Finite Group Databasebeta

← #6495371

Subgroups — #6495371

Order 512.

1851 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35343C211yesmaximal, characteristic, complemented
2#35278C211yesmaximal, characteristic, complemented
3#33304C211yesmaximal, characteristic, complemented
4#33299C211yesmaximal, characteristic, complemented
5#33237C211yesmaximal, characteristic, complemented
6#33231C211yesmaximal, characteristic, complemented
7#29811C212yesmaximal, complemented
8#29810C212yesmaximal, complemented
9#28630C214yesmaximal, complemented
10#28618C214yesmaximal, complemented
11#25611C214yesmaximal, complemented
12#25605C214yesmaximal, complemented
13#22276C212yesmaximal, complemented
14#22275C212yesmaximal, complemented
15#20465C211yesmaximal, characteristic, complemented
16C2×#540C2212yes—
17C2×#529C2212yes—
18C2×#496C2214yes—
19C2×#496C2214yes—
20C2×#493C2212yes—
21C2×#492C2212yes—
22C2×#490C2214yes—
23C2×#488C2214yes—
24C2×#433C2212yes—
25#2867C2211yescharacteristic
26#2861C2212yes—
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2856C2212yes—
31#2641C2212yes—
32#2621C2211yescharacteristic
33#2620C2212yes—
34#2619C2211yescharacteristic
35#2619C2212yes—
36#2602—24no—
37#2601—24no—
38#2445—24no—
39#2428—24no—
40#2328C2212yes—
41#2328C2211yescharacteristic
42#2327C2211yescharacteristic
43#2326C2212yes—
44C4×#127C2212yes—
45C4×#117C2211yescharacteristic
46#2055C2218yes—
47#2054C2218yes—
48#2000C2218yes—
49#1999C2218yes—
50#1992C2214yes—
51#1992C2214yes—
52#1961C2212yes—
53#1961C2212yes—
54#1961C2212yes—
55#1960C2212yes—
56#1959C2214yes—
57#1957C2214yes—
58#1880C2218yes—
59#1879C2218yes—
60#1879C2218yes—
61#1877C2218yes—
62#1835C2212yes—
63#1835C2212yes—
64#1768C2212yes—
65#1767C2212yes—
66#1750C2212yes—
67#1749C2212yes—
68#1746C2212yes—
69C4×#97C2211yescharacteristic
70C4×#97C2211yescharacteristic
71C22×SD16C2318yes—
72C22×D8C2318yes—
73#540—48no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×#127—24no—
76C2×#127C2312yes—
77C2×#127C2311yescharacteristic
78C2×#127C2312yes—
79C2×#127C2312yes—
80#529—48no—
81C2×#126—22no—
82C2×#125C2312yes—
83C2×#125C2312yes—
84C2×#124C2312yes—
85C2×#124C2312yes—
86C2×#123—22no—
87C2×#123—24no—
88C2×C22⋊Q8C2312yes—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121C2314yes—
92C2×#121C2312yes—
93C2×C4×Q8C2312yes—
94C2×C4×D4—22no—
95C2×C4×D4—24no—
96C2×C4×D4C2314yes—
97C2×C4×D4C2312yes—
98C2×#117—22no—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#496—48no—
103#496—48no—
104#493—28no—
105#492—28no—
106#490—48no—
107#488—48no—
108#433—28no—
109C2×C4⋊C8C2314yes—
110C2×#103C2318yes—
111C2×#102C2318yes—
112C2×#102C2318yes—
113C2×#102C2318yes—
114C2×C22⋊C8C2314yes—
115C2×#97C2312yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119C2×C4×C8C2312yes—
120C2×C4×C8C2312yes—
121#398—22no—
122#397—22no—
123#396—24no—
124#394—22no—
125#394—24no—
126#394—24no—
127#394—28no—
128#392—22no—
129#392—24no—
130#391C2312yes—
131#391C2312yes—
132#390—24no—
133#390—24no—
134#390—24no—
135#390C2314yes—
136#390C2312yes—
137#390C2314yes—
138#390C2312yes—
139#389—24no—
140#389—24no—
141#388—24no—
142#388—24no—
143#388—24no—
144#387—24no—
145#386—24no—
146#386—28no—
147#385—24no—
148#384C2312yes—
149#384C2312yes—
150#382C2311yescharacteristic
151#382C2312yes—
152#382C2311yescharacteristic
153C4×C4⋊C4D414yes—
154C4×C22⋊C4D414yes—
155#376C2311yescharacteristic
156#376C2311yescharacteristic
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#340C2318yes—
160#340C2318yes—
161#336C2314yes—
162#336C2314yes—
163C22×Q8C2412yes—
164C22×D4—22no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4C2412yes—
171C22×D4C2414yes—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4C2×D412yes—
175C23×C4C2411yescharacteristic
176C2×SD16—216no—
177C2×SD16—48no—
178C2×SD16—48no—
