Finite Group Databasebeta

← #657004

Subgroups — #657004

Order 512.

11457 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2846C211yesmaximal, characteristic
2C2×#2483C211yesmaximal, characteristic, complemented
3C2×#2412C212yesmaximal, complemented
4C22×#379C212yesmaximal, complemented
5#15480C214yesmaximal, complemented
6#15283C214yesmaximal, complemented
7#14911C214yesmaximal, complemented
8#14734C214yesmaximal, complemented
9#13995C214yesmaximal, complemented
10#13903C214yesmaximal, complemented
11C22⋊C42C211yesmaximal, characteristic, complemented
12C27C2211yescharacteristic
13C22×#123C2212yescomplemented
14C22×#120C2211yescharacteristic
15C22×#120C2214yes—
16C22×#120C2212yes—
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2214yes—
19C23×C22⋊C4C2214yes—
20C23×C22⋊C4C2212yes—
21#2846—28no—
22#2483—28no—
23#2412—216no—
24C2×#399C2214yescomplemented
25C2×#397C2214yescomplemented
26C2×#397C2214yescomplemented
27C2×#394C2214yescomplemented
28C2×#390C2214yescomplemented
29C2×#386C2214yescomplemented
30C2×#386C2212yescomplemented
31C2×#385C2212yescomplemented
32C2×#385C2214yescomplemented
33C2×#381C2214yescomplemented
34C2×#381C2212yescomplemented
35C2×#380C2214yescomplemented
36C2×#379—216no—
37C2×#379C2214yes—
38C2×#379C2212yes—
39C2×#379C2214yes—
40C2×#379C2214yes—
41C2×#379C2218yes—
42C2×#379C2211yescharacteristic
43C2×C4×C22⋊C4C2214yescomplemented
44#1439C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1438C2214yescomplemented
49#1438C2218yescomplemented
50#1437C2214yescomplemented
51#1437C2218yescomplemented
52#1437C2214yescomplemented
53C22⋊#95C2214yescomplemented
54C26—28no—
55C26—216no—
56C26—24no—
57C26—48no—
58C26—48no—
59C26C2314yes—
60C26C2312yes—
61C26C2311yescharacteristic
62C23×D4C2312yes—
63C23×D4C2314yes—
64C23×D4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2314yes—
68C2×#123—28no—
69C2×#123—48no—
70C2×#120—24no—
71C2×#120—28no—
72C2×#120—216no—
73C2×#120—216no—
74C2×#120—28no—
75C2×#120—48no—
76C2×#120—48no—
77C22×C4⋊C4C2314yescomplemented
78C22×C4⋊C4C2314yescomplemented
79C22×C4⋊C4C2314yescomplemented
80C22×C22⋊C4—28no—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—28no—
83C22×C22⋊C4—216no—
84C22×C22⋊C4—216no—
85C22×C22⋊C4—216no—
86C22×C22⋊C4—48no—
87C22×C22⋊C4—416no—
88C22×C22⋊C4—48no—
89C22×C22⋊C4D4116yescomplemented
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2318yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2318yes—
97C22×C22⋊C4C2318yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yescomplemented
106C22×C22⋊C4C2314yescomplemented
107C22×C42C2314yescomplemented
108#399—48no—
109#397—216no—
110#397—216no—
111#394—216no—
112#390—48no—
113#386—216no—
114#386—28no—
115#385—28no—
116#385—216no—
117#381—28no—
118#381—216no—
119#380—216no—
120#379—216no—
121#379—28no—
122#379—216no—
123#379—216no—
124#379—232no—
125#379D418yes—
126C4×C22⋊C4—48no—
127C2×#95C2314yescomplemented
128C2×#95C2314yescomplemented
129C2×#95C2314yescomplemented
130C2×#95C2314yescomplemented
131C2×#95C2318yescomplemented
132C2×#95C2314yescomplemented
133C2×#95C2314yescomplemented
134C2×#95C2318yescomplemented
135C25—28no—
136C25—216no—
137C25—28no—
138C25—216no—
139C25—28no—
140C25—28no—
141C25—264no—
142C25—24no—
143C25—28no—
144C25—28no—
145C25—28no—
146C25—216no—
147C25—28no—
148C25—28no—
149C25—28no—
150C25—44no—
151C25—48no—
152C25—48no—
153C25—48no—
154C25—44no—
155C25—464no—
