Finite Group Databasebeta

← #238966

Subgroups — #238966

Order 512.

941 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 941 conjugacy classes of subgroups — too many to draw. Only its 174 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#60968C211yesmaximal, characteristic
2#20827C211yesmaximal, characteristic, complemented
3#17743C212yesmaximal, complemented
4#17647C212yesmaximal, complemented
5#10525C214yesmaximal, complemented
6#10524C214yesmaximal, complemented
7#8122C211yesmaximal, characteristic, complemented
8C2×#532C2212yes—
9#3253C2214yes—
10#3227—22no—
11#3205—22no—
12#3192—22no—
13#3184—22no—
14#3109—24no—
15#2920C2212yes—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#1849C2212yescomplemented
20#1838C2212yescomplemented
21#1811—22no—
22C4×#108—22no—
23#1557C2218yescomplemented
24#1553—24no—
25#1546—22no—
26#1545—22no—
27#1530C2214yescomplemented
28#1530C2214yescomplemented
29#1527—24no—
30#1525—24no—
31#1307C2214yescomplemented
32C22×#41—22no—
33C22×M16—21no—
34#532—22no—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40#532D414yes—
41#532C2314yes—
42C2×C4⋊Q8D412yes—
43C2×#127D412yes—
44C2×#124—22no—
45C2×#121—22no—
46C2×#121—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C22×C42C2311yescharacteristic
50#498—22no—
51#493—22no—
52#486—24no—
53#481—28no—
54#476—28no—
55#469—28no—
56#459—28no—
57#447—28no—
58#442—28no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432C2314yes—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67C2×#108—24no—
68#423—22no—
69#423C2312yes—
70#423C2312yes—
71#407—22no—
72#407—21no—
73#407D412yes—
74C4×M16—22no—
75C4×M16—22no—
76C4×M16—21no—
77C4×M16D412yes—
78C4×M16D414yes—
79C2×#97C2311yescharacteristic
80C2×#97C2311yescharacteristic
81C2×C4×C8—21no—
82#341—24no—
83#332—28no—
84#331—28no—
85#329—28no—
86#329—28no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—42no—
90C2×#41—42no—
91C2×#41—88no—
92C22×Q8—22no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—42no—
97C23×C4—42no—
98C23×C4C2×D411yescharacteristic
99C23×C4#4112yes—
100C2×D8—28no—
101C2×M16—24no—
102C2×M16—22no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16C2×D414yes—
106C22×C8—21no—
107C22×C8—21no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8C2×D414yes—
112#127—22no—
113#127—22no—
114#127—22no—
115#127C2×D414yes—
116#124—24no—
117#124—24no—
118#124—42no—
119#124—42no—
120#121—24no—
121#121—24no—
122#121—24no—
123#121—24no—
124#121—42no—
125#121—42no—
126#121—42no—
127#121—42no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135C4×D4—42no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42#4112yes—
145C2×C42#4112yes—
146C2×C42C2411yescharacteristic
147#108—48no—
148#106—28no—
149C4⋊C8—24no—
150C4⋊C8—24no—
151C4⋊C8—24no—
152C4⋊C8—42no—
153#102—216no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#97—22no—
159#97—24no—
160#97—21no—
161#97—21no—
162#97—41no—
163#97C2×D412yes—
164#97C2×D412yes—
165C4×C8—22no—
166C4×C8—24no—
167C4×C8—41no—
168C4×C8—42no—
169C24—42no—
170C24#12411yescharacteristic
171#41—88no—
172#41—88no—
173#41—88no—
174#41—88no—
175C2×Q8—24no—
176C2×Q8—42no—
177C2×Q8—84no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—84no—
187C2×D4—84no—
188C2×D4—84no—
189C2×D4—88no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—88no—
196C22×C4#12012yes—
197C22×C4#12112yes—
198C22×C4#12112yes—
199C22×C4#12112yes—
200C22×C4#12312yes—
201C22×C4#12312yes—
202C22×C4#12412yes—
203C22×C4#12411yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C2×#4112yes—
206D8—88no—
207M16—48no—
208M16—44no—
209M16—44no—
210M16—84no—
211C2×C8—24no—
212C2×C8—22no—
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—44no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C42#12011yescharacteristic
227C42#12011yescharacteristic
228C42#12112yes—
229C42#12112yes—
230C42#12112yes—
231C42#12312yes—
232C42#12312yes—
233C42C22×D412yes—
234C42C22×D411yescharacteristic
235C42C2×#4111yescharacteristic
236C23—42no—
237C23—41no—
238C23—44no—
239C23—84no—
240C23—84no—
241C23—84no—
242C23#39412yes—
243C23C2×#12411yescharacteristic
244Q8—84no—
245Q8—84no—
246D4—48no—
247D4—84no—
248D4—84no—
249D4—84no—
250D4—88no—
251D4—88no—
252D4—84no—
253D4—84no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—88no—
266C2×C4—88no—
267C2×C4—84no—
268C2×C4#39412yes—
269C2×C4#39412yes—
270C2×C4#39412yes—
271C2×C4C2×#12011yescharacteristic
272C2×C4C2×#12112yes—
273C2×C4C2×#12111yescharacteristic
274C2×C4C2×#12312yes—
275C8—24no—
276C8—42no—
277C8—44no—
278C8—84no—
279C8—82no—
280C22—22no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—82no—
286C22—84no—
287C22—84no—
288C22—84no—
289C22—168no—
290C22C2×#39411yescharacteristic
291C4—21no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—84no—
297C4#204814yes—
298C4#301812yes—
299C2—41no—
300C2—42no—
301C2—164no—
302C2#2349811yescharacteristic
303C2C2×#204812yes—
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radical#60968C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20827C2semidirect
3. #17743C2semidirect
4. #17647C2semidirect
5. #10525C2semidirect
6. #10524C2semidirect
7. #8122C2semidirect
19. #1849C22semidirect
20. #1838C22semidirect
23. #1557C22semidirect
27. #1530C22semidirect
28. #1530C22semidirect
31. #1307C22semidirect

Maximal subgroups

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

subgroupindex
#609682
#208272
#177432
#176472
#105252
#105242
#81222