Finite Group Databasebeta

← C2×#32584 #7510377

Subgroups — C2×#32584 #7510377

Order 512.

1976 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic
3#32584C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2088C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2058C211yesmaximal, characteristic, complemented
9C2×#2023C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1943C211yesmaximal, characteristic, complemented
12C2×#1936C211yesmaximal, characteristic
13C2×#1925C211yesmaximal, characteristic
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1864C211yesmaximal, characteristic
16C2×#1864C211yesmaximal, characteristic
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#97C2211yescharacteristic
31#2580—24no—
32#2580C2218yes—
33#2546—24no—
34#2546C2218yescomplemented
35C2×#399—21no—
36C2×#398—21no—
37C2×#398—21no—
38C2×#393—21no—
39C2×#391—21no—
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#385—21no—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2088C2218yescomplemented
54#2085C2218yescomplemented
55#2065C2218yes—
56#2058C2218yescomplemented
57#2023C2218yes—
58#1985C2218yescomplemented
59#1943C2218yescomplemented
60#1936C2218yes—
61#1925C2218yes—
62#1920C2218yescomplemented
63#1864C2218yes—
64#1864C2218yes—
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×Q8—21no—
78C23×Q8D411yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8—42no—
89C2×C4⋊Q8—48no—
90C2×C4⋊Q8C2314yes—
91C2×C4×Q8—42no—
92C2×C4×Q8—42no—
93C2×C4×Q8D414yes—
94C2×C4×Q8C2314yes—
95C2×C4×D4—22no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C2×#107—24no—
126C2×#107C2314yes—
127C2×#106—24no—
128C2×#106—42no—
129C2×#106—42no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8—42no—
134C2×C4⋊C8C2314yes—
135C2×#102—24no—
136C2×#102—42no—
137C2×#102—42no—
138C2×#102Q818yescomplemented
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#97—42no—
144C2×#97C2314yes—
145#399—28no—
146#398—28no—
147#398—28no—
148#393—28no—
149#391—28no—
150#390—22no—
151#390—22no—
152#390C2314yes—
153#390C2314yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#389C2314yes—
166#385—28no—
167#385—28no—
168#382—22no—
169#382C2314yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4C2314yes—
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—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2314yes—
193C25—21no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—44no—
201C22×Q8C2×D414yes—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C2×Q814yes—
209C22×D4C2412yes—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—82no—
220C23×C4—82no—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8—22no—
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×C8C2×Q814yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—48no—
247C4⋊Q8—432no—
248C4×Q8—48no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—48no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
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—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
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—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×Q814yes—
294C2×C4⋊C4#4114yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4—82no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326#107—28no—
327#106—28no—
328#106—44no—
329#106—44no—
330C4⋊C8—44no—
331#102—28no—
332#102—44no—
333#102—44no—
334#102C2×Q8116yescomplemented
335#97—44no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345C24—24no—
346C24—22no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24—42no—
353C24C22×D411yescharacteristic
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—416no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—816no—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4C22×Q814yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
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—22no—
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—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
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—84no—
422C22×C4—84no—
423C22×C4—816no—
424C22×C4AGL(1,ℤ/8)14yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×Q812yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4H(5,𝔽2)12yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C4C2511yescharacteristic
439C2×C8—28no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
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—44no—
452C2×C8—44no—
453C2×C8—44no—
454C2×C8C22×Q814yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
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—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4C22×Q814yes—
485C4⋊C4C2×#4114yes—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—48no—
501C23—24no—
502C23—48no—
503C23—48no—
504C23—42no—
505C23—42no—
506C23—48no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23#53412yes—
516C23#54812yes—
517C23D4×Q812yes—
518C23C2×AGL(1,ℤ/8)12yes—
519C23#57512yes—
520C23#57712yes—
521C23#57712yes—
522C23C23×D411yescharacteristic
523Q8—416no—
524Q8—816no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—48no—
529D4—42no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
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—816no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#53412yes—
583C2×C4#54214yes—
584C2×C4#54414yes—
585C2×C4#54812yes—
586C2×C4D4×Q812yes—
587C2×C4#55414yes—
588C2×C4#56314yes—
589C2×C4C2×AGL(1,ℤ/8)14yes—
590C2×C4C23×Q811yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4C2×H(5,𝔽2)11yescharacteristic
594C2×C4C2×2−1+411yescharacteristic
595C2×C4C2×2−1+411yescharacteristic
596C8—24no—
597C8—42no—
598C8—42no—
599C8—42no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—82no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22#272914yes—
609C22#319414yes—
610C22#320014yes—
611C22#337114yes—
612C22#337314yes—
613C22#338414yes—
614C22#338814yes—
615C22C2×#53411yescharacteristic
616C22C2×#54811yescharacteristic
617C22C2×D4×Q811yescharacteristic
618C22C22×AGL(1,ℤ/8)11yescharacteristic
619C22C2×#57511yescharacteristic
620C22C2×#57711yescharacteristic
621C22C2×#57711yescharacteristic
622C4—24no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#272914yes—
636C4C2×#54211yescharacteristic
637C4C2×#54411yescharacteristic
638C4C2×#55411yescharacteristic
639C4C2×#56311yescharacteristic
640C2—84no—
641C2#3258418yesdirect product factor
642C2C2×#272911yescharacteristic
643C2C2×#319411yescharacteristic
644C2C2×#320011yescharacteristic
645C2C2×#337311yescharacteristic
646C2C2×#337111yescharacteristic
647C2C2×#338411yescharacteristic
648C2C2×#338811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2546C2semidirect
3. #32584C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2088C2semidirect
6. C2×#2085C2semidirect
8. C2×#2058C2semidirect
10. C2×#1985C2semidirect
11. C2×#1943C2semidirect
14. C2×#1920C2semidirect
34. #2546C22semidirect
52. #2101C22semidirect
53. #2088C22semidirect
54. #2085C22semidirect
56. #2058C22semidirect
58. #1985C22semidirect
59. #1943C22semidirect
62. #1920C22semidirect
138. C2×#102Q8semidirect
334. #102C2×Q8semidirect
641. C2#32584direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25802
#325842
C2×#21012
C2×#20882
C2×#20852
C2×#20652
C2×#20582
C2×#20232
C2×#19852
C2×#19432
C2×#19362
C2×#19252
C2×#19202
C2×#18642