156C25—464no—
157C25—464no—
158C25—464no—
159C25—4128no—
160C25—432no—
161C25—432no—
162C25—464no—
163C25—416no—
164C25C2×D418yes—
165C25C2×D414yes—
166C25#41116yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C25C2414yes—
170C25C2412yes—
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—216no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—28no—
181C22×D4—48no—
182C22×D4—44no—
183C22×D4—44no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4C2×D4116yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205#123—432no—
206#120—464no—
207#120—432no—
208#120—416no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—44no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—216no—
233C2×C22⋊C4—216no—
234C2×C22⋊C4—216no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—216no—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—432no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—464no—
256C2×C22⋊C4—464no—
257C2×C22⋊C4—432no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4C2×D418yes—
260C2×C22⋊C4C2×D418yes—
261C2×C22⋊C4C2×D4116yes—
262C2×C22⋊C4C2×D4116yes—
263C2×C22⋊C4C2×D418yes—
264C2×C42—48no—
265C2×C42—44no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—216no—
270#95—28no—
271#95—28no—
272#95—48no—
273#95—44no—
274C24—24no—
275C24—24no—
276C24—28no—
277C24—232no—
278C24—232no—
279C24—264no—
280C24—216no—
281C24—216no—
282C24—232no—
283C24—232no—
284C24—232no—
285C24—264no—
286C24—28no—
287C24—24no—
288C24—24no—
289C24—28no—
290C24—24no—
291C24—28no—
292C24—28no—
293C24—28no—
294C24—24no—
295C24—24no—
296C24—216no—
297C24—232no—
298C24—232no—
299C24—232no—
300C24—22no—
301C24—21no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—416no—
306C24—48no—
307C24—432no—
308C24—432no—
309C24—432no—
310C24—416no—
311C24—432no—
312C24—432no—
313C24—432no—
314C24—432no—
315C24—432no—
316C24—432no—
317C24—416no—
318C24—432no—
319C24—432no—
320C24—432no—
321C24—416no—
322C24—432no—
323C24—432no—
324C24—416no—
325C24—416no—
326C24—432no—
327C24—4512no—
328C24—416no—
329C24—48no—
330C24—432no—
331C24—416no—
332C24—4256no—
333C24—4128no—
334C24—4512no—
335C24—4512no—
336C24—416no—
337C24—48no—
338C24—416no—
339C24—416no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24#127116yes—
344C24C22×D414yes—
345C24C22×D412yes—
346C24C22×D414yes—
347C24C22×D414yes—
348C24C22×D414yes—
349C24C2×#41116yes—
350C24C2×#4118yes—
351C24H(5,𝔽2)18yes—
352C24H(5,𝔽2)18yes—
353C24C2511yescharacteristic
354C2×D4—432no—
355C2×D4—416no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—816no—
365C2×D4—88no—
366C2×D4—88no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—816no—
408C22×C4—816no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C22×D418yes—
412C22×C4C22×D418yes—
413C22×C4C22×D418yes—
414C4⋊C4—416no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C22⋊C4—216no—
418C22⋊C4—216no—
419C22⋊C4—232no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—432no—
423C22⋊C4—432no—
424C22⋊C4—432no—
425C22⋊C4—432no—
426C22⋊C4—432no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—88no—
434C42—88no—
435C23—24no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—216no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—216no—
444C23—216no—
445C23—216no—
446C23—2128no—
447C23—24no—
448C23—216no—
449C23—24no—
450C23—216no—
451C23—216no—
452C23—28no—
453C23—28no—
454C23—216no—
455C23—216no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—216no—
