Finite Group Databasebeta

← #326642

Subgroups — #326642

Order 512.

522 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 122 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#21710C211yesmaximal, characteristic
2#21344C211yesmaximal, characteristic, complemented
3#10688C211yesmaximal, characteristic, complemented
4#10662C211yesmaximal, characteristic, complemented
5#10546C211yesmaximal, characteristic
6#10034C211yesmaximal, characteristic
7#9577C212yesmaximal, complemented
8#9238C212yesmaximal, complemented
9#9047C212yesmaximal, complemented
10#8878C212yesmaximal, complemented
11#8800C211yesmaximal, characteristic, complemented
12#3150C2211yescharacteristic
13#3103C2212yes—
14C2×#444—21no—
15C2×#443—21no—
16#2932C2212yes—
17#2295C2211yescharacteristic
18#1985C2211yescharacteristic
19#1985C2211yescharacteristic
20#1975—21no—
21#1974—21no—
22#1962C2212yes—
23#1940C2211yescharacteristic
24#1940C2211yescharacteristic
25#1916—21no—
26#1916—21no—
27#1849C2211yescharacteristic
28#1768—21no—
29C4×C4⋊C8C2211yescharacteristic
30#1746—21no—
31C4×#97C2211yescharacteristic
32#1571C2212yes—
33#1540C2212yes—
34#1502C2212yes—
35#1487C2212yes—
36#1480C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1281C2212yes—
40#1276C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43C2×#135—21no—
44C2×#135—21no—
45C2×#131—22no—
46#532C2312yes—
47C2×C4×Q8C2311yescharacteristic
48#497—22no—
49#496—22no—
50#481—22no—
51#475—22no—
52#460—22no—
53#448—22no—
54#444—44no—
55#443—44no—
56C4×SD16—42no—
57#435—42no—
58#435—42no—
59#431C2312yes—
60#431C2312yes—
61C2×#108—21no—
62C2×#108—21no—
63#423C2312yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×#104—22no—
69C2×#104—22no—
70#408—22no—
71C2×#97D411yescharacteristic
72C2×#97D411yescharacteristic
73C2×#97C2311yescharacteristic
74C2×#97C2311yescharacteristic
75C2×C4×C8—21no—
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78#391—21no—
79#389—21no—
80#387—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2311yescharacteristic
83#376—21no—
84C43C2311yescharacteristic
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#335—42no—
93#332—22no—
94#331—22no—
95#327—24no—
96Q8⋊C8—42no—
97D4⋊C8—42no—
98C2×#41—22no—
99C22×Q8#4111yescharacteristic
100#135—42no—
101#135—42no—
102#135—42no—
103#135—42no—
104C2×Q16—21no—
105C2×Q16—21no—
106C2×SD16—22no—
107C2×SD16—22no—
108C2×SD16—42no—
109C2×D8—21no—
110C2×D8—21no—
111#131—44no—
112#131—44no—
113C2×M16—22no—
114C2×M16—42no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—42no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4⋊Q8#4112yes—
123#127#4112yes—
124#124—22no—
125#121—22no—
126#121—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×D4—22no—
130C4×D4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4#4111yescharacteristic
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2411yescharacteristic
142#108—42no—
143#108—42no—
144#106—42no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—42no—
152C4⋊C8—42no—
153#104—44no—
154#104—44no—
155#103—42no—
156#102—42no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—44no—
160#97—22no—
161#97—22no—
162#97—22no—
163#97—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—42no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95#4112yes—
173#41—42no—
174#41—42no—
175#41—42no—
176#41—42no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—81no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—42no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C2×#4111yescharacteristic
192C22×C4C2×#4111yescharacteristic
193Q16—42no—
194Q16—42no—
195SD16—44no—
196SD16—44no—
197SD16—84no—
198D8—42no—
199D8—42no—
200M16—42no—
201M16—42no—
202M16—84no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—84no—
222C2×C8#13512yes—
223C2×C8AGL(1,ℤ/8)12yes—
224C2×C8#13712yes—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—42no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—81no—
242C42AGL(1,ℤ/8)12yes—
243C42C2×#4112yes—
244C42H(5,𝔽2)12yes—
245C23—42no—
246C23#52511yescharacteristic
247Q8—42no—
248Q8—42no—
249Q8—42no—
250Q8—42no—
251D4—42no—
252D4—42no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—82no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4#52511yescharacteristic
276C2×C4#53511yescharacteristic
277C2×C4#55112yes—
278C2×C4#55312yes—
279C2×C4C2×#13511yescharacteristic
280C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
281C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
282C2×C4C2×#13711yescharacteristic
283C8—22no—
284C8—22no—
285C8—22no—
286C8—41no—
287C8—82no—
288C8—82no—
289C8—81no—
290C8—81no—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22#259811yescharacteristic
298C22#310611yescharacteristic
299C22#311211yescharacteristic
300C4—22no—
301C4—22no—
302C4—42no—
303C4—42no—
304C4—82no—
305C4—82no—
306C4—81no—
307C4—81no—
308C4#310511yescharacteristic
309C4#311011yescharacteristic
310C4#321011yescharacteristic
311C4#321711yescharacteristic
312C2—21no—
313C2—21no—
314C2—162no—
315C2#2936711yescharacteristic
316C2#2938511yescharacteristic
317C2#2938611yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2598
prime radical#3150C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21344C2semidirect
3. #10688C2semidirect
4. #10662C2semidirect
7. #9577C2semidirect
8. #9238C2semidirect
9. #9047C2semidirect
10. #8878C2semidirect
11. #8800C2semidirect

Maximal subgroups

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

subgroupindex
#217102
#213442
#106882
#106622
#105462
#100342
#95772
#92382
#90472
#88782
#88002