Finite Group Databasebeta

← #5955354

Subgroups — #5955354

Order 512.

4646 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2315C211yesmaximal, characteristic
2C22×#385C212yesmaximal
3C22×#380C211yesmaximal, characteristic
4#15014C214yesmaximal, complemented
5#14920C218yesmaximal, complemented
6#14821C214yesmaximal, complemented
7#14728C218yesmaximal, complemented
8#13794C212yesmaximal
9#13794C211yesmaximal, characteristic
10C25×C4C2211yescharacteristic
11C23×C4⋊C4C2214yes—
12C23×C4⋊C4C2212yes—
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2212yes—
17C2×#398C2218yescomplemented
18#2315—28no—
19C2×#385—216no—
20C2×#385C2214yes—
21C2×#385C2214yes—
22C2×#385C2218yes—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#380—28no—
26C2×#380C2214yes—
27C2×#380C2218yes—
28C22×#95C2214yes—
29C22×#95C2212yes—
30#1445C2218yescomplemented
31#1444C2218yescomplemented
32#1442C22116yescomplemented
33#1441C2218yescomplemented
34#1441C22116yescomplemented
35C22⋊#95C2214yes—
36C22⋊#95C2218yes—
37C22⋊#95C2218yes—
38C22⋊#95C2214yes—
39C22⋊#95C2218yes—
40C22⋊#95C2214yes—
41C26C2311yescharacteristic
42C24×C4—24no—
43C24×C4—24no—
44C24×C4—48no—
45C24×C4D418yes—
46C24×C4C2311yescharacteristic
47C24×C4C2314yes—
48C24×C4C2312yes—
49C24×C4C2312yes—
50C24×C4C2312yes—
51C24×C4C2311yescharacteristic
52C24×C4C2312yes—
53C24×C4C2314yes—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—216no—
56C22×C4⋊C4—48no—
57C22×C4⋊C4—416no—
58C22×C4⋊C4C2318yes—
59C22×C4⋊C4C2318yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2318yes—
62C22×C4⋊C4C2314yes—
63C22×C22⋊C4—28no—
64C22×C22⋊C4—28no—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—28no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—28no—
70C22×C22⋊C4—28no—
71C22×C22⋊C4D418yes—
72C22×C22⋊C4Q8116yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79#398—416no—
80#385—216no—
81#385—232no—
82#385—216no—
83#385—216no—
84#385—216no—
85#380—216no—
86#380—232no—
87C2×#95—28no—
88C2×#95—216no—
89C2×#95C2318yes—
90C2×#95C2318yes—
91C2×#95C2318yes—
92C2×#95C2318yes—
93C2×#95C2318yes—
94C2×#95C23116yes—
95C2×#95C23116yes—
96C2×#95C2318yes—
97C25—44no—
98C25—44no—
99C25C2×D414yes—
100C25C2×D414yes—
101C25C2×Q818yes—
102C25#4114yes—
103C25#4118yes—
104C25C2411yescharacteristic
105C25C2412yes—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—28no—
109C23×C4—232no—
110C23×C4—28no—
111C23×C4—22no—
112C23×C4—28no—
113C23×C4—28no—
114C23×C4—28no—
115C23×C4—22no—
116C23×C4—28no—
117C23×C4—216no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—28no—
126C23×C4—432no—
127C23×C4—48no—
128C23×C4—48no—
129C23×C4—44no—
130C23×C4—48no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—48no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—48no—
137C23×C4—432no—
138C23×C4—432no—
139C23×C4C2×D414yes—
140C23×C4C2×D4116yes—
141C23×C4C2×D418yes—
142C23×C4C2×D414yes—
143C23×C4C2×Q8116yes—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2412yes—
149C23×C4C2418yes—
150C23×C4C2412yes—
151C2×C4⋊C4—216no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—216no—
154C2×C4⋊C4—216no—
155C2×C4⋊C4—232no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—416no—
160C2×C4⋊C4—432no—
161C2×C4⋊C4—464no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C22⋊C4—264no—
165C2×C22⋊C4—232no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—432no—
182C2×C22⋊C4—416no—
183C2×C22⋊C4C2×D4116yes—
184C2×C22⋊C4C2×Q8116yes—
185C2×C22⋊C4C2×Q8116yes—
186#95—216no—
187#95—216no—
188#95—216no—
189#95—216no—
190#95—48no—
191#95—416no—
192#95—48no—
193#95—416no—
194C24—28no—
195C24—24no—
196C24—28no—
197C24—24no—
198C24—48no—
199C24—48no—
200C24—48no—
201C24—48no—
202C24—48no—
203C24—432no—
204C24—48no—
205C24—432no—
206C24#121116yes—
207C24C22⋊Q8132yes—
208C24#123116yes—
209C24#125132yes—
210C24C4⋊Q8116yes—
211C24C22×D412yes—
