Finite Group Databasebeta

← C2×#29444 #6461600

Subgroups — C2×#29444 #6461600

Order 512.

2236 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic, complemented
3#29444C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2067C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2054C211yesmaximal, characteristic
8C2×#1972C211yesmaximal, characteristic
9C2×#1943C211yesmaximal, characteristic, complemented
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1919C211yesmaximal, characteristic, complemented
14C2×#1879C211yesmaximal, characteristic
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1767C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#126—21no—
19C22×C4×D4C2211yescharacteristic
20C22×#117—21no—
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#97C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2860—24no—
35#2860C2218yescomplemented
36#2293—24no—
37#2293C2218yescomplemented
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#388—21no—
51C2×#388—21no—
52C2×#388—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2088C2218yescomplemented
57#2067C2218yescomplemented
58#2065C2218yescomplemented
59#2054C2218yes—
60#1972C2218yes—
61#1943C2218yescomplemented
62#1936C2218yescomplemented
63#1936C2218yescomplemented
64#1926C2218yescomplemented
65#1919C2218yescomplemented
66#1879C2218yes—
67#1862C2218yescomplemented
68#1767C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×D8—24no—
81C22×D8—24no—
82C22×D8—42no—
83C22×D8C2314yes—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×#126—28no—
89C2×#126—48no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—22no—
93C2×C4×D4C2314yes—
94C2×#117—28no—
95C2×#117—48no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C2×#107—24no—
127C2×#107C2314yes—
128C2×#106—24no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8—24no—
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—24no—
140C2×#102—42no—
141C2×#102—42no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#97—42no—
149C2×#97—42no—
150C2×#97C2314yes—
151C2×#97C2314yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#390C2314yes—
159#390C2314yes—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#388—28no—
173#388—28no—
174#388—28no—
175#388—28no—
176#388—28no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—28no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C43D414yes—
187C43C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2314yes—
196C25—21no—
197C25—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4#4114yes—
212C22×D4C2412yes—
213C22×D4C2412yes—
214C23×C4—28no—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×D8—28no—
232C2×D8—28no—
233C2×D8—44no—
234C2×D8—416no—
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×C8#4114yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251#126—432no—
252C4×D4—216no—
253C4×D4—28no—
254C4×D4—28no—
255#117—432no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D412yes—
326C2×C42C2×D414yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#106—28no—
336#106—44no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#102—28no—
340#102—28no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345#97—44no—
346#97—44no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24C22×D411yescharacteristic
370C2×D4—24no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4C2×#4114yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—22no—
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—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—432no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—48no—
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×C4AGL(1,ℤ/8)14yes—
441C22×C4AGL(1,ℤ/8)14yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456D8—416no—
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×C8C2×#4114yes—
472C2×C8C2×#4114yes—
473C4⋊C4—24no—
474C4⋊C4—22no—
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—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4C2×#4114yes—
496C22⋊C4—216no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—28no—
511C42—28no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—44no—
519C42—44no—
520C42—48no—
521C23—24no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23#52512yes—
550C23#53212yes—
551C23#53712yes—
552C23C2×AGL(1,ℤ/8)12yes—
553C23C2×AGL(1,ℤ/8)12yes—
554C23#57512yes—
555C23#57512yes—
556C23C23×D411yescharacteristic
557D4—24no—
558D4—24no—
559D4—48no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—84no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—432no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
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—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#52512yes—
615C2×C4#53212yes—
616C2×C4#53712yes—
617C2×C4#55114yes—
618C2×C4#55214yes—
619C2×C4#55314yes—
620C2×C4#55614yes—
621C2×C4C2×AGL(1,ℤ/8)14yes—
622C2×C4C2×AGL(1,ℤ/8)14yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C8—24no—
630C8—24no—
631C8—42no—
632C8—42no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—82no—
637C22—816no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—168no—
649C22#260114yes—
650C22#310914yes—
651C22#311114yes—
652C22#317114yes—
653C22#318014yes—
654C22#325114yes—
655C22#325114yes—
656C22C2×#52511yescharacteristic
657C22C2×#53211yescharacteristic
658C22C2×#53711yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57511yescharacteristic
663C4—24no—
664C4—24no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#260114yes—
676C4C2×#55111yescharacteristic
677C4C2×#55211yescharacteristic
678C4C2×#55311yescharacteristic
679C4C2×#55611yescharacteristic
680C2—84no—
681C2—162no—
682C2#2944418yesdirect product factor
683C2C2×#260111yescharacteristic
684C2C2×#310911yescharacteristic
685C2C2×#311111yescharacteristic
686C2C2×#317111yescharacteristic
687C2C2×#318011yescharacteristic
688C2C2×#325111yescharacteristic
689C2C2×#325111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2860C2semidirect
2. C2×#2293C2semidirect
3. #29444C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2067C2semidirect
6. C2×#2065C2semidirect
9. C2×#1943C2semidirect
10. C2×#1936C2semidirect
11. C2×#1936C2semidirect
12. C2×#1926C2semidirect
13. C2×#1919C2semidirect
15. C2×#1862C2semidirect
16. C2×#1767C2semidirect
35. #2860C22semidirect
37. #2293C22semidirect
56. #2088C22semidirect
57. #2067C22semidirect
58. #2065C22semidirect
61. #1943C22semidirect
62. #1936C22semidirect
63. #1936C22semidirect
64. #1926C22semidirect
65. #1919C22semidirect
67. #1862C22semidirect
68. #1767C22semidirect
682. C2#29444direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28602
C2×#22932
#294442
C2×#20882
C2×#20672
C2×#20652
C2×#20542
C2×#19722
C2×#19432
C2×#19362
C2×#19262
C2×#19192
C2×#18792
C2×#18622
C2×#17672