Finite Group Databasebeta

← C2×#26542 #6447280

Subgroups — C2×#26542 #6447280

Order 512.

2116 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic
3#26542C2116yesmaximal, direct product factor
4C2×#2095C211yesmaximal, characteristic, complemented
5C2×#2093C211yesmaximal, characteristic
6C2×#2068C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#1983C211yesmaximal, characteristic
9C2×#1959C211yesmaximal, characteristic
10C2×#1947C211yesmaximal, characteristic, complemented
11C2×#1940C211yesmaximal, characteristic, complemented
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1864C211yesmaximal, characteristic
14C2×#1803C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×#125—21no—
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#97C2211yescharacteristic
37#2552—24no—
38#2552C2218yescomplemented
39C2×#398—21no—
40C2×#398—21no—
41#2367—24no—
42#2367C2218yes—
43C2×#391C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#385—21no—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4—21no—
58#2095C2218yescomplemented
59#2093C2218yes—
60#2068C2218yescomplemented
61#2067C2218yes—
62#1983C2218yes—
63#1959C2218yes—
64#1947C2218yescomplemented
65#1940C2218yescomplemented
66#1865C2218yescomplemented
67#1864C2218yes—
68#1803C2218yescomplemented
69#1802C2218yes—
70#1776C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#125—28no—
85C2×#125—48no—
86C2×#124—22no—
87C2×#124—24no—
88C2×#124Q818yes—
89C2×#124C2314yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—42no—
93C2×C4×Q8C2314yes—
94C2×C4×D4—24no—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4C2314yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—24no—
127C2×#107—42no—
128C2×#107—42no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106—24no—
133C2×#106C2314yes—
134C2×C4⋊C8—24no—
135C2×C4⋊C8—42no—
136C2×C4⋊C8—42no—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×#103—24no—
141C2×#103—42no—
142C2×#103—42no—
143C2×#103—42no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102—42no—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97—24no—
157C2×#97C2314yes—
158#398—28no—
159#398—28no—
160#391—22no—
161#391C2314yes—
162#390—22no—
163#390C2314yes—
164#389—28no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#388—28no—
170#388—28no—
171#385—28no—
172#384—22no—
173#384—22no—
174#384—22no—
175#384C2314yes—
176#384C2314yes—
177#384C2314yes—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D414yes—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2314yes—
184C4×C22⋊C4—28no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95D411yescharacteristic
192C2×#95D411yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C22×Q8—21no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8C2×Q814yes—
207C22×Q8C2412yes—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×Q814yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—24no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—82no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
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#4114yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#125—432no—
251#124—216no—
252#124—28no—
253#124C2×Q8116yes—
254C4×Q8—28no—
255C4×Q8—44no—
256C4×Q8—48no—
257C4×D4—28no—
258C4×D4—44no—
259C4×D4—48no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
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—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—82no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—82no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×Q814yes—
330C2×C42#4114yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#107—44no—
336#107—44no—
337#106—28no—
338C4⋊C8—28no—
339C4⋊C8—44no—
340C4⋊C8—44no—
341#103—28no—
342#103—44no—
343#103—44no—
344#103—44no—
345#102—28no—
346#102—44no—
347#102—44no—
348#102—44no—
349#97—28no—
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—24no—
359C24—22no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24C22×D411yescharacteristic
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8C22×Q814yes—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4C22×Q814yes—
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—24no—
402C22×C4—22no—
403C22×C4—22no—
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—48no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
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—48no—
437C22×C4—84no—
438C22×C4—816no—
439C22×C4C22⋊Q814yes—
440C22×C4C22⋊Q814yes—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4#13714yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4C2511yescharacteristic
457C2×C8—28no—
458C2×C8—22no—
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×C8—44no—
472C2×C8C2×#4114yes—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—28no—
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—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4C2×#4114yes—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—48no—
514C42—48no—
515C42—84no—
516C42C22⋊Q818yes—
517C42C22×Q814yes—
518C42C2×#4114yes—
519C23—24no—
520C23—48no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23C2×C22⋊Q812yes—
534C23#54612yes—
535C23#55012yes—
536C23C2×AGL(1,ℤ/8)12yes—
537C23C2×#13712yes—
538C23#57612yes—
539C23#57812yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—48no—
546D4—24no—
547D4—24no—
548D4—42no—
549D4—42no—
550D4—48no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—48no—
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—816no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4C2×C22⋊Q814yes—
601C2×C4C2×C22⋊Q814yes—
602C2×C4C2×C22⋊Q812yes—
603C2×C4#52914yes—
604C2×C4#54014yes—
605C2×C4#54612yes—
606C2×C4#55012yes—
607C2×C4#55414yes—
608C2×C4#55414yes—
609C2×C4C2×AGL(1,ℤ/8)14yes—
610C2×C4C2×#13714yes—
611C2×C4C23×Q811yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C8—24no—
618C8—42no—
619C8—42no—
620C8—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—816no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#245414yes—
630C22#307914yes—
631C22#308314yes—
632C22#330814yes—
633C22#331414yes—
634C22#339814yes—
635C22#339914yes—
636C22C22×C22⋊Q811yescharacteristic
637C22C2×#54611yescharacteristic
638C22C2×#55011yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C22×#13711yescharacteristic
641C22C2×#57611yescharacteristic
642C22C2×#57811yescharacteristic
643C4—24no—
644C4—24no—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4#245414yes—
657C4C2×#52911yescharacteristic
658C4C2×#54011yescharacteristic
659C4C2×#55411yescharacteristic
660C4C2×#55411yescharacteristic
661C2—84no—
662C2#2654218yesdirect product factor
663C2C2×#245411yescharacteristic
664C2C2×#307911yescharacteristic
665C2C2×#308311yescharacteristic
666C2C2×#331411yescharacteristic
667C2C2×#330811yescharacteristic
668C2C2×#339811yescharacteristic
669C2C2×#339911yescharacteristic
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
1. C2×#2552C2semidirect
3. #26542C2direct (N × H)
4. C2×#2095C2semidirect
6. C2×#2068C2semidirect
10. C2×#1947C2semidirect
11. C2×#1940C2semidirect
12. C2×#1865C2semidirect
14. C2×#1803C2semidirect
16. C2×#1776C2semidirect
38. #2552C22semidirect
58. #2095C22semidirect
60. #2068C22semidirect
64. #1947C22semidirect
65. #1940C22semidirect
66. #1865C22semidirect
68. #1803C22semidirect
70. #1776C22semidirect
662. C2#26542direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23672
#265422
C2×#20952
C2×#20932
C2×#20682
C2×#20672
C2×#19832
C2×#19592
C2×#19472
C2×#19402
C2×#18652
C2×#18642
C2×#18032
C2×#18022
C2×#17762