Finite Group Databasebeta

← C2×#27574 #6441055

Subgroups — C2×#27574 #6441055

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2512C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic, complemented
3#27574C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2099C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#1936C211yesmaximal, characteristic, complemented
9C2×#1919C211yesmaximal, characteristic, complemented
10C2×#1917C211yesmaximal, characteristic, complemented
11C2×#1879C211yesmaximal, characteristic
12C2×#1879C211yesmaximal, characteristic
13C2×#1879C211yesmaximal, characteristic
14C2×#1864C211yesmaximal, characteristic, complemented
15C2×#1802C211yesmaximal, characteristic, complemented
16C2×#1762C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×#126—21no—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2512—24no—
32#2512C2218yescomplemented
33C2×#397—21no—
34C2×#396—21no—
35#2349—24no—
36#2349C2218yescomplemented
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#380—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C22×#95—21no—
52#2101C2218yescomplemented
53#2099C2218yescomplemented
54#2073C2218yescomplemented
55#2065C2218yescomplemented
56#1936C2218yescomplemented
57#1919C2218yescomplemented
58#1917C2218yescomplemented
59#1879C2218yes—
60#1879C2218yes—
61#1879C2218yes—
62#1864C2218yescomplemented
63#1802C2218yescomplemented
64#1762C2218yescomplemented
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×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4—21no—
81C24×C4—21no—
82C22×D8—24no—
83C22×D8—24no—
84C22×D8—24no—
85C22×D8C2314yes—
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#126—28no—
91C2×#126—48no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4C2314yes—
102C2×C4×D4C2314yes—
103C2×C4×D4C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C2×#106—24no—
131C2×#106C2314yes—
132C2×C4⋊C8—42no—
133C2×C4⋊C8C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136C2×#102—24no—
137C2×#102—24no—
138C2×#102—42no—
139C2×#102—42no—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97—42no—
147C2×#97C2314yes—
148#397—28no—
149#396—28no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#390C2314yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#388—28no—
163#385—28no—
164#385—28no—
165#384—22no—
166#384C2314yes—
167#382—22no—
168#382C2314yes—
169#380—28no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4D414yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—28no—
181C2×#95D411yescharacteristic
182C2×#95D411yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4#4114yes—
209C22×D4#4114yes—
210C22×D4#4114yes—
211C22×D4C2412yes—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—82no—
228C23×C4—82no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×D8—28no—
239C2×D8—28no—
240C2×D8—28no—
241C2×D8—416no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258#126—432no—
259C4×D4—216no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
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—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4#4114yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338#106—28no—
339C4⋊C8—44no—
340#102—28no—
341#102—28no—
342#102—28no—
343#102—28no—
344#102—44no—
345#102—44no—
346#97—44no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95C2×D414yes—
354#95C2×D414yes—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24C22×D411yescharacteristic
372C2×D4—24no—
373C2×D4—28no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4#126116yescomplemented
395C2×D4C2×#4114yes—
396C2×D4C2×#4114yes—
397C2×D4C2×#4114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
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—216no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
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—48no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—816no—
449C22×C4—816no—
450C22×C4—816no—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+412yes—
465C22×C4C2511yescharacteristic
466D8—416no—
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—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—44no—
482C2×C8—44no—
483C4⋊C4—24no—
484C4⋊C4—28no—
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—22no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4C2×#4114yes—
508C22⋊C4—216no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—48no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—216no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23#52512yes—
556C23#54612yes—
557C23#54812yes—
558C23C2×AGL(1,ℤ/8)12yes—
559C23#57512yes—
560C23#57612yes—
561C23#57712yes—
562C23C23×D411yescharacteristic
563D4—24no—
564D4—24no—
565D4—24no—
566D4—216no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—84no—
572D4C2×#12618yescomplemented
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—816no—
617C2×C4—816no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4#52512yes—
624C2×C4C2×#12614yes—
625C2×C4#53914yes—
626C2×C4#54612yes—
627C2×C4#54812yes—
628C2×C4#55214yes—
629C2×C4#55314yes—
630C2×C4C2×AGL(1,ℤ/8)14yes—
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C22—28no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—432no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—168no—
657C22#248114yes—
658C22#311514yes—
659C22#311814yes—
660C22#330614yes—
661C22#332214yes—
662C22#336614yes—
663C22#337614yes—
664C22C2×#52511yescharacteristic
665C22C2×#54611yescharacteristic
666C22C2×#54811yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22C2×#57511yescharacteristic
669C22C2×#57611yescharacteristic
670C22C2×#57711yescharacteristic
671C4—24no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#248114yes—
684C4C22×#12611yescharacteristic
685C4C2×#53911yescharacteristic
686C4C2×#55211yescharacteristic
687C4C2×#55311yescharacteristic
688C2—48no—
689C2—162no—
690C2#2757418yesdirect product factor
691C2C2×#248111yescharacteristic
692C2C2×#311511yescharacteristic
693C2C2×#311811yescharacteristic
694C2C2×#330611yescharacteristic
695C2C2×#332211yescharacteristic
696C2C2×#336611yescharacteristic
697C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
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
1. C2×#2512C2semidirect
2. C2×#2349C2semidirect
3. #27574C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2099C2semidirect
6. C2×#2073C2semidirect
7. C2×#2065C2semidirect
8. C2×#1936C2semidirect
9. C2×#1919C2semidirect
10. C2×#1917C2semidirect
14. C2×#1864C2semidirect
15. C2×#1802C2semidirect
16. C2×#1762C2semidirect
32. #2512C22semidirect
36. #2349C22semidirect
52. #2101C22semidirect
53. #2099C22semidirect
54. #2073C22semidirect
55. #2065C22semidirect
56. #1936C22semidirect
57. #1919C22semidirect
58. #1917C22semidirect
62. #1864C22semidirect
63. #1802C22semidirect
64. #1762C22semidirect
394. C2×D4#126semidirect
572. D4C2×#126semidirect
690. C2#27574direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#23492
#275742
C2×#21012
C2×#20992
C2×#20732
C2×#20652
C2×#19362
C2×#19192
C2×#19172
C2×#18792
C2×#18642
C2×#18022
C2×#17622