Finite Group Databasebeta

← C2×#30792 #7501883

Subgroups — C2×#30792 #7501883

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#485C211yesmaximal, characteristic
3C2×#2492C211yesmaximal, characteristic
4C2×#2441C211yesmaximal, characteristic
5#30792C2116yesmaximal, direct product factor
6C2×#2096C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#2035C211yesmaximal, characteristic
11C2×#2026C211yesmaximal, characteristic, complemented
12C2×#1948C211yesmaximal, characteristic, complemented
13C2×#1935C211yesmaximal, characteristic, complemented
14C2×#1924C211yesmaximal, characteristic
15C2×#1921C211yesmaximal, characteristic, complemented
16C2×#1854C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#501—28no—
22C2×#501C2218yescomplemented
23C2×#485—28no—
24C2×#485C2218yes—
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×C22⋊C8C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×#97C2211yescharacteristic
39#2492—24no—
40#2492C2218yes—
41#2441—24no—
42#2441C2218yes—
43C2×#400—21no—
44C2×#399—21no—
45C2×#396—21no—
46C2×#393—21no—
47C2×#391—21no—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#388—21no—
54C2×#387—21no—
55C2×#384C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2096C2218yescomplemented
61#2077C2218yes—
62#2068C2218yescomplemented
63#2058C2218yescomplemented
64#2035C2218yes—
65#2026C2218yescomplemented
66#1948C2218yescomplemented
67#1935C2218yescomplemented
68#1924C2218yes—
69#1921C2218yescomplemented
70#1854C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8C2314yes—
90C2×#125—24no—
91C2×#125—22no—
92C2×#125—42no—
93C2×#125C2314yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—42no—
97C2×C22⋊Q8C2314yes—
98C2×#117—22no—
99C2×#117—24no—
100C2×#117—42no—
101C2×#117C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#501—232no—
128#485—232no—
129C2×#107—24no—
130C2×#107D418yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106—24no—
134C2×#106—24no—
135C2×#106—24no—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8—24no—
142C2×C4⋊C8C2314yes—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103—42no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×C22⋊C8—24no—
152C2×C22⋊C8—24no—
153C2×C22⋊C8C2314yes—
154C2×C22⋊C8C2314yes—
155C2×#97—24no—
156C2×#97C2314yes—
157#400—28no—
158#399—28no—
159#396—28no—
160#393—28no—
161#391—22no—
162#391—22no—
163#391—28no—
164#391C2314yes—
165#391C2314yes—
166#389—22no—
167#389—22no—
168#389—28no—
169#389C2314yes—
170#389C2314yes—
171#388—28no—
172#387—28no—
173#384—22no—
174#384C2314yes—
175#380—22no—
176#380—22no—
177#380C2314yes—
178#380C2314yes—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4D414yes—
182C4×C22⋊C4C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95D411yescharacteristic
193C2×#95D411yescharacteristic
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—82no—
214C22×Q8C2412yes—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8C2×D414yes—
248C22×C8#4114yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256#125—28no—
257#125—44no—
258#125—48no—
259C22⋊Q8—28no—
260C22⋊Q8—44no—
261C22⋊Q8—48no—
262#117—28no—
263#117—44no—
264#117—48no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—82no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—82no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333#107—28no—
334#107C2×D4116yes—
335#106—28no—
336#106—28no—
337#106—28no—
338#106—44no—
339C4⋊C8—28no—
340#103—28no—
341#103—28no—
342#103—28no—
343#103—44no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346#97—28no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C22×D412yes—
366C24C22×D411yescharacteristic
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—816no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4#13714yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+412yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C22×C4C2511yescharacteristic
457C2×C8—28no—
458C2×C8—28no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8C22×D414yes—
472C2×C8C2×#4114yes—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4C22×D414yes—
501C4⋊C4C22×D414yes—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—48no—
515C42—84no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#53612yes—
531C23D4212yes—
532C23#54712yes—
533C23C2×#13712yes—
534C23#57512yes—
535C23#57712yes—
536C23#57712yes—
537C23C23×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—816no—
543Q8—84no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4#53612yes—
601C2×C4#53614yes—
602C2×C4#54214yes—
603C2×C4D4212yes—
604C2×C4#54712yes—
605C2×C4#54814yes—
606C2×C4#56314yes—
607C2×C4C2×#13714yes—
608C2×C4C23×D411yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—24no—
615C8—24no—
616C8—42no—
617C8—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—88no—
626C22#267414yes—
627C22#323714yes—
628C22#324414yes—
629C22#326714yes—
630C22#327814yes—
631C22#334414yes—
632C22#335014yes—
633C22C2×#53611yescharacteristic
634C22C2×D4211yescharacteristic
635C22C2×#54711yescharacteristic
636C22C22×#13711yescharacteristic
637C22C2×#57511yescharacteristic
638C22C2×#57711yescharacteristic
639C22C2×#57711yescharacteristic
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#267414yes—
654C4C2×#53611yescharacteristic
655C4C2×#54211yescharacteristic
656C4C2×#54811yescharacteristic
657C4C2×#56311yescharacteristic
658C2—82no—
659C2#3079218yesdirect product factor
660C2C2×#267411yescharacteristic
661C2C2×#323711yescharacteristic
662C2C2×#324411yescharacteristic
663C2C2×#326711yescharacteristic
664C2C2×#327811yescharacteristic
665C2C2×#334411yescharacteristic
666C2C2×#335011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
5. #30792C2direct (N × H)
6. C2×#2096C2semidirect
8. C2×#2068C2semidirect
9. C2×#2058C2semidirect
11. C2×#2026C2semidirect
12. C2×#1948C2semidirect
13. C2×#1935C2semidirect
15. C2×#1921C2semidirect
22. C2×#501C22semidirect
60. #2096C22semidirect
62. #2068C22semidirect
63. #2058C22semidirect
65. #2026C22semidirect
66. #1948C22semidirect
67. #1935C22semidirect
69. #1921C22semidirect
659. C2#30792direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4852
C2×#24922
C2×#24412
#307922
C2×#20962
C2×#20772
C2×#20682
C2×#20582
C2×#20352
C2×#20262
C2×#19482
C2×#19352
C2×#19242
C2×#19212
C2×#18542