460C23—216no—
461C23—216no—
462C23—28no—
463C23—264no—
464C23—416no—
465C23—464no—
466C23—4128no—
467C23—464no—
468C23—4128no—
469C23—464no—
470C23—464no—
471C23—464no—
472C23—464no—
473C23—4128no—
474C23—464no—
475C23—464no—
476C23—4128no—
477C23—464no—
478C23—432no—
479C23—4128no—
480C23—464no—
481C23—41024no—
482C23—464no—
483C23—4128no—
484C23—44no—
485C23—48no—
486C23—48no—
487C23—48no—
488C23—44no—
489C23—44no—
490C23—48no—
491C23—44no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—44no—
496C23—44no—
497C23—464no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—48no—
507C23—44no—
508C23#52518yes—
509C23C2×#12714yes—
510C23C2×#12718yes—
511C23#53418yes—
512C23#546116yes—
513C23#546116yes—
514C23#551116yes—
515C23C23×D412yes—
516C23C23×D411yescharacteristic
517C23C22×#4114yes—
518C23C2×H(5,𝔽2)14yes—
519C23C2×H(5,𝔽2)14yes—
520D4—832no—
521D4—88no—
522D4—816no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—216no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—832no—
557C2×C4—816no—
558C2×C4—816no—
559C22—28no—
560C22—28no—
561C22—216no—
562C22—216no—
563C22—232no—
564C22—216no—
565C22—216no—
566C22—28no—
567C22—24no—
568C22—232no—
569C22—216no—
570C22—464no—
571C22—4256no—
572C22—4128no—
573C22—416no—
574C22—48no—
575C22—48no—
576C22—416no—
577C22—416no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—416no—
582C22—416no—
583C22—82no—
584C22—81no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22#2539116yes—
589C22C2×#52512yes—
590C22C22×#12711yescharacteristic
591C22C2×#53412yes—
592C22C2×#54614yes—
593C22C2×#54614yes—
594C22C2×#55114yes—
595C22#348414yes—
596C22#348412yes—
597C22#349018yes—
598C22#353214yes—
599C4—48no—
600C4—48no—
601C4—48no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4—81no—
606C2—216no—
607C2—44no—
608C2—416no—
609C2—161no—
610C2C2×#253912yes—
611C2#3539011yescharacteristic
612C2#3615814yes—
613C2#3783214yes—
614C2#5900614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2846C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2483C2semidirect
3. C2×#2412C2semidirect
4. C22×#379C2semidirect
5. #15480C2semidirect
6. #15283C2semidirect
7. #14911C2semidirect
8. #14734C2semidirect
9. #13995C2semidirect
10. #13903C2semidirect
11. C22⋊C42C2semidirect
13. C22×#123C22semidirect
24. C2×#399C22semidirect
25. C2×#397C22semidirect
26. C2×#397C22semidirect
27. C2×#394C22semidirect
28. C2×#390C22semidirect
29. C2×#386C22semidirect
30. C2×#386C22semidirect
31. C2×#385C22semidirect
32. C2×#385C22semidirect
33. C2×#381C22semidirect
34. C2×#381C22semidirect
35. C2×#380C22semidirect
43. C2×C4×C22⋊C4C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1438C22semidirect
49. #1438C22semidirect
50. #1437C22semidirect
51. #1437C22semidirect
52. #1437C22semidirect
53. C22⋊#95C22semidirect
77. C22×C4⋊C4C23semidirect
78. C22×C4⋊C4C23semidirect
79. C22×C4⋊C4C23semidirect
89. C22×C22⋊C4D4semidirect
105. C22×C22⋊C4C23semidirect
106. C22×C22⋊C4C23semidirect
107. C22×C42C23semidirect
127. C2×#95C23semidirect
128. C2×#95C23semidirect
129. C2×#95C23semidirect
130. C2×#95C23semidirect
131. C2×#95C23semidirect
132. C2×#95C23semidirect
133. C2×#95C23semidirect
134. C2×#95C23semidirect

Maximal subgroups

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

subgroupindex
C2×#28462
C2×#24832
C2×#24122
C22×#3792
#154802
#152832
#149112
#147342
#139952
#139032
C22⋊C422