Finite Group Databasebeta

← #135034

Subgroups — #135034

Order 512.

319 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 102 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#17886C211yesmaximal, characteristic
2#12816C212yesmaximal
3#11996C211yesmaximal, characteristic, complemented
4#8814C211yesmaximal, characteristic, complemented
5#7278C212yesmaximal, complemented
6#2965—21no—
7C4×#135C2211yescharacteristic
8#2150C2212yes—
9C2×#360—22no—
10#1845—21no—
11#1844C2212yes—
12#1834—21no—
13#1591C412yescomplemented
14#1590—22no—
15#1589C412yescomplemented
16#1564C2211yescharacteristic
17#1476—21no—
18C2×C82C2211yescharacteristic
19#1386—22no—
20#1367—22no—
21C2×#135D411yescharacteristic
22C4×#41—21no—
23C2×M32—22no—
24#446—22no—
25C4×Q16C2×C411yescharacteristic
26C4×Q16Q811yescharacteristic
27C4×SD16—21no—
28C4×SD16—21no—
29C4×D8C2×C411yescharacteristic
30C4×D8Q811yescharacteristic
31#436—22no—
32#435—42no—
33#431—21no—
34#431—41no—
35#427D411yescharacteristic
36C2×#107—42no—
37C2×#106—22no—
38C2×#106—22no—
39#416—21no—
40C4×M16—41no—
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44#384—22no—
45C4⋊C16—22no—
46C4⋊C16—22no—
47#360—44no—
48#340—42no—
49#335C2×C412yes—
50#334—21no—
51Q8⋊C8—22no—
52D4⋊C8—22no—
53C82C2×C412yes—
54C82D412yes—
55C2×#41—21no—
56#135—22no—
57#135—22no—
58C2×Q16C4⋊C411yescharacteristic
59C2×SD16—21no—
60C2×D8C4⋊C411yescharacteristic
61C2×M16—42no—
62C22×C8—21no—
63C22×C8D812yes—
64C22×C8C2×D411yescharacteristic
65C22×C8#4111yescharacteristic
66C4×Q8—21no—
67C4×Q8—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4—41no—
71C4×D4—41no—
72#117—21no—
73#117—41no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C42—41no—
78C2×C42C2×D411yescharacteristic
79M32—42no—
80M32—42no—
81C2×C16—42no—
82#107—84no—
83#107C4⋊C411yescharacteristic
84#107C4⋊C411yescharacteristic
85#106—21no—
86#106—42no—
87#106—42no—
88#106—82no—
89C4⋊C8—22no—
90C4⋊C8—42no—
91#103—21no—
92#103—21no—
93#102—21no—
94#102—21no—
95C22⋊C8—41no—
96C22⋊C8—42no—
97#97—42no—
98C4×C8—21no—
99C4×C8—41no—
100C4×C8C22⋊C412yes—
101C4×C8C22⋊C412yes—
102C4×C8SD1611yescharacteristic
103C4×C8SD1611yescharacteristic
104C4×C8C22×C411yescharacteristic
105C4×C8C2×D411yescharacteristic
106C4×C8C2×Q811yescharacteristic
107C4×C8#4111yescharacteristic
108#41—41no—
109#41—41no—
110#41—41no—
111#41—41no—
112C2×Q8—21no—
113C2×D4—21no—
114C2×D4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—82no—
119C22×C4#12111yescharacteristic
120C22×C4C2×D811yescharacteristic
121C22×C4AGL(1,ℤ/8)11yescharacteristic
122Q16—22no—
123SD16—42no—
124D8—22no—
125M16—84no—
126C2×C8—24no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8#10212yes—
133C2×C8#10212yes—
134C2×C8C2×C4⋊C411yescharacteristic
135C2×C8#11711yescharacteristic
136C2×C8#12111yescharacteristic
137C2×C8C22⋊Q811yescharacteristic
138C2×C8AGL(1,ℤ/8)12yes—
139C2×C8#13712yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C22⋊C4—41no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C42—41no—
152C42—41no—
153C42—41no—
154C42C2×C22⋊C411yescharacteristic
155C42C22⋊Q811yescharacteristic
156C42C2×SD1611yescharacteristic
157C42#13711yescharacteristic
158C16—82no—
159C23—41no—
160C23#45911yescharacteristic
161Q8—41no—
162Q8—41no—
163D4—41no—
164D4—41no—
165D4—81no—
166D4—81no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—82no—
176C2×C4—162no—
177C2×C4—162no—
178C2×C4—162no—
179C2×C4#38011yescharacteristic
180C2×C4C2×#10211yescharacteristic
181C2×C4#41711yescharacteristic
182C2×C4#46111yescharacteristic
183C2×C4#46911yescharacteristic
184C2×C4#47011yescharacteristic
185C2×C4#47211yescharacteristic
186C2×C4#47311yescharacteristic
187C2×C4#47411yescharacteristic
188C2×C4#47611yescharacteristic
189C8—24no—
190C8—21no—
191C8—21no—
192C8—22no—
193C8—22no—
194C8—82no—
195C8#42812yes—
196C8#46812yes—
197C22—21no—
198C22—21no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22#168711yescharacteristic
203C22#168911yescharacteristic
204C22#180211yescharacteristic
205C4—21no—
206C4—21no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4—322no—
211C4#169311yescharacteristic
212C4#169511yescharacteristic
213C4#180911yescharacteristic
214C4#181611yescharacteristic
215C4#214512yes—
216C4#223012yes—
217C2—21no—
218C2—21no—
219C2—161no—
220C2#865211yescharacteristic
221C2