Finite Group Databasebeta

← #379878

Subgroups — #379878

Order 512.

1749 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3C2×#1897C212yesmaximal
4#20682C211yesmaximal, characteristic, complemented
5#11226C214yesmaximal, complemented
6#11075C214yesmaximal, complemented
7#10720C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C22×#117C2212yes—
10C23×C42C2211yescharacteristic
11C22×#104C2214yes—
12C2×#407C2212yes—
13#2855C414yescomplemented
14#2854C414yescomplemented
15#2853—21no—
16#2853—21no—
17#2849—22no—
18C4×#123—24no—
19#2297C2212yes—
20#2296C2212yes—
21#2285—28no—
22C2×#384—21no—
23C2×#382—21no—
24#2280C2211yescharacteristic
25C2×#380—22no—
26C2×C4×C4⋊C4C2214yes—
27#1897—28no—
28#1817—24no—
29C2×#339C2214yes—
30C2×#339C2218yes—
31#1728C2214yes—
32C22×#41C2311yescharacteristic
33C24×C4D412yes—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36#525—24no—
37#525—24no—
38C2×#123—24no—
39C2×#123—21no—
40C2×C22⋊Q8C2×C412yes—
41C2×#121C2×C412yes—
42C2×#120—21no—
43C2×C4×D4—22no—
44C2×#117—24no—
45C2×#117—48no—
46C2×#117D418yes—
47C2×#117C2311yescharacteristic
48C2×#117C2312yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C22⋊C4—22no—
52C22×C42—24no—
53C22×C42—24no—
54C22×C42C2×C414yes—
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D414yes—
58C22×C42C2314yes—
59C22×C42C2312yes—
60C2×#104—28no—
61C2×#104—48no—
62#407—24no—
63#407—24no—
64C2×C22⋊C8—24no—
65#391C2×C414yes—
66#390C2×C414yes—
67#389C2×C414yes—
68#389C2×C414yes—
69#388—24no—
70#388—24no—
71#387—24no—
72#387—24no—
73#386—24no—
74#386—22no—
75#386—22no—
76#385—24no—
77#385—22no—
78#385—22no—
79#384—28no—
80#382—28no—
81#381—24no—
82#381—21no—
83#380—216no—
84#380C2×C414yes—
85#379—21no—
86C4×C4⋊C4—216no—
87C4×C4⋊C4—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—24no—
90C2×#95—24no—
91#339—28no—
92#339—48no—
93#336—28no—
94C25C2×D411yescharacteristic
95C2×#41—22no—
96C2×#41—22no—
97C2×#41—24no—
98C2×#41—24no—
99C22×Q8C22×C411yescharacteristic
100C22×D4—22no—
101C22×D4—21no—
102C22×D4C22×C411yescharacteristic
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—48no—
110C23×C4C22⋊C414yes—
111C23×C4C22⋊C414yes—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D414yes—
117C23×C4C2×D412yes—
118C23×C4#4114yes—
119C23×C4C2411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—28no—
123C2×M16—28no—
124C22×C8—24no—
125#123—416no—
126#123—42no—
127#123—42no—
128C22⋊Q8—28no—
129#121—28no—
130#120—42no—
131#120—42no—
132C4×D4—48no—
133#117—22no—
134#117—22no—
135#117—24no—
136#117—24no—
137#117—416no—
138#117—416no—
139C2×C4⋊C4—216no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—416no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—48no—
157C2×C42—216no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—216no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—28no—
166C2×C42—28no—
167C2×C42—24no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42C22⋊C414yes—
171C2×C42C22⋊C414yes—
172C2×C42C22×C414yes—
173C2×C42C2×D418yes—
174C2×C42C2×D414yes—
175#104—416no—
176C22⋊C8—48no—
177#95—216no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183C24—22no—
184C24—22no—
185C24—21no—
186C24—44no—
187C24C2×C22⋊C412yes—
188C24#12012yes—
189C24#12312yes—
190C24C22×D411yescharacteristic
191#41—44no—
192#41—44no—
193#41—48no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×D4—22no—
197C2×D4—24no—
198C2×D4—48no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—44no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—216no—
213C22×C4—28no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—28no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—416no—
228C22×C4—416no—
229C22×C4—416no—
230C22×C4—48no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C4×D418yes—
237C22×C4#12012yes—
238C22×C4#12118yes—
239C22×C4#12312yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D412yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4H(5,𝔽2)14yes—
245C22×C42−1+414yes—
246M16—44no—
247M16—44no—
248M16—48no—
249C2×C8—44no—
250C2×C8—48no—
251C4⋊C4—24no—
252C4⋊C4—28no—
253C4⋊C4—432no—
254C4⋊C4—416no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—416no—
260C22⋊C4—48no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—48no—
268C42—28no—
269C42—28no—
270C42—28no—
271C42—28no—
272C42—28no—
273C42—28no—
274C42—42no—
275C42—48no—
276C42—44no—
277C23—24no—
278C23—24no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—48no—
288C23—42no—
289C23—44no—
290C23—41no—
291C23#38614yes—
292C23C22×C22⋊C411yescharacteristic
293C23C2×#12011yescharacteristic
294C23C2×#12311yescharacteristic
295C23#53512yes—
296C23#53612yes—
297Q8—44no—
298D4—44no—
299D4—88no—
300D4—81no—
301D4—81no—
302D4—82no—
303C2×C4—28no—
304C2×C4—28no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—216no—
312C2×C4—216no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—416no—
317C2×C4—416no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—416no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—816no—
330C2×C4—84no—
331C2×C4#38614yes—
332C2×C4#42118yes—
333C2×C4C2×C4×D412yes—
334C2×C4#52014yes—
335C2×C4C2×#12112yes—
336C2×C4#53414yes—
337C2×C4#53614yes—
338C2×C4#53714yes—
339C2×C4#53918yes—
340C8—84no—
341C22—24no—
342C22—24no—
343C22—22no—
344C22—22no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—44no—
349C22—44no—
350C22—48no—
351C22—48no—
352C22—84no—
353C22—81no—
354C22—82no—
355C22C2×#38611yescharacteristic
356C22#231112yes—
357C22#261311yescharacteristic
358C22#261411yescharacteristic
359C22#262312yes—
360C22C2×#42114yes—
361C4—24no—
362C4—28no—
363C4—44no—
364C4—88no—
365C4—84no—
366C4—81no—
367C4—82no—
368C4#232214yes—
369C4#261814yes—
370C2—21no—
371C2—21no—
372C2—22no—
373C2—44no—
374C2—161no—
375C2#1425611yescharacteristic
376C2#2097912yes—
377C2#2190014yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
4. #20682C2semidirect
5. #11226C2semidirect
6. #11075C2semidirect
7. #10720C2semidirect
13. #2855C4semidirect
14. #2854C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22852
C2×#18972
#206822
#112262
#110752
#107202