Finite Group Databasebeta

← #7569862

Subgroups — #7569862

Order 512.

1953 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#32037C214yesmaximal, complemented
4#28748C212yesmaximal, complemented
5#28608C214yesmaximal, complemented
6#26234C214yesmaximal, complemented
7#25886C214yesmaximal, complemented
8#24055C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic, complemented
10#22431C214yesmaximal, complemented
11C4×#476C212yesmaximal, complemented
12C2×#547C2214yes—
13C2×#547C2212yes—
14C2×#487C2214yes—
15C2×#483C2214yes—
16C2×#476C2212yes—
17C2×#436C2214yes—
18C2×#417C2214yes—
19#2868C2211yescharacteristic
20#2856—21no—
21#2804—22no—
22#2580—24no—
23#2571—21no—
24#2563C2214yes—
25#2542C2212yes—
26#2542C2214yes—
27#2501—24no—
28#2501—22no—
29#2487C2214yes—
30#2366—21no—
31D4×C4⋊C4C2214yes—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34C2×#384—21no—
35C42×D4C2211yescharacteristic
36#2043C2218yes—
37#2035C2218yes—
38#2035C2218yes—
39#2035C2218yes—
40#1987C2212yes—
41#1981C2214yes—
42#1981C2214yes—
43#1956C2214yes—
44#1954C2214yes—
45#1948C2212yes—
46#1920C2214yes—
47#1894C2218yes—
48#1894C2214yes—
49#1872C2218yes—
50#1870C2214yes—
51#1864C2214yes—
52#1802C2212yes—
53#1788C2214yes—
54#1788C2218yes—
55C4×#106C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#102C2214yes—
58C22×#41C2314yes—
59C22×M16C2314yes—
60#547—216no—
61#547—48no—
62#547—44no—
63#547—44no—
64C2×C4⋊Q8C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#127D411yescharacteristic
68C2×#123—28no—
69C2×#123—24no—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2318yes—
72C2×#121—24no—
73C2×#121—22no—
74C2×C4×Q8—22no—
75C2×C4×D4—24no—
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2312yes—
78C2×C4×D4C2314yes—
79C2×C4×D4C2312yes—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4C2314yes—
83C22×C42—21no—
84#487—48no—
85#483—48no—
86#476—28no—
87#436—48no—
88C2×#106C2314yes—
89C2×#106C2314yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92#417—48no—
93C2×#103C2318yes—
94C2×#103C2318yes—
95C2×#103C2318yes—
96C2×#103C2318yes—
97C2×#102C2314yes—
98C2×#102C2314yes—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2318yes—
101C2×C4×C8C2314yes—
102#400—24no—
103#399—24no—
104#399—28no—
105#399—24no—
106#393—28no—
107#391—22no—
108#390—24no—
109#390—24no—
110#390—22no—
111#390D414yes—
112#390D412yes—
113#389—24no—
114#389—22no—
115#389C2314yes—
116#386—24no—
117#385—24no—
118#385—28no—
119#384—24no—
120#384—22no—
121#384—21no—
122#384—22no—
123#384—21no—
124#384D411yescharacteristic
125#384C2312yes—
126#382—24no—
127#380C2314yes—
128#380C2318yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2314yes—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C43C2311yescharacteristic
136#340C2314yes—
137#340C2314yes—
138#340C2314yes—
139#340C2314yes—
140#336C2314yes—
141C2×#41—216no—
142C2×#41—28no—
143C2×#41—44no—
144C2×#41—48no—
145C22×Q8—22no—
146C22×Q8C2414yes—
147C22×D4—24no—
148C22×D4C2×D414yes—
149C22×D4C2×D412yes—
150C22×D4C2411yescharacteristic
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4C2414yes—
156C2×M16—44no—
157C2×M16—48no—
158C22×C8C2414yes—
159C22×C8C2414yes—
160C4⋊Q8—24no—
161C4⋊Q8—28no—
162C4⋊Q8—24no—
163C4⋊Q8—28no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166#127—24no—
167#123—432no—
168#123—416no—
169C22⋊Q8—216no—
170C22⋊Q8—28no—
171C22⋊Q8—48no—
172C22⋊Q8—44no—
173#121—416no—
174#121—48no—
175C4×Q8—84no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—28no—
179C4×D4—24no—
180C4×D4—28no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—88no—
