Finite Group Databasebeta

← #238809

Subgroups — #238809

Order 512.

974 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 182 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#20809C211yesmaximal, characteristic, complemented
3#17714C212yesmaximal, complemented
4#17544C212yesmaximal, complemented
5#10513C214yesmaximal, complemented
6#10512C214yesmaximal, complemented
7#7987C211yesmaximal, characteristic, complemented
8C2×#532C2212yes—
9#3245C2214yes—
10#3226—22no—
11#3205—22no—
12#3186—22no—
13#3169—22no—
14#3100—24no—
15#2920C2212yes—
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1843C2212yescomplemented
20#1832C2212yescomplemented
21#1798—22no—
22C4×C4⋊C8—22no—
23#1551C2218yescomplemented
24#1548—22no—
25#1545—22no—
26#1543—24no—
27#1507—24no—
28#1505C2214yescomplemented
29#1505C2214yescomplemented
30#1502—24no—
31#1290C2214yescomplemented
32C22×#41—22no—
33#532—22no—
34#532—22no—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39#532D414yes—
40#532C2314yes—
41C2×C4⋊Q8D412yes—
42C2×#127D412yes—
43C2×#124—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#500—22no—
51#493—22no—
52#486—24no—
53#480—28no—
54#474—28no—
55#466—28no—
56#459—28no—
57#447—28no—
58C4×D8—28no—
59#432—22no—
60#432C2314yes—
61#431—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#423—22no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71#407—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D412yes—
74C4×M16—22no—
75C2×C4×C8—21no—
76C2×C4×C8D412yes—
77C4×C4⋊C4—21no—
78C4×C4⋊C4D412yes—
79C4×C22⋊C4—21no—
80C4×C22⋊C4D412yes—
81#376C2311yescharacteristic
82#376C2311yescharacteristic
83C43—21no—
84#336—24no—
85#333—28no—
86#331—28no—
87#327—28no—
88#327—28no—
89C2×#41—24no—
90C2×#41—24no—
91C2×#41—42no—
92C2×#41—42no—
93C2×#41—88no—
94C22×Q8—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—42no—
99C23×C4—42no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C2×D8—28no—
104C2×M16—42no—
105C22×C8—22no—
106C22×C8C2×D414yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8C2×D414yes—
111#127—22no—
112#127—22no—
113#127—22no—
114#127C2×D414yes—
115#124—24no—
116#124—24no—
117#124—42no—
118#124—42no—
119#121—24no—
120#121—24no—
121#121—24no—
122#121—24no—
123#121—42no—
124#121—42no—
125#121—42no—
126#121—42no—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—42no—
132C4×D4—42no—
133C4×D4—42no—
134C4×D4—42no—
135#117—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—21no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×D411yescharacteristic
150C2×C42#4112yes—
151C2×C42#4112yes—
152C2×C42#4111yescharacteristic
153C2×C42C2411yescharacteristic
154#107—28no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—24no—
158C4⋊C8—48no—
159C4⋊C8—42no—
160#102—216no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165#97—24no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—41no—
169C4×C8—41no—
170C4×C8—42no—
171#95—21no—
172#95—21no—
173#95C2×D412yes—
174#95C2×D412yes—
175C24—42no—
176C24#12111yescharacteristic
177#41—88no—
178#41—88no—
179#41—88no—
180#41—88no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—84no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—84no—
193C2×D4—84no—
194C2×D4—84no—
195C2×D4—88no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—88no—
205C22×C4#12012yes—
206C22×C4#12112yes—
207C22×C4#12111yescharacteristic
208C22×C4#12112yes—
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12312yes—
212C22×C4#12312yes—
213C22×C4#12412yes—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C2×#4111yescharacteristic
217D8—88no—
218M16—84no—
219C2×C8—24no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—42no—
223C2×C8—44no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C42—24no—
235C42—44no—
236C42—42no—
237C42—44no—
238C42#12011yescharacteristic
239C42#12011yescharacteristic
240C42#12112yes—
241C42#12112yes—
242C42#12312yes—
243C42#12312yes—
244C42#12411yescharacteristic
245C42#12411yescharacteristic
246C42C22×D412yes—
247C42C2×#4112yes—
248C23—22no—
249C23—41no—
250C23—41no—
251C23—44no—
252C23—84no—
253C23—84no—
254C23—84no—
255C23#39412yes—
256C23C2×#12111yescharacteristic
257Q8—84no—
258Q8—84no—
259D4—48no—
260D4—84no—
261D4—84no—
262D4—84no—
263D4—88no—
264D4—88no—
265D4—84no—
266D4—84no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—88no—
286C2×C4—84no—
287C2×C4—88no—
288C2×C4#39411yescharacteristic
289C2×C4#39412yes—
290C2×C4#39411yescharacteristic
291C2×C4#39411yescharacteristic
292C2×C4#39411yescharacteristic
293C2×C4#47214yes—
294C2×C4C2×#12011yescharacteristic
295C2×C4C2×#12112yes—
296C2×C4C2×#12312yes—
297C2×C4C2×#12411yescharacteristic
298C8—44no—
299C8—84no—
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—82no—
307C22—84no—
308C22—84no—
309C22—84no—
310C22—168no—
311C22C2×#39411yescharacteristic
312C22C2×#47212yes—
313C4—21no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—21no—
318C4—44no—
319C4—82no—
320C4—84no—
321C4#204512yes—
322C4#204512yes—
323C4#205312yes—
324C4#301812yes—
325C2—22no—
326C2—42no—
327C2—164no—
328C2#2349811yescharacteristic
329C2C2×#204511yescharacteristic
330C2C2×#205311yescharacteristic
subgroupG/H
centerC2×C4#394
commutator, Frattini, cosocleC2×C42C24
socleC22C2×#394
prime radical#60940C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20809C2semidirect
3. #17714C2semidirect
4. #17544C2semidirect
5. #10513C2semidirect
6. #10512C2semidirect
7. #7987C2semidirect
19. #1843C22semidirect
20. #1832C22semidirect
23. #1551C22semidirect
28. #1505C22semidirect
29. #1505C22semidirect
31. #1290C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#208092
#177142
#175442
#105132
#105122
#79872