Finite Group Databasebeta

← #4453297

Subgroups — #4453297

Order 512.

4134 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2483C212yesmaximal
3C22×#397C211yesmaximal, characteristic
4#16364C218yesmaximal, complemented
5#16262C218yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14934C212yesmaximal, complemented
8#14914C212yesmaximal, complemented
9#14720C212yesmaximal, complemented
10#14341C211yesmaximal, characteristic
11#13869C212yesmaximal
12C2×#543—28no—
13C2×#543—216no—
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2214yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23#2483—216no—
24C2×#397—28no—
25C2×#397C2218yes—
26C2×#397C2218yes—
27C2×#395C2214yescomplemented
28C2×#385C2214yes—
29C2×#385C2214yes—
30C2×#381C2214yes—
31C2×#381C2214yes—
32C2×#381C2214yes—
33C2×#379C2212yes—
34C22×#95C2211yescharacteristic
35C22×#95C2211yescharacteristic
36C22×#95C2211yescharacteristic
37#1446C2214yescomplemented
38#1446C2214yescomplemented
39#1446C2214yescomplemented
40#1444C2214yescomplemented
41#1444C2218yescomplemented
42#1442C2214yescomplemented
43#1442C2218yescomplemented
44#1442C2218yescomplemented
45#1442C2214yescomplemented
46#1441C2218yescomplemented
47#1441C2214yescomplemented
48#1440C2218yes—
49#1440C2218yes—
50C22⋊#95C2212yes—
51C22⋊#95C2212yes—
52C22⋊#95C2214yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C26C2311yescharacteristic
56C23×Q8C2312yes—
57C23×Q8C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2312yes—
65C24×C4C2312yes—
66#543—2128no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—216no—
70C2×C22⋊Q8—216no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—216no—
76C2×C22⋊Q8—216no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2318yes—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4D418yes—
93C22×C22⋊C4D4116yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102#397—232no—
103#397—232no—
104#395—48no—
105#385—216no—
106#385—216no—
107#381—216no—
108#381—216no—
109#381—48no—
110#379—28no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2314yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C25—24no—
129C25—24no—
130C25—42no—
131C25—42no—
132C25—42no—
133C25—42no—
134C25C2×D414yes—
135C25C2×D418yes—
136C25C2412yes—
137C25C2411yescharacteristic
138C22×Q8—24no—
139C22×Q8—28no—
140C22×Q8—24no—
141C22×Q8—28no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—44no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4C2×D418yes—
179C23×C4C2×D418yes—
180C23×C4C2418yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C22⋊Q8—2128no—
190C22⋊Q8—2128no—
191C22⋊Q8—264no—
192C22⋊Q8—264no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—416no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—48no—
210C2×C22⋊C4—232no—
211C2×C22⋊C4—264no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
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—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—416no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C2×D4116yes—
236C2×C22⋊C4C2×D4116yes—
237C2×C22⋊C4C2×D4116yes—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—216no—
242#95—216no—
243#95—216no—
244#95—216no—
245#95—48no—
246#95—44no—
247#95—44no—
248#95—48no—
249#95—44no—
250#95—48no—
251#95—44no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—232no—
263C24—28no—
264C24—28no—
265C24—44no—
266C24—44no—
267C24—44no—
268C24—44no—
269C24—416no—
270C24—416no—
271C24—88no—
272C24—88no—
273C24—88no—
274C24—88no—
275C24#120116yes—
276C24C22×D414yes—
277C24C22×D414yes—
278C24C22×D414yes—
279C24C22×D414yes—
280C24C22×D412yes—
281C242−1+414yes—
282C242−1+414yes—
283C24C2511yescharacteristic
284C2×Q8—216no—
285C2×Q8—216no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—816no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—816no—
292C22×C4—28no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—28no—
311C22×C4—216no—
312C22×C4—216no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—28no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—416no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—416no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4C22×D414yes—
364C22×C4C22×D418yes—
365C22×C4C22×D414yes—
366C22×C4C22×D418yes—
367C22×C4H(5,𝔽2)14yes—
368C4⋊C4—232no—
369C4⋊C4—216no—
370C4⋊C4—216no—
371C4⋊C4—232no—
372C4⋊C4—416no—
373C4⋊C4—816no—
374C22⋊C4—232no—
375C22⋊C4—232no—
376C22⋊C4—232no—
377C22⋊C4—416no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—88no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—28no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—28no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—28no—
398C23—24no—
399C23—28no—
400C23—24no—
401C23—24no—
402C23—28no—
403C23—28no—
404C23—232no—
405C23—416no—
406C23—416no—
407C23—48no—
408C23—48no—
409C23—48no—
410C23—48no—
411C23—864no—
412C23—88no—
413C23—88no—
414C23—88no—
415C23—88no—
416C23C2×#12018yes—
417C23C2×#12014yes—
418C23#536116yes—
419C23#53614yes—
420C23#53614yes—
421C23C23×D411yescharacteristic
422C23C23×D412yes—
423C23C2×H(5,𝔽2)12yes—
424C23C2×2−1+412yes—
425C23C2×2−1+412yes—
426C23#58512yes—
427C23#58512yes—
428C23#58512yes—
429Q8—816no—
430Q8—816no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—216no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—216no—
442C2×C4—28no—
443C2×C4—416no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—48no—
474C2×C4—88no—
475C2×C4—816no—
476C2×C4—88no—
477C2×C4—816no—
478C2×C4—88no—
479C2×C4—88no—
480C2×C4—88no—
481C22—24no—
482C22—232no—
483C22—24no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—28no—
489C22—28no—
490C22—28no—
491C22—416no—
492C22—416no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—832no—
498C22#261418yes—
499C22C22×#12011yescharacteristic
500C22C2×#53614yes—
501C22C2×#53611yescharacteristic
502C22C2×#53611yescharacteristic
503C22#348411yescharacteristic
504C22#348511yescharacteristic
505C22#348514yes—
506C22#348511yescharacteristic
507C22#348511yescharacteristic
508C22#349614yes—
509C22#353514yes—
510C22#356114yes—
511C22#356414yes—
512C22#356414yes—
513C4—44no—
514C4—48no—
515C4—44no—
516C4—48no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4—88no—
521C4—82no—
522C2—28no—
523C2—48no—
524C2C2×#261411yescharacteristic
525C2#3534711yescharacteristic
526C2#3544611yescharacteristic
527C2#3672011yescharacteristic
528C2#3696611yescharacteristic
529C2#3697011yescharacteristic
530C2#3697011yescharacteristic
531C2#4051118yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #16364C2semidirect
5. #16262C2semidirect
6. #14934C2semidirect
7. #14934C2semidirect
8. #14914C2semidirect
9. #14720C2semidirect
27. C2×#395C22semidirect
37. #1446C22semidirect
38. #1446C22semidirect
39. #1446C22semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#24832
C22×#3972
#163642
#162622
#149342
#149142
#147202
#143412
#138692