Finite Group Databasebeta

← #636926

Subgroups — #636926

Order 512.

5510 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2483C212yesmaximal
3C22×#385C211yesmaximal, characteristic
4#16262C218yesmaximal, complemented
5#15073C212yesmaximal, complemented
6#14934C214yesmaximal, complemented
7#14874C212yesmaximal, complemented
8#14720C214yesmaximal, complemented
9#14681C211yesmaximal, characteristic
10#14263C214yesmaximal, complemented
11#13869C212yesmaximal
12C24×Q8C2211yescharacteristic
13C25×C4C2211yescharacteristic
14C22×C22⋊Q8—28no—
15C22×C22⋊Q8—216no—
16C22×C22⋊Q8C2214yes—
17C22×C22⋊Q8C2214yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24#2483—216no—
25C2×#397C2218yes—
26C2×#395C2218yescomplemented
27C2×#385—28no—
28C2×#385C2214yes—
29C2×#385C2212yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#381C2214yes—
34C2×#381C2214yes—
35C2×#381C2214yes—
36C2×#381C2214yes—
37C2×#379C2212yes—
38C22×#95C2212yes—
39#1446C2218yescomplemented
40#1444C22116yescomplemented
41#1442C22116yescomplemented
42#1442C2218yescomplemented
43#1441C2218yescomplemented
44#1440C2218yes—
45C22⋊#95C2214yes—
46C22⋊#95C2212yes—
47C22⋊#95C2218yes—
48C22⋊#95C2214yes—
49C22⋊#95C2212yes—
50C26C2311yescharacteristic
51C23×Q8—28no—
52C23×Q8—24no—
53C23×Q8—48no—
54C23×Q8C2314yes—
55C24×C4—28no—
56C24×C4—48no—
57C24×C4D418yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2312yes—
61C24×C4C2312yes—
62C24×C4C2312yes—
63C24×C4C2312yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2312yes—
67C2×C22⋊Q8—2128no—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8—216no—
70C2×C22⋊Q8—232no—
71C2×C22⋊Q8—232no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—28no—
75C22×C4⋊C4—28no—
76C22×C4⋊C4—28no—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—48no—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C2318yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—24no—
87C22×C22⋊C4—28no—
88C22×C22⋊C4—28no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4D4116yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100#397—232no—
101#395—416no—
102#385—216no—
103#385—28no—
104#385—216no—
105#385—28no—
106#385—216no—
107#381—216no—
108#381—216no—
109#381—216no—
110#381—48no—
111#379—28no—
112C2×#95—28no—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C23116yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C23116yes—
121C2×#95C2318yes—
122C25—24no—
123C25—28no—
124C25—24no—
125C25—44no—
126C25C2×D414yes—
127C25C2×D418yes—
128C25C2411yescharacteristic
129C25C2411yescharacteristic
130C25C2411yescharacteristic
131C22×Q8—232no—
132C22×Q8—216no—
133C22×Q8—28no—
134C22×Q8—216no—
135C22×Q8—48no—
136C22×Q8—464no—
137C22×Q8—432no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—232no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—28no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—24no—
160C23×C4—432no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—48no—
166C23×C4—44no—
167C23×C4—464no—
168C23×C4C2×D418yes—
169C23×C4C2×D414yes—
170C23×C4C2×D418yes—
171C23×C4C2×D418yes—
172C23×C4C2×D418yes—
173C23×C4C2×D414yes—
174C23×C4C2418yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C22⋊Q8—2128no—
183C22⋊Q8—2128no—
184C2×C4⋊C4—232no—
185C2×C4⋊C4—264no—
186C2×C4⋊C4—216no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—216no—
189C2×C4⋊C4—216no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—216no—
194C2×C4⋊C4—216no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—432no—
197C2×C4⋊C4—416no—
198C2×C22⋊C4—232no—
199C2×C22⋊C4—264no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—416no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4C2×D4116yes—
221C2×C22⋊C4C2×D4116yes—
222C2×C22⋊C4#4118yes—
223#95—28no—
224#95—28no—
225#95—216no—
226#95—216no—
227#95—216no—
228#95—232no—
229#95—216no—
230#95—48no—
231#95—416no—
232C24—22no—
233C24—24no—
234C24—24no—
