8—28no—
369C4⋊C8—44no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—44no—
376#102—28no—
377#102—44no—
378#97—44no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—44no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8C22×D414yes—
438C2×Q8C22×D414yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—44no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
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—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4#13714yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4H(5,𝔽2)12yes—
510C22×C42−1+412yes—
511C22×C4C2511yescharacteristic
512Q16—416no—
513Q16—416no—
514SD16—416no—
515SD16—416no—
516C2×C8—28no—
517C2×C8—28no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—44no—
529C2×C8—44no—
530C2×C8C22×D414yes—
531C2×C8C2×#4114yes—
532C4⋊C4—28no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—48no—
546C4⋊C4—48no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—84no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—24no—
559C42—24no—
560C42—24no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—48no—
568C42—48no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23#53712yes—
584C23D4212yes—
585C23#55012yes—
586C23C2×#13712yes—
587C23#57512yes—
588C23#57712yes—
589C23#57812yes—
590C23C23×D411yescharacteristic
591Q8—24no—
592Q8—24no—
593Q8—24no—
594Q8—24no—
595Q8—48no—
596Q8—48no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—84no—
606D4—42no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—84no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
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—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4#53414yes—
663C2×C4#53712yes—
664C2×C4#53814yes—
665C2×C4D4212yes—
666C2×C4#54714yes—
667C2×C4#55012yes—
668C2×C4#56214yes—
669C2×C4C2×#13714yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C8—24no—
677C8—24no—
678C8—42no—
679C8—42no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—168no—
688C22#268214yes—
689C22#325214yes—
690C22#325814yes—
691C22#327014yes—
692C22#327114yes—
693C22#340014yes—
694C22#340214yes—
695C22C2×#53711yescharacteristic
696C22C2×D4211yescharacteristic
697C22C2×#55011yescharacteristic
698C22C22×#13711yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57711yescharacteristic
701C22C2×#57811yescharacteristic
702C4—24no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4#268214yes—
716C4C2×#53411yescharacteristic
717C4C2×#53811yescharacteristic
718C4C2×#54711yescharacteristic
719C4C2×#56211yescharacteristic
720C2—162no—
721C2#3122618yesdirect product factor
722C2C2×#268211yescharacteristic
723C2C2×#325211yescharacteristic
724C2C2×#325811yescharacteristic
725C2C2×#327011yescharacteristic
726C2C2×#327111yescharacteristic
727C2C2×#340011yescharacteristic
728C2C2×#340211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#462C2semidirect
4. C2×#2446C2semidirect
5. #31226C2direct (N × H)
6. C2×#2066C2semidirect
8. C2×#2056C2semidirect
13. C2×#1971C2semidirect
14. C2×#1940C2semidirect
15. C2×#1863C2semidirect
16. C2×#1863C2semidirect
28. C2×#462C22semidirect
46. #2446C22semidirect
64. #2066C22semidirect
66. #2056C22semidirect
71. #1971C22semidirect
72. #1940C22semidirect
73. #1863C22semidirect
74. #1863C22semidirect
721. C2#31226direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4642
C22×#4622
C2×#24482
C2×#24462
#312262
C2×#20662
C2×#20652
C2×#20562
C2×#20552
C2×#20132
C2×#20152
C2×#19732
C2×#19712
C2×#19402
C2×#18632