Finite Group Databasebeta

← C2×#30823 #7501961

Subgroups — C2×#30823 #7501961

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic
3C2×#2550C211yesmaximal, characteristic, complemented
4C2×#2447C211yesmaximal, characteristic
5#30823C2116yesmaximal, direct product factor
6C2×#2097C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic
8C2×#2066C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#2042C211yesmaximal, characteristic
11C2×#2026C211yesmaximal, characteristic, complemented
12C2×#1944C211yesmaximal, characteristic, complemented
13C2×#1926C211yesmaximal, characteristic
14C2×#1927C211yesmaximal, characteristic, complemented
15C2×#1917C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#489—28no—
22C2×#489C2218yes—
23C2×#478—28no—
24C2×#478C2218yes—
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#97C2211yescharacteristic
39#2550—24no—
40#2550C2218yescomplemented
41#2447—24no—
42#2447C2218yes—
43C2×#398—21no—
44C2×#397—21no—
45C2×#397—21no—
46C2×#395—21no—
47C2×#391—21no—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#385—21no—
56C2×#382C2211yescharacteristic
57C2×#381—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2097C2218yescomplemented
61#2073C2218yes—
62#2066C2218yescomplemented
63#2058C2218yescomplemented
64#2042C2218yes—
65#2026C2218yescomplemented
66#1944C2218yescomplemented
67#1927C2218yescomplemented
68#1926C2218yes—
69#1917C2218yescomplemented
70#1864C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2314yes—
91C2×#125—24no—
92C2×#125—22no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×#124—24no—
96C2×#124—22no—
97C2×#124—42no—
98C2×#124C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
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⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
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⋊C4—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#489—232no—
127#478—232no—
128C2×#107—24no—
129C2×#107C2314yes—
130C2×#106—24no—
131C2×#106—24no—
132C2×#106—42no—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—24no—
137C2×C4⋊C8C2314yes—
138C2×#103—24no—
139C2×#103—24no—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102D418yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—24no—
155C2×#97C2314yes—
156#398—28no—
157#397—28no—
158#397—28no—
159#395—28no—
160#391—22no—
161#391—22no—
162#391—28no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390C2314yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—28no—
171#389C2314yes—
172#389C2314yes—
173#389C2314yes—
174#385—28no—
175#382—22no—
176#382C2314yes—
177#381—28no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4D414yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2314yes—
182C2×#95—21no—
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#336C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202C25—21no—
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—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×D414yes—
222C22×D4C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8C2×D414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260#125—28no—
261#125—44no—
262#125—48no—
263#124—28no—
264#124—44no—
265#124—48no—
266C4×D4—28no—
267C4×D4—44no—
268C4×D4—48no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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—22no—
280C2×C4⋊C4—22no—
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—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4#4114yes—
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—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—82no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333#107—28no—
334#106—28no—
335#106—28no—
336#106—44no—
337C4⋊C8—28no—
338#103—28no—
339#103—28no—
340#103—28no—
341#103—44no—
342#102—28no—
343#102—28no—
344#102—28no—
345#102C2×D4116yes—
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#95—24no—
357#95C2×D414yes—
358#95C2×D414yes—
359C24—24no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24C22×D411yescharacteristic
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—84no—
384C2×Q8—816no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4C22×D414yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—48no—
426C22×C4—48no—
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—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C2×#4112yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462C2×C8—28no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—44no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8C22×D414yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4C22×D414yes—
503C4⋊C4C2×#4114yes—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C42—24no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—84no—
519C23—24no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23#53612yes—
534C23D4212yes—
535C23#54812yes—
536C23#57512yes—
537C23#57512yes—
538C23#57712yes—
539C23#57712yes—
540C23C23×D411yescharacteristic
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—816no—
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—42no—
551D4—42no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
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—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—816no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4#53612yes—
606C2×C4#53614yes—
607C2×C4#54214yes—
608C2×C4D4212yes—
609C2×C4#54714yes—
610C2×C4#54812yes—
611C2×C4#56314yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×2−1+411yescharacteristic
616C2×C4C2×2−1+411yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—42no—
620C8—42no—
621C8—42no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—816no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#267414yes—
631C22#324014yes—
632C22#324414yes—
633C22#326714yes—
634C22#327514yes—
635C22#337114yes—
636C22#337614yes—
637C22C2×#53611yescharacteristic
638C22C2×D4211yescharacteristic
639C22C2×#54811yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57711yescharacteristic
644C4—42no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#267414yes—
658C4C2×#53611yescharacteristic
659C4C2×#54211yescharacteristic
660C4C2×#54711yescharacteristic
661C4C2×#56311yescharacteristic
662C2—84no—
663C2#3082318yesdirect product factor
664C2C2×#267411yescharacteristic
665C2C2×#324011yescharacteristic
666C2C2×#324411yescharacteristic
667C2C2×#326711yescharacteristic
668C2C2×#327511yescharacteristic
669C2C2×#337611yescharacteristic
670C2C2×#337111yescharacteristic
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
3. C2×#2550C2semidirect
5. #30823C2direct (N × H)
6. C2×#2097C2semidirect
8. C2×#2066C2semidirect
9. C2×#2058C2semidirect
11. C2×#2026C2semidirect
12. C2×#1944C2semidirect
14. C2×#1927C2semidirect
15. C2×#1917C2semidirect
40. #2550C22semidirect
60. #2097C22semidirect
62. #2066C22semidirect
63. #2058C22semidirect
65. #2026C22semidirect
66. #1944C22semidirect
67. #1927C22semidirect
69. #1917C22semidirect
663. C2#30823direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4782
C2×#25502
C2×#24472
#308232
C2×#20972
C2×#20732
C2×#20662
C2×#20582
C2×#20422
C2×#20262
C2×#19442
C2×#19262
C2×#19272
C2×#19172
C2×#18642