Finite Group Databasebeta

← C2×#25582 #6449040

Subgroups — C2×#25582 #6449040

Order 512.

3852 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
1C22×#553C211yesmaximal, characteristic
2C22×#491C211yesmaximal, characteristic, complemented
3C22×#461C211yesmaximal, characteristic, complemented
4C2×#2542C211yesmaximal, characteristic, complemented
5#25582C2116yesmaximal, direct product factor
6C2×#2000C212yesmaximal, complemented
7C2×#2000C212yesmaximal, complemented
8C2×#1989C211yesmaximal, characteristic, complemented
9C2×#1955C211yesmaximal, characteristic, complemented
10C2×#1873C212yesmaximal, complemented
11C2×#1872C212yesmaximal, complemented
12C2×#1762C211yesmaximal, characteristic, complemented
13C23×SD16C2212yes—
14C2×#553—28no—
15C2×#553—24no—
16C2×#553—28no—
17C2×#553C2218yes—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126—21no—
20C22×#125C2211yescharacteristic
21C22×#123—21no—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×#121—21no—
25C22×#121C2211yescharacteristic
26C22×#121C2211yescharacteristic
27C22×C4×D4—21no—
28C22×C4×D4C2211yescharacteristic
29C22×C4×D4C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C2×#491—28no—
33C2×#491C2218yescomplemented
34C2×#461—28no—
35C2×#461C2218yescomplemented
36C22×C4⋊C8C2211yescharacteristic
37C22×#103C2212yes—
38C22×#103C2212yes—
39C22×#103C2212yes—
40C22×#102C2212yes—
41C22×C22⋊C8C2212yes—
42C22×C22⋊C8C2212yes—
43C22×#97C2211yescharacteristic
44#2542—24no—
45#2542C2218yescomplemented
46C2×#399—21no—
47C2×#399—21no—
48C2×#390—21no—
49C2×#384—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2000C22116yescomplemented
52#2000C22116yescomplemented
53#1989C2218yescomplemented
54#1955C2218yescomplemented
55#1873C22116yescomplemented
56#1872C22116yescomplemented
57#1762C2218yescomplemented
58C2×#340C2212yes—
59C2×#340C2212yes—
60C2×#340C2212yes—
61C23×Q8C2312yes—
62C23×D4—21no—
63C23×D4—21no—
64C23×D4D411yescharacteristic
65C23×D4C2311yescharacteristic
66C24×C4—21no—
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×SD16—28no—
71C22×SD16—44no—
72C22×SD16—44no—
73C22×SD16C2318yes—
74C23×C8C2312yes—
75C23×C8C2312yes—
76#553—264no—
77#553—232no—
78#553—232no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2314yes—
83C2×#126—216no—
84C2×#126—28no—
85C2×#125—22no—
86C2×#125—24no—
87C2×#125D418yes—
88C2×#125C2314yes—
89C2×#123—216no—
90C2×#123—28no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8C2318yes—
98C2×C22⋊Q8C2318yes—
99C2×#121—216no—
100C2×#121—28no—
101C2×#121—24no—
102C2×#121—24no—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121C2314yes—
108C2×#121C2314yes—
109C2×C4×D4—216no—
110C2×C4×D4—28no—
111C2×C4×D4—22no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4D418yes—
116C2×C4×D4D414yes—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C2×#117—22no—
120C2×#117—24no—
121C2×#117—24no—
122C2×#117C2314yes—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—21no—
142C22×C22⋊C4—21no—
143C22×C22⋊C4—22no—
144C22×C22⋊C4—22no—
145C22×C22⋊C4D411yescharacteristic
146C22×C22⋊C4D411yescharacteristic
147C22×C42—21no—
148C22×C42—21no—
149C22×C42C2311yescharacteristic
150#491—232no—
151#461—232no—
152C2×C4⋊C8—24no—
153C2×C4⋊C8C2314yes—
154C2×#103—28no—
155C2×#103—28no—
156C2×#103—28no—
157C2×#103C2318yes—
158C2×#103C2318yes—
159C2×#103C2318yes—
160C2×#102—28no—
161C2×#102C2318yes—
162C2×C22⋊C8—44no—
163C2×C22⋊C8—44no—
164C2×C22⋊C8C2318yes—
165C2×C22⋊C8C2318yes—
166C2×#97—24no—
167C2×#97C2314yes—
168#399—28no—
169#399—28no—
170#390—28no—
171#384—28no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C2×#95—21no—
175#340C2318yes—
176#340C2318yes—
177#340C2318yes—
178C25—21no—
179C25C2×D411yescharacteristic
180C25C2×D411yescharacteristic
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8C2414yes—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—84no—
203C22×D4C2×D414yes—
204C22×D4C2×D414yes—
205C22×D4C2412yes—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—84no—
224C23×C4—84no—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C2×SD16—216no—
235C2×SD16—48no—
236C2×SD16—48no—
237C2×SD16—432no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—48no—
249#126—264no—
250#125—216no—
251#125—28no—
252#125C2×D4116yes—
253#123—264no—
254C22⋊Q8—216no—
