Finite Group Databasebeta

← #145610

Subgroups — #145610

Order 512.

285 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 66 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#19638C211yesmaximal, characteristic
2#11986C211yesmaximal, characteristic
3#11971C211yesmaximal, characteristic, complemented
4#11941C211yesmaximal, characteristic, complemented
5#11649C211yesmaximal, characteristic
6#9634C211yesmaximal, characteristic, complemented
7#7212C211yesmaximal, characteristic, complemented
8#3206—21no—
9#3169C2211yescharacteristic
10C2×#500C2211yescharacteristic
11#3141—21no—
12#2162C2211yescharacteristic
13#2150C2211yescharacteristic
14#1719—22no—
15#1710—21no—
16#1706—21no—
17#1574—21no—
18#1567C2211yescharacteristic
19#1564C2211yescharacteristic
20C8×M16C2211yescharacteristic
21#1365—21no—
22#1365—22no—
23#1364—22no—
24#1361—21no—
25#1331—22no—
26#1299—21no—
27#532—21no—
28C2×C4⋊Q8—21no—
29C2×M32—21no—
30#500—21no—
31#500—21no—
32#500—21no—
33#500D412yes—
34#500D411yescharacteristic
35#493D411yescharacteristic
36#492—22no—
37#489—42no—
38#487—22no—
39#487—22no—
40#486—22no—
41#486—21no—
42#466—21no—
43#466—21no—
44#461—41no—
45#461—41no—
46#448—42no—
47C4×D8—21no—
48C4×D8—21no—
49#431—21no—
50#431D411yescharacteristic
51C2×#107—21no—
52C2×#107—21no—
53#423—21no—
54#417—42no—
55C4×M16D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16—21no—
60C22⋊C16—21no—
61C4×C16—21no—
62#335—21no—
63#335—21no—
64#335—41no—
65#335D412yes—
66#334—21no—
67#334—41no—
68#322—21no—
69C82—21no—
70C2×#41—41no—
71C22×Q8—41no—
72C2×SD16—42no—
73C2×SD16—42no—
74C2×D8—21no—
75C2×D8—21no—
76C2×M16—21no—
77C2×M16—41no—
78C22×C8#4111yescharacteristic
79C22×C8#4111yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—21no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85#127—21no—
86#124—41no—
87#121—41no—
88#121—41no—
89C4×D4—41no—
90C4×D4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C42C2×D411yescharacteristic
94M32—81no—
95C2×C16—41no—
96C2×C16—41no—
97#107—22no—
98#107—21no—
99#107—22no—
100#107—21no—
101#107—41no—
102#107—41no—
103C4⋊C8—22no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8D811yescharacteristic
107C4⋊C8D811yescharacteristic
108#103—44no—
109#103—42no—
110#103—42no—
111#102—44no—
112#102—41no—
113#102—41no—
114C22⋊C8—21no—
115C22⋊C8—41no—
116#97SD1612yes—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—41no—
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8C2×D411yescharacteristic
123C4×C8#4111yescharacteristic
124#41—84no—
125C2×Q8—42no—
126C2×Q8—41no—
127C2×Q8—81no—
128C2×Q8—161no—
129C2×Q8—161no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—81no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4#12411yescharacteristic
137C22×C4AGL(1,ℤ/8)11yescharacteristic
138C22×C4AGL(1,ℤ/8)11yescharacteristic
139SD16—162no—
140SD16—162no—
141D8—41no—
142D8—41no—
143D8—81no—
144M16—42no—
145M16—161no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—81no—
154C2×C8#12111yescharacteristic
155C2×C8#12111yescharacteristic
156C2×C8#12311yescharacteristic
157C2×C8#12311yescharacteristic
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C22⋊C4—81no—
165C22⋊C4—81no—
166C42#12011yescharacteristic
167C42#12111yescharacteristic
168C42C2×D811yescharacteristic
169C42C2×SD1611yescharacteristic
170C16—81no—
171C16—81no—
172C23—81no—
173C23#49011yescharacteristic
174Q8—82no—
175Q8—82no—
176Q8—82no—
177Q8—162no—
178D4—82no—
179D4—81no—
180D4—81no—
181D4—81no—
182D4—81no—
183D4—162no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—81no—
190C2×C4—82no—
191C2×C4—162no—
192C2×C4—162no—
193C2×C4#39411yescharacteristic
194C2×C4#44711yescharacteristic
195C2×C4#45011yescharacteristic
196C2×C4#45911yescharacteristic
197C2×C4#46011yescharacteristic
198C2×C4#48611yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—42no—
203C8—41no—
204C8—41no—
205C8—82no—
206C8—161no—
207C8#47111yescharacteristic
208C8#47311yescharacteristic
209C22—21no—
210C22—21no—
211C22—21no—
212C22—161no—
213C22—161no—
214C22—161no—
215C22#204111yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—162no—
220C4—161no—
221C4#204911yescharacteristic
222C4#205211yescharacteristic
223C4#219211yescharacteristic
224C4#219411yescharacteristic
225C4#221212yes—
226C2—41no—
227C2—321no—
228C2#1055011yescharacteristic
229C2#1341111yescharacteristic
230C2#1341411yescharacteristic
subgroupG/H
center, socleC22#2041
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11971C2semidirect
4. #11941C2semidirect
6. #9634C2semidirect
7. #7212C2semidirect

Maximal subgroups

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

subgroupindex
#196382
#119862
#119712
#119412
#116492
#96342
#72122