Finite Group Databasebeta

← #460353

Subgroups — #460353

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2096C212yesmaximal, complemented
3C2×#2035C214yesmaximal
4#14686C211yesmaximal, characteristic
5#11323C212yesmaximal, complemented
6#11050C212yesmaximal, complemented
7#11010C212yesmaximal, complemented
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C23×C42C2211yescharacteristic
12C22×#106C2214yes—
13C22×#103C2214yes—
14C22×#103C2214yes—
15C22×C22⋊C8C2212yes—
16#2857—28no—
17C2×#391C2212yes—
18C2×#389—22no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#380—22no—
23C2×#380—22no—
24C2×#380C2212yes—
25#2096—28no—
26#2035—216no—
27C2×#340C2214yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C4⋊#95—22no—
31C4⋊#95C2212yes—
32C23×Q8C2312yes—
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C23×C8C2314yes—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—24no—
38C2×C4⋊Q8—42no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—28no—
42C2×C22⋊Q8—44no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C42—24no—
54C22×C42—24no—
55C22×C42—42no—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#106—28no—
60C2×#106—28no—
61C2×#106—44no—
62C2×#103—28no—
63C2×#103—28no—
64C2×#103—28no—
65C2×#103—44no—
66C2×#103—44no—
67C2×#103—44no—
68C2×C22⋊C8—42no—
69C2×C22⋊C8—42no—
70C2×C22⋊C8D418yes—
71#391—24no—
72#391D418yes—
73#389—216no—
74#384—24no—
75#384—24no—
76#384—42no—
77#380—216no—
78#380—28no—
79#380—48no—
80C2×#95—24no—
81C2×#95—22no—
82C2×#95—24no—
83#340—28no—
84#336—24no—
85#336—24no—
86C25C2×D411yescharacteristic
87C22×Q8—22no—
88C22×Q8—24no—
89C22×Q8—24no—
90C22×Q8—24no—
91C22×Q8—24no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8C2×D414yes—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4SD1618yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D414yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—24no—
125C22×C8—24no—
126C22×C8C2×D418yes—
127C4⋊Q8—416no—
128C22⋊Q8—416no—
129C22⋊Q8—416no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4C2×D414yes—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—48no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—416no—
169C2×C42—42no—
170C2×C42C2×D414yes—
171C2×C42#4114yes—
172#106—416no—
173#103—416no—
174#103—416no—
175C22⋊C8—48no—
176#95—216no—
177#95—44no—
178#95—48no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—22no—
186C24—22no—
187C24—22no—
188C24#12712yes—
189C24C2×SD1614yes—
190C24C22×D411yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—28no—
194C2×Q8—28no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—44no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—28no—
225C22×C4—216no—
226C22×C4—216no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—48no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4#12014yes—
261C22×C4#12314yes—
262C22×C4#12712yes—
263C22×C4C2×SD1614yes—
264C22×C4C2×SD1614yes—
265C22×C4#13714yes—
266C22×C4#13714yes—
267C22×C4C22×D412yes—
268C22×C4C22×D412yes—
269C22×C4C2×#4112yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)12yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C42−1+411yescharacteristic
274C22×C42−1+412yes—
275C2×C8—28no—
276C2×C8—28no—
277C2×C8—28no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C4⋊C4—28no—
283C4⋊C4—416no—
284C4⋊C4—416no—
285C4⋊C4—416no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—88no—
292C4⋊C4—88no—
293C22⋊C4—416no—
294C22⋊C4—88no—
295C42—28no—
296C42—28no—
297C42—48no—
298C23—24no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—24no—
306C23—24no—
307C23—24no—
308C23—24no—
309C23—22no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#49214yes—
321C23C2×#12011yescharacteristic
322C23C2×#12311yescharacteristic
323C23C2×#12711yescharacteristic
324C23#53411yescharacteristic
325C23#53511yescharacteristic
326C23#53611yescharacteristic
327C23#53711yescharacteristic
328C23C22×SD1612yes—
329C23C2×#13712yes—
330C23C2×#13712yes—
331C23#57512yes—
332Q8—48no—
333Q8—48no—
334Q8—84no—
335Q8—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—432no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—84no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4#44818yes—
381C2×C4#48318yes—
382C2×C4#53412yes—
383C2×C4#53612yes—
384C2×C4#53712yes—
385C2×C4#53712yes—
386C2×C4#54012yes—
387C2×C4#54212yes—
388C2×C4D4214yes—
389C2×C4#54614yes—
390C2×C4#54714yes—
391C2×C4#56214yes—
392C8—48no—
393C22—24no—
394C22—24no—
395C22—24no—
396C22—44no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—44no—
402C22—42no—
403C22—44no—
404C22#240511yescharacteristic
405C22#243311yescharacteristic
406C22#250111yescharacteristic
407C22#261611yescharacteristic
408C22#262411yescharacteristic
409C22#265611yescharacteristic
410C22#281611yescharacteristic
411C22C2×#44812yes—
412C22#300712yes—
413C22C2×#48312yes—
414C22#309512yes—
415C22C2×#49211yescharacteristic
416C22#315011yescharacteristic
417C22#315211yescharacteristic
418C22#315211yescharacteristic
419C22#318712yes—
420C22#319512yes—
421C22#320612yes—
422C22#321612yes—
423C22#322912yes—
424C22#323712yes—
425C22#324612yes—
426C22#325512yes—
427C4—28no—
428C4—28no—
429C4—42no—
430C4—84no—
431C4—84no—
432C4—162no—
433C4—162no—
434C4#267114yes—
435C4#267214yes—
436C2—44no—
437C2#1636311yescharacteristic
438C2#2400312yes—
439C2#2571811yescharacteristic
440C2#2572811yescharacteristic
441C2#2790511yescharacteristic
442C2#2791611yescharacteristic
443C2#2968812yes—
444C2#2992212yes—
445C2#3024011yescharacteristic
446C2#3025011yescharacteristic
447C2#3277112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2096C2semidirect
5. #11323C2semidirect
6. #11050C2semidirect
7. #11010C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20962
C2×#20352
#146862
#113232
#110502
#110102
#107122