255C22⋊Q8—216no—
256C22⋊Q8—48no—
257C22⋊Q8—48no—
258C22⋊Q8—416no—
259C22⋊Q8—416no—
260#121—264no—
261#121—216no—
262#121—216no—
263#121—28no—
264#121—28no—
265#121—28no—
266#121—28no—
267C4×D4—264no—
268C4×D4—216no—
269C4×D4—216no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4C2×D4116yes—
274#117—216no—
275#117—28no—
276#117—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—216no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4#4118yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D414yes—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42C2×D412yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2412yes—
336C4⋊C8—28no—
337#103—216no—
338#103—216no—
339#103—216no—
340#102—216no—
341C22⋊C8—48no—
342C22⋊C8—48no—
343#97—28no—
344#95—24no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—44no—
357C24—42no—
358C24—42no—
359C24—44no—
360C24#13714yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C22×D411yescharacteristic
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—28no—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—44no—
388C2×D4—416no—
389C2×D4—432no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—88no—
395C2×D4—816no—
396C2×D4C22×D414yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—816no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—816no—
446C22×C4#12014yes—
447C22×C4#12014yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4C2511yescharacteristic
458SD16—432no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—48no—
466C2×C8—48no—
467C4⋊C4—216no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4C22×D414yes—
484C4⋊C4C22×D414yes—
485C4⋊C4C22×D414yes—
486C4⋊C4C2×#4118yes—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C42—216no—
496C42—48no—
497C42#12018yes—
498C42C22×D414yes—
499C42C22×D414yes—
500C23—22no—
501C23—24no—
502C23—28no—
503C23—216no—
504C23—22no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—44no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—88no—
528C23—88no—
529C23C2×#12012yes—
530C23#54614yes—
531C23C2×#13714yes—
532C23C2×#13712yes—
533C23#57512yes—
534C23#57714yes—
535C23C23×D411yescharacteristic
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—88no—
541D4—24no—
542D4—24no—
543D4—416no—
544D4—432no—
545D4—48no—
546D4—44no—
547D4—816no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—816no—
585C2×C4—816no—
586C2×C4—88no—
587C2×C4C2×#12014yes—
588C2×C4C2×#12014yes—
589C2×C4C2×#12012yes—
590C2×C4#53214yes—
591C2×C4#53514yes—
592C2×C4#54614yes—
593C2×C4#55018yes—
594C2×C4C23×D411yescharacteristic
595C2×C4C23×D411yescharacteristic
596C2×C4C22×#4112yes—
597C2×C4C2×H(5,𝔽2)12yes—
598C8—44no—
599C8—44no—
600C22—24no—
601C22—28no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—416no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—84no—
615C22—84no—
616C22—88no—
617C22—816no—
618C22#243114yes—
619C22#300714yes—
620C22#300914yes—
621C22#330718yes—
622C22#332018yes—
623C22C22×#12011yescharacteristic
624C22C2×#54612yes—
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57712yes—
628C4—24no—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—48no—
636C4—84no—
637C4#243114yes—
638C4C2×#53211yescharacteristic
639C4C2×#53511yescharacteristic
640C4C2×#55012yes—
641C2—44no—
642C2—82no—
643C2—84no—
644C2#2558218yesdirect product factor
645C2C2×#243111yescharacteristic
646C2C2×#300711yescharacteristic
647C2C2×#300911yescharacteristic
648C2C2×#332012yes—
649C2C2×#330712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#553C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#491C2semidirect
3. C22×#461C2semidirect
4. C2×#2542C2semidirect
5. #25582C2direct (N × H)
6. C2×#2000C2semidirect
7. C2×#2000C2semidirect
8. C2×#1989C2semidirect
9. C2×#1955C2semidirect
10. C2×#1873C2semidirect
11. C2×#1872C2semidirect
12. C2×#1762C2semidirect
33. C2×#491C22semidirect
35. C2×#461C22semidirect
45. #2542C22semidirect
51. #2000C22semidirect
52. #2000C22semidirect
53. #1989C22semidirect
54. #1955C22semidirect
55. #1873C22semidirect
56. #1872C22semidirect
57. #1762C22semidirect
644. C2#25582direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5532
C22×#4912
C22×#4612
C2×#25422
#255822
C2×#20002
C2×#19892
C2×#19552
C2×#18732
C2×#18722
C2×#17622