Finite Group Databasebeta

← #6408955

Subgroups — #6408955

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#23372C214yesmaximal, complemented
3#23153C214yesmaximal, complemented
4#22825C214yesmaximal, complemented
5#22722C214yesmaximal, complemented
6#22104C218yesmaximal
7#21926C214yesmaximal
8#20808C212yesmaximal
9C2×#533C411yescharacteristic
10C2×#446C2218yes—
11C2×#431C2214yes—
12C4×C4⋊Q8C412yes—
13C4×C4⋊Q8C2212yes—
14C4×#125C412yes—
15C4×#125C2212yes—
16C4×C22⋊Q8C414yes—
17C4×C22⋊Q8C2214yes—
18#2296C412yes—
19#2295C414yes—
20#2285C411yescharacteristic
21C42×Q8C2214yes—
22C4×#117C2212yes—
23C2×C43C2211yescharacteristic
24#1984C2218yes—
25#1940C22116yes—
26#1939C2218yes—
27#1924C22116yes—
28#1917C22116yes—
29#1917C22116yes—
30#1842C2212yes—
31#1842C2212yes—
32#1842C2214yes—
33#1831C2214yes—
34#1831C2214yes—
35#1797C2214yes—
36#1797C2214yes—
37#1770C2218yes—
38#1767C2218yes—
39C4×C4⋊C8C2214yes—
40C4×C4⋊C8C2214yes—
41#533—28no—
42C2×C4⋊Q8C2×C412yes—
43C2×#125C2×C412yes—
44C2×C22⋊Q8C2×C414yes—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8C2314yes—
48C2×#117C2×C412yes—
49C2×#117C2×C412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55#446—416no—
56#431—48no—
57C2×C4⋊C8C2314yes—
58C2×C4⋊C8C2314yes—
59C2×C4⋊C8C2314yes—
60C2×C4⋊C8C2314yes—
61C2×C4⋊C8C2314yes—
62C2×C4⋊C8C2314yes—
63C2×C4⋊C8C23116yes—
64C2×C22⋊C8C2318yes—
65C2×#97C23116yes—
66C2×C4×C8C2318yes—
67#391C2×C414yes—
68#391C2×C414yes—
69#389C2×C414yes—
70#389C2×C414yes—
71#389C2×C414yes—
72#389C2×C414yes—
73#387C2×C418yes—
74#387C2×C418yes—
75#385C2×C418yes—
76#384C2×C412yes—
77#382C2×C412yes—
78#380C2×C414yes—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2×C418yes—
81C4×C4⋊C4C2×C414yes—
82C4×C4⋊C4C2×C414yes—
83C4×C4⋊C4C2×C418yes—
84C4×C4⋊C4C2314yes—
85C4×C4⋊C4C2318yes—
86C4×C4⋊C4C2314yes—
87C4×C22⋊C4C2×C414yes—
88C4×C22⋊C4C2314yes—
89#376C2×C414yes—
90C43C2×C411yescharacteristic
91C43C2×C411yescharacteristic
92C43C2×C412yes—
93C43C2311yescharacteristic
94C43C2311yescharacteristic
95C43C2314yes—
96C43C2312yes—
97#336C2318yes—
98#336C2318yes—
99#336C23116yes—
100#336C23116yes—
101C22×Q8C22×C414yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C412yes—
106C23×C4C2411yescharacteristic
107C22×C8C2418yes—
108C22×C8C2418yes—
109C4⋊Q8—28no—
110#125—28no—
111C22⋊Q8—216no—
112C4×Q8—216no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—48no—
116#117—28no—
117#117—44no—
118C2×C4⋊C4C22×C414yes—
119C2×C4⋊C4C22×C418yes—
120C2×C4⋊C4C22×C414yes—
121C2×C4⋊C4C22×C414yes—
122C2×C4⋊C4C22×C414yes—
123C2×C4⋊C4C22×C418yes—
124C2×C4⋊C4C22×C414yes—
125C2×C4⋊C4C22×C418yes—
126C2×C22⋊C4C22×C414yes—
127C2×C22⋊C4C22×C414yes—
128C2×C42—24no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—42no—
133C2×C42—44no—
134C2×C42M1618yes—
135C2×C42C22×C414yes—
136C2×C42C22×C414yes—
137C2×C42C22×C412yes—
138C2×C42C22×C411yescharacteristic
139C2×C42C22×C411yescharacteristic
140C2×C42C22×C412yes—
141C2×C42C22×C412yes—
142C2×C42C22×C412yes—
143C2×C42C22×C412yes—
144C2×C42C22×C418yes—
145C2×C42C22×C412yes—
146C2×C42C22×C412yes—
147C2×C42C22×C411yescharacteristic
