Finite Group Databasebeta

← C2×#32573 #7510367

Subgroups — C2×#32573 #7510367

Order 512.

1916 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic
3#32573C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2087C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic
8C2×#2057C211yesmaximal, characteristic, complemented
9C2×#2023C211yesmaximal, characteristic
10C2×#1984C211yesmaximal, characteristic, complemented
11C2×#1942C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic
13C2×#1925C211yesmaximal, characteristic
14C2×#1920C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic
16C2×#1862C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2580—24no—
31#2580C2218yes—
32#2567—24no—
33#2567C2218yescomplemented
34C2×#399—21no—
35C2×#395—21no—
36C2×#395—21no—
37C2×#393—21no—
38C2×#391—21no—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#385—21no—
48C2×#385—21no—
49C2×#384—21no—
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2218yescomplemented
53#2087C2218yescomplemented
54#2085C2218yescomplemented
55#2065C2218yes—
56#2057C2218yescomplemented
57#2023C2218yes—
58#1984C2218yescomplemented
59#1942C2218yescomplemented
60#1934C2218yes—
61#1925C2218yes—
62#1920C2218yescomplemented
63#1862C2218yes—
64#1862C2218yes—
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—22no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×C4⋊Q8C2314yes—
89C2×C4×Q8—42no—
90C2×C4×Q8—42no—
91C2×C4×Q8D414yes—
92C2×C4×Q8C2314yes—
93C2×C4×D4—22no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4D411yescharacteristic
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×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C2×#107—24no—
124C2×#107—42no—
125C2×#107—42no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106—24no—
130C2×#106C2314yes—
131C2×C4⋊C8—42no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—42no—
135C2×#102—42no—
136C2×#102Q818yescomplemented
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C4×C8—42no—
142C2×C4×C8C2314yes—
143#399—28no—
144#395—28no—
145#395—28no—
146#393—28no—
147#391—28no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#389C2314yes—
161#389C2314yes—
162#389C2314yes—
163#389C2314yes—
164#385—28no—
165#385—28no—
166#384—28no—
167#382—22no—
168#382C2314yes—
169C4×C4⋊C4—22no—
170C4×C4⋊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—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—82no—
199C22×Q8—82no—
200C22×Q8C2×D414yes—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2×Q814yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—82no—
219C23×C4—82no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2411yescharacteristic
227C22×C8—22no—
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×C8C2×Q814yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—44no—
244C4⋊Q8—44no—
245C4⋊Q8—48no—
246C4×Q8—48no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—48no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
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—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×Q814yes—
292C2×C4⋊C4#4114yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323C2×C42C2412yes—
324#107—28no—
325#107—44no—
326#107—44no—
327#106—28no—
328C4⋊C8—44no—
329#102—28no—
330#102—44no—
331#102—44no—
332#102C2×Q8116yescomplemented
333C4×C8—44no—
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—
343C24—24no—
344C24—22no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24C22×D411yescharacteristic
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—44no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—84no—
358C2×Q8—84no—
359C2×Q8—816no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4C22×Q814yes—
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—24no—
382C22×C4—24no—
383C22×C4—22no—
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—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
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—84no—
419C22×C4—84no—
420C22×C4—816no—
421C22×C4#13514yes—
422C22×C4#13714yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×Q812yes—
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+412yes—
435C22×C42−1+412yes—
436C22×C4C2511yescharacteristic
437C2×C8—28no—
438C2×C8—22no—
439C2×C8—22no—
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—44no—
450C2×C8—44no—
451C2×C8—44no—
452C2×C8C22×Q814yes—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
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—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4C22×Q814yes—
483C4⋊C4C2×#4114yes—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—48no—
499C23—24no—
500C23—48no—
501C23—48no—
502C23—42no—
503C23—42no—
504C23—48no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23#53412yes—
514C23#54812yes—
515C23D4×Q812yes—
516C23C2×#13512yes—
517C23C2×#13712yes—
518C23#57612yes—
519C23#57612yes—
520C23C23×D411yescharacteristic
521Q8—416no—
522Q8—816no—
523D4—24no—
524D4—24no—
525D4—42no—
526D4—48no—
527D4—42no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—28no—
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—48no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
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—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—816no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4#53412yes—
580C2×C4#54214yes—
581C2×C4#54414yes—
582C2×C4#54812yes—
583C2×C4D4×Q812yes—
584C2×C4#55414yes—
585C2×C4#56314yes—
586C2×C4C2×#13514yes—
587C2×C4C2×#13714yes—
588C2×C4C23×Q811yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)11yescharacteristic
591C2×C4C2×H(5,𝔽2)11yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4C2×2−1+411yescharacteristic
594C8—24no—
595C8—42no—
596C8—42no—
597C8—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—82no—
602C22—816no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#272914yes—
607C22#319014yes—
608C22#319914yes—
609C22#336114yes—
610C22#337514yes—
611C22#338014yes—
612C22#338514yes—
613C22C2×#53411yescharacteristic
614C22C2×#54811yescharacteristic
615C22C2×D4×Q811yescharacteristic
616C22C22×#13511yescharacteristic
617C22C22×#13711yescharacteristic
618C22C2×#57611yescharacteristic
619C22C2×#57611yescharacteristic
620C4—24no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#272914yes—
634C4C2×#54211yescharacteristic
635C4C2×#54411yescharacteristic
636C4C2×#55411yescharacteristic
637C4C2×#56311yescharacteristic
638C2—84no—
639C2#3257318yesdirect product factor
640C2C2×#272911yescharacteristic
641C2C2×#319011yescharacteristic
642C2C2×#319911yescharacteristic
643C2C2×#336111yescharacteristic
644C2C2×#337511yescharacteristic
645C2C2×#338011yescharacteristic
646C2C2×#338511yescharacteristic
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×#2567C2semidirect
3. #32573C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2087C2semidirect
6. C2×#2085C2semidirect
8. C2×#2057C2semidirect
10. C2×#1984C2semidirect
11. C2×#1942C2semidirect
14. C2×#1920C2semidirect
33. #2567C22semidirect
52. #2101C22semidirect
53. #2087C22semidirect
54. #2085C22semidirect
56. #2057C22semidirect
58. #1984C22semidirect
59. #1942C22semidirect
62. #1920C22semidirect
136. C2×#102Q8semidirect
332. #102C2×Q8semidirect
639. C2#32573direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25802
#325732
C2×#21012
C2×#20872
C2×#20852
C2×#20652
C2×#20572
C2×#20232
C2×#19842
C2×#19422
C2×#19342
C2×#19252
C2×#19202
C2×#18622