Finite Group Databasebeta

← C2×#23778 #6432517

Subgroups — C2×#23778 #6432517

Order 512.

2162 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#23778C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2092C211yesmaximal, characteristic
6C2×#2087C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic
12C2×C4×#107C211yesmaximal, characteristic, complemented
13C2×C4×#106C211yesmaximal, characteristic, complemented
14C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
15C2×C4×#102C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C4×#125—24no—
33C4×#125C2218yescomplemented
34#2293—24no—
35#2293C2218yes—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#388—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#382C2211yescharacteristic
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C43C2211yescharacteristic
54#2101C2218yescomplemented
55#2092C2218yes—
56#2088C2218yescomplemented
57#2087C2218yescomplemented
58#2067C2218yes—
59#2065C2218yes—
60#2059C2218yescomplemented
61#2025C2218yes—
62C4×#107C2218yescomplemented
63C4×#106C2218yescomplemented
64C4×C4⋊C8C2218yescomplemented
65C4×#102C2218yes—
66C4×#102C2218yes—
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×#125—24no—
84C2×#125—24no—
85C2×#125—22no—
86C2×#125C2314yes—
87C2×C4×D4—22no—
88C2×C4×D4—24no—
89C2×C4×D4Q818yes—
90C2×C4×D4C2314yes—
91C2×#117—28no—
92C2×#117—48no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C2×#107—24no—
122C2×#107—24no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106—24no—
126C2×#106—24no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8—24no—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C4×C8—24no—
142C2×C4×C8—24no—
143C2×C4×C8C2314yes—
144C2×C4×C8C2314yes—
145#390—22no—
146#390—22no—
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#388—28no—
158#388—28no—
159#387—28no—
160#387—28no—
161#382—22no—
162#382C2314yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—28no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C43D414yes—
182C43C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4C2×Q814yes—
201C22×D4C2412yes—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8#4114yes—
231C22×C8#4114yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236#125—216no—
237#125—28no—
238#125—28no—
239C4×D4—216no—
240C4×D4—28no—
241C4×D4C2×Q8116yes—
242#117—432no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4#4114yes—
275C2×C4⋊C4#4114yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C2×D414yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D412yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2412yes—
319#107—28no—
320#107—28no—
321#106—28no—
322#106—28no—
323C4⋊C8—28no—
324C4⋊C8—28no—
325#102—28no—
326#102—28no—
327#102—28no—
328#102—28no—
329C4×C8—28no—
330C4×C8—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340C24—24no—
341C24—22no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24C22×D411yescharacteristic
349C2×D4—24no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4C22×Q814yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—432no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4C22⋊Q814yes—
408C22×C4C22⋊Q814yes—
409C22×C4#13514yes—
410C22×C4#13514yes—
411C22×C4AGL(1,ℤ/8)14yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×Q812yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2511yescharacteristic
426C2×C8—28no—
427C2×C8—28no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—44no—
439C2×C8—44no—
440C2×C8C2×#4114yes—
441C2×C8C2×#4114yes—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4C22×Q814yes—
467C4⋊C4C2×#4114yes—
468C4⋊C4C2×#4114yes—
469C22⋊C4—216no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C42—24no—
475C42—24no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—44no—
484C42—48no—
485C42—48no—
486C42—48no—
487C42C22⋊Q818yes—
488C42C22×Q814yes—
489C42C2×#4114yes—
490C23—24no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23C2×C22⋊Q812yes—
505C23#52512yes—
506C23#53212yes—
507C23C2×#13512yes—
508C23C2×#13512yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23#57512yes—
511C23C23×D411yescharacteristic
512D4—24no—
513D4—24no—
514D4—48no—
515D4—42no—
516D4—42no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—432no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4#478116yes—
565C2×C4C2×C22⋊Q814yes—
566C2×C4C2×C22⋊Q814yes—
567C2×C4C2×C22⋊Q812yes—
568C2×C4#52512yes—
569C2×C4#52914yes—
570C2×C4#52914yes—
571C2×C4#53212yes—
572C2×C4#53314yes—
573C2×C4#53314yes—
574C2×C4C2×#13514yes—
575C2×C4C2×#13514yes—
576C2×C4C2×AGL(1,ℤ/8)14yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C8—24no—
584C8—24no—
585C8—42no—
586C8—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—816no—
594C22—82no—
595C22#239614yes—
596C22C2×#47814yes—
597C22#307614yes—
598C22#310214yes—
599C22#310614yes—
600C22#316814yes—
601C22#317114yes—
602C22C22×C22⋊Q811yescharacteristic
603C22C2×#52511yescharacteristic
604C22C2×#53211yescharacteristic
605C22C22×#13511yescharacteristic
606C22C22×#13511yescharacteristic
607C22C22×AGL(1,ℤ/8)11yescharacteristic
608C22C2×#57511yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#239614yes—
622C4C2×#47818yes—
623C4C2×#52911yescharacteristic
624C4C2×#52911yescharacteristic
625C4C2×#53311yescharacteristic
626C4C2×#53311yescharacteristic
627C2—84no—
628C2#2377818yesdirect product factor
629C2C2×#239611yescharacteristic
630C2C22×#47811yescharacteristic
631C2C2×#307611yescharacteristic
632C2C2×#310211yescharacteristic
633C2C2×#310611yescharacteristic
634C2C2×#316811yescharacteristic
635C2C2×#317111yescharacteristic
subgroupG/H
centerC23×C4C2×D4
commutator, Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#125C2semidirect
3. #23778C2direct (N × H)
4. C2×#2101C2semidirect
6. C2×#2087C2semidirect
7. C2×#2088C2semidirect
10. C2×#2059C2semidirect
12. C2×C4×#107C2semidirect
13. C2×C4×#106C2semidirect
14. C2×C4×C4⋊C8C2semidirect
33. C4×#125C22semidirect
54. #2101C22semidirect
56. #2088C22semidirect
57. #2087C22semidirect
60. #2059C22semidirect
62. C4×#107C22semidirect
63. C4×#106C22semidirect
64. C4×C4⋊C8C22semidirect
628. C2#23778direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#22932
#237782
C2×#21012
C2×#20922
C2×#20872
C2×#20882
C2×#20672
C2×#20652
C2×#20592
C2×#20252
C2×C4×#1072
C2×C4×#1062
C2×C4×C4⋊C82
C2×C4×#1022