Finite Group Databasebeta

← C2×#26052 #6443370

Subgroups — C2×#26052 #6443370

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C22×#474C211yesmaximal, characteristic
3C2×#2550C211yesmaximal, characteristic, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic
5#26052C2116yesmaximal, direct product factor
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2057C211yesmaximal, characteristic, complemented
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1864C211yesmaximal, characteristic
13C2×#1862C211yesmaximal, characteristic
14C2×#1847C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×#1762C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4—21no—
23C2×#478—28no—
24C2×#478C2218yes—
25C2×#474—28no—
26C2×#474C2218yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×#106C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×#102C2211yescharacteristic
40C22×#97C2211yescharacteristic
41#2550—24no—
42#2550C2218yescomplemented
43C2×#398—21no—
44C2×#395—21no—
45D4×C4⋊C4—24no—
46D4×C4⋊C4C2218yes—
47C2×#391—21no—
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#385—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60#2088C2218yescomplemented
61#2087C2218yescomplemented
62#2059C2218yescomplemented
63#2057C2218yescomplemented
64#1947C2218yescomplemented
65#1940C2218yescomplemented
66#1864C2218yes—
67#1862C2218yes—
68#1847C2218yescomplemented
69#1804C2218yes—
70#1762C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C24×C4—21no—
83C24×C4—21no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—42no—
91C2×C4⋊Q8C2314yes—
92C2×#125—24no—
93C2×#125—22no—
94C2×#125—42no—
95C2×#125C2314yes—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—24no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4Q818yescomplemented
105C2×C4×D4C2314yes—
106C2×C4×D4C2314yes—
107C2×C4×D4C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—28no—
118C22×C4⋊C4—48no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#478—232no—
136#474—232no—
137C2×#107—24no—
138C2×#107—24no—
139C2×#107—24no—
140C2×#107—42no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106—24no—
146C2×#106—42no—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8—24no—
150C2×C4⋊C8—24no—
151C2×C4⋊C8—24no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×#102—24no—
156C2×#102—24no—
157C2×#102D418yes—
158C2×#102Q818yescomplemented
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#97—24no—
164C2×#97C2314yes—
165#398—28no—
166#395—28no—
167#391—28no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—28no—
173#389C2314yes—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#385—28no—
178#385—28no—
179#384—22no—
180#384C2314yes—
181#382—22no—
182#382C2314yes—
183#380—28no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4D414yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95D411yescharacteristic
196C2×#95D411yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205C25—21no—
206C22×Q8—24no—
207C22×Q8—44no—
208C22×Q8—44no—
209C22×Q8—82no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×D414yes—
215C22×D4C2×Q814yes—
216C22×D4C2×Q814yes—
217C22×D4C2412yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
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×C8—22no—
253C22×C8—22no—
254C22×C8C2×D414yes—
255C22×C8C2×Q814yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—44no—
262C4⋊Q8—48no—
263#125—28no—
264#125—44no—
265#125—48no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—216no—
269C4×D4—28no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—48no—
274C4×D4C2×Q8116yescomplemented
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—432no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×Q814yes—
318C2×C4⋊C4C2×Q814yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—82no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346#107—28no—
347#107—28no—
348#107—28no—
349#107—44no—
350#106—28no—
351#106—44no—
352C4⋊C8—28no—
353C4⋊C8—28no—
354C4⋊C8—28no—
355#102—28no—
356#102—28no—
357#102C2×D4116yes—
358#102C2×Q8116yescomplemented
359#97—28no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95C2×D414yes—
368#95C2×D414yes—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22×D411yescharacteristic
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—84no—
381C2×Q8—816no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4C4⋊Q8116yes—
392C2×D4C22×D414yes—
393C2×D4C22×Q814yes—
394C2×D4C22×Q814yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
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—216no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—48no—
422C22×C4—432no—
423C22×C4—48no—
424C22×C4—48no—
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—48no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—816no—
449C22×C4—816no—
450C22×C4C22⋊Q814yes—
451C22×C4C22⋊Q814yes—
452C22×C4AGL(1,ℤ/8)14yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×Q812yes—
462C22×C4C22×Q812yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+412yes—
466C22×C4C2511yescharacteristic
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8C22×D414yes—
482C2×C8C22×Q814yes—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4C22×D414yes—
513C4⋊C4C22×Q814yes—
514C4⋊C4C22×Q814yes—
515C22⋊C4—216no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—84no—
525C42C22⋊Q818yes—
526C42C22×Q814yes—
527C42C2×#4114yes—
528C23—24no—
529C23—24no—
530C23—216no—
531C23—24no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C2×C22⋊Q812yes—
543C23D4212yes—
544C23D4×Q812yes—
545C23C2×AGL(1,ℤ/8)12yes—
546C23#57512yes—
547C23#57612yes—
548C23#57812yes—
549C23C23×D411yescharacteristic
550Q8—816no—
551D4—24no—
552D4—216no—
553D4—24no—
554D4—24no—
555D4C2×C4⋊Q818yes—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—432no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
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—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4—816no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4C2×C22⋊Q814yes—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q812yes—
611C2×C4C2×C4⋊Q814yes—
612C2×C4#54414yes—
613C2×C4D4212yes—
614C2×C4#54714yes—
615C2×C4D4×Q812yes—
616C2×C4#55414yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C23×Q811yescharacteristic
620C2×C4C23×Q811yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C8—24no—
625C8—24no—
626C8—42no—
627C8—42no—
628C22—28no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—432no—
636C22#244014yes—
637C22#307714yes—
638C22#308314yes—
639C22#326414yes—
640C22#326914yes—
641C22#338614yes—
642C22#338914yes—
643C22C22×C22⋊Q811yescharacteristic
644C22C2×D4211yescharacteristic
645C22C2×D4×Q811yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C2×#57511yescharacteristic
648C22C2×#57611yescharacteristic
649C22C2×#57811yescharacteristic
650C4—24no—
651C4—24no—
652C4—24no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#244014yes—
664C4C22×C4⋊Q811yescharacteristic
665C4C2×#54411yescharacteristic
666C4C2×#54711yescharacteristic
667C4C2×#55411yescharacteristic
668C2—48no—
669C2#2605218yesdirect product factor
670C2C2×#244011yescharacteristic
671C2C2×#307711yescharacteristic
672C2C2×#308311yescharacteristic
673C2C2×#326411yescharacteristic
674C2C2×#326911yescharacteristic
675C2C2×#338611yescharacteristic
676C2C2×#338911yescharacteristic
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. #26052C2direct (N × H)
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
8. C2×#2059C2semidirect
9. C2×#2057C2semidirect
10. C2×#1947C2semidirect
11. C2×#1940C2semidirect
14. C2×#1847C2semidirect
42. #2550C22semidirect
60. #2088C22semidirect
61. #2087C22semidirect
62. #2059C22semidirect
63. #2057C22semidirect
64. #1947C22semidirect
65. #1940C22semidirect
68. #1847C22semidirect
104. C2×C4×D4Q8semidirect
158. C2×#102Q8semidirect
274. C4×D4C2×Q8semidirect
358. #102C2×Q8semidirect
669. C2#26052direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4782
C22×#4742
C2×#25502
C2×D4×C4⋊C42
#260522
C2×#20872
C2×#20882
C2×#20592
C2×#20572
C2×#19472
C2×#19402
C2×#18642
C2×#18622
C2×#18472
C2×#18042
C2×#17622