235C24—24no—
236C24—22no—
237C24—28no—
238C24—232no—
239C24—216no—
240C24—28no—
241C24—232no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—22no—
246C24—48no—
247C24—416no—
248C24—432no—
249C24—432no—
250C24#120116yes—
251C24C22×D412yes—
252C24C22×D414yes—
253C24C22×D412yes—
254C24C22×D414yes—
255C24C22×D412yes—
256C24C22×D414yes—
257C24C2×#4114yes—
258C242−1+418yes—
259C242−1+414yes—
260C24C2511yescharacteristic
261C2×Q8—232no—
262C2×Q8—464no—
263C2×Q8—4128no—
264C2×Q8—416no—
265C2×Q8—416no—
266C22×C4—232no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—216no—
271C22×C4—216no—
272C22×C4—216no—
273C22×C4—216no—
274C22×C4—28no—
275C22×C4—216no—
276C22×C4—216no—
277C22×C4—232no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—24no—
288C22×C4—216no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—216no—
293C22×C4—216no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—216no—
301C22×C4—28no—
302C22×C4—216no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—416no—
309C22×C4—432no—
310C22×C4—432no—
311C22×C4—432no—
312C22×C4—4128no—
313C22×C4—48no—
314C22×C4—416no—
315C22×C4—416no—
316C22×C4—416no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—416no—
321C22×C4—48no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—416no—
325C22×C4—48no—
326C22×C4—88no—
327C22×C4C22×D418yes—
328C22×C4C22×D418yes—
329C22×C4C22×D414yes—
330C22×C4C22×D414yes—
331C22×C4C2×#4118yes—
332C22×C4H(5,𝔽2)18yes—
333C4⋊C4—232no—
334C4⋊C4—264no—
335C4⋊C4—416no—
336C4⋊C4—416no—
337C4⋊C4—816no—
338C22⋊C4—216no—
339C22⋊C4—232no—
340C22⋊C4—232no—
341C22⋊C4—216no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—416no—
345C22⋊C4—416no—
346C22⋊C4—88no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—28no—
352C23—22no—
353C23—28no—
354C23—28no—
355C23—28no—
356C23—216no—
357C23—216no—
358C23—216no—
359C23—216no—
360C23—232no—
361C23—232no—
362C23—22no—
363C23—24no—
364C23—24no—
365C23—28no—
366C23—22no—
367C23—28no—
368C23—28no—
369C23—416no—
370C23—432no—
371C23—432no—
372C23—4128no—
373C23—48no—
374C23—48no—
375C23C2×#12014yes—
376C23C2×#12014yes—
377C23C2×#12014yes—
378C23#52514yes—
379C23#536116yes—
380C23#53618yes—
381C23#53614yes—
382C23#547116yes—
383C23C23×D411yescharacteristic
384C23C23×D412yes—
385C23C22×#4112yes—
386C23C2×H(5,𝔽2)14yes—
387C23C2×2−1+414yes—
388C23C2×2−1+412yes—
389Q8—464no—
390C2×C4—232no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—216no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—216no—
401C2×C4—216no—
402C2×C4—216no—
403C2×C4—432no—
404C2×C4—432no—
405C2×C4—432no—
406C2×C4—416no—
407C2×C4—432no—
408C2×C4—432no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—416no—
428C2×C4—416no—
429C2×C4—88no—
430C2×C4—88no—
431C2×C4#53418yes—
432C22—28no—
433C22—28no—
434C22—24no—
435C22—28no—
436C22—28no—
437C22—232no—
438C22—232no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—464no—
446C22—44no—
447C22—44no—
448C22—416no—
449C22#240618yes—
450C22#261418yes—
451C22C22×#12011yescharacteristic
452C22C2×#52511yescharacteristic
453C22C2×#53412yes—
454C22C2×#53612yes—
455C22C2×#53614yes—
456C22C2×#53611yescharacteristic
457C22C2×#54714yes—
458C22#348414yes—
459C22#352018yes—
460C22#353818yes—
461C4—28no—
462C4—48no—
463C4—44no—
464C4—48no—
465C4—48no—
466C4—82no—
467C4—88no—
468C4—82no—
469C2—28no—
470C2—28no—
471C2—44no—
472C2C2×#261411yescharacteristic
473C2C2×#240611yescharacteristic
474C2#3498011yescharacteristic
475C2#3508412yes—
476C2#3745312yes—
477C2#3803718yes—
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. #16262C2semidirect
5. #15073C2semidirect
6. #14934C2semidirect
7. #14874C2semidirect
8. #14720C2semidirect
10. #14263C2semidirect
26. C2×#395C22semidirect
39. #1446C22semidirect
40. #1444C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#24832
C22×#3852
#162622
#150732
#149342
#148742
#147202
#146812
#142632
#138692