148C2×C42C22×C411yescharacteristic
149C2×C42C22×C412yes—
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152C2×C42C2418yes—
153C2×C42C2411yescharacteristic
154C2×C42C2412yes—
155C2×C42C2412yes—
156C2×C42C2411yescharacteristic
157C4⋊C8—28no—
158C4⋊C8—28no—
159C4⋊C8—28no—
160C4⋊C8—28no—
161C4⋊C8—44no—
162C4⋊C8—44no—
163C4⋊C8—416no—
164C22⋊C8—48no—
165#97—416no—
166C4×C8—216no—
167#95C22×C418yes—
168#95C22×C418yes—
169#95C22×C418yes—
170#95C22×C418yes—
171#95C22×C418yes—
172C24C23×C411yescharacteristic
173C2×Q8—28no—
174C2×Q8—28no—
175C2×Q8—28no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C2×M1614yes—
190C22×C4C2×M1618yes—
191C22×C4C23×C411yescharacteristic
192C22×C4C23×C411yescharacteristic
193C22×C4C23×C418yes—
194C22×C4C23×C412yes—
195C22×C4C23×C418yes—
196C22×C4C23×C411yescharacteristic
197C22×C4C23×C412yes—
198C22×C4C23×C412yes—
199C22×C4C23×C411yescharacteristic
200C22×C4C23×C411yescharacteristic
201C22×C4C23×C412yes—
202C22×C4C2511yescharacteristic
203C2×C8—28no—
204C2×C8—28no—
205C2×C8—28no—
206C2×C8—28no—
207C2×C8—28no—
208C2×C8—28no—
209C4⋊C4—28no—
210C4⋊C4—216no—
211C4⋊C4—28no—
212C4⋊C4—216no—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C22⋊C4—28no—
218C22⋊C4—44no—
219C42—22no—
220C42—22no—
221C42—24no—
222C42—24no—
223C42—28no—
224C42—24no—
225C42—24no—
226C42—24no—
227C42—28no—
228C42—28no—
229C42—22no—
230C42—21no—
231C42—21no—
232C42—22no—
233C42—28no—
234C42—42no—
235C42—41no—
236C42—41no—
237C42—42no—
238C42—48no—
239C42C2×M1618yes—
240C42C2×M1614yes—
241C42C2×M1614yes—
242C42H(5,𝔽2)14yes—
243C422−1+412yes—
244C422−1+412yes—
245C422−1+414yes—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23C22×M1612yes—
251C23C24×C411yescharacteristic
252Q8—416no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—28no—
261C2×C4—216no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—28no—
273C2×C4—28no—
274C2×C4—28no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4#51912yes—
281C2×C4#51912yes—
282C2×C4#52014yes—
283C2×C4#52014yes—
284C2×C4C22×M1612yes—
285C2×C4C22×M1614yes—
286C2×C4C22×M1612yes—
287C2×C4C22×M1614yes—
288C2×C4#56812yes—
289C2×C4#56814yes—
290C2×C4C2×H(5,𝔽2)12yes—
291C2×C4C2×2−1+411yescharacteristic
292C2×C4C2×2−1+411yescharacteristic
293C2×C4C2×2−1+412yes—
294C8—48no—
295C8—48no—
296C22—22no—
297C22—24no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22C2×#51911yescharacteristic
302C22C2×#52012yes—
303C22C23×M1611yescharacteristic
304C22C2×#56811yescharacteristic
305C22C2×#56812yes—
306C4—22no—
307C4—22no—
308C4—24no—
309C4—24no—
310C4—41no—
311C4—41no—
312C4—48no—
313C4—48no—
314C4—42no—
315C4#298314yes—
316C4#298918yes—
317C4#354811yescharacteristic
318C4#355012yes—
319C4#355611yescharacteristic
320C2—42no—
321C2#3437411yescharacteristic
322C2C2×#298311yescharacteristic
323C2#6024911yescharacteristic
324C2C2×#298912yes—
325C2#6026612yes—
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. #23372C2semidirect
3. #23153C2semidirect
4. #22825C2semidirect
5. #22722C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#233722
#231532
#228252
#227222
#221042
#219262
#208082