Finite Group Databasebeta

← #239315

Subgroups — #239315

Order 512.

997 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 997 conjugacy classes of subgroups — too many to draw. Only its 180 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
1#60940C211yesmaximal, characteristic
2#21424C212yesmaximal, complemented
3#21354C211yesmaximal, characteristic, complemented
4#17583C212yesmaximal, complemented
5#10557C218yesmaximal, complemented
6#8006C211yesmaximal, characteristic, complemented
7C2×#532C2212yes—
8#3245—22no—
9#3226—22no—
10#3205—22no—
11#3186C2214yes—
12#3169—22no—
13#3100—24no—
14#2920C2212yes—
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2283C2211yescharacteristic
18#1997—22no—
19#1965—22no—
20#1939—22no—
21#1900C2218yes—
22#1851C2214yes—
23#1768—22no—
24C4×#100—24no—
25#1517—24no—
26#1514—24no—
27#1507C2218yes—
28#1293C2214yes—
29C22×#41—22no—
30C22×M16C2312yes—
31#532—22no—
32#532—22no—
33#532—22no—
34#532—22no—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39C2×C4⋊Q8D412yes—
40C2×#127D412yes—
41C2×#124—22no—
42C2×#121—22no—
43C2×#121C2314yes—
44C2×C4×D4—22no—
45C2×C4×D4D414yes—
46C22×C42C2311yescharacteristic
47#500—22no—
48#493—22no—
49#486—24no—
50#480—28no—
51#474—28no—
52#466—28no—
53#459—28no—
54#447—28no—
55C4×D8—28no—
56#432—22no—
57#432—22no—
58#431—22no—
59#431—22no—
60#431—22no—
61#431—22no—
62#423—22no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8D412yes—
65C2×#100—28no—
66#407C2314yes—
67C2×C22⋊C8C2312yes—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—22no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8D412yes—
74#383D412yes—
75#382D412yes—
76#381—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C22⋊C4C2312yes—
80#376—21no—
81#343—28no—
82#336—24no—
83#327—216no—
84C2×#41—42no—
85C2×#41—42no—
86C2×#41—42no—
87C2×#41—42no—
88C2×#41—88no—
89C22×Q8—22no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—22no—
93C22×D4C2×D414yes—
94C23×C4—24no—
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4C2411yescharacteristic
98C2×D8—28no—
99C2×M16—24no—
100C2×M16—24no—
101C2×M16—44no—
102C22×C8—22no—
103C22×C8C2×D414yes—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108#127—22no—
109#127—22no—
110#127—22no—
111#127—22no—
112#124—42no—
113#124—42no—
114#124—42no—
115#124—42no—
116#121—24no—
117#121—24no—
118#121—24no—
119#121—24no—
120#121—42no—
121#121—42no—
122#121—42no—
123#121—42no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—24no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C2×D414yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—22no—
138C2×C42—22no—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4112yes—
146C2×C42#4112yes—
147#107—28no—
148C4⋊C8—24no—
149C4⋊C8—48no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152#102—216no—
153#100—416no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156C22⋊C8—24no—
157#97—41no—
158#97—41no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163C4×C8—42no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95C2×D414yes—
168C24—24no—
169C24C2×#4111yescharacteristic
170#41—88no—
171#41—88no—
172#41—88no—
173#41—88no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—84no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—48no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—84no—
187C2×D4—84no—
188C2×D4—88no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—88no—
196C22×C4—81no—
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12312yes—
202C22×C4#12312yes—
203C22×C4#12312yes—
204C22×C4#12312yes—
205C22×C4C22×D412yes—
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C22×C4C2×#4112yes—
209D8—88no—
210M16—28no—
211M16—44no—
212M16—48no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—44no—
218C2×C8—44no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C42—44no—
230C42#12011yescharacteristic
231C42#12011yescharacteristic
232C42#12012yes—
233C42#12112yes—
234C42#12112yes—
235C42#12112yes—
236C42#12112yes—
237C42#12411yescharacteristic
238C42#12412yes—
239C42#12411yescharacteristic
240C23—22no—
241C23—41no—
242C23—41no—
243C23—48no—
244C23—44no—
245C23—84no—
246C23—84no—
247C23C2×#12111yescharacteristic
248C23C2×#12312yes—
249Q8—84no—
250Q8—84no—
251D4—48no—
252D4—84no—
253D4—84no—
254D4—84no—
255D4—84no—
256D4—88no—
257D4—88no—
258D4—84no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—88no—
275C2×C4—88no—
276C2×C4—84no—
277C2×C4#39411yescharacteristic
278C2×C4#39412yes—
279C2×C4#39411yescharacteristic
280C2×C4#39411yescharacteristic
281C2×C4#39411yescharacteristic
282C2×C4#39412yes—
283C2×C4#47214yes—
284C2×C4C2×#12011yescharacteristic
285C2×C4C2×#12112yes—
286C2×C4C2×#12411yescharacteristic
287C8—44no—
288C8—44no—
289C8—84no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—82no—
296C22—84no—
297C22—84no—
298C22—84no—
299C22—168no—
300C22C2×#39411yescharacteristic
301C22C2×#47212yes—
302C4—22no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—21no—
309C4—84no—
310C4—84no—
311C4#204514yes—
312C4#302512yes—
313C2—22no—
314C2—42no—
315C2—164no—
316C2#2351411yescharacteristic
317C2C2×#204511yescharacteristic
318C2#2352711yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC23×C4C24
prime radical#60940C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21424C2semidirect
3. #21354C2semidirect
4. #17583C2semidirect
5. #10557C2semidirect
6. #8006C2semidirect

Maximal subgroups

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

subgroupindex
#609402
#214242
#213542
#175832
#105572
#80062