212C24C22×D414yes—
213C24C22×D414yes—
214C24C22×D412yes—
215C24C22×Q814yes—
216C24C22×Q814yes—
217C24C22×Q814yes—
218C24C2×#4112yes—
219C24C2×#4114yes—
220C24C2×#4114yes—
221C24C2×#4114yes—
222C24C2×#4114yes—
223C24C2511yescharacteristic
224C22×C4—216no—
225C22×C4—216no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—232no—
232C22×C4—28no—
233C22×C4—232no—
234C22×C4—22no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—216no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—216no—
243C22×C4—28no—
244C22×C4—216no—
245C22×C4—232no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—216no—
256C22×C4—28no—
257C22×C4—216no—
258C22×C4—416no—
259C22×C4—48no—
260C22×C4—432no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—432no—
264C22×C4—432no—
265C22×C4—4128no—
266C22×C4—416no—
267C22×C4—416no—
268C22×C4—432no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—416no—
272C22×C4—48no—
273C22×C4—416no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—816no—
279C22×C4—88no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—88no—
283C22×C4—816no—
284C22×C4C22×D418yes—
285C22×C4C22×D414yes—
286C22×C4C22×D414yes—
287C22×C4C22×Q8116yes—
288C22×C4H(5,𝔽2)18yes—
289C4⋊C4—432no—
290C4⋊C4—432no—
291C4⋊C4—432no—
292C4⋊C4—88no—
293C4⋊C4—88no—
294C22⋊C4—232no—
295C22⋊C4—232no—
296C22⋊C4—232no—
297C22⋊C4—416no—
298C22⋊C4—416no—
299C22⋊C4—416no—
300C23—28no—
301C23—24no—
302C23—28no—
303C23—24no—
304C23—24no—
305C23—28no—
306C23—24no—
307C23—28no—
308C23—24no—
309C23—28no—
310C23—28no—
311C23—24no—
312C23—216no—
313C23—216no—
314C23—216no—
315C23—216no—
316C23—216no—
317C23—216no—
318C23—416no—
319C23—416no—
320C23—464no—
321C23—432no—
322C23—464no—
323C23—432no—
324C23#398164yescomplemented
325C23C2×#12114yes—
326C23C2×#12118yes—
327C23C2×C22⋊Q818yes—
328C23C2×C22⋊Q818yes—
329C23C2×C22⋊Q818yes—
330C23C2×#12318yes—
331C23C2×#12314yes—
332C23C2×#12518yes—
333C23C2×#12518yes—
334C23C2×#12518yes—
335C23C2×C4⋊Q818yes—
336C23C2×C4⋊Q814yes—
337C23C23×D411yescharacteristic
338C23C23×D411yescharacteristic
339C23C23×Q812yes—
340C23C22×#4111yescharacteristic
341C23C22×#4112yes—
342C23C2×H(5,𝔽2)14yes—
343C23#58514yes—
344C2×C4—232no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—216no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—232no—
356C2×C4—464no—
357C2×C4—464no—
358C2×C4—416no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—416no—
364C2×C4—416no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—416no—
377C2×C4—816no—
378C2×C4—816no—
379C2×C4—816no—
380C2×C4—816no—
381C2×C4#53418yes—
382C22—28no—
383C22—24no—
384C22—28no—
385C22—216no—
386C22—216no—
387C22—216no—
388C22—216no—
389C22—28no—
390C22—216no—
391C22—24no—
392C22—28no—
393C22—216no—
394C22—216no—
395C22—216no—
396C22—464no—
397C22C2×#39818yes—
398C22C2×#398116yes—
399C22C22×#12111yescharacteristic
400C22C22×C22⋊Q812yes—
401C22C22×#12311yescharacteristic
402C22C22×#12512yes—
403C22C22×C4⋊Q811yescharacteristic
404C22C2×#53412yes—
405C22#344912yes—
406C22#348412yes—
407C22#348512yes—
408C22#349012yes—
409C22#349112yes—
410C22#350118yes—
411C22#350518yes—
412C4—28no—
413C4—48no—
414C4—44no—
415C4—416no—
416C4—84no—
417C4—84no—
418C4—84no—
419C2—216no—
420C2—28no—
421C2C22×#39811yescharacteristic
422C2#3498111yescharacteristic
423C2#3501014yes—
424C2#3508911yescharacteristic
425C2#3535112yes—
426C2#3536814yes—
427C2#3539112yes—
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. #15014C2semidirect
5. #14920C2semidirect
6. #14821C2semidirect
7. #14728C2semidirect
17. C2×#398C22semidirect
30. #1445C22semidirect
31. #1444C22semidirect
32. #1442C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
324. C23#398semidirect

Maximal subgroups

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

subgroupindex
C2×#23152
C22×#3852
C22×#3802
#150142
#149202
#148212
#147282
#137942