Finite Group Databasebeta

← #363356

Subgroups — #363356

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2#14339C211yesmaximal, characteristic
3#13776C211yesmaximal, characteristic
4#11197C214yesmaximal
5#10854C212yesmaximal
6#10854C212yesmaximal
7#10801C214yesmaximal
8C23×C4⋊C4C2211yescharacteristic
9#2313—24no—
10#2313—24no—
11C2×#387C414yescomplemented
12C2×#387C414yescomplemented
13C2×#381C2212yes—
14C2×#381C2212yes—
15C2×#381C2212yes—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×#342C2218yes—
22C2×#342C2218yes—
23#1441—24no—
24#1436—24no—
25C22⋊#95C2212yes—
26C22⋊#95C2212yes—
27C24×C4C2311yescharacteristic
28C24×C4C2311yescharacteristic
29C24×C4C2311yescharacteristic
30C22×C4⋊C4—24no—
31C22×C4⋊C4—24no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4D414yes—
38C22×C4⋊C4D414yes—
39C22×C22⋊C4C2314yes—
40C22×C22⋊C4C2312yes—
41C22×C22⋊C4C2312yes—
42C22×C22⋊C4C2314yes—
43C22×C42—22no—
44C22×C42C2×C414yes—
45#387—216no—
46#387—216no—
47#381—24no—
48#381—24no—
49#381—28no—
50#381—44no—
51#380—22no—
52#380—22no—
53#380—22no—
54#380—22no—
55#380—24no—
56#380—24no—
57C4×C22⋊C4—24no—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—24no—
60C4×C22⋊C4—24no—
61C2×#95—24no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—24no—
65C2×#95—24no—
66C2×#95—24no—
67C2×#95C2×C414yes—
68C2×#95C2×C412yes—
69C2×#95C2×C412yes—
70C2×#95C2×C418yes—
71#342—216no—
72#342—48no—
73C25C2411yescharacteristic
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—24no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—82no—
84C23×C4—82no—
85C23×C4C22×C412yes—
86C23×C4C22×C414yes—
87C23×C4C22×C414yes—
88C23×C4C22×C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4#4114yes—
98C23×C4#4114yes—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—84no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C22⋊C418yes—
114C2×C4⋊C4C22⋊C418yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—28no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4#4118yes—
132C2×C42—28no—
133C2×C42—44no—
134C2×C42—84no—
135C2×C42—84no—
136#95—216no—
137#95—28no—
138#95—24no—
139#95—24no—
140#95—216no—
141#95—48no—
142#95—44no—
143#95—44no—
144#95—84no—
145#95—84no—
146C24—42no—
147C24—42no—
148C24C23×C411yescharacteristic
149C24C22×D411yescharacteristic
150C24C22×D411yescharacteristic
151C24C2×#4112yes—
152C24C2×#4112yes—
153C24C2×#4114yes—
154C24H(5,𝔽2)12yes—
155C242−1+412yes—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—48no—
188C22×C4—48no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×C22⋊C414yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4#11712yes—
209C22×C4#11712yes—
210C22×C4#11712yes—
211C22×C4#11712yes—
212C22×C4#12314yes—
213C22×C4#12314yes—
214C22×C4#12314yes—
215C22×C4#12314yes—
216C4⋊C4—216no—
217C4⋊C4—216no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—88no—
221C22⋊C4—28no—
222C22⋊C4—28no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C42—88no—
233C42—88no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—48no—
243C23—42no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23C22×C22⋊C411yescharacteristic
248C23C2×#11711yescharacteristic
249C23C2×#11711yescharacteristic
250C23C4×#4112yes—
251C23#52012yes—
252C23C2×#12312yes—
253C23C2×#12312yes—
254C23#52512yes—
255C23#53212yes—
256C23#53511yescharacteristic
257C23#53611yescharacteristic
258C23#53712yes—
259C23#55214yes—
260C23#55314yes—
261C23#55514yes—
262C23#55614yes—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—28no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—416no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—88no—
297C2×C4—84no—
298C2×C4—88no—
299C2×C4—88no—
300C2×C4—84no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4#38114yes—
306C2×C4#38114yes—
307C22—22no—
308C22—24no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—24no—
313C22—24no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—21no—
321C22—21no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22C2×#38111yescharacteristic
327C22#229711yescharacteristic
328C22#231111yescharacteristic
329C22#234612yes—
330C22#234712yes—
331C22#248514yes—
332C22#249912yes—
333C22#250012yes—
334C22#288312yes—
335C22#288412yes—
336C4—24no—
337C4—24no—
338C4—44no—
339C4—44no—
340C4—88no—
341C4—84no—
342C4—84no—
343C4—164no—
344C2—24no—
345C2—22no—
346C2—22no—
347C2—22no—
348C2—22no—
349C2#1393911yescharacteristic
350C2#2068911yescharacteristic
351C2#2069111yescharacteristic
352C2#2142312yes—
353C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C2×#387C4semidirect
12. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
#143392
#137762
#111972
#108542
#108012