Finite Group Databasebeta

← C2×#26049 #6443361

Subgroups — C2×#26049 #6443361

Order 512.

2424 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×#478C211yesmaximal, characteristic
2C22×#474C211yesmaximal, characteristic
3C2×#2550C211yesmaximal, characteristic, complemented
4C2×#2349C211yesmaximal, characteristic
5#26049C2116yesmaximal, direct product factor
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2058C211yesmaximal, characteristic, complemented
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1939C211yesmaximal, characteristic, complemented
12C2×#1864C211yesmaximal, characteristic
13C2×#1864C211yesmaximal, characteristic
14C2×#1846C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#478—28no—
23C2×#478C2218yes—
24C2×#474—28no—
25C2×#474C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×C4⋊C8C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2550—24no—
41#2550C2218yescomplemented
42C2×#398—21no—
43C2×#395—21no—
44#2349—24no—
45#2349C2218yes—
46C2×#391—21no—
47C2×#389—21no—
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#385—21no—
53C2×#385—21no—
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#380—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C22×#95—21no—
60#2087C2218yescomplemented
61#2086C2218yescomplemented
62#2059C2218yescomplemented
63#2058C2218yescomplemented
64#1949C2218yescomplemented
65#1939C2218yescomplemented
66#1864C2218yes—
67#1864C2218yes—
68#1846C2218yescomplemented
69#1804C2218yes—
70C4×#102C2218yes—
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⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#478—232no—
132#474—232no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106—24no—
140C2×#106—42no—
141C2×#106—42no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8—24no—
146C2×C4⋊C8—24no—
147C2×C4⋊C8—24no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102D418yes—
154C2×#102Q818yescomplemented
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161#398—28no—
162#395—28no—
163#391—28no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—28no—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#389C2314yes—
173#385—28no—
174#385—28no—
175#382—22no—
176#382—22no—
177#382C2314yes—
178#382C2314yes—
179#380—28no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D414yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
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×#95—21no—
193C2×#95—28no—
194C2×#95D411yescharacteristic
195C2×#95D411yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2314yes—
203#336C2314yes—
204C25—21no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—44no—
208C22×Q8—82no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4C2×D414yes—
214C22×D4C2×Q814yes—
215C22×D4C2×Q814yes—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—82no—
233C23×C4—82no—
234C23×C4—82no—
235C23×C4—82no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
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×C8—22no—
253C22×C8C2×D414yes—
254C22×C8C2×Q814yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—44no—
261C4⋊Q8—48no—
262#125—28no—
263#125—44no—
264#125—48no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—216no—
268C4×D4—28no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—48no—
272C4×D4—48no—
273C4×D4C2×Q8116yescomplemented
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
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—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×Q814yes—
312C2×C4⋊C4C2×Q814yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—82no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×Q814yes—
336C2×C42#4114yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#107—28no—
342#107—28no—
343#106—28no—
344#106—44no—
345#106—44no—
346C4⋊C8—28no—
347C4⋊C8—28no—
348C4⋊C8—28no—
349#102—28no—
350#102—28no—
351#102C2×D4116yes—
352#102C2×Q8116yescomplemented
353C4×C8—28no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24C22×D411yescharacteristic
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—84no—
377C2×Q8—816no—
378C2×D4—24no—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4C4⋊Q8116yes—
388C2×D4C22×D414yes—
389C2×D4C22×Q814yes—
390C2×D4C22×Q814yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
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—48no—
439C22×C4—816no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—816no—
445C22×C4—816no—
446C22×C4C22⋊Q814yes—
447C22×C4C22⋊Q814yes—
448C22×C4#13514yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q812yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463C2×C8—28no—
464C2×C8—28no—
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—22no—
475C2×C8—44no—
476C2×C8—44no—
477C2×C8C22×D414yes—
478C2×C8C22×Q814yes—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—28no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4C22×D414yes—
507C4⋊C4C22×Q814yes—
508C4⋊C4C22×Q814yes—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—84no—
519C42C22⋊Q818yes—
520C42C22×Q814yes—
521C42C2×#4114yes—
522C23—24no—
523C23—24no—
524C23—216no—
525C23—24no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×C22⋊Q812yes—
537C23D4212yes—
538C23D4×Q812yes—
539C23C2×#13512yes—
540C23#57512yes—
541C23#57612yes—
542C23#57712yes—
543C23C23×D411yescharacteristic
544Q8—816no—
545D4—24no—
546D4—216no—
547D4—24no—
548D4—24no—
549D4C2×C4⋊Q818yes—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
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—42no—
595C2×C4—816no—
596C2×C4—816no—
597C2×C4—816no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C2×C22⋊Q814yes—
603C2×C4C2×C22⋊Q814yes—
604C2×C4C2×C22⋊Q812yes—
605C2×C4C2×C4⋊Q814yes—
606C2×C4#54414yes—
607C2×C4D4212yes—
608C2×C4#54714yes—
609C2×C4D4×Q812yes—
610C2×C4#55414yes—
611C2×C4C2×#13514yes—
612C2×C4C23×D411yescharacteristic
613C2×C4C23×Q811yescharacteristic
614C2×C4C23×Q811yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C8—24no—
619C8—24no—
620C8—42no—
621C8—42no—
622C22—28no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—432no—
630C22#244014yes—
631C22#307614yes—
632C22#308114yes—
633C22#327514yes—
634C22#328014yes—
635C22#338114yes—
636C22#338914yes—
637C22C22×C22⋊Q811yescharacteristic
638C22C2×D4211yescharacteristic
639C22C2×D4×Q811yescharacteristic
640C22C22×#13511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57711yescharacteristic
644C4—24no—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4#244014yes—
658C4C22×C4⋊Q811yescharacteristic
659C4C2×#54411yescharacteristic
660C4C2×#54711yescharacteristic
661C4C2×#55411yescharacteristic
662C2—48no—
663C2#2604918yesdirect product factor
664C2C2×#244011yescharacteristic
665C2C2×#307611yescharacteristic
666C2C2×#308111yescharacteristic
667C2C2×#327511yescharacteristic
668C2C2×#328011yescharacteristic
669C2C2×#338111yescharacteristic
670C2C2×#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. #26049C2direct (N × H)
6. C2×#2087C2semidirect
7. C2×#2086C2semidirect
8. C2×#2059C2semidirect
9. C2×#2058C2semidirect
10. C2×#1949C2semidirect
11. C2×#1939C2semidirect
14. C2×#1846C2semidirect
41. #2550C22semidirect
60. #2087C22semidirect
61. #2086C22semidirect
62. #2059C22semidirect
63. #2058C22semidirect
64. #1949C22semidirect
65. #1939C22semidirect
68. #1846C22semidirect
104. C2×C4×D4Q8semidirect
154. C2×#102Q8semidirect
273. C4×D4C2×Q8semidirect
352. #102C2×Q8semidirect
663. C2#26049direct (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×#23492
#260492
C2×#20872
C2×#20862
C2×#20592
C2×#20582
C2×#19492
C2×#19392
C2×#18642
C2×#18462
C2×#18042
C2×C4×#1022