Finite Group Databasebeta

← #480365

Subgroups — #480365

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#11534C214yesmaximal, complemented
4#11532C214yesmaximal, complemented
5#11450C212yesmaximal, complemented
6#11431C212yesmaximal
7C2×#536C411yescharacteristic
8C2×#520C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—22no—
11#2362—22no—
12#2360—22no—
13#2359—24no—
14#2327C412yes—
15#2326C2212yes—
16#2317C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18C4×#124C412yes—
19#2297C2211yescharacteristic
20C2×#391C411yescharacteristic
21C4×#117C411yescharacteristic
22#1467C2218yes—
23#1464—24no—
24#1461—22no—
25#1458C2214yes—
26#1325C2218yes—
27#1319C2214yes—
28#1318C2214yes—
29C22×#41C2×C411yescharacteristic
30C23×Q8C2×C411yescharacteristic
31#536—24no—
32#536—24no—
33C2×C4⋊Q8C2×C412yes—
34C2×#124C2×C412yes—
35C2×#123—22no—
36C2×C22⋊Q8—22no—
37#520—44no—
38#519—44no—
39C2×C4×Q8C2×C412yes—
40C2×C4×Q8C2×C412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×C4×Q8C2312yes—
44C2×C4×D4—22no—
45C2×C4×D4C2×C412yes—
46C2×#117—21no—
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D414yes—
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54#446—48no—
55#432—28no—
56#431—24no—
57#391—22no—
58#391C2×C412yes—
59#391C2×C412yes—
60#391C2×C411yescharacteristic
61#391C2311yescharacteristic
62#390C2×C411yescharacteristic
63#390C2311yescharacteristic
64#389—24no—
65#389—22no—
66#388—24no—
67#388—24no—
68#387—24no—
69#387—24no—
70#386—22no—
71#385—24no—
72#385—22no—
73#384C2312yes—
74#382C2×C412yes—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—24no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—22no—
81#376C2×C412yes—
82C2×#95—22no—
83C43C2×C412yes—
84#324—28no—
85#324C2318yes—
86#323—24no—
87C2×#41—22no—
88C2×#41—42no—
89C2×#41—42no—
90C2×#41M1614yes—
91C22×Q8—22no—
92C22×Q8—22no—
93C22×Q8—42no—
94C22×Q8—42no—
95C22×Q8C22⋊C412yes—
96C22×Q8C22×C411yescharacteristic
97C22×Q8C22×C412yes—
98C22×D4—21no—
99C22×D4C22×C411yescharacteristic
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4C22⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108#124—24no—
109#124—24no—
110#123—216no—
111C22⋊Q8—216no—
112C4×Q8—28no—
113C4×Q8—44no—
114C4×Q8—44no—
115C4×Q8—44no—
116C4×Q8—44no—
117C4×D4—44no—
118C4×D4—48no—
119#117—22no—
120#117—24no—
121#117—42no—
122#117—48no—
123#117—44no—
124#117M1614yes—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C414yes—
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C22×C412yes—
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2411yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C2×D412yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—42no—
151C2×C42C22⋊C414yes—
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D414yes—
157C2×C42C2×D412yes—
158C4⋊C8—44no—
159C4⋊C8—48no—
160C22⋊C8—28no—
161#97—48no—
162C4×C8—44no—
163#95—24no—
164#95—24no—
165#95—22no—
166#95—22no—
167#95—24no—
168#95—22no—
169#95—22no—
170C24—21no—
171C24C2×C22⋊C411yescharacteristic
172#41—48no—
173#41—48no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—21no—
177C2×Q8—24no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—48no—
183C2×Q8—48no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8C2×M1612yes—
187C2×D4—24no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4C2×M1612yes—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4—84no—
213C22×C4—82no—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×M1612yes—
220C22×C4C2×M1612yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D412yes—
223C2×C8—48no—
224C4⋊C4—216no—
225C4⋊C4—24no—
226C4⋊C4—22no—
227C4⋊C4—28no—
228C4⋊C4—41no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—84no—
235C4⋊C4#9914yes—
236C22⋊C4—216no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—82no—
243C42—22no—
244C42—24no—
245C42—28no—
246C42—24no—
247C42—44no—
248C42—41no—
249C42—41no—
250C42—44no—
251C42—44no—
252C42—42no—
253C42—82no—
254C42C2×M1614yes—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23#40712yes—
265C23C22×C22⋊C411yescharacteristic
266Q8—42no—
267Q8—44no—
268Q8—44no—
269Q8—42no—
270Q8—48no—
271D4—42no—
272D4—42no—
273D4—48no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—48no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—48no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—84no—
305C2×C4—164no—
306C2×C4—162no—
307C2×C4#40712yes—
308C2×C4C2×#9914yes—
309C2×C4C2×#9912yes—
310C2×C4#41312yes—
311C2×C4C22×M1611yescharacteristic
312C2×C4C22×M1611yescharacteristic
313C8—88no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22—44no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—82no—
324C22—81no—
325C22—81no—
326C22C2×#40711yescharacteristic
327C22C22×#9911yescharacteristic
328C22C2×#41311yescharacteristic
329C4—24no—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—44no—
338C4—42no—
339C4—81no—
340C4—82no—
341C4#146514yes—
342C2—21no—
343C2—21no—
344C2—42no—
345C2—82no—
346C2C2×#146511yescharacteristic
347C2#3336211yescharacteristic
348C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #11534C2semidirect
4. #11532C2semidirect
5. #11450C2semidirect

Maximal subgroups

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

subgroupindex
#342762
#173712
#115342
#115322
#114502
#114312