Finite Group Databasebeta

← C2×#27116 #6446839

Subgroups — C2×#27116 #6446839

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic
3#27116C2116yesmaximal, direct product factor
4C2×#2088C211yesmaximal, characteristic, complemented
5C2×#2085C211yesmaximal, characteristic, complemented
6C2×#2036C211yesmaximal, characteristic
7C2×#2035C211yesmaximal, characteristic
8C2×#1935C211yesmaximal, characteristic, complemented
9C2×#1927C211yesmaximal, characteristic, complemented
10C2×#1924C211yesmaximal, characteristic
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1865C211yesmaximal, characteristic, complemented
13C2×#1853C211yesmaximal, characteristic
14C2×#1824C211yesmaximal, characteristic
15C2×#1805C211yesmaximal, characteristic, complemented
16C2×#1776C211yesmaximal, characteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×#97C2211yescharacteristic
32#2494—24no—
33#2494C2218yes—
34C2×#401—21no—
35C2×#397—21no—
36#2360—24no—
37#2360C2218yes—
38C2×#391C2211yescharacteristic
39C2×#389—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#387—21no—
45C2×#387—21no—
46C2×#385—21no—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4C2211yescharacteristic
54#2088C2218yescomplemented
55#2085C2218yescomplemented
56#2036C2218yes—
57#2035C2218yes—
58#1935C2218yescomplemented
59#1927C2218yescomplemented
60#1924C2218yes—
61#1920C2218yescomplemented
62#1865C2218yescomplemented
63#1853C2218yes—
64#1824C2218yes—
65#1805C2218yescomplemented
66#1776C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2314yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—24no—
90C2×C4×Q8—42no—
91C2×C4×Q8C2314yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—24no—
120C2×#107C2314yes—
121C2×#106—24no—
122C2×#106—24no—
123C2×#106—42no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8—42no—
128C2×C4⋊C8C2314yes—
129C2×#103—24no—
130C2×#103—24no—
131C2×#103—24no—
132C2×#103—42no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8—24no—
138C2×C22⋊C8—42no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97—24no—
142C2×#97C2314yes—
143#401—28no—
144#397—28no—
145#391—22no—
146#391C2314yes—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—28no—
151#389C2314yes—
152#389C2314yes—
153#389C2314yes—
154#388—28no—
155#387—28no—
156#387—28no—
157#385—28no—
158#382—22no—
159#382C2314yes—
160#380—22no—
161#380—22no—
162#380C2314yes—
163#380C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4C2314yes—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—28no—
169C4×C22⋊C4D414yes—
170C4×C22⋊C4C2314yes—
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×#95—21no—
181C2×#95—21no—
182C2×#95D411yescharacteristic
183C2×#95D411yescharacteristic
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8#4114yes—
202C22×Q8C2412yes—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—82no—
221C23×C4C2×D412yes—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8#4114yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22⋊Q8—216no—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C4×Q8—28no—
249C4×Q8—44no—
250C4×Q8—48no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
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—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—82no—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—82no—
313C2×C42—82no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318#107—28no—
319#106—28no—
320#106—28no—
321#106—44no—
322C4⋊C8—44no—
323#103—28no—
324#103—28no—
325#103—28no—
326#103—44no—
327C22⋊C8—28no—
328C22⋊C8—44no—
329#97—28no—
330#95—24no—
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—
340#95—24no—
341#95C2×D414yes—
342#95C2×D414yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24#13714yes—
350C24C22×D412yes—
351C24C22×D411yescharacteristic
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8C2×#4114yes—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
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—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
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—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—48no—
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—84no—
428C22×C4—816no—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+412yes—
441C22×C4C2511yescharacteristic
442C2×C8—28no—
443C2×C8—22no—
444C2×C8—22no—
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—44no—
455C2×C8—44no—
456C2×C8—44no—
457C2×C8C2×#4114yes—
458C4⋊C4—216no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
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—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4C2×#4114yes—
484C4⋊C4C2×#4114yes—
485C22⋊C4—216no—
486C22⋊C4—28no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—84no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—48no—
497C42—84no—
498C42—84no—
499C23—28no—
500C23—28no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23#52512yes—
513C23#54612yes—
514C23#54712yes—
515C23C2×#13714yes—
516C23C2×#13712yes—
517C23#57512yes—
518C23#57712yes—
519C23#57712yes—
520C23C23×D411yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—42no—
524Q8—42no—
525Q8—48no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
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—816no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4#52512yes—
585C2×C4#52914yes—
586C2×C4#54214yes—
587C2×C4#54612yes—
588C2×C4#54712yes—
589C2×C4#55214yes—
590C2×C4#55314yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×2−1+411yescharacteristic
597C8—24no—
598C8—42no—
599C8—42no—
600C8—42no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—416no—
609C22#246814yes—
610C22#311714yes—
611C22#312514yes—
612C22#331114yes—
613C22#332014yes—
614C22#334414yes—
615C22#335114yes—
616C22C2×#52511yescharacteristic
617C22C2×#54611yescharacteristic
618C22C2×#54711yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57511yescharacteristic
621C22C2×#57711yescharacteristic
622C22C2×#57711yescharacteristic
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#246814yes—
637C4C2×#52911yescharacteristic
638C4C2×#54211yescharacteristic
639C4C2×#55211yescharacteristic
640C4C2×#55311yescharacteristic
641C2—44no—
642C2#2711618yesdirect product factor
643C2C2×#246811yescharacteristic
644C2C2×#311711yescharacteristic
645C2C2×#312511yescharacteristic
646C2C2×#332011yescharacteristic
647C2C2×#331111yescharacteristic
648C2C2×#334411yescharacteristic
649C2C2×#335111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27116C2direct (N × H)
4. C2×#2088C2semidirect
5. C2×#2085C2semidirect
8. C2×#1935C2semidirect
9. C2×#1927C2semidirect
11. C2×#1920C2semidirect
12. C2×#1865C2semidirect
15. C2×#1805C2semidirect
16. C2×#1776C2semidirect
54. #2088C22semidirect
55. #2085C22semidirect
58. #1935C22semidirect
59. #1927C22semidirect
61. #1920C22semidirect
62. #1865C22semidirect
65. #1805C22semidirect
66. #1776C22semidirect
642. C2#27116direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24942
C2×#23602
#271162
C2×#20882
C2×#20852
C2×#20362
C2×#20352
C2×#19352
C2×#19272
C2×#19242
C2×#19202
C2×#18652
C2×#18532
C2×#18242
C2×#18052
C2×#17762