Finite Group Databasebeta

← C2×#27175 #6450828

Subgroups — C2×#27175 #6450828

Order 512.

2218 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2422C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic, complemented
3#27175C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2075C211yesmaximal, characteristic, complemented
6C2×#2065C211yesmaximal, characteristic, complemented
7C2×#2033C211yesmaximal, characteristic
8C2×#1947C211yesmaximal, characteristic, complemented
9C2×#1944C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic, complemented
11C2×#1924C211yesmaximal, characteristic, complemented
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1874C211yesmaximal, characteristic
14C2×#1862C211yesmaximal, characteristic, complemented
15C2×#1853C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C22×#123—21no—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#2422—24no—
34#2422C2218yes—
35C2×#397—21no—
36C2×#394—21no—
37#2345—24no—
38#2345C2218yescomplemented
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#387—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4—21no—
53C2×C4×C22⋊C4C2211yescharacteristic
54#2100C2218yescomplemented
55#2075C2218yescomplemented
56#2065C2218yescomplemented
57#2033C2218yes—
58#1947C2218yescomplemented
59#1944C2218yescomplemented
60#1926C2218yescomplemented
61#1924C2218yescomplemented
62#1920C2218yescomplemented
63#1874C2218yes—
64#1862C2218yescomplemented
65#1853C2218yescomplemented
66#1762C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C2×#123—28no—
87C2×#123—48no—
88C2×#121—22no—
89C2×#121—24no—
90C2×#121—42no—
91C2×#121C2314yes—
92C2×C4×D4—24no—
93C2×C4×D4—22no—
94C2×C4×D4—42no—
95C2×C4×D4C2314yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
119C22×C22⋊C4D411yescharacteristic
120C22×C22⋊C4D411yescharacteristic
121C22×C42—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—42no—
126C2×#107—42no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106—24no—
130C2×#106—24no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—24no—
134C2×C4⋊C8C2314yes—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8—24no—
144C2×C22⋊C8—42no—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2314yes—
147C2×#97—42no—
148C2×#97C2314yes—
149#397—28no—
150#394—28no—
151#390—22no—
152#390C2314yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389C2314yes—
157#389C2314yes—
158#389C2314yes—
159#387—28no—
160#387—28no—
161#385—28no—
162#384—22no—
163#384C2314yes—
164#380—22no—
165#380—22no—
166#380—28no—
167#380C2314yes—
168#380C2314yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2314yes—
173C4×C22⋊C4—28no—
174C4×C22⋊C4D414yes—
175C4×C22⋊C4C2314yes—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95D411yescharacteristic
184C2×#95D411yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—82no—
207C22×D4#4114yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4—82no—
229C23×C4—82no—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254#123—432no—
255#121—28no—
256#121—44no—
257#121—48no—
258C4×D4—28no—
259C4×D4—44no—
260C4×D4—48no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4#4114yes—
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—82no—
327C2×C42—82no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332#107—44no—
333#107—44no—
334#106—28no—
335#106—28no—
336C4⋊C8—28no—
337#102—28no—
338#102—28no—
339#102—28no—
340#102—28no—
341C22⋊C8—28no—
342C22⋊C8—44no—
343#97—44no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24AGL(1,ℤ/8)14yes—
368C24C22×D412yes—
369C24C22×D411yescharacteristic
370C2×D4—28no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—42no—
383C2×D4—44no—
384C2×D4—84no—
385C2×D4—816no—
386C2×D4C2×#4114yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
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—216no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—816no—
449C22×C4—816no—
450C22×C4#13714yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464C2×C8—28no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—44no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8C2×#4114yes—
480C4⋊C4—216no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4C2×#4114yes—
506C4⋊C4C2×#4114yes—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—84no—
522C42—84no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—48no—
538C23—42no—
539C23—42no—
540C23—48no—
541C23—48no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—84no—
547C23—84no—
548C23—84no—
549C23—84no—
550C23#52512yes—
551C23#54612yes—
552C23#54712yes—
553C23C2×AGL(1,ℤ/8)14yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23C2×#13712yes—
556C23#57612yes—
557C23#57712yes—
558C23C23×D411yescharacteristic
559D4—24no—
560D4—24no—
561D4—42no—
562D4—42no—
563D4—48no—
564D4—816no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—816no—
613C2×C4—816no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4#52512yes—
620C2×C4#52914yes—
621C2×C4#54214yes—
622C2×C4#54612yes—
623C2×C4#54712yes—
624C2×C4#55214yes—
625C2×C4#55314yes—
626C2×C4C2×#13714yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)11yescharacteristic
632C2×C4C2×2−1+411yescharacteristic
633C8—24no—
634C8—42no—
635C8—42no—
636C8—42no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—416no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22#247014yes—
654C22#311214yes—
655C22#311814yes—
656C22#331814yes—
657C22#333114yes—
658C22#334514yes—
659C22#334614yes—
660C22C2×#52511yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#54711yescharacteristic
663C22C22×AGL(1,ℤ/8)11yescharacteristic
664C22C22×#13711yescharacteristic
665C22C2×#57611yescharacteristic
666C22C2×#57711yescharacteristic
667C4—24no—
668C4—42no—
669C4—44no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#247014yes—
680C4C2×#52911yescharacteristic
681C4C2×#54211yescharacteristic
682C4C2×#55211yescharacteristic
683C4C2×#55311yescharacteristic
684C2—44no—
685C2—84no—
686C2#2717518yesdirect product factor
687C2C2×#247011yescharacteristic
688C2C2×#311211yescharacteristic
689C2C2×#311811yescharacteristic
690C2C2×#331811yescharacteristic
691C2C2×#333111yescharacteristic
692C2C2×#334511yescharacteristic
693C2C2×#334611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C2×#2345C2semidirect
3. #27175C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2075C2semidirect
6. C2×#2065C2semidirect
8. C2×#1947C2semidirect
9. C2×#1944C2semidirect
10. C2×#1926C2semidirect
11. C2×#1924C2semidirect
12. C2×#1920C2semidirect
14. C2×#1862C2semidirect
15. C2×#1853C2semidirect
16. C2×#1762C2semidirect
38. #2345C22semidirect
54. #2100C22semidirect
55. #2075C22semidirect
56. #2065C22semidirect
58. #1947C22semidirect
59. #1944C22semidirect
60. #1926C22semidirect
61. #1924C22semidirect
62. #1920C22semidirect
64. #1862C22semidirect
65. #1853C22semidirect
66. #1762C22semidirect
686. C2#27175direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#24222
C2×#23452
#271752
C2×#21002
C2×#20752
C2×#20652
C2×#20332
C2×#19472
C2×#19442
C2×#19262
C2×#19242
C2×#19202
C2×#18742
C2×#18622
C2×#18532
C2×#17622