Finite Group Databasebeta

← #208459

Subgroups — #208459

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35086C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic
3#13796C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9714C212yesmaximal, complemented
6#9146C212yesmaximal
7#9143C212yesmaximal
8#8221C212yesmaximal, complemented
9#8217C212yesmaximal, complemented
10#2857C2211yescharacteristic
11#2851C2211yescharacteristic
12#2848C2212yes—
13#2652—22no—
14#2505—22no—
15#2496C412yescomplemented
16#2489C412yescomplemented
17#2480—22no—
18#2429C412yescomplemented
19#2421C412yescomplemented
20#2334—22no—
21C4×C22⋊Q8C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C22×#95C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1797C2212yes—
28#1779C2212yes—
29#1749C2211yescharacteristic
30C4×C22⋊C8C2211yescharacteristic
31C2×#328C2212yes—
32C2×#328C2212yes—
33C2×#328C2212yes—
34C2×#328C2212yes—
35#1434—22no—
36#1434C2212yes—
37#1434C2212yes—
38C22⋊#95—22no—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#126—22no—
47C2×#125C2×C411yescharacteristic, complemented
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8C2×C412yescomplemented
52C22×C22⋊C4—21no—
53C22×C22⋊C4D412yes—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2312yes—
61C2×C22⋊C8C2312yes—
62C2×#97—21no—
63C2×C4×C8—21no—
64#400—22no—
65#400—22no—
66#400—22no—
67#398—22no—
68#397—22no—
69#397—22no—
70#397—22no—
71#396—22no—
72#391C2×C412yescomplemented
73#391C2×C412yescomplemented
74#389—22no—
75#389—22no—
76#389C2×C412yescomplemented
77#389C2×C412yescomplemented
78#388—22no—
79#388—22no—
80#388—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#384C2×C411yescharacteristic, complemented
91#382—22no—
92#382C2×C411yescharacteristic, complemented
93#380—22no—
94#380C2×C412yes—
95#380C2×C412yes—
96#380C2312yes—
97#379—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yescomplemented
100C4×C22⋊C4—22no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2×C412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106C2×#95—21no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#336—22no—
116#336—21no—
117#336—21no—
118#328—24no—
119#328—24no—
120#328—24no—
121#328—24no—
122C25C2×D411yescharacteristic
123C22×Q8C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—44no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—82no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C4⋊Q8—81no—
144#126—48no—
145#125—81no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—82no—
151C22⋊Q8—82no—
152C4×Q8—82no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C4⋊C8—82no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198#97—81no—
199C4×C8—81no—
200#95—24no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95#4114yes—
223#95#4114yes—
224C24—21no—
225C24—21no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—41no—
230C24—41no—
231C24C2×C22⋊C412yes—
232C24C22×D411yescharacteristic
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—48no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—41no—
325C42—41no—
326C42—42no—
327C42—42no—
328C42—41no—
329C42—41no—
330C42—44no—
331C42—82no—
332C42—84no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23#42114yes—
354C23C22×C22⋊C411yescharacteristic
355C23#52511yescharacteristic
356C23#52511yescharacteristic
357C23#57712yes—
358C23#57812yes—
359Q8—82no—
360Q8—82no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—816no—
417C2×C4C4×#4112yes—
418C2×C4#53212yes—
419C8—82no—
420C8—82no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—44no—
433C22—48no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22#229711yescharacteristic
440C22#288311yescharacteristic
441C22#288311yescharacteristic
442C22#289812yes—
443C22C2×#42112yes—
444C22#312012yes—
445C22#312311yescharacteristic
446C22#312711yescharacteristic
447C4—22no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—84no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C2—21no—
461C2—22no—
462C2—21no—
463C2—44no—
464C2#1759812yes—
465C2#1759812yes—
466C2#2139811yescharacteristic
467C2#2141711yescharacteristic
468C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9714C2semidirect
8. #8221C2semidirect
9. #8217C2semidirect
15. #2496C4semidirect
16. #2489C4semidirect
18. #2429C4semidirect
19. #2421C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
47. C2×#125C2×C4semidirect
51. C2×C4×Q8C2×C4semidirect
72. #391C2×C4semidirect
73. #391C2×C4semidirect
76. #389C2×C4semidirect
77. #389C2×C4semidirect
90. #384C2×C4semidirect
92. #382C2×C4semidirect
99. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350862
#219042
#137962
#99132
#97142
#91462
#91432
#82212
#82172