179C2×D8—216no—
180C2×D8—48no—
181C2×D8—48no—
182C22×C8C2414yes—
183C22×C8C2414yes—
184C4⋊Q8—24no—
185#127—24no—
186#127—24no—
187#127—24no—
188#127—28no—
189#127—28no—
190#127—88no—
191#126—48no—
192#125—28no—
193#125—28no—
194#124—28no—
195#124—28no—
196#123—416no—
197#123—48no—
198C22⋊Q8—44no—
199#121—416no—
200#121—44no—
201#121—48no—
202#121—88no—
203C4×Q8—44no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—88no—
208#117—24no—
209#117—24no—
210#117—48no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2414yes—
222C2×C4⋊C4C2412yes—
223C2×C4⋊C4C2412yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42C2×D412yes—
241C2×C42C2×D414yes—
242C2×C42C2×D414yes—
243C2×C42C2411yescharacteristic
244C2×C42C2411yescharacteristic
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247C2×C42C2411yescharacteristic
248C2×C42C2411yescharacteristic
249C4⋊C8—28no—
250#103—48no—
251#102—48no—
252#102—48no—
253#102—48no—
254C22⋊C8—28no—
255#97—24no—
256#97—24no—
257#97—24no—
258#97—24no—
259C4×C8—24no—
260C4×C8#4118yes—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95C2×D414yes—
267#95C2×D414yes—
268C24—22no—
269C24—24no—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—28no—
279C2×D4—24no—
280C2×D4—28no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—88no—
290C2×D4—88no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—88no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—84no—
306C22×C4—88no—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×D412yes—
314C22×C4C2511yescharacteristic
315SD16—816no—
316D8—816no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8C2×#4118yes—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—84no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—44no—
339C22⋊C4—48no—
340C22⋊C4—44no—
341C22⋊C4—48no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—48no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C42—24no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—44no—
355C42—44no—
356C42—44no—
357C42C2×#4114yes—
358C42H(5,𝔽2)14yes—
359C42H(5,𝔽2)14yes—
360C23—24no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—48no—
365C23—44no—
366C23—48no—
367C23—44no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23—44no—
372C23—42no—
373C23—44no—
374C23C23×D411yescharacteristic
375Q8—48no—
376D4—48no—
377D4—48no—
378D4—48no—
379D4—816no—
380D4—88no—
381D4—168no—
382D4—164no—
383D4—168no—
384D4—168no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—84no—
408C2×C4—88no—
409C2×C4—168no—
410C2×C4#53214yes—
411C2×C4#53514yes—
412C2×C4#53514yes—
413C2×C4C22×#4112yes—
414C2×C4C2×H(5,𝔽2)12yes—
415C2×C4C2×H(5,𝔽2)12yes—
416C8—28no—
417C8—44no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—88no—
423C22—84no—
424C22—82no—
425C22—88no—
426C22—82no—
427C22—84no—
428C22—84no—
429C22—84no—
430C22—82no—
431C22—84no—
432C22C2×#53211yescharacteristic
433C22C2×#53511yescharacteristic
434C22C2×#53511yescharacteristic
435C22#358514yes—
436C4—24no—
437C4—24no—
438C4—24no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4—84no—
446C4—82no—
447C4#350814yes—
448C2—81no—
449C2—162no—
450C2—164no—
451C2#3529311yescharacteristic
452C2#6071912yes—
453C2#6080812yes—
454C2#6083012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #35343C2semidirect
2. #35278C2semidirect
3. #33304C2semidirect
4. #33299C2semidirect
5. #33237C2semidirect
6. #33231C2semidirect
7. #29811C2semidirect
8. #29810C2semidirect
9. #28630C2semidirect
10. #28618C2semidirect
11. #25611C2semidirect
12. #25605C2semidirect
13. #22276C2semidirect
14. #22275C2semidirect
15. #20465C2semidirect

Maximal subgroups

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

subgroupindex
#353432
#352782
#333042
#332992
#332372
#332312
#298112
#298102
#286302
#286182
#256112
#256052
#222762
#222752
#204652