Finite Group Databasebeta

← C2×#30153 #6434821

Subgroups — C2×#30153 #6434821

Order 512.

1800 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×C4×#125C211yesmaximal, characteristic, complemented
3#30153C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2086C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic
8C2×#2067C211yesmaximal, characteristic
9C2×#2065C211yesmaximal, characteristic
10C2×#2059C211yesmaximal, characteristic, complemented
11C2×#2025C211yesmaximal, characteristic
12C2×#1926C211yesmaximal, characteristic
13C2×#1925C211yesmaximal, characteristic
14C2×#1921C211yesmaximal, characteristic, complemented
15C2×#1920C211yesmaximal, characteristic, complemented
16C2×#1831C211yesmaximal, characteristic, complemented
17C22×#126—21no—
18C22×#125C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29#2860—24no—
30#2860C2218yes—
31C4×#125—24no—
32C4×#125C2218yescomplemented
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388—21no—
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×C43C2211yescharacteristic
52#2088C2218yescomplemented
53#2086C2218yescomplemented
54#2085C2218yescomplemented
55#2073C2218yes—
56#2067C2218yes—
57#2065C2218yes—
58#2059C2218yescomplemented
59#2025C2218yes—
60#1926C2218yes—
61#1925C2218yes—
62#1921C2218yescomplemented
63#1920C2218yescomplemented
64#1831C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#126—28no—
83C2×#126—48no—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—22no—
87C2×#125C2314yes—
88C22×C4⋊C4—21no—
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×C22⋊C4—21no—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C2×#107—24no—
118C2×#107—42no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106—24no—
122C2×#106—42no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8—42no—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#102—24no—
130C2×#102—24no—
131C2×#102—42no—
132C2×#102—42no—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137#390—22no—
138#390—22no—
139#390—22no—
140#390C2314yes—
141#390C2314yes—
142#390C2314yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389C2314yes—
149#389C2314yes—
150#389C2314yes—
151#389C2314yes—
152#389C2314yes—
153#388—28no—
154#388—28no—
155#388—28no—
156#387—28no—
157#387—28no—
158#382—22no—
159#382C2314yes—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4C2314yes—
165C4×C4⋊C4C2314yes—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C43D414yes—
176C43C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C25—21no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2411yescharacteristic
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228#126—432no—
229#125—216no—
230#125—28no—
231#125—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
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—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4#4114yes—
266C2×C4⋊C4#4114yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C2×D412yes—
301C2×C42C2×D414yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309C2×C42C2412yes—
310#107—28no—
311#107—44no—
312#106—28no—
313#106—44no—
314C4⋊C8—44no—
315C4⋊C8—44no—
316#102—28no—
317#102—28no—
318#102—44no—
319#102—44no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329C24—22no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24C22×D411yescharacteristic
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—44no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
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—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
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#13514yes—
396C22×C4AGL(1,ℤ/8)14yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408C22×C42−1+412yes—
409C22×C42−1+412yes—
410C22×C4C2511yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—44no—
427C4⋊C4—216no—
428C4⋊C4—216no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4C2×#4114yes—
454C4⋊C4C2×#4114yes—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—28no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42—44no—
474C42—48no—
475C42—48no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23#52512yes—
491C23#53612yes—
492C23#53712yes—
493C23C2×#13512yes—
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496C23#57512yes—
497C23C23×D411yescharacteristic
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—84no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
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—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4#52512yes—
554C2×C4#53612yes—
555C2×C4#53712yes—
556C2×C4#54114yes—
557C2×C4#54114yes—
558C2×C4#54214yes—
559C2×C4#54214yes—
560C2×C4C2×#13514yes—
561C2×C4C2×AGL(1,ℤ/8)14yes—
562C2×C4C22×#4111yescharacteristic
563C2×C4C22×#4111yescharacteristic
564C2×C4C2×H(5,𝔽2)11yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4C2×2−1+411yescharacteristic
568C8—42no—
569C8—42no—
570C8—42no—
571C8—42no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—168no—
580C22#263714yes—
581C22#310214yes—
582C22#311514yes—
583C22#323214yes—
584C22#323514yes—
585C22#324714yes—
586C22#325614yes—
587C22C2×#52511yescharacteristic
588C22C2×#53611yescharacteristic
589C22C2×#53711yescharacteristic
590C22C22×#13511yescharacteristic
591C22C22×AGL(1,ℤ/8)11yescharacteristic
592C22C2×#57511yescharacteristic
593C22C2×#57511yescharacteristic
594C4—24no—
595C4—24no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—44no—
600C4—42no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4#263714yes—
608C4C2×#54111yescharacteristic
609C4C2×#54111yescharacteristic
610C4C2×#54211yescharacteristic
611C4C2×#54211yescharacteristic
612C2—162no—
613C2#3015318yesdirect product factor
614C2C2×#263711yescharacteristic
615C2C2×#310211yescharacteristic
616C2C2×#311511yescharacteristic
617C2C2×#323211yescharacteristic
618C2C2×#323511yescharacteristic
619C2C2×#324711yescharacteristic
620C2C2×#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
2. C2×C4×#125C2semidirect
3. #30153C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2086C2semidirect
6. C2×#2085C2semidirect
10. C2×#2059C2semidirect
14. C2×#1921C2semidirect
15. C2×#1920C2semidirect
16. C2×#1831C2semidirect
32. C4×#125C22semidirect
52. #2088C22semidirect
53. #2086C22semidirect
54. #2085C22semidirect
58. #2059C22semidirect
62. #1921C22semidirect
63. #1920C22semidirect
64. #1831C22semidirect
613. C2#30153direct (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×C4×#1252
#301532
C2×#20882
C2×#20862
C2×#20852
C2×#20732
C2×#20672
C2×#20652
C2×#20592
C2×#20252
C2×#19262
C2×#19252
C2×#19212
C2×#19202
C2×#18312