Finite Group Databasebeta

← #7535001

Subgroups — #7535001

Order 512.

1955 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#31272C214yesmaximal, complemented
4#29141C212yesmaximal, complemented
5#28150C214yesmaximal, complemented
6#24972C214yesmaximal, complemented
7#24239C214yesmaximal, complemented
8#23977C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic, complemented
10#22430C214yesmaximal, complemented
11C4×#461C212yesmaximal, complemented
12C2×#555C2212yes—
13C2×D4×Q8C2214yes—
14C2×#461C2212yes—
15C2×#448C2214yes—
16C2×C4×SD16C2214yes—
17C2×#436C2214yes—
18C2×#419C2214yes—
19#2868C2211yescharacteristic
20#2856—21no—
21#2683—22no—
22#2563C2212yes—
23#2551—21no—
24#2538C2214yes—
25#2537C2214yes—
26#2500—22no—
27#2491—24no—
28#2452—24no—
29#2409C2214yes—
30#2366—21no—
31#2354C2214yes—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34#2285—21no—
35C42×D4C2211yescharacteristic
36#2077C2218yes—
37#2035C2218yes—
38#2024C2218yes—
39#2000C2218yes—
40#1981C2212yes—
41#1956C2214yes—
42#1954C2214yes—
43#1948C2212yes—
44#1927C2214yes—
45#1895C2214yes—
46#1894C2218yes—
47#1877C2214yes—
48#1873C2218yes—
49#1870C2214yes—
50#1834C2214yes—
51#1825C2214yes—
52#1803C2214yes—
53#1803C2214yes—
54#1802C2212yes—
55#1787C2218yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C23×Q8C2314yes—
59C22×SD16C2314yes—
60C22×M16C2314yes—
61#555—44no—
62#555—44no—
63D4×Q8—216no—
64D4×Q8—48no—
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#127D411yescharacteristic
70C2×#124—24no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8C2318yes—
73C2×#121—22no—
74C2×#121—24no—
75C2×#121C2314yes—
76C2×C4×Q8C2314yes—
77C2×C4×D4—24no—
78C2×C4×D4C2312yes—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2314yes—
81C2×C4×D4C2312yes—
82C2×#117—22no—
83C2×#117C2314yes—
84C22×C42—21no—
85#461—28no—
86#448—48no—
87C4×SD16—48no—
88#436—48no—
89C2×#106C2314yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92#419—48no—
93C2×#103C2314yes—
94C2×#103C2314yes—
95C2×#103C2318yes—
96C2×#103C2318yes—
97C2×#103C2314yes—
98C2×#102C2314yes—
99C2×C22⋊C8C2318yes—
100C2×C22⋊C8C2314yes—
101C2×C4×C8C2314yes—
102#396—28no—
103#395—24no—
104#394—28no—
105#393—24no—
106#393—24no—
107#391C2314yes—
108#390—22no—
109#390—24no—
110#390—24no—
111#390D412yes—
112#390D414yes—
113#389—24no—
114#389—22no—
115#389—22no—
116#388—28no—
117#386—24no—
118#385—24no—
119#384—21no—
120#384—21no—
121#384D411yescharacteristic
122#384C2312yes—
123#384C2314yes—
124#382—22no—
125#380—24no—
126#380C2318yes—
127#380C2314yes—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2318yes—
138#340C2318yes—
139#340C2314yes—
140#336C2314yes—
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—216no—
144C22×Q8—48no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8C2414yes—
148C22×D4—24no—
149C22×D4C2×D414yes—
150C22×D4C2×D412yes—
151C22×D4C2411yescharacteristic
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4C2414yes—
156C2×SD16—28no—
157C2×SD16—28no—
158C2×SD16—44no—
159C2×M16—44no—
160C2×M16—48no—
161C22×C8C2414yes—
162C22×C8C2414yes—
163C4⋊Q8—24no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—416no—
167C4⋊Q8—84no—
168C4⋊Q8#4118yes—
169#127—24no—
170#124—416no—
171C22⋊Q8—216no—
172C22⋊Q8—432no—
173C22⋊Q8—48no—
174#121—28no—
175#121—44no—
176#121—48no—
177#121—88no—
178C4×Q8—28no—
179C4×Q8—28no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—28no—
183C4×D4—24no—
184C4×D4—28no—
185C4×D4—24no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—416no—
189#117—28no—
