Finite Group Databasebeta

← #492676

Subgroups — #492676

Order 512.

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

normal subgroups only

This group has 421 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#33908C211yesmaximal, characteristic, complemented
2#33901C211yesmaximal, characteristic, complemented
3#19989C212yesmaximal, complemented
4#12522C211yesmaximal, characteristic, complemented
5#12452C211yesmaximal, characteristic, complemented
6#12451C211yesmaximal, characteristic, complemented
7#12449C212yesmaximal, complemented
8#12396C212yesmaximal, complemented
9#12395C212yesmaximal, complemented
10C8×D16C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic, complemented
12#3341C2212yes—
13#3334C2212yes—
14#3303—22no—
15#2254C2212yes—
16#2252C2211yescharacteristic
17#2252C2211yescharacteristic
18#2234—22no—
19#2232—22no—
20#2226C2212yes—
21#2224C2212yes—
22#2215—21no—
23#2215—21no—
24C4×D16C2211yescharacteristic
25D4×C16C2211yescharacteristic
26D4×C16C2211yescharacteristic
27#2158—21no—
28C2×#366—21no—
29#1710C2212yes—
30#1683—22no—
31#1675C2212yes—
32#1673C2212yes—
33#1646—22no—
34#1635—22no—
35C8×D8C2212yes—
36C8×D8C2211yescharacteristic
37#1371C2211yescharacteristic
38#1370C2211yescharacteristic
39#1365C2212yes—
40#1364C2212yes—
41#1331C2212yes—
42D4⋊C16C2211yescharacteristic
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic
45#555—22no—
46#547—22no—
47C2×D16D411yescharacteristic
48C22×C16—21no—
49C22×C16—21no—
50#500C2312yes—
51#500C2312yes—
52#487—22no—
53#487—22no—
54#485—42no—
55#482—42no—
56#480—42no—
57#476—22no—
58#476—22no—
59#474—22no—
60#466—22no—
61#466—22no—
62#465—42no—
63#461—22no—
64#449—42no—
65C4×D8D411yescharacteristic
66C4×D8C2312yes—
67C8×D4—22no—
68C8×D4C2311yescharacteristic
69C8×D4C2311yescharacteristic
70#427—22no—
71C2×#107—22no—
72#416—42no—
73#367—22no—
74#367—42no—
75#366—21no—
76#366—21no—
77#366—21no—
78#366—21no—
79#366—22no—
80#366D411yescharacteristic
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357D412yes—
86C22⋊C16—21no—
87C22⋊C16—21no—
88C4×C16C2311yescharacteristic
89C4×C16C2311yescharacteristic
90#335D411yescharacteristic
91#335C2312yes—
92#333—22no—
93#332—22no—
94D4⋊C8—22no—
95D4⋊C8D411yescharacteristic
96D4⋊C8D411yescharacteristic
97C82C2311yescharacteristic
98C2×#41—42no—
99C2×SD16—42no—
100C2×D8D811yescharacteristic
101C2×D8C2×D412yes—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—42no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107#124—82no—
108#123—82no—
109C22⋊Q8—42no—
110#121—42no—
111#121—42no—
112#121—42no—
113C4×D4—22no—
114C4×D4C2×D411yescharacteristic
115C4×D4C2×D411yescharacteristic
116#117—42no—
117#117—42no—
118C2×C4⋊C4—42no—
119D16—22no—
120C2×C16—21no—
121C2×C16—21no—
122C2×C16—21no—
123C2×C16—42no—
124C2×C16—42no—
125C2×C16C2×D411yescharacteristic
126C2×C16C2×D411yescharacteristic
127#107—22no—
128#107—22no—
129#107—22no—
130#107—22no—
131#107—42no—
132#107D811yescharacteristic
133#107C2×D412yes—
134#106—42no—
135#106—42no—
136C4⋊C8—22no—
137C4⋊C8C2×D411yescharacteristic
138C4⋊C8C2×D411yescharacteristic
139#103—42no—
140#103—42no—
141#103—42no—
142#103—82no—
143#102—22no—
144#102—42no—
145#102—42no—
146#102—42no—
147#102D811yescharacteristic
148#102D811yescharacteristic
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8—42no—
152C4×C8C2×D411yescharacteristic
153C4×C8C2×D411yescharacteristic
154C4×C8C2411yescharacteristic
155#41—84no—
156#41—162no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×D4—22no—
160C2×D4—82no—
161C2×D4—82no—
162C2×D4C2×D811yescharacteristic
163C2×D4C2×D811yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—82no—
168SD16—162no—
169D8—22no—
170D8—22no—
171D8—22no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—162no—
181C2×C8C2×D811yescharacteristic
182C2×C8C2×D811yescharacteristic
183C2×C8C22×D411yescharacteristic
184C2×C8H(5,𝔽2)11yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—82no—
191C4⋊C4C2×D811yescharacteristic
192C4⋊C4C2×D811yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—42no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C22⋊C4—82no—
199C42—82no—
200C42C22×D411yescharacteristic
201C16—22no—
202C16—21no—
203C16—41no—
204C16—41no—
205C16#13512yes—
206C23—21no—
207C23—21no—
208C23—42no—
209Q8—82no—
210Q8—82no—
211Q8—162no—
212D4—21no—
213D4—21no—
214D4—21no—
215D4—42no—
216D4—42no—
217D4—164no—
218D4—322no—
219D4#50812yes—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—164no—
232C2×C4D4211yescharacteristic
233C2×C4C22×D811yescharacteristic
234C2×C4#57611yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—41no—
239C8—41no—
240C8—41no—
241C8—82no—
242C8C2×#13511yescharacteristic
243C8#57811yescharacteristic
244C22—22no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—82no—
251C22—82no—
252C22—82no—
253C22D4×D811yescharacteristic
254C4—21no—
255C4—41no—
256C4—41no—
257C4—82no—
258C4—162no—
259C4—162no—
260C4#328311yescharacteristic
261C4#329311yescharacteristic
262C4C2×#50811yescharacteristic
263C4#341511yescharacteristic
264C2—42no—
265C2—81no—
266C2—162no—
267C2#1997111yescharacteristic
268C2#3384611yescharacteristic
269C2#3385911yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33908C2semidirect
2. #33901C2semidirect
3. #19989C2semidirect
4. #12522C2semidirect
5. #12452C2semidirect
6. #12451C2semidirect
7. #12449C2semidirect
8. #12396C2semidirect
9. #12395C2semidirect
10. C8×D16C2semidirect
11. C16×D8C2semidirect

Maximal subgroups

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

subgroupindex
#339082
#339012
#199892
#125222
#124522
#124512
#124492
#123962
#123952
C8×D162
C16×D82