184C4×D4#4118yes—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—416no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2412yes—
205C2×C4⋊C4C2411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4C2×D418yes—
213C2×C22⋊C4C2×D414yes—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—44no—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C2×C42C2412yes—
223#106—28no—
224#106—28no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227#103—48no—
228#103—48no—
229#103—48no—
230#103—48no—
231#102—28no—
232#102—28no—
233C22⋊C8—48no—
234C22⋊C8—44no—
235C4×C8—28no—
236#95—21no—
237#95—22no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—28no—
242C24—21no—
243C24C22×D414yes—
244#41—416no—
245#41—816no—
246#41—88no—
247C2×Q8—28no—
248C2×Q8—24no—
249C2×Q8—28no—
250C2×Q8—24no—
251C2×Q8—44no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—24no—
257C2×D4—28no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—416no—
263C2×D4—48no—
264C2×D4C2×#4114yes—
265C2×D4H(5,𝔽2)12yes—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—416no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—84no—
284C22×C4—88no—
285C22×C4C22×D414yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D412yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C2511yescharacteristic
291M16—88no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C4⋊C4—28no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—416no—
307C4⋊C4—48no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—84no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4C2×#4114yes—
318C4⋊C4H(5,𝔽2)12yes—
319C22⋊C4—24no—
320C22⋊C4—28no—
321C22⋊C4—216no—
322C22⋊C4—44no—
323C22⋊C4—416no—
324C22⋊C4—48no—
325C22⋊C4—84no—
326C22⋊C4—88no—
327C42—24no—
328C42—21no—
329C42—21no—
330C42—41no—
331C42—42no—
332C42—44no—
333C42—48no—
334C42#13714yes—
335C42C2×#4114yes—
336C42H(5,𝔽2)14yes—
337C23—24no—
338C23—22no—
339C23—24no—
340C23—28no—
341C23—24no—
342C23—24no—
343C23—28no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23C23×D411yescharacteristic
348Q8—44no—
349Q8—48no—
350Q8—44no—
351Q8—88no—
352D4—24no—
353D4—22no—
354D4—416no—
355D4—84no—
356D4—816no—
357D4#55914yes—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—416no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—816no—
386C2×C4—168no—
387C2×C4#54614yes—
388C2×C4#54614yes—
389C2×C4C2×#13712yes—
390C2×C4C2×#13714yes—
391C2×C4#57714yes—
392C2×C4C22×#4112yes—
393C2×C4C2×H(5,𝔽2)11yescharacteristic
394C2×C4C2×H(5,𝔽2)11yescharacteristic
395C2×C4C2×H(5,𝔽2)12yes—
396C8—44no—
397C8—44no—
398C22—24no—
399C22—44no—
400C22—42no—
401C22—44no—
402C22—42no—
403C22—44no—
404C22—48no—
405C22—44no—
406C22—48no—
407C22—44no—
408C22C2×#54612yes—
409C22#348411yescharacteristic
410C22C22×#13711yescharacteristic
411C22C2×#57712yes—
412C22#358611yescharacteristic
413C4—24no—
414C4—22no—
415C4—44no—
416C4—42no—
417C4—44no—
418C4—41no—
419C4—84no—
420C4—84no—
421C4—84no—
422C4#333118yes—
423C4C2×#55911yescharacteristic
424C4#350812yes—
425C4#354311yescharacteristic
426C2—44no—
427C2—84no—
428C2#5914311yescharacteristic
429C2C2×#333112yes—
430C2#6120912yes—
431C2#6158611yescharacteristic
432C2#6166411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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. #59864C2semidirect
3. #32037C2semidirect
4. #28748C2semidirect
5. #28608C2semidirect
6. #26234C2semidirect
7. #25886C2semidirect
8. #24055C2semidirect
9. #23176C2semidirect
10. #22431C2semidirect
11. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#361792
#320372
#287482
#286082
#262342
#258862
#240552
#231762
#224312
C4×#4762