Finite Group Databasebeta

← #194903

Subgroups — #194903

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#21833C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9027C218yesmaximal
6#8014C212yesmaximal
7C22×C4⋊Q8—21no—
8C22×C4⋊Q8C2211yescharacteristic
9C2×#519—21no—
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2212yes—
12#2546—24no—
13#2546C418yescomplemented
14C2×#398C2214yes—
15#2351—24no—
16Q8×C4⋊C4—24no—
17#2316—21no—
18#2316C2214yes—
19C2×#391—21no—
20C2×#389—22no—
21#2285—22no—
22#2285C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1990—24no—
25#1988—24no—
26#1823C2218yes—
27#1461—24no—
28C2×#324C2214yes—
29#1294C2218yes—
30C23×Q8—21no—
31C23×Q8D411yescharacteristic
32C2×C4⋊Q8—28no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×C4⋊Q8—44no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8C2×C414yes—
40#519—44no—
41#519—44no—
42C2×C4×Q8—28no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—24no—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8D414yes—
48C2×#117—22no—
49C2×#117—24no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#431—24no—
60#431—24no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8C2314yes—
63C2×C4×C8—22no—
64#398—28no—
65#398C2×C4116yes—
66#391—28no—
67#389—28no—
68#389—216no—
69#389—216no—
70#384—24no—
71#384D414yes—
72#382—24no—
73#382—22no—
74#380—24no—
75#380—22no—
76#380C2314yes—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2×C414yes—
82C4×C4⋊C4D414yes—
83C2×#95—24no—
84C2×#95C2312yes—
85#340—216no—
86#324—28no—
87C22×Q8—24no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8C22⋊C414yes—
99C22×Q8D814yes—
100C22×Q8SD1614yes—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—24no—
109C4⋊Q8—28no—
110C4⋊Q8—28no—
111C4⋊Q8—432no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—432no—
119#117—48no—
120#117—48no—
121#117—44no—
122#117—44no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—216no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D418yes—
148C2×C22⋊C4—24no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42—28no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—42no—
156C2×C42—42no—
157C2×C42C22⋊C418yes—
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42#4114yes—
163C4⋊C8—44no—
164C22⋊C8—24no—
165C22⋊C8—24no—
166C4×C8—44no—
167#95—28no—
168#95—216no—
169#95—24no—
170#95—22no—
171#95—24no—
172#95C22×C414yes—
173C24C22×D411yescharacteristic
174C2×Q8—28no—
175C2×Q8—28no—
176C2×Q8—28no—
177C2×Q8—28no—
178C2×Q8—28no—
179C2×Q8—28no—
180C2×Q8—48no—
181C2×Q8—48no—
182C2×Q8—48no—
183C2×Q8—84no—
184C2×Q8—84no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8#102116yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C414yes—
212C22×C4C2×C22⋊C414yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4#12011yescharacteristic
215C22×C4#12012yes—
216C22×C4#12011yescharacteristic
217C22×C4#12312yes—
218C22×C4#12312yes—
219C22×C4C2×D814yes—
220C22×C4C2×D812yes—
221C22×C4C2×SD1614yes—
222C22×C4C2×SD1612yes—
223C22×C4AGL(1,ℤ/8)12yes—
224C22×C4#13712yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C2×#4112yes—
229C2×C8—48no—
230C2×C8—44no—
231C4⋊C4—216no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—48no—
237C4⋊C4—416no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—432no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C4⋊C4—88no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C42—28no—
248C42—28no—
249C42—416no—
250C42—48no—
251C42—48no—
252C42C4×D414yes—
253C42C4×D414yes—
254C42#12114yes—
255C42#12114yes—
256C23—22no—
257C23—22no—
258C23C22×C22⋊C411yescharacteristic
259C23C2×#12011yescharacteristic
260C23C2×#12311yescharacteristic
261C23C22×D811yescharacteristic
262C23C22×SD1611yescharacteristic
263C23C2×AGL(1,ℤ/8)11yescharacteristic
264C23C2×#13711yescharacteristic
265Q8—232no—
266Q8—816no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—28no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#38612yes—
300C2×C4#38614yes—
301C2×C4#38612yes—
302C2×C4C2×#10218yes—
303C2×C4C2×#10214yes—
304C2×C4#41714yes—
305C2×C4#44714yes—
306C2×C4#44814yes—
307C2×C4#45614yes—
308C2×C4#48014yes—
309C2×C4#48114yes—
310C2×C4C2×C4×D412yes—
311C2×C4C2×#12112yes—
312C8—84no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22C2×#38611yescharacteristic
322C22C22×#10211yescharacteristic
323C22C2×#41711yescharacteristic
324C22C2×#44711yescharacteristic
325C22C2×#44811yescharacteristic
326C22C2×#45612yes—
327C22#302812yes—
328C22C2×#48011yescharacteristic
329C22C2×#48111yescharacteristic
330C4—28no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—48no—
335C4—44no—
336C4—84no—
337C4—88no—
338C4#187418yes—
339C4#188614yes—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1840812yes—
345C2#1841012yes—
346C2C2×#187411yescharacteristic
347C2C2×#188611yescharacteristic
348C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#341822
#218332
C2×#14612
#90272
#80142