Finite Group Databasebeta

← #6489370

Subgroups — #6489370

Order 512.

1616 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic, complemented
2#34208C211yesmaximal, characteristic
3#32968C214yesmaximal, complemented
4#30086C212yesmaximal, complemented
5#30084C212yesmaximal, complemented
6#28721C212yesmaximal, complemented
7#28513C212yesmaximal, complemented
8#28216C212yesmaximal, complemented
9#28214C212yesmaximal, complemented
10#27952C212yesmaximal, complemented
11#27597C212yesmaximal, complemented
12#27595C212yesmaximal, complemented
13#25858C212yesmaximal
14#22222C214yesmaximal, complemented
15#20782C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#520C2212yes—
18C2×#488C2214yes—
19C2×#479C2214yes—
20C2×#478C2214yes—
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2851C2212yes—
24#2647C2212yes—
25#2629C2212yes—
26#2590—24no—
27#2587—24no—
28#2364—24no—
29#2352—24no—
30#2330C2212yes—
31#2328C2211yescharacteristic
32#2328C2211yescharacteristic
33#2327C2211yescharacteristic
34#2327C2211yescharacteristic
35#2314C2212yes—
36#2311C2212yes—
37C4×#125C2212yes—
38C4×#124C2212yes—
39C4×#124C2212yes—
40C2×#382C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2096C2212yes—
43#2095C2212yes—
44#2094C2212yes—
45#2093C2212yes—
46#2085C2214yes—
47#2085C2214yes—
48#2074C2212yes—
49#2074C2212yes—
50#2073C2212yes—
51#2073C2212yes—
52#2043C2214yes—
53#2041C2214yes—
54#1991C2214yes—
55#1958C2214yes—
56#1928C2214yes—
57#1927C2214yes—
58#1926C2214yes—
59#1925C2214yes—
60#1865C2214yes—
61#1864C2214yes—
62#1863C2214yes—
63#1862C2214yes—
64#1853C2214yes—
65#1852C2214yes—
66#1846C2214yes—
67#1840C2212yes—
68#1832C2212yes—
69#1818C2211yescharacteristic
70#1818C2211yescharacteristic
71#1803C2214yes—
72#1802C2214yes—
73#1798C2212yes—
74#1788C2214yes—
75#1788C2214yes—
76C4×C4⋊C8C2212yes—
77C4×C22⋊C8C2212yes—
78#1744C2212yes—
79C22×#41C2312yes—
80#532—28no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#125C2312yes—
84C2×#125C2312yes—
85C2×#124C2312yes—
86C2×#121—22no—
87#520—48no—
88C2×C4×Q8C2312yes—
89C2×C4×Q8C2312yes—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4C2312yes—
94C2×C4×D4C2312yes—
95C2×#117—22no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101#488—216no—
102#479—48no—
103#478—48no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×#103C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×C4×C8C2314yes—
120#400—24no—
121#398—24no—
122#398—24no—
123#396—24no—
124#391C2312yes—
125#391C2312yes—
126#390C2312yes—
127#390C2312yes—
128#389—24no—
129#389—24no—
130#389—24no—
131#389C2314yes—
132#389C2314yes—
133#388—24no—
134#388—24no—
135#388—24no—
136#387—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#387—24no—
141#387—24no—
142#386—24no—
143#386—24no—
144#386—24no—
145#385—24no—
146#384C2312yes—
147#384C2312yes—
148#382—22no—
149#382C2312yes—
150#382C2312yes—
151#382C2311yescharacteristic
152#382C2311yescharacteristic
153#382C2312yes—
154#380—24no—
155#380C2314yes—
156C4×C4⋊C4—24no—
157C4×C4⋊C4D414yes—
158C4×C4⋊C4C2314yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4D414yes—
162#376C2311yescharacteristic
163#376C2311yescharacteristic
164C2×#95—22no—
165C43C2311yescharacteristic
166C43C2311yescharacteristic
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171#336C2314yes—
172#336C2312yes—
173#336C2312yes—
174#336C2312yes—
175#336C2312yes—
176#336C2314yes—
177C2×#41—24no—
178C2×#41—24no—
179C2×#41—28no—
180C2×#41—44no—