190#117—44no—
191#117—48no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2412yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4C2×D418yes—
215C2×C22⋊C4C2×D414yes—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—44no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C2×C42C2412yes—
226#106—44no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229#103—28no—
230#103—28no—
231#103—48no—
232#103—44no—
233#103—48no—
234#102—44no—
235C22⋊C8—48no—
236C22⋊C8—44no—
237C4×C8—28no—
238#95—22no—
239#95—21no—
240#95—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244C24—21no—
245C24C22×D414yes—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—416no—
252C2×Q8—44no—
253C2×Q8—416no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—816no—
257C2×Q8—88no—
258C2×Q8C2×#4118yes—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—24no—
262C2×D4—28no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—48no—
269C2×D4—88no—
270C2×D4H(5,𝔽2)12yes—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—416no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—84no—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D414yes—
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C2511yescharacteristic
294SD16—416no—
295M16—88no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—416no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—82no—
319C4⋊C4—84no—
320C4⋊C4H(5,𝔽2)12yes—
321C22⋊C4—24no—
322C22⋊C4—216no—
323C22⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—416no—
327C22⋊C4—44no—
328C22⋊C4—84no—
329C22⋊C4—88no—
330C42—24no—
331C42—21no—
332C42—21no—
333C42—41no—
334C42—48no—
335C42—48no—
336C42—44no—
337C42—42no—
338C42#13714yes—
339C42C2×#4114yes—
340C42H(5,𝔽2)14yes—
341C23—24no—
342C23—28no—
343C23—22no—
344C23—24no—
345C23—24no—
346C23—28no—
347C23—41no—
348C23—42no—
349C23—41no—
350C23—42no—
351C23C23×D411yescharacteristic
352Q8—28no—
353Q8—28no—
354Q8—44no—
355Q8—44no—
356Q8—816no—
357Q8—164no—
358D4—24no—
359D4—416no—
360D4—41no—
361D4—41no—
362D4—84no—
363D4—168no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—28no—
369C2×C4—28no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—48no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—816no—
391C2×C4—88no—
392C2×C4#54614yes—
393C2×C4#54614yes—
394C2×C4C2×#13712yes—
395C2×C4C2×#13714yes—
396C2×C4#57714yes—
397C2×C4C22×#4112yes—
398C2×C4C2×H(5,𝔽2)11yescharacteristic
399C2×C4C2×H(5,𝔽2)11yescharacteristic
400C2×C4C2×H(5,𝔽2)12yes—
401C8—44no—
402C8—44no—
403C22—48no—
404C22—42no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—48no—
409C22—42no—
410C22—44no—
411C22—82no—
412C22—82no—
413C22C2×#54612yes—
414C22#348411yescharacteristic
415C22C22×#13711yescharacteristic
416C22C2×#57712yes—
417C22#358611yescharacteristic
418C4—24no—
419C4—22no—
420C4—42no—
421C4—44no—
422C4—48no—
423C4—41no—
424C4—84no—
425C4—82no—
426C4—84no—
427C4#332918yes—
428C4#348111yescharacteristic
429C4#351912yes—
430C4#354911yescharacteristic
431C2—84no—
432C2—82no—
433C2#5904111yescharacteristic
434C2C2×#332912yes—
435C2#6120212yes—
436C2#6158611yescharacteristic
437C2#6166211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59817C2semidirect
3. #31272C2semidirect
4. #29141C2semidirect
5. #28150C2semidirect
6. #24972C2semidirect
7. #24239C2semidirect
8. #23977C2semidirect
9. #23176C2semidirect
10. #22430C2semidirect
11. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#598172
#361872
#312722
#291412
#281502
#249722
#242392
#239772
#231762
#224302
C4×#4612