Finite Group Databasebeta

← C2×#30000 #6435601

Subgroups — C2×#30000 #6435601

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#30000C2116yesmaximal, direct product factor
4C2×#2087C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2086C211yesmaximal, characteristic, complemented
8C2×#2067C211yesmaximal, characteristic
9C2×#2067C211yesmaximal, characteristic
10C2×#2065C211yesmaximal, characteristic
11C2×#2065C211yesmaximal, characteristic
12C2×#1949C211yesmaximal, characteristic, complemented
13C2×#1949C211yesmaximal, characteristic, complemented
14C2×#1842C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×#1804C211yesmaximal, characteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117—21no—
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33#2864—24no—
34#2864C2218yescomplemented
35#2293—24no—
36#2293C2218yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#388—21no—
47C2×#387—21no—
48C2×#387—21no—
49C2×#387—21no—
50C2×#387—21no—
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2088C2218yescomplemented
57#2088C2218yescomplemented
58#2087C2218yescomplemented
59#2086C2218yescomplemented
60#2067C2218yes—
61#2067C2218yes—
62#2065C2218yes—
63#2065C2218yes—
64#1949C2218yescomplemented
65#1949C2218yescomplemented
66#1842C2218yescomplemented
67#1804C2218yes—
68#1804C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4—21no—
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C4×D4—22no—
84C2×C4×D4—24no—
85C2×C4×D4Q818yes—
86C2×C4×D4C2314yes—
87C2×#117—28no—
88C2×#117—48no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C42—21no—
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C2×#107—42no—
116C2×#107—42no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106—42no—
120C2×#106—42no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8—24no—
124C2×C4⋊C8—24no—
125C2×C4⋊C8—42no—
126C2×C4⋊C8—42no—
127C2×C4⋊C8—42no—
128C2×C4⋊C8—42no—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#102—42no—
136C2×#102—42no—
137C2×#102—42no—
138C2×#102—42no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#388—28no—
160#388—28no—
161#387—28no—
162#387—28no—
163#387—28no—
164#387—28no—
165#382—22no—
166#382—22no—
167#382C2314yes—
168#382C2314yes—
169C4×C4⋊C4—22no—
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—
181C2×#95—21no—
182C2×#95—21no—
183C43D414yes—
184C43C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
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—28no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—21no—
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—22no—
231C22×C8—22no—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C4×D4—216no—
237C4×D4—28no—
238C4×D4C2×Q8116yes—
239#117—432no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42C2×D414yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D412yes—
307C2×C42C2×Q814yes—
308C2×C42#4114yes—
309C2×C42C2412yes—
310C2×C42C2412yes—
311C2×C42C2412yes—
312#107—44no—
313#107—44no—
314#106—44no—
315#106—44no—
316C4⋊C8—28no—
317C4⋊C8—28no—
318C4⋊C8—44no—
319C4⋊C8—44no—
320C4⋊C8—44no—
321C4⋊C8—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337C24—24no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C22×D411yescharacteristic
346C2×D4—24no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4C22×Q814yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—48no—
383C22×C4—432no—
384C22×C4—48no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
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×C4C22⋊Q814yes—
406C22×C4C22⋊Q814yes—
407C22×C4#13514yes—
408C22×C4AGL(1,ℤ/8)14yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q812yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+412yes—
421C22×C42−1+412yes—
422C22×C4C2511yescharacteristic
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4C22×Q814yes—
466C22⋊C4—216no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C42—24no—
472C42—24no—
473C42—28no—
474C42—28no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—44no—
480C42—44no—
481C42—48no—
482C42C22⋊Q818yes—
483C42C22×Q814yes—
484C42C2×#4114yes—
485C23—24no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23C2×C22⋊Q812yes—
500C23#53712yes—
501C23#53712yes—
502C23C2×#13512yes—
503C23C2×AGL(1,ℤ/8)12yes—
504C23#57512yes—
505C23#57512yes—
506C23C23×D411yescharacteristic
507D4—24no—
508D4—24no—
509D4—48no—
510D4—42no—
511D4—42no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—432no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4C2×C22⋊Q814yes—
562C2×C4C2×C22⋊Q814yes—
563C2×C4C2×C22⋊Q812yes—
564C2×C4#53712yes—
565C2×C4#53712yes—
566C2×C4#54014yes—
567C2×C4#54114yes—
568C2×C4#54414yes—
569C2×C4#54414yes—
570C2×C4C2×#13514yes—
571C2×C4C2×AGL(1,ℤ/8)14yes—
572C2×C4C23×Q811yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4C2×H(5,𝔽2)11yescharacteristic
575C2×C4C2×H(5,𝔽2)11yescharacteristic
576C2×C4C2×2−1+411yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C8—42no—
579C8—42no—
580C8—42no—
581C8—42no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—816no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22#262914yes—
591C22#307614yes—
592C22#307714yes—
593C22#324714yes—
594C22#325014yes—
595C22#325114yes—
596C22#325614yes—
597C22C22×C22⋊Q811yescharacteristic
598C22C2×#53711yescharacteristic
599C22C2×#53711yescharacteristic
600C22C22×#13511yescharacteristic
601C22C22×AGL(1,ℤ/8)11yescharacteristic
602C22C2×#57511yescharacteristic
603C22C2×#57511yescharacteristic
604C4—24no—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4#262914yes—
618C4C2×#54011yescharacteristic
619C4C2×#54111yescharacteristic
620C4C2×#54411yescharacteristic
621C4C2×#54411yescharacteristic
622C2—84no—
623C2#3000018yesdirect product factor
624C2C2×#262911yescharacteristic
625C2C2×#307611yescharacteristic
626C2C2×#307711yescharacteristic
627C2C2×#324711yescharacteristic
628C2C2×#325011yescharacteristic
629C2C2×#325111yescharacteristic
630C2C2×#325611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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×#2864C2semidirect
3. #30000C2direct (N × H)
4. C2×#2087C2semidirect
5. C2×#2088C2semidirect
6. C2×#2088C2semidirect
7. C2×#2086C2semidirect
12. C2×#1949C2semidirect
13. C2×#1949C2semidirect
14. C2×#1842C2semidirect
34. #2864C22semidirect
56. #2088C22semidirect
57. #2088C22semidirect
58. #2087C22semidirect
59. #2086C22semidirect
64. #1949C22semidirect
65. #1949C22semidirect
66. #1842C22semidirect
623. C2#30000direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#22932
#300002
C2×#20872
C2×#20882
C2×#20862
C2×#20672
C2×#20652
C2×#19492
C2×#18422
C2×#18042