181C22×Q8C2412yes—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4C2412yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4C2×D412yes—
190C23×C4C2411yescharacteristic
191C22×C8C2414yes—
192C22×C8C2414yes—
193C4⋊Q8—24no—
194#127—24no—
195#125—28no—
196#125—24no—
197#125—24no—
198#124—24no—
199#124#4118yes—
200#121—216no—
201C4×Q8—28no—
202C4×Q8—44no—
203C4×D4—216no—
204C4×D4—28no—
205C4×D4—44no—
206C4×D4—48no—
207C4×D4—48no—
208#117—24no—
209#117—24no—
210#117—48no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C2×D412yes—
236C2×C22⋊C4C2×D412yes—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42C2×D412yes—
242C2×C42C2×D414yes—
243C2×C42C2×D414yes—
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249C2×C42C2411yescharacteristic
250#107—44no—
251#107—44no—
252#106—44no—
253#106—44no—
254C4⋊C8—28no—
255#103—28no—
256#103—28no—
257#103—44no—
258#103—44no—
259#102—28no—
260#102—28no—
261#102—44no—
262#102—44no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C4×C8—28no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95C2×D414yes—
278#95C2×D414yes—
279C24—22no—
280C24C22×D411yescharacteristic
281#41—416no—
282#41—416no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8C2×#4114yes—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—84no—
295C2×D4C2×#4114yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—28no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C22×D412yes—
315C22×C4C22×D412yes—
316C22×C4C22×D412yes—
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D412yes—
320C22×C4C22×D411yescharacteristic
321C22×C42−1+414yes—
322C22×C4C2511yescharacteristic
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C4⋊C4—28no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4—84no—
346C4⋊C4—84no—
347C22⋊C4—24no—
348C22⋊C4—216no—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—24no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—44no—
361C42—44no—
362C42—44no—
363C42—44no—
364C42—84no—
365C42C2×#4114yes—
366C42H(5,𝔽2)14yes—
367C23—24no—
368C23—21no—
369C23—21no—
370C23—24no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23#53612yes—
376C23C23×D411yescharacteristic
377Q8—28no—
378Q8—44no—
379D4—28no—
380D4—416no—
381D4—44no—
382D4—88no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—416no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—84no—
411C2×C4—164no—
412C2×C4—164no—
413C2×C4—164no—
414C2×C4#52514yes—
415C2×C4#53514yes—
416C2×C4#53612yes—
417C2×C4C22×#4112yes—
418C2×C4C2×H(5,𝔽2)12yes—
419C2×C4C2×2−1+412yes—
420C8—44no—
421C8—44no—
422C22—22no—
423C22—24no—
424C22—42no—
425C22—42no—
426C22—48no—
427C22—44no—
428C22—82no—
429C22—82no—
430C22—84no—
431C22C2×#52511yescharacteristic
432C22C2×#53511yescharacteristic
433C22C2×#53611yescharacteristic
434C22#358414yes—
435C4—24no—
436C4—24no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4#352014yes—
448C2—42no—
449C2—84no—
450C2#3524011yescharacteristic
451C2#6055712yes—
452C2#6082312yes—
453C2#6088212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35336C2semidirect
3. #32968C2semidirect
4. #30086C2semidirect
5. #30084C2semidirect
6. #28721C2semidirect
7. #28513C2semidirect
8. #28216C2semidirect
9. #28214C2semidirect
10. #27952C2semidirect
11. #27597C2semidirect
12. #27595C2semidirect
14. #22222C2semidirect
15. #20782C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#342082
#329682
#300862
#300842
#287212
#285132
#282162
#282142
#279522
#275972
#275952
#258582
#222222
#207822