Finite Group Databasebeta

← #492752

Subgroups — #492752

Order 512.

410 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 118 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#33942C212yesmaximal, complemented
2#20021C211yesmaximal, characteristic
3Q162C211yesmaximal, characteristic, complemented
4#12509C211yesmaximal, characteristic
5#12480C212yesmaximal, complemented
6#12448C211yesmaximal, characteristic, complemented
7#12425C211yesmaximal, characteristic
8#12385C212yesmaximal, complemented
9#12354C212yesmaximal, complemented
10#12054C211yesmaximal, characteristic, complemented
11C8×Q32C211yesmaximal, characteristic, complemented
12#3396—21no—
13#3396C2212yes—
14Q8×Q16—21no—
15Q8×Q16C2212yes—
16#2243—22no—
17#2240C2212yes—
18#2240C2212yes—
19#2227C2212yes—
20#2224—22no—
21#2211—22no—
22#2207C2212yes—
23#2183C2212yes—
24#2175—22no—
25C4×Q32C2211yescharacteristic
26#1713C2211yescharacteristic
27#1712C2211yescharacteristic
28#1676—21no—
29#1676C2212yes—
30#1668—21no—
31#1668C2212yes—
32#1633—22no—
33#1620—22no—
34C8×Q16C2211yescharacteristic
35C8×Q16C2211yescharacteristic
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1361C2212yes—
41Q8⋊C16C2212yes—
42#1332C2211yescharacteristic
43#1332C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#559—22no—
46Q82—22no—
47C2×Q32D411yescharacteristic
48C2×SD32—22no—
49#500—22no—
50#500—21no—
51#500—41no—
52#500C2312yes—
53#495—22no—
54#495—41no—
55#494—21no—
56#493—21no—
57#493—21no—
58#493C2312yes—
59#477—22no—
60#477—22no—
61#477—42no—
62#477—42no—
63#463—42no—
64#463—42no—
65#460—22no—
66#460—22no—
67C8×Q8—21no—
68C8×Q8—21no—
69C8×Q8C2312yes—
70C4×Q16—22no—
71C4×Q16—41no—
72C4×Q16D411yescharacteristic
73C4×Q16C2311yescharacteristic
74C4×Q16C2311yescharacteristic
75C4×D8—22no—
76C4×D8—41no—
77#366—22no—
78#366D411yescharacteristic
79C4⋊C16—22no—
80C4⋊C16C2312yes—
81#358—22no—
82#358D411yescharacteristic
83#358D411yescharacteristic
84#357—22no—
85#357—22no—
86#357—22no—
87#357—42no—
88#346—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#331—22no—
96Q8⋊C8—21no—
97Q8⋊C8—21no—
98Q8⋊C8D412yes—
99C82C2311yescharacteristic
100C2×Q16—81no—
101C2×Q16D811yescharacteristic
102C2×Q16C2×D411yescharacteristic
103C2×Q16C2×D411yescharacteristic
104C2×SD16—42no—
105C2×SD16—42no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—41no—
109C2×D8—41no—
110C2×D8—81no—
111C4⋊Q8—22no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114C4⋊Q8—82no—
115#127—22no—
116#124—42no—
117#124—42no—
118#123—82no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—42no—
122C4×Q8—42no—
123C4×Q8C2×D412yes—
124C4×D4—42no—
125C4×D4—42no—
126Q32—22no—
127SD32—44no—
128C2×C16—22no—
129C2×C16—21no—
130C2×C16C2×D411yescharacteristic
131C2×C16C2×D411yescharacteristic
132#107—22no—
133#107—22no—
134#107—42no—
135#107—41no—
136#107—41no—
137#107—42no—
138#107D811yescharacteristic
139#107C2×D411yescharacteristic
140#107C2×D411yescharacteristic
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8C2×D412yes—
147#103—21no—
148#103—21no—
149#103—42no—
150#103—42no—
151#103D812yes—
152#102—42no—
153#102—42no—
154C4×C8—22no—
155C4×C8—41no—
156C4×C8—41no—
157C4×C8C2×D411yescharacteristic
158C4×C8C2×D411yescharacteristic
159C4×C8C2411yescharacteristic
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8C2×D812yes—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—42no—
168C22×C4—82no—
169Q16—21no—
170Q16—21no—
171Q16—21no—
172Q16—22no—
173Q16—162no—
174Q16#13512yes—
175SD16—162no—
176SD16—162no—
177D8—42no—
178D8—42no—
179D8—82no—
180D8—82no—
181D8—161no—
182D8—162no—
183C2×C8—21no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8C2×D811yescharacteristic
188C2×C8C2×D811yescharacteristic
189C2×C8C22×D411yescharacteristic
190C2×C8H(5,𝔽2)11yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4C2×D812yes—
201C22⋊C4—82no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C42—22no—
205C42—41no—
206C42—41no—
207C42—82no—
208C42C22×D411yescharacteristic
209C16—21no—
210C16—21no—
211C16—41no—
212C16—42no—
213C23—82no—
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—162no—
221Q8—322no—
222D4—82no—
223D4—82no—
224D4—82no—
225D4—82no—
226D4—162no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—322no—
235C2×C4D4211yescharacteristic
236C2×C4C22×D811yescharacteristic
237C2×C4#57611yescharacteristic
238C8—22no—
239C8—21no—
240C8—21no—
241C8—42no—
242C8—81no—
243C8—81no—
244C8#51012yes—
245C8C2×#13511yescharacteristic
246C8#57811yescharacteristic
247C22—162no—
248C22—162no—
249C22—162no—
250C22D4×D811yescharacteristic
251C4—21no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C4—81no—
258C4—162no—
259C4#328511yescharacteristic
260C4#329011yescharacteristic
261C4C2×#51011yescharacteristic
262C4#341611yescharacteristic
263C2—322no—
264C2#1997311yescharacteristic
265C2#3384411yescharacteristic
266C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radical#1712C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33942C2semidirect
3. Q162C2semidirect
5. #12480C2semidirect
6. #12448C2semidirect
8. #12385C2semidirect
9. #12354C2semidirect
10. #12054C2semidirect
11. C8×Q32C2semidirect

Maximal subgroups

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

subgroupindex
#339422
#200212
Q1622
#125092
#124802
#124482
#124252
#123852
#123542
#120542
C8×Q322