Finite Group Databasebeta

← C2×#24006 #6440299

Subgroups — C2×#24006 #6440299

Order 512.

5238 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C22×#484C212yesmaximal, complemented
3C22×#380C211yesmaximal, characteristic
4#24006C2116yesmaximal, direct product factor
5C2×#2036C214yesmaximal, complemented
6C2×#1872C212yesmaximal, complemented
7C2×#1852C212yesmaximal, complemented
8C2×#1787C212yesmaximal, complemented
9C2×#1779C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C25×C4C2211yescharacteristic
12C22×C22⋊Q8—21no—
13C22×C22⋊Q8—24no—
14C22×C22⋊Q8—21no—
15C22×C22⋊Q8—216no—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2218yes—
18C22×C22⋊Q8C2212yes—
19C23×C4⋊C4—21no—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4—21no—
24C23×C22⋊C4—21no—
25C2×#484—216no—
26C2×#484C22116yescomplemented
27C22×#107C2214yescomplemented
28C22×#103C2214yescomplemented
29C22×#103C2214yescomplemented
30C22×C22⋊C8C2212yes—
31C22×C22⋊C8C2212yes—
32C22×C22⋊C8C2212yes—
33C2×#380—21no—
34C2×#380—24no—
35C2×#380—21no—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2218yes—
39C22×#95—21no—
40#2036C22132yescomplemented
41#1872C22116yescomplemented
42#1852C22116yescomplemented
43#1787C22116yescomplemented
44#1779C2218yes—
45C2×#340C2214yescomplemented
46C26D411yescharacteristic
47C23×Q8—28no—
48C23×Q8—22no—
49C23×Q8—21no—
50C23×Q8—28no—
51C23×Q8—44no—
52C23×Q8C2312yes—
53C23×Q8C2314yes—
54C24×C4—21no—
55C24×C4—21no—
56C24×C4—28no—
57C24×C4—28no—
58C24×C4—21no—
59C24×C4—21no—
60C24×C4—44no—
61C24×C4D411yescharacteristic
62C24×C4D411yescharacteristic
63C24×C4D414yes—
64C24×C4D411yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2314yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C23×C8C2314yes—
70C2×C22⋊Q8—264no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—28no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—264no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8D4116yes—
82C2×C22⋊Q8C2318yes—
83C2×C22⋊Q8C2318yes—
84C22×C4⋊C4—216no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—216no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—48no—
111#484—264no—
112C2×#107—216no—
113C2×#107C23116yescomplemented
114C2×#103—216no—
115C2×#103—48no—
116C2×#103C23116yescomplemented
117C2×#103C23116yescomplemented
118C2×C22⋊C8—28no—
119C2×C22⋊C8—28no—
120C2×C22⋊C8—44no—
121C2×C22⋊C8C2318yes—
122C2×C22⋊C8C2318yes—
123C2×C22⋊C8C2318yes—
124#380—24no—
125#380—28no—
126#380—28no—
127#380—24no—
128#380C2318yes—
129#380C2318yes—
130C2×#95—28no—
131C2×#95—22no—
132C2×#95—22no—
133#340C23116yescomplemented
134C25—24no—
135C25—24no—
136C25—24no—
137C25—24no—
138C25—42no—
139C25—42no—
140C25Q1618yes—
141C25C2×D411yescharacteristic
142C25C2×D414yes—
143C25C2×D411yescharacteristic
144C25C2×D411yescharacteristic
145C22×Q8—28no—
146C22×Q8—216no—
147C22×Q8—216no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—28no—
152C22×Q8—216no—
153C22×Q8—28no—
154C22×Q8—48no—
155C22×Q8—464no—
156C22×Q8—44no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—44no—
160C22×Q8C2×D418yes—
161C22×Q8C2414yes—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—216no—
177C23×C4—24no—
178C23×C4—216no—
179C23×C4—24no—
180C23×C4—216no—
181C23×C4—24no—
182C23×C4—216no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—48no—
190C23×C4—48no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—464no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D412yes—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D418yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4#4118yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C22×C8—28no—
221C22×C8—28no—
222C22×C8—28no—
223C22×C8C2418yes—
224C22⋊Q8—232no—
225C22⋊Q8—264no—
226C22⋊Q8—264no—
227C22⋊Q8—216no—
228C22⋊Q8—232no—
229C22⋊Q8—216no—
230C22⋊Q8—216no—
231C22⋊Q8C2×D4132yes—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—264no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2414yes—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—264no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4—48no—
269#107—232no—
270#103—232no—
271#103—416no—
272C22⋊C8—216no—
273C22⋊C8—216no—
274C22⋊C8—48no—
275#95—28no—
276#95—28no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—28no—
286C24—22no—
287C24—28no—
288C24—22no—
289C24—22no—
290C24—28no—
291C24—22no—
292C24—28no—
293C24—22no—
294C24—22no—
295C24—232no—
296C24—232no—
297C24—44no—
298C24—416no—
299C24—44no—
300C24—416no—
301C24—416no—
302C24—416no—
303C24—416no—
304C24#12014yes—
