Finite Group Databasebeta

← #379840

Subgroups — #379840

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1899C212yesmaximal
4#14684C211yesmaximal, characteristic, complemented
5#11203C214yesmaximal, complemented
6#11043C214yesmaximal
7#10736C212yesmaximal, complemented
8C22×#123C2211yescharacteristic
9C22×#117C2212yes—
10C23×C42C2211yescharacteristic
11C22×#99C2214yes—
12#2853—24no—
13#2853—22no—
14#2853—21no—
15#2853—21no—
16C2×#388C414yescomplemented
17C2×#387C414yescomplemented
18C2×#386C2212yes—
19C2×#385C2212yes—
20#2285—28no—
21C2×#384—21no—
22C2×#382—21no—
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#380C2214yes—
26C2×C4×C4⋊C4—22no—
27#1899—28no—
28C2×#342C2214yes—
29C2×#342C2218yes—
30#1434—24no—
31C22⋊#95C2214yes—
32C23×D4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C24×C4C2312yes—
35C2×#123—21no—
36C2×#123—21no—
37C2×#123—22no—
38C2×#123—24no—
39C2×#123—24no—
40C2×#117—24no—
41C2×#117—48no—
42C2×#117D418yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—24no—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C414yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2×C414yes—
49C22×C22⋊C4C2314yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2314yes—
53C22×C42—24no—
54C22×C42—24no—
55C22×C42—22no—
56C22×C42C2×C414yes—
57C22×C42D414yes—
58C22×C42D412yes—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C2×#99—28no—
62C2×#99—48no—
63#388—24no—
64#388—24no—
65#388—24no—
66#388—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#386—22no—
72#386—24no—
73#386—22no—
74#385—22no—
75#385—24no—
76#385—22no—
77#384—28no—
78#382—28no—
79#381—21no—
80#381—21no—
81#381—22no—
82#381—24no—
83#381—24no—
84#380—216no—
85C4×C4⋊C4—216no—
86C2×#95—24no—
87C2×#95—28no—
88C2×#95—24no—
89C2×#95C2×C414yes—
90C2×#95C2×C414yes—
91C2×#95C2×C414yes—
92#342—28no—
93#342—48no—
94C25C22×C411yescharacteristic
95C25C2411yescharacteristic
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—48no—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C414yes—
111C23×C4C22×C412yes—
112C23×C4C22×C412yes—
113C23×C4C22×C414yes—
114C23×C4C2×D412yes—
115C23×C4C2×D414yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D414yes—
120C23×C4C2×D412yes—
121C23×C4#4114yes—
122#123—42no—
123#123—42no—
124#123—48no—
125#123—42no—
126#123—42no—
127#117—416no—
128#117—416no—
129C2×C4⋊C4—216no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—416no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—48no—
139C2×C22⋊C4—28no—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4C2×D418yes—
161C2×C42—216no—
162C2×C42—24no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—216no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—28no—
170C2×C42—48no—
171C2×C42C22⋊C414yes—
172C2×C42C22⋊C414yes—
173C2×C42C22⋊C414yes—
174C2×C42C22⋊C414yes—
175#99—416no—
176#95—216no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—416no—
184#95—48no—
185C24—21no—
186C24—22no—
187C24—24no—
188C24—44no—
189C24—44no—
190C24C23×C411yescharacteristic
191C24C22×D411yescharacteristic
192C24C22×D411yescharacteristic
193C24C22×D412yes—
194C24C2×#4112yes—
195C24H(5,𝔽2)14yes—
196C242−1+414yes—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—41no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—48no—
206C22×C4—28no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—28no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—216no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—22no—
220C22×C4—28no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—28no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—416no—
236C22×C4—416no—
237C22×C4—416no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—416no—
243C22×C4—44no—
244C22×C4—48no—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C414yes—
249C22×C4C4×D418yes—
250C22×C4#12012yes—
251C22×C4#12012yes—
252C22×C4#12118yes—
253C22×C4#12312yes—
254C22×C4#12312yes—
255C4⋊C4—432no—
256C4⋊C4—48no—
257C4⋊C4—416no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—48no—
261C22⋊C4—28no—
262C22⋊C4—28no—
263C22⋊C4—48no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—48no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—48no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C42—28no—
278C42—28no—
279C42—28no—
280C42—28no—
281C42—416no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—44no—
286C23—44no—
287C23—48no—
288C23—48no—
289C23—42no—
290C23—44no—
291C23—41no—
292C23—44no—
293C23—88no—
294C23—84no—
295C23C22×C22⋊C411yescharacteristic
296C23C2×C4×D412yes—
297C23#52014yes—
298C23C2×#12011yescharacteristic
299C23C2×#12112yes—
300C23C2×#12311yescharacteristic
301C23#53414yes—
302C23#53512yes—
303C23#53612yes—
304C23#53614yes—
305C23#53714yes—
306C23#53918yes—
307D4—81no—
308D4—81no—
309D4—84no—
310D4—82no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—216no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—216no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—416no—
329C2×C4—416no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—416no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—816no—
343C2×C4—88no—
344C2×C4—84no—
345C2×C4—816no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4#38614yes—
349C2×C4#38614yes—
350C2×C4#41018yes—
351C22—24no—
352C22—28no—
353C22—28no—
354C22—24no—
355C22—22no—
356C22—22no—
357C22—24no—
358C22—24no—
359C22—44no—
360C22—44no—
361C22—48no—
362C22—84no—
363C22—81no—
364C22—82no—
365C22—164no—
366C22C2×#38611yescharacteristic
367C22#231112yes—
368C22#232214yes—
369C22#261311yescharacteristic
370C22#261411yescharacteristic
371C22#261814yes—
372C22#262312yes—
373C22C2×#41014yes—
374C4—24no—
375C4—24no—
376C4—28no—
377C4—44no—
378C4—88no—
379C4—84no—
380C4—81no—
381C4—82no—
382C4—164no—
383C2—21no—
384C2—24no—
385C2—24no—
386C2—21no—
387C2—22no—
388C2—161no—
389C2#1425611yescharacteristic
390C2#2098312yes—
391C2#2187514yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
4. #14684C2semidirect
5. #11203C2semidirect
7. #10736C2semidirect
16. C2×#388C4semidirect
17. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#22852
C2×#18992
#146842
#112032
#110432
#107362