Finite Group Databasebeta

← C2×#27451 #6450201

Subgroups — C2×#27451 #6450201

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2454C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic, complemented
3#27451C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2073C211yesmaximal, characteristic
6C2×#2055C211yesmaximal, characteristic
7C2×#2026C211yesmaximal, characteristic, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1935C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic
11C2×#1919C211yesmaximal, characteristic, complemented
12C2×#1917C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2454—24no—
31#2454C2218yes—
32C2×#399—21no—
33C2×#396—21no—
34#2356—24no—
35#2356C2218yescomplemented
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#387—21no—
44C2×#385—21no—
45C2×#383—21no—
46C2×#382C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376—21no—
52#2101C2218yescomplemented
53#2073C2218yes—
54#2055C2218yes—
55#2026C2218yescomplemented
56#1989C2218yescomplemented
57#1935C2218yescomplemented
58#1926C2218yes—
59#1919C2218yescomplemented
60#1917C2218yescomplemented
61#1880C2218yes—
62#1877C2218yes—
63#1865C2218yescomplemented
64#1762C2218yes—
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×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×SD16—24no—
81C22×SD16—24no—
82C22×SD16—42no—
83C22×SD16C2314yes—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C2×#125—24no—
89C2×#125—22no—
90C2×#125—42no—
91C2×#125C2314yes—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—22no—
95C2×C4×Q8C2314yes—
96C2×C4×D4—24no—
97C2×C4×D4—22no—
98C2×C4×D4—42no—
99C2×C4×D4C2314yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C2×#106—24no—
125C2×#106C2314yes—
126C2×C4⋊C8—42no—
127C2×C4⋊C8C2314yes—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—42no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#97—42no—
141C2×#97C2314yes—
142#399—28no—
143#396—28no—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2314yes—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#387—28no—
157#387—28no—
158#385—28no—
159#383—28no—
160#382—22no—
161#382C2314yes—
162#380—28no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4D414yes—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C4⋊C4C2314yes—
169#376—28no—
170C2×#95—21no—
171C2×#95—21no—
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×#95D411yescharacteristic
181C2×#95D411yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8#4114yes—
202C22×Q8C2412yes—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4#4114yes—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—82no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×SD16—28no—
230C2×SD16—28no—
231C2×SD16—44no—
232C2×SD16—416no—
233C22×C8—22no—
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×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249#125—28no—
250#125—44no—
251#125—48no—
252C4×Q8—216no—
253C4×Q8—28no—
254C4×Q8—28no—
255C4×D4—28no—
256C4×D4—44no—
257C4×D4—48no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
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—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4#4114yes—
289C2×C4⋊C4#4114yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323C2×C42C2412yes—
324#106—28no—
325C4⋊C8—44no—
326#103—28no—
327#103—28no—
328#103—44no—
329#102—28no—
330#102—28no—
331#102—28no—
332#97—44no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95C2×D414yes—
344#95C2×D414yes—
345C24—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24C22×D411yescharacteristic
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8C2×#4114yes—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—44no—
379C2×D4C2×#4114yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
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—84no—
429C22×C4—816no—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C22×C4C2511yescharacteristic
444SD16—416no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C4⋊C4—216no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4114yes—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C42—216no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—48no—
502C42—48no—
503C23—24no—
504C23—48no—
505C23—48no—
506C23—42no—
507C23—42no—
508C23—48no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—84no—
514C23—84no—
515C23—84no—
516C23—84no—
517C23#52512yes—
518C23#54612yes—
519C23#54812yes—
520C23#57512yes—
521C23#57512yes—
522C23#57712yes—
523C23#57712yes—
524C23C23×D411yescharacteristic
525Q8—24no—
526Q8—24no—
527Q8—42no—
528Q8—42no—
529Q8—48no—
530D4—24no—
531D4—24no—
532D4—42no—
533D4—42no—
534D4—48no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—816no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#52512yes—
589C2×C4#52914yes—
590C2×C4#54214yes—
591C2×C4#54612yes—
592C2×C4#54812yes—
593C2×C4#55114yes—
594C2×C4#55214yes—
595C2×C4C22×#4111yescharacteristic
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4C2×H(5,𝔽2)11yescharacteristic
600C2×C4C2×2−1+411yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—816no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#247714yes—
614C22#311114yes—
615C22#312614yes—
616C22#331714yes—
617C22#332314yes—
618C22#337614yes—
619C22#337614yes—
620C22C2×#52511yescharacteristic
621C22C2×#54611yescharacteristic
622C22C2×#54811yescharacteristic
623C22C2×#57511yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57711yescharacteristic
626C22C2×#57711yescharacteristic
627C4—42no—
628C4—44no—
629C4—42no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#247714yes—
641C4C2×#52911yescharacteristic
642C4C2×#54211yescharacteristic
643C4C2×#55111yescharacteristic
644C4C2×#55211yescharacteristic
645C2—84no—
646C2#2745118yesdirect product factor
647C2C2×#247711yescharacteristic
648C2C2×#311111yescharacteristic
649C2C2×#312611yescharacteristic
650C2C2×#331711yescharacteristic
651C2C2×#332311yescharacteristic
652C2C2×#337611yescharacteristic
653C2C2×#337611yescharacteristic
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
2. C2×#2356C2semidirect
3. #27451C2direct (N × H)
4. C2×#2101C2semidirect
7. C2×#2026C2semidirect
8. C2×#1989C2semidirect
9. C2×#1935C2semidirect
11. C2×#1919C2semidirect
12. C2×#1917C2semidirect
15. C2×#1865C2semidirect
35. #2356C22semidirect
52. #2101C22semidirect
55. #2026C22semidirect
56. #1989C22semidirect
57. #1935C22semidirect
59. #1919C22semidirect
60. #1917C22semidirect
63. #1865C22semidirect
646. C2#27451direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24542
C2×#23562
#274512
C2×#21012
C2×#20732
C2×#20552
C2×#20262
C2×#19892
C2×#19352
C2×#19262
C2×#19192
C2×#19172
C2×#18802
C2×#18772
C2×#18652
C2×#17622