305C24#12314yes—
306C24C2×Q1614yes—
307C24C2×Q16116yes—
308C24C2×Q1614yes—
309C24C2×Q1614yes—
310C24AGL(1,ℤ/8)14yes—
311C24#13714yes—
312C24C22×D412yes—
313C24C22×D412yes—
314C24C22×D412yes—
315C24C22×D411yescharacteristic
316C2×Q8—216no—
317C2×Q8—28no—
318C2×Q8—24no—
319C2×Q8—232no—
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—464no—
323C2×Q8—432no—
324C2×Q8—48no—
325C2×Q8—464no—
326C2×Q8—48no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—416no—
330C2×Q8C22×D418yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—232no—
340C22×C4—232no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—464no—
373C22×C4—464no—
374C22×C4—416no—
375C22×C4—416no—
376C22×C4—432no—
377C22×C4—48no—
378C22×C4—432no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—816no—
400C22×C4—816no—
401C22×C4#12014yes—
402C22×C4#12018yes—
403C22×C4#12314yes—
404C22×C4#12318yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D418yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D414yes—
411C22×C4C2×#4118yes—
412C22×C4C2×#4114yes—
413C22×C4H(5,𝔽2)14yes—
414C22×C4C2511yescharacteristic
415C2×C8—28no—
416C2×C8—28no—
417C2×C8—28no—
418C2×C8—416no—
419C4⋊C4—216no—
420C4⋊C4—232no—
421C4⋊C4—232no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4C22×D418yes—
431C22⋊C4—232no—
432C22⋊C4—232no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—28no—
439C23—28no—
440C23—22no—
441C23—22no—
442C23—28no—
443C23—28no—
444C23—22no—
445C23—22no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—28no—
456C23—232no—
457C23—22no—
458C23—22no—
459C23—28no—
460C23—22no—
461C23—232no—
462C23—28no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—416no—
470C23—416no—
471C23—4128no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23#451116yes—
477C23#484116yescomplemented
478C23C2×#12014yes—
479C23C2×#12012yes—
480C23C2×#12314yes—
481C23C2×#12312yes—
482C23#53412yes—
483C23C22×Q1614yes—
484C23C22×Q1614yes—
485C23C22×Q1614yes—
486C23C22×Q1612yes—
487C23C2×AGL(1,ℤ/8)14yes—
488C23C2×AGL(1,ℤ/8)12yes—
489C23C2×#13714yes—
490C23C2×#13712yes—
491C23#57512yes—
492C23C23×D411yescharacteristic
493Q8—216no—
494Q8—432no—
495Q8—416no—
496Q8—48no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—216no—
504C2×C4—216no—
505C2×C4—216no—
506C2×C4—216no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—432no—
516C2×C4—432no—
517C2×C4—44no—
518C2×C4—416no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—416no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—816no—
533C2×C4—816no—
534C2×C4—88no—
535C2×C4—88no—
536C2×C4C2×#12014yes—
537C2×C4C2×#12012yes—
538C2×C4C2×#12314yes—
539C2×C4C2×#12312yes—
540C2×C4#53412yes—
541C2×C4#546116yes—
542C2×C4C23×D412yes—
543C2×C4C22×#4112yes—
544C2×C4C2×H(5,𝔽2)12yes—
545C8—48no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—232no—
559C22—232no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—464no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—416no—
569C22#240514yes—
570C22C2×#45118yes—
571C22C2×#45114yes—
572C22#300614yes—
573C22C2×#48414yes—
574C22C2×#48418yescomplemented
575C22#309514yes—
576C22#318914yes—
577C22#319314yes—
578C22C22×#12011yescharacteristic
579C22C22×#12311yescharacteristic
580C22C2×#53411yescharacteristic
581C22C23×Q1611yescharacteristic
582C22C22×AGL(1,ℤ/8)11yescharacteristic
583C22C22×#13711yescharacteristic
584C22C2×#57511yescharacteristic
585C4—24no—
586C4—24no—
587C4—42no—
588C4—48no—
589C4—48no—
590C4—84no—
591C4—84no—
592C4#240514yes—
593C4C2×#54614yes—
594C2—28no—
595C2—28no—
596C2—44no—
597C2#2400618yesdirect product factor
598C2C2×#240511yescharacteristic
599C2C22×#45111yescharacteristic
600C2C2×#300611yescharacteristic
601C2C22×#48411yescharacteristic
602C2C2×#309511yescharacteristic
603C2C2×#318911yescharacteristic
604C2C2×#319311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#484C2semidirect
4. #24006C2direct (N × H)
5. C2×#2036C2semidirect
6. C2×#1872C2semidirect
7. C2×#1852C2semidirect
8. C2×#1787C2semidirect
26. C2×#484C22semidirect
27. C22×#107C22semidirect
28. C22×#103C22semidirect
29. C22×#103C22semidirect
40. #2036C22semidirect
41. #1872C22semidirect
42. #1852C22semidirect
43. #1787C22semidirect
45. C2×#340C22semidirect
113. C2×#107C23semidirect
116. C2×#103C23semidirect
117. C2×#103C23semidirect
133. #340C23semidirect
477. C23#484semidirect
574. C22C2×#484semidirect
597. C2#24006direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C22×#4842
C22×#3802
#240062
C2×#20362
C2×#18722
C2×#18522
C2×#17872
C2×#17792