Finite Group Databasebeta

← #379874

Subgroups — #379874

Order 512.

1505 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35157C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic, complemented
3#20979C212yesmaximal
4C2×#1822C211yesmaximal, characteristic, complemented
5#11224C214yesmaximal, complemented
6#11072C214yesmaximal, complemented
7#10716C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11C2×#421C2214yes—
12C2×#407C2212yes—
13#2853—22no—
14#2622—22no—
15#2500C414yescomplemented
16#2499C414yescomplemented
17#2485—24no—
18#2317—24no—
19#2317C2212yes—
20#2316C2214yes—
21#2315—22no—
22#2311C2212yes—
23#2296—24no—
24#2285—22no—
25#2280C2211yescharacteristic
26#1897—24no—
27#1896—24no—
28#1822—24no—
29C2×#343C2214yes—
30#1733C2214yes—
31C2×H(3,ℤ/4)C2218yes—
32C22×#41C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35#525—22no—
36#525—22no—
37#525—24no—
38C2×#123—21no—
39C2×C22⋊Q8C2×C412yes—
40C2×#121C2×C412yes—
41C2×#120—21no—
42C2×C4×D4—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—24no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—48no—
49C2×#117C2×C414yes—
50C2×#117D414yes—
51C2×#117D414yes—
52C2×#117C2311yescharacteristic
53C2×#117C2312yes—
54C2×#117C2314yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42D412yes—
60C22×C42D414yes—
61#421—48no—
62C2×#104—28no—
63#407—24no—
64#407—22no—
65#407—22no—
66C2×C22⋊C8—24no—
67#397—28no—
68#394—28no—
69#391C2×C414yescomplemented
70#390C2×C414yescomplemented
71#389—24no—
72#389—216no—
73#388—28no—
74#387—28no—
75#386—24no—
76#385—216no—
77#385—24no—
78#384—24no—
79#384C2×C414yescomplemented
80#382—24no—
81#382C2×C414yescomplemented
82#381—21no—
83#380—28no—
84#380—22no—
85#380—24no—
86#379—21no—
87C4×C4⋊C4—28no—
88C4×C22⋊C4C2×C414yes—
89#343—28no—
90#339—28no—
91H(3,ℤ/4)—48no—
92C25C2×D411yescharacteristic
93C2×#41—22no—
94C2×#41—22no—
95C2×#41—24no—
96C2×#41—24no—
97C22×Q8C22×C411yescharacteristic
98C22×D4—21no—
99C22×D4—22no—
100C22×D4C22×C411yescharacteristic
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—21no—
105C23×C4C22×C412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D414yes—
111C23×C4#4114yes—
112C23×C4C2411yescharacteristic
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—28no—
116C2×M16—28no—
117C22×C8—24no—
118#123—41no—
119#123—41no—
120#123—42no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123#121—24no—
124#121—24no—
125#120—42no—
126#120—42no—
127C4×D4—44no—
128C4×D4—44no—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—28no—
133#117—28no—
134#117—28no—
135#117—24no—
136#117—24no—
137#117—22no—
138#117—22no—
139#117—48no—
140#117—48no—
141#117—48no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—216no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—48no—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C2×D414yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4C22⋊C418yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—21no—
171C2×C42—48no—
172C2×C42C22⋊C418yes—
173C2×C42C22×C414yes—
174C2×C42C2×D418yes—
175#104—416no—
176C22⋊C8—44no—
177C22⋊C8—44no—
178#95—216no—
179#95—24no—
180#95—28no—
181#95—28no—
182C24—22no—
183C24—22no—
184C24—21no—
185C24—82no—
186C24C2×C22⋊C412yes—
187C24#12012yes—
188C24#12312yes—
189C24C22×D411yescharacteristic
190#41—44no—
191#41—44no—
192#41—48no—
193C2×Q8—22no—
194C2×Q8—24no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—84no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—216no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—84no—
222C22×C4—88no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C418yes—
225C22×C4C4×D414yes—
226C22×C4C4×D414yes—
227C22×C4#12011yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12114yes—
230C22×C4#12114yes—
231C22×C4#12312yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C2×#4112yes—
236C22×C4H(5,𝔽2)14yes—
237C22×C42−1+414yes—
238M16—44no—
239M16—44no—
240M16—48no—
241C2×C8—44no—
242C2×C8—48no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—816no—
252C4⋊C4—88no—
253C22⋊C4—216no—
254C22⋊C4—48no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—48no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—88no—
264C22⋊C4—84no—
265C42—28no—
266C42—28no—
267C42—28no—
268C42—28no—
269C42—28no—
270C42—48no—
271C42—42no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—24no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—44no—
281C23—44no—
282C23—42no—
283C23—41no—
284C23—84no—
285C23—84no—
286C23#38614yes—
287C23C22×C22⋊C411yescharacteristic
288C23C2×#12011yescharacteristic
289C23C2×#12311yescharacteristic
290C23#53512yes—
291C23#53612yes—
292Q8—44no—
293D4—44no—
294D4—88no—
295D4—81no—
296D4—81no—
297D4—82no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—84no—
321C2×C4—816no—
322C2×C4—84no—
323C2×C4#38612yes—
324C2×C4#38612yes—
325C2×C4C2×C4×D412yes—
326C2×C4#52014yes—
327C2×C4C2×#12112yes—
328C2×C4#53414yes—
329C2×C4#53614yes—
330C2×C4#53714yes—
331C2×C4#53918yes—
332C8—84no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—24no—
338C22—24no—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—44no—
343C22—84no—
344C22—84no—
345C22—84no—
346C22—82no—
347C22—81no—
348C22C2×#38611yescharacteristic
349C22#231112yes—
350C22#261311yescharacteristic
351C22#261411yescharacteristic
352C22#262312yes—
353C4—22no—
354C4—22no—
355C4—48no—
356C4—84no—
357C4—88no—
358C4—82no—
359C4—81no—
360C4#232214yes—
361C4#261814yes—
362C2—21no—
363C2—22no—
364C2—21no—
365C2—22no—
366C2—44no—
367C2—161no—
368C2#1425611yescharacteristic
369C2#2101212yes—
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34052C2semidirect
4. C2×#1822C2semidirect
5. #11224C2semidirect
6. #11072C2semidirect
7. #10716C2semidirect
15. #2500C4semidirect
16. #2499C4semidirect
69. #391C2×C4semidirect
70. #390C2×C4semidirect
79. #384C2×C4semidirect
81. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351572
#340522
#209792
C2×#18222
#112242
#110722
#107162