Finite Group Databasebeta

← #194870

Subgroups — #194870

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21811C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9015C214yesmaximal
6#9014C214yesmaximal
7#8021C212yesmaximal
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2567—24no—
12#2567C418yescomplemented
13C2×#395C2212yes—
14C2×#395C2212yes—
15#2320—24no—
16#2316C2212yes—
17#2313—21no—
18#2313C2212yes—
19#2310—21no—
20#2295—24no—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#384—21no—
25#2283—21no—
26#2283C2212yes—
27C4×#117—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1981—24no—
30#1960—24no—
31#1787C2218yes—
32#1467—24no—
33C2×#324C2212yes—
34C2×#324C2212yes—
35#1294C2218yes—
36C23×Q8D411yescharacteristic
37C23×Q8C2312yes—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C414yes—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8—24no—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8D414yes—
48C2×#117—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61#432—24no—
62#432—24no—
63C2×C4⋊C8—22no—
64C2×#103—216no—
65C2×C22⋊C8C2314yes—
66C2×#97—22no—
67#395—24no—
68#395—24no—
69#395C2×C418yes—
70#395C2×C418yes—
71#391—28no—
72#391—28no—
73#389—28no—
74#389—28no—
75#389—28no—
76#387—28no—
77#387—28no—
78#384—28no—
79#383—22no—
80#383—22no—
81#382—22no—
82#382D414yes—
83#381—22no—
84#381—22no—
85#380—22no—
86#380C2314yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95#376—22no—
96#376—22no—
97#376—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101C43—22no—
102#324—24no—
103#324—24no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—24no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8—42no—
111C22×Q8—84no—
112C22×Q8C22⋊C414yes—
113C22×Q8C22×C414yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—24no—
124C4⋊Q8—28no—
125C4⋊Q8—28no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4×Q8—28no—
129C4×Q8—28no—
130C4×Q8—28no—
131C4×Q8—28no—
132#117—44no—
133#117—44no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—84no—
154C2×C4⋊C4—84no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D418yes—
161C2×C22⋊C4—24no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42C22⋊C414yes—
174C2×C42C22⋊C414yes—
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42#4114yes—
180C4⋊C8—44no—
181#103—816no—
182C22⋊C8—24no—
183C22⋊C8—24no—
184#97—44no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—28no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—24no—
196#95C22×C414yes—
197C24C22×D411yescharacteristic
198C2×Q8—28no—
199C2×Q8—28no—
200C2×Q8—28no—
201C2×Q8—44no—
202C2×Q8—48no—
203C2×Q8—48no—
204C2×Q8—416no—
205C2×Q8—44no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8—88no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—28no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4#12011yescharacteristic
243C22×C4#12012yes—
244C22×C4#12011yescharacteristic
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#13714yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4112yes—
252C2×C8—48no—
253C2×C8—44no—
254C4⋊C4—216no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—28no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—44no—
265C4⋊C4—84no—
266C4⋊C4—88no—
267C4⋊C4—88no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—24no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—48no—
275C42—48no—
276C42—48no—
277C42—84no—
278C42C4×D414yes—
279C42C4×D414yes—
280C42#12114yes—
281C42#12114yes—
282C23—22no—
283C23—22no—
284C23C22×C22⋊C411yescharacteristic
285C23C2×#12011yescharacteristic
286C23C2×#12311yescharacteristic
287C23C2×#13712yes—
288C23#57512yes—
289Q8—416no—
290Q8—416no—
291Q8—88no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—416no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—44no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4#38612yes—
333C2×C4#38614yes—
334C2×C4#38612yes—
335C2×C4#41914yes—
336C2×C4#45614yes—
337C2×C4C2×C4×D412yes—
338C2×C4C2×#12112yes—
339C8—84no—
340C22—24no—
341C22—24no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22C2×#38611yescharacteristic
347C22C2×#41911yescharacteristic
348C22#289611yescharacteristic
349C22#300712yes—
350C22C2×#45612yes—
351C22#302712yes—
352C22#309512yes—
353C4—24no—
354C4—24no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—48no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—88no—
363C4—84no—
364C4#188614yes—
365C2—21no—
366C2—22no—
367C2—21no—
368C2#1833814yes—
369C2#2091111yescharacteristic
370C2C2×#188611yescharacteristic
371C2#2096111yescharacteristic
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
12. #2567C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#340592
#218112
C2×#14672
#90152
#90142
#80212