Finite Group Databasebeta

← #194774

Subgroups — #194774

Order 512.

1474 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
1C2×#2580C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#20734C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9024C214yesmaximal, complemented
6#9016C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C22×C4×Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#431C2212yes—
11#2580—24no—
12#2580C418yescomplemented
13C2×#399C2212yes—
14C2×#393C2212yes—
15#2357—24no—
16#2354—24no—
17#2312C2214yes—
18#2309—21no—
19C2×#390—21no—
20C2×#389—21no—
21C2×#385—22no—
22#2285C2212yes—
23C2×#382C2211yescharacteristic
24#2280—22no—
25#2274—21no—
26#1824C2214yes—
27#1803—24no—
28#1787C2214yes—
29C4×#103—24no—
30#1460—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—24no—
40C2×C4×Q8C2×C414yes—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C414yes—
45C2×C4×D4D414yes—
46C2×#117—22no—
47C2×#117—24no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#431—24no—
59#431—24no—
60C2×C4⋊C8—22no—
61C2×#103—28no—
62C2×C22⋊C8C2314yes—
63C2×C4×C8—22no—
64#399—24no—
65#399C2×C418yes—
66#393—24no—
67#393C2×C418yes—
68#390—28no—
69#390—28no—
70#389—28no—
71#388—216no—
72#385—216no—
73#384D414yes—
74#382—22no—
75#382—22no—
76#382C2×C414yes—
77#381—24no—
78#380—24no—
79#380—22no—
80#380C2314yes—
81#379—24no—
82C4×C4⋊C4—28no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yes—
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2312yes—
90#340—28no—
91#323—28no—
92C25C2×D412yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—44no—
97C22×Q8C22×C412yes—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4C22⋊C414yes—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—84no—
110C23×C4C22⋊C418yes—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—24no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×D4—28no—
121C4×D4—28no—
122C4×D4—28no—
123C4×D4—28no—
124#117—44no—
125#117—44no—
126#117—88no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—84no—
142C2×C4⋊C4C22⋊C414yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C2×D418yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—48no—
153C2×C22⋊C4—84no—
154C2×C22⋊C4C22×C414yes—
155C2×C22⋊C4C2×D414yes—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—28no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C22×C412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42#4114yes—
166C4⋊C8—44no—
167#103—416no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C4×C8—44no—
171#95—28no—
172#95—24no—
173#95—28no—
174#95—24no—
175#95—22no—
176#95—24no—
177#95C22×C414yes—
178C24—22no—
179C24—24no—
180C24—22no—
181C24—24no—
182C24—44no—
183C24—44no—
184C24C2×C22⋊C414yes—
185C24C22×D411yescharacteristic
186C2×Q8—28no—
187C2×Q8—24no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—48no—
192C2×Q8—42no—
193C2×D4—28no—
194C2×D4—28no—
195C2×D4—28no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—48no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—48no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—84no—
223C22×C4—88no—
224C22×C4—88no—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4#12011yescharacteristic
229C22×C4#12011yescharacteristic
230C22×C4#12012yes—
231C22×C4#12312yes—
232C22×C4#12312yes—
233C22×C4#13514yes—
234C22×C4#13714yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C22×D411yescharacteristic
238C22×C4C2×#4112yes—
239C2×C8—48no—
240C2×C8—44no—
241C4⋊C4—28no—
242C4⋊C4—24no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—88no—
251C22⋊C4—28no—
252C22⋊C4—28no—
253C22⋊C4—416no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—88no—
257C22⋊C4—88no—
258C22⋊C4—88no—
259C42—48no—
260C42—48no—
261C42—88no—
262C42C4×D414yes—
263C42C4×D414yes—
264C42#12114yes—
265C42#12114yes—
266C23—22no—
267C23—22no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—44no—
272C23—44no—
273C23—416no—
274C23—44no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—82no—
279C23—82no—
280C23—88no—
281C23C22×C22⋊C411yescharacteristic
282C23C2×#12011yescharacteristic
283C23C2×#12311yescharacteristic
284C23C2×#13512yes—
285C23C2×#13712yes—
286Q8—28no—
287Q8—48no—
288Q8—44no—
289D4—416no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—416no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—42no—
316C2×C4—88no—
317C2×C4—82no—
318C2×C4—88no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4#38612yes—
322C2×C4#38612yes—
323C2×C4#38614yes—
324C2×C4#41614yes—
325C2×C4#41914yes—
326C2×C4#45214yes—
327C2×C4#45214yes—
328C2×C4#48518yes—
329C2×C4C2×C4×D412yes—
330C2×C4C2×#12112yes—
331C8—84no—
332C22—24no—
333C22—24no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—48no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—84no—
343C22—88no—
344C22—88no—
345C22C2×#38611yescharacteristic
346C22C2×#41611yescharacteristic
347C22C2×#41911yescharacteristic
348C22C2×#45212yes—
349C22#301912yes—
350C22#302512yes—
351C22C2×#48512yes—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—44no—
357C4—44no—
358C4—84no—
359C4—88no—
360C4—82no—
361C4#187318yes—
362C2—21no—
363C2—21no—
364C2—22no—
365C2—84no—
366C2#1822314yes—
367C2C2×#187311yescharacteristic
368C2#2095311yescharacteristic
369C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. #20734C2semidirect
5. #9024C2semidirect
6. #9016C2semidirect
7. #8026C2semidirect
12. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
#341862
#207342
C2×#14602
#90242
#90162
#80262