Finite Group Databasebeta

← #6383715

Subgroups — #6383715

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#23266C212yesmaximal, complemented
3#23263C212yesmaximal, complemented
4#23152C214yesmaximal, complemented
5#23052C218yesmaximal
6#21435C214yesmaximal
7#21346C214yesmaximal, complemented
8#20520C214yesmaximal, complemented
9#20463C212yesmaximal
10C2×#533C411yescharacteristic
11C2×#446C2218yes—
12C2×#431C2214yes—
13C4×C4⋊Q8C412yes—
14C4×C4⋊Q8C2212yes—
15C4×#125C411yescharacteristic
16C4×#125C411yescharacteristic
17C4×#125C2211yescharacteristic
18C4×#125C2211yescharacteristic
19C4×C22⋊Q8C414yes—
20C4×C22⋊Q8C2214yes—
21#2296C412yes—
22#2295C414yes—
23#2285C411yescharacteristic
24C42×Q8C2214yes—
25C4×#117C2212yes—
26C2×C43C2211yescharacteristic
27#1985C2218yes—
28#1940C2218yes—
29#1924C2218yes—
30#1923C2218yes—
31#1917C2218yes—
32#1916C2218yes—
33#1916C2218yes—
34#1916C2218yes—
35#1771C2218yes—
36#1771C2218yes—
37C8×C4⋊C4C2218yes—
38#1767C2212yes—
39#1767C2214yes—
40#1767C2214yes—
41#1767C2212yes—
42#1767C2214yes—
43#1749C2214yes—
44#1749C2214yes—
45#1745C2214yes—
46#1744C2214yes—
47C4×#97C2214yes—
48C4×#97C2214yes—
49#533—28no—
50C2×C4⋊Q8C2×C412yes—
51C2×#125C2×C411yescharacteristic
52C2×#125C2×C411yescharacteristic
53C2×C22⋊Q8C2×C414yes—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8C2×C414yes—
56C2×C4×Q8C2314yes—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C412yes—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64#446—232no—
65#431—216no—
66C2×C4⋊C8C2314yes—
67C2×C4⋊C8C2318yes—
68C2×C4⋊C8C2318yes—
69C2×C4⋊C8C2314yes—
70C2×C22⋊C8C2318yes—
71C2×#97C2318yes—
72C2×#97C2318yes—
73C2×#97C2314yes—
74C2×#97C2314yes—
75C2×C4×C8C2318yes—
76C2×C4×C8C2318yes—
77#391C2×C414yes—
78#391C2×C414yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C414yes—
84#389C2×C414yes—
85#387C2×C414yes—
86#387C2×C414yes—
87#387C2×C414yes—
88#387C2×C414yes—
89#385C2×C414yes—
90#385C2×C414yes—
91#384C2×C412yes—
92#382C2×C411yescharacteristic
93#382C2×C411yescharacteristic
94#380C2×C414yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C414yes—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4C2×C414yes—
101C4×C4⋊C4C2×C414yes—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2312yes—
104C4×C4⋊C4C2312yes—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2314yes—
107C4×C4⋊C4C2314yes—
108C4×C22⋊C4C2×C414yes—
109C4×C22⋊C4C2314yes—
110#376C2×C414yes—
111C43C2×C411yescharacteristic
112C43C2×C411yescharacteristic
113C43C2×C412yes—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116C43C2312yes—
117C43C2314yes—
118#336C2314yes—
119#336C2318yes—
120#336C2318yes—
121#336C2318yes—
122#336C2318yes—
123#336C2318yes—
124#336C2314yes—
125C22×Q8C22×C414yes—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8C2418yes—
132C22×C8C2418yes—
133C4⋊Q8—28no—
134#125—24no—
135#125M1618yes—
136C22⋊Q8—216no—
137C4×Q8—216no—
138C4×Q8—216no—
139C4×Q8—48no—
140#117—28no—
141#117—44no—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4C22×C414yes—
156C2×C22⋊C4C22×C414yes—
157C2×C42—24no—
158C2×C42—28no—
159C2×C42—24no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42C22×C414yes—
163C2×C42C22×C412yes—
164C2×C42C22×C412yes—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C414yes—
172C2×C42C22×C414yes—
173C2×C42C22×C414yes—
174C2×C42C22×C412yes—
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C2414yes—
181C2×C42C2414yes—
182C2×C42C2411yescharacteristic
183C2×C42C2412yes—
184C2×C42C2412yes—
185C2×C42C2411yescharacteristic
186C4⋊C8—216no—
187C4⋊C8—216no—
188C4⋊C8—28no—
189C4⋊C8#41116yes—
190C22⋊C8—216no—
191#97—216no—
192#97—216no—
193#97—28no—
194#97—28no—
195C4×C8—216no—
196C4×C8—216no—
197#95C22×C414yes—
198#95C22×C414yes—
199#95C22×C414yes—
200#95C22×C414yes—
201#95C22×C414yes—
202#95C22×C414yes—
203#95C22×C414yes—
204#95C22×C414yes—
205#95C22×C414yes—
206#95C22×C414yes—
207C24C23×C411yescharacteristic
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—44no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C23×C411yescharacteristic
225C22×C4C23×C411yescharacteristic
226C22×C4C23×C414yes—
227C22×C4C23×C414yes—
228C22×C4C23×C414yes—
229C22×C4C23×C412yes—
230C22×C4C23×C412yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C23×C411yescharacteristic
233C22×C4C23×C412yes—
234C22×C4C23×C414yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C23×C412yes—
237C22×C4C2511yescharacteristic
238C2×C8—28no—
239C2×C8—28no—
240C2×C8—28no—
241C2×C8—28no—
242C2×C8—28no—
243C2×C8C2×#41116yes—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—24no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4C2×M1618yes—
256C4⋊C4C2×M16116yes—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C42—24no—
260C42—24no—
261C42—22no—
262C42—22no—
263C42—24no—
264C42—22no—
265C42—28no—
266C42—24no—
267C42—24no—
268C42—24no—
269C42—28no—
270C42—28no—
271C42—22no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—44no—
278C42—44no—
279C42—44no—
280C42C2×M1614yes—
281C42C2×#4118yes—
282C422−1+414yes—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23#56812yes—
288C23C24×C411yescharacteristic
289Q8—416no—
290C2×C4—216no—
291C2×C4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4C4×#4114yes—
324C2×C4C4×#4114yes—
325C2×C4#51912yes—
326C2×C4#51912yes—
327C2×C4C22×M1614yes—
328C2×C4C22×M1612yes—
329C2×C4C22×M1618yes—
330C2×C4#56814yes—
331C2×C4C22×#4114yes—
332C2×C4C2×2−1+412yes—
333C8—216no—
334C8—48no—
335C22—24no—
336C22—22no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22C2×C4×#4112yes—
341C22C2×#51911yescharacteristic
342C22C23×M1611yescharacteristic
343C22C2×#56811yescharacteristic
344C22C2×#56812yes—
345C4—24no—
346C4—24no—
347C4—24no—
348C4—24no—
349C4—28no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—44no—
355C4—44no—
356C4—41no—
357C4—41no—
358C4#348214yes—
359C2—42no—
360C2#3415411yescharacteristic
361C2#6020712yes—
362C2M16×#4112yes—
363C2#6024811yescharacteristic
364C2#6024911yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23266C2semidirect
3. #23263C2semidirect
4. #23152C2semidirect
7. #21346C2semidirect
8. #20520C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#232662
#232632
#231522
#230522
#214352
#213462
#205202
#204632