Finite Group Databasebeta

← #492477

Subgroups — #492477

Order 512.

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

normal subgroups only

This group has 399 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#33960C211yesmaximal, characteristic
2#33947C211yesmaximal, characteristic
3#19959C212yesmaximal, complemented
4#12504C211yesmaximal, characteristic, complemented
5#12487C212yesmaximal, complemented
6#12458C211yesmaximal, characteristic, complemented
7#12440C211yesmaximal, characteristic, complemented
8#12387C212yesmaximal, complemented
9#12359C212yesmaximal, complemented
10#12081C211yesmaximal, characteristic
11C16×Q16C211yesmaximal, characteristic, complemented
12#3393C2212yes—
13#3380C2212yes—
14#3356—22no—
15#2253—21no—
16#2252C2211yescharacteristic
17#2249—21no—
18#2247C2211yescharacteristic
19#2241C2212yes—
20#2228—22no—
21#2226C2212yes—
22#2211—22no—
23#2208C2212yes—
24Q8×C16C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26C4×Q32—21no—
27C4×D16—21no—
28C4×D16C2211yescharacteristic
29#1707C2212yes—
30#1691—22no—
31#1676C2212yes—
32#1670C2212yes—
33#1643—22no—
34#1638—22no—
35#1589C2212yes—
36C8×Q16C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1362C2212yes—
41Q8⋊C16C2211yescharacteristic
42Q8⋊C16C2211yescharacteristic
43#1331C2212yes—
44C8×C16C2211yescharacteristic
45#556—22no—
46#554—22no—
47C2×Q32—21no—
48C2×Q32—21no—
49C2×SD32—22no—
50C2×D16D411yescharacteristic
51#500C2312yes—
52#499—22no—
53#498—22no—
54#495—22no—
55#494C2312yes—
56#487—22no—
57#485—42no—
58#483—42no—
59#478—42no—
60#477—22no—
61#476—22no—
62#475—22no—
63#470—42no—
64#464—42no—
65#461—22no—
66C8×Q8C2311yescharacteristic
67C8×Q8C2311yescharacteristic
68C4×Q16—22no—
69C4×Q16D411yescharacteristic
70C4×D8—22no—
71C4×D8C2312yes—
72#436—22no—
73#428—42no—
74#367—22no—
75#366—21no—
76#366—21no—
77#366D411yescharacteristic
78C4⋊C16—21no—
79C4⋊C16—21no—
80C4⋊C16C2311yescharacteristic
81C4⋊C16C2311yescharacteristic
82#358—22no—
83#358—42no—
84#357—22no—
85#357D412yes—
86C4×C16—21no—
87C4×C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94Q8⋊C8D411yescharacteristic
95Q8⋊C8D411yescharacteristic
96D4⋊C8—22no—
97C82C2311yescharacteristic
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16D811yescharacteristic
101C2×SD16—42no—
102C2×D8C2×D412yes—
103C2×M16—42no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106#126—82no—
107#125—42no—
108#124—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C22⋊Q8—82no—
112C4×Q8—42no—
113C4×Q8C2×D411yescharacteristic
114C4×Q8C2×D411yescharacteristic
115C4×D4—22no—
116C4×D4—42no—
117#117—42no—
118C2×C4⋊C4—42no—
119Q32—42no—
120Q32—81no—
121SD32—82no—
122D16—22no—
123C2×C16—21no—
124C2×C16—21no—
125C2×C16—21no—
126C2×C16C2×D411yescharacteristic
127C2×C16C2×D411yescharacteristic
128#107—22no—
129#107—22no—
130#107—42no—
131#107D811yescharacteristic
132#107C2×D412yes—
133#106—42no—
134#106—42no—
135#106—42no—
136#106—42no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8C2×D411yescharacteristic
141C4⋊C8C2×D411yescharacteristic
142#103—42no—
143#103—42no—
144#103—42no—
145#103—82no—
146#103D811yescharacteristic
147#103D811yescharacteristic
148#102—22no—
149#102—42no—
150C22⋊C8—42no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8C2×D411yescharacteristic
154C4×C8C2×D411yescharacteristic
155C4×C8C2411yescharacteristic
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8C2×D811yescharacteristic
159C2×Q8C2×D811yescharacteristic
160C2×D4—22no—
161C22×C4—42no—
162C22×C4—82no—
163Q16—22no—
164Q16—42no—
165Q16—42no—
166Q16—82no—
167SD16—162no—
168D8—22no—
169D8—22no—
170M16—84no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—42no—
175C2×C8C2×D811yescharacteristic
176C2×C8C2×D811yescharacteristic
177C2×C8C22×D411yescharacteristic
178C2×C8H(5,𝔽2)11yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4C2×D811yescharacteristic
190C4⋊C4C2×D811yescharacteristic
191C22⋊C4—42no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—21no—
196C42—21no—
197C42—82no—
198C42C22×D411yescharacteristic
199C16—21no—
200C16—41no—
201C16—41no—
202C16—41no—
203C16#13712yes—
204C23—42no—
205Q8—21no—
206Q8—21no—
207Q8—21no—
208Q8—21no—
209Q8—82no—
210Q8—82no—
211Q8—162no—
212D4—42no—
213D4—42no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—322no—
224C2×C4D4211yescharacteristic
225C2×C4C22×D811yescharacteristic
226C2×C4#57611yescharacteristic
227C8—21no—
228C8—21no—
229C8—41no—
230C8—41no—
231C8—41no—
232C8—82no—
233C8#50812yes—
234C8C2×#13711yescharacteristic
235C8#57711yescharacteristic
236C22—82no—
237C22—82no—
238C22—82no—
239C22D4×D811yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—162no—
247C4—162no—
248C4#327811yescharacteristic
249C4#329011yescharacteristic
250C4C2×#50811yescharacteristic
251C4#341711yescharacteristic
252C2—162no—
253C2#1988911yescharacteristic
254C2#3384911yescharacteristic
255C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalC2×D16D4
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
3. #19959C2semidirect
4. #12504C2semidirect
5. #12487C2semidirect
6. #12458C2semidirect
7. #12440C2semidirect
8. #12387C2semidirect
9. #12359C2semidirect
11. C16×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#339602
#339472
#199592
#125042
#124872
#124582
#124402
#123872
#123592
#120812
C16×Q162