Finite Group Databasebeta

← #355143

Subgroups — #355143

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34248C211yesmaximal, characteristic
2C2×#1828C212yesmaximal
3#20392C212yesmaximal
4#10918C214yesmaximal, complemented
5#10915C214yesmaximal, complemented
6#10761C212yesmaximal
7C2×#518C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C22×#99C2214yes—
11#2349—24no—
12D4×C4⋊C4—24no—
13#2317C412yescomplemented
14#2315C412yescomplemented
15#2315C2212yes—
16#2296C412yescomplemented
17C2×#385C412yescomplemented
18C2×#385C2212yes—
19#2273C2214yes—
20#1828C4116yescomplemented
21C4×#99—24no—
22C4×#99—24no—
23C2×#342C2214yes—
24C2×#342C2214yes—
25C2×#342C2218yes—
26C22⋊#95C2214yes—
27C23×D4C2311yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2311yescharacteristic
30#518—24no—
31#518—44no—
32#518C2×C418yes—
33C2×C4×D4—24no—
34C2×C4×D4—24no—
35C2×C4×D4—44no—
36C2×C4×D4D414yes—
37C2×C4×D4D414yes—
38C2×#117C2×C412yes—
39C2×#117C2×C412yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C22⋊C4C2318yes—
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C42C2×C411yescharacteristic
51C2×#99—28no—
52C2×#99—28no—
53C2×#99C2×C4116yes—
54#389—22no—
55#389—22no—
56#389—24no—
57#385—22no—
58#385—22no—
59#385—22no—
60#385—22no—
61#385—24no—
62#385—24no—
63#385C2×C414yes—
64#385C2×C414yes—
65#385C2×C414yes—
66#385C2×C414yes—
67#384—22no—
68#382—22no—
69#380—24no—
70#380—24no—
71C4×C4⋊C4—24no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—48no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—44no—
78#342—28no—
79#342—44no—
80#342C2×C4132yescomplemented
81C25C22×C411yescharacteristic
82C25C2411yescharacteristic
83C22×D4—24no—
84C22×D4—24no—
85C22×D4C22×C414yes—
86C22×D4C2×D414yes—
87C22×D4#4114yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—28no—
94C23×C4—42no—
95C23×C4—82no—
96C23×C4—82no—
97C23×C4C22⋊C412yes—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C4×D4—232no—
107C4×D4—48no—
108C4×D4—48no—
109C4×D4—416no—
110#117—24no—
111#117—44no—
112#117C4218yescomplemented
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—84no—
121C2×C4⋊C4—84no—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C22⋊C412yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—28no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4C4218yescomplemented
140C2×C22⋊C4C22⋊C414yes—
141C2×C22⋊C4C22×C4116yes—
142C2×C22⋊C4C22×C414yes—
143C2×C22⋊C4C22×C414yes—
144C2×C22⋊C4C22×C418yes—
145C2×C22⋊C4C2×D414yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—42no—
149C2×C42—44no—
150C2×C42—44no—
151C2×C42C22⋊C414yes—
152#99—232no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—44no—
158#95—44no—
159#95—88no—
160C24—22no—
161C24—22no—
162C24—81no—
163C24—81no—
164C24—81no—
165C24—81no—
166C24C2×C4212yes—
167C24C2×C22⋊C412yes—
168C24#11712yes—
169C24C23×C411yescharacteristic
170C24C23×C412yes—
171C24C22×D412yes—
172C24C2×#4112yes—
173C24H(5,𝔽2)14yes—
174C242−1+414yes—
175C2×D4—216no—
176C2×D4—28no—
177C2×D4—28no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4C4×D4116yes—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—28no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—216no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—48no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—48no—
205C22×C4—44no—
206C22×C4—48no—
207C22×C4—84no—
208C22×C4—81no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4—88no—
224C22×C4—88no—
225C22×C4C2×C4212yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C414yes—
229C22×C4#11712yes—
230C4⋊C4—28no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C22⋊C4—24no—
239C22⋊C4—28no—
240C22⋊C4—28no—
241C22⋊C4—28no—
242C22⋊C4—28no—
243C22⋊C4—216no—
244C22⋊C4—216no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—48no—
249C22⋊C4—48no—
250C22⋊C4C2×C4218yes—
251C42—28no—
252C42—44no—
253C42—44no—
254C42—164no—
255C42—164no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—24no—
260C23—42no—
261C23—44no—
262C23—41no—
263C23—42no—
264C23—44no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—82no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—84no—
273C23—82no—
274C23—82no—
275C23—84no—
276C23C4×C22⋊C414yes—
277C23C22×C4211yescharacteristic
278C23C22×C22⋊C411yescharacteristic
279C23C2×#11711yescharacteristic
280C23C2×C4×D414yes—
281C23#51812yes—
282C23#51912yes—
283C23#52018yes—
284C23#53414yes—
285C23#53614yes—
286C23#53914yes—
287C23#53914yes—
288D4—232no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—416no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4—88no—
317C2×C4—88no—
318C2×C4—88no—
319C2×C4—88no—
320C2×C4—88no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—816no—
324C2×C4—162no—
325C2×C4—162no—
326C2×C4—168no—
327C2×C4C4×C22⋊C414yes—
328C22—21no—
329C22—22no—
330C22—24no—
331C22—21no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—44no—
336C22—48no—
337C22—44no—
338C22—41no—
339C22—42no—
340C22—41no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22—84no—
348C22—84no—
349C22C2×C4×C22⋊C411yescharacteristic
350C22#227412yes—
351C22#230811yescharacteristic
352C22#231011yescharacteristic
353C22#231312yes—
354C22#232218yes—
355C4—24no—
356C4—48no—
357C4—42no—
358C4—44no—
359C4—44no—
360C4—816no—
361C4—81no—
362C4—81no—
363C4—81no—
364C2—21no—
365C2—28no—
366C2—21no—
367C2—22no—
368C2—22no—
369C2—44no—
370C2#1377611yescharacteristic
371C2#2177611yescharacteristic
372C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10918C2semidirect
5. #10915C2semidirect
13. #2317C4semidirect
14. #2315C4semidirect
16. #2296C4semidirect
17. C2×#385C4semidirect
20. #1828C4semidirect
80. #342C2×C4semidirect
112. #117C42semidirect
139. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#342482
C2×#18282
#203922
#109182
#109152
#107612