Finite Group Databasebeta

← #480732

Subgroups — #480732

Order 512.

1250 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2548C211yesmaximal, characteristic
2#17365C212yesmaximal, complemented
3#11567C218yesmaximal, complemented
4#11505C214yesmaximal, complemented
5C22×#127C411yescharacteristic
6C22×#124C412yes—
7C22×C4×Q8C2211yescharacteristic
8#2920—22no—
9#2548—22no—
10#2548—22no—
11#2548C414yes—
12#2548C2214yes—
13C2×#394—22no—
14C2×#390—22no—
15C2×#390C2212yes—
16C2×C4×C4⋊C4C411yescharacteristic
17#1467—22no—
18#1467C2214yescomplemented
19#1463—22no—
20#1463C2214yes—
21#1461—22no—
22#1461C2214yescomplemented
23#1318C22116yescomplemented
24C23×Q8C2×C411yescharacteristic
25C23×D4—21no—
26C23×D4—21no—
27C23×D4C2×C412yes—
28C2×#127—22no—
29C2×#127—41no—
30C2×#127—42no—
31C2×#127—41no—
32C2×#127C2×C414yes—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124—48no—
36C2×#124C2×C418yes—
37C2×C4×Q8—42no—
38C2×C4×Q8—41no—
39C2×C4×Q8—41no—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C412yes—
51C22×C42D412yes—
52#432—24no—
53#432—42no—
54#431—24no—
55#431—42no—
56#423—42no—
57#407—24no—
58C4×M16—42no—
59#394—24no—
60#394—24no—
61#394—24no—
62#394—24no—
63#390—24no—
64#390—24no—
65#390—24no—
66#390—24no—
67#390—24no—
68#390C2×C414yes—
69#390C2314yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2×C414yes—
72C2×#95—21no—
73#324—24no—
74#324—24no—
75#324—24no—
76#324C2318yes—
77C25—22no—
78C22×Q8—24no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—42no—
82C22×Q8C22⋊C411yescharacteristic
83C22×Q8C22⋊C411yescharacteristic
84C22×Q8C22×C412yes—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—44no—
89C22×D4—42no—
90C22×D4—42no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—48no—
94C22×D4—44no—
95C22×D4—41no—
96C22×D4—42no—
97C22×D4—41no—
98C22×D4—84no—
99C22×D4C22×C414yes—
100C23×C4—22no—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C2×M16—44no—
107#127—44no—
108#127—44no—
109#127—44no—
110#127—44no—
111#124—48no—
112#124—48no—
113#124—416no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—42no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22×C411yescharacteristic
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2411yescharacteristic
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—416no—
140C2×C22⋊C4—88no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—44no—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C414yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C4⋊C8—44no—
158C4⋊C8—44no—
159C22⋊C8—28no—
160C22⋊C8—44no—
161#97—44no—
162C4×C8—44no—
163#95—24no—
164C24—24no—
165C24—44no—
166C24—48no—
167C24—44no—
168C24—48no—
169C24—42no—
170C24—42no—
171C24C2×C22⋊C411yescharacteristic
172C2×Q8—24no—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—48no—
181C2×Q8—48no—
182C2×Q8#10014yes—
183C2×D4—28no—
184C2×D4—44no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—48no—
188C2×D4—44no—
189C2×D4—416no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—88no—
194C2×D4—88no—
195C2×D4—84no—
196C2×D4—88no—
197C2×D4—88no—
198C2×D4—88no—
199C2×D4—88no—
200C2×D4—84no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4#9911yescharacteristic
221C22×C4#9912yes—
222C22×C4#9911yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231M16—88no—
232C2×C8—48no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—48no—
241C22⋊C4—416no—
242C22⋊C4—416no—
243C22⋊C4—88no—
244C22⋊C4—88no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—44no—
258C42—44no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—48no—
264C23—48no—
265C23—44no—
266C23—44no—
267C23—44no—
268C23—84no—
269C23—84no—
270C23—84no—
271C23—84no—
272C23—816no—
273C23—816no—
274C23—84no—
275C23—84no—
276C23C2×#9911yescharacteristic
277C23C2×#9911yescharacteristic
278C23C22×C22⋊C411yescharacteristic
279Q8—44no—
280Q8—48no—
281Q8—44no—
282D4—88no—
283D4—88no—
284D4—88no—
285D4—816no—
286D4—88no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—82no—
316C2×C4—88no—
317C2×C4#35514yes—
318C2×C4C2×#9912yes—
319C2×C4C2×#9912yes—
320C2×C4C2×#10012yes—
321C2×C4C2×#10014yes—
322C2×C4#56812yes—
323C8—88no—
324C22—24no—
325C22—24no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—88no—
333C22—88no—
334C22—84no—
335C22—84no—
336C22—84no—
337C22—168no—
338C22—168no—
339C22#212812yes—
340C22C2×#35512yes—
341C22#287911yescharacteristic
342C22C22×#9911yescharacteristic
343C22C22×#10011yescharacteristic
344C4—22no—
345C4—22no—
346C4—22no—
347C4—24no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—84no—
353C4C2×#35514yes—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—22no—
358C2—164no—
359C2#1739611yescharacteristic
360C2C2×#212811yescharacteristic
361C2C22×#35511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17365C2semidirect
3. #11567C2semidirect
4. #11505C2semidirect
18. #1467C22semidirect
22. #1461C22semidirect
23. #1318C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25482
#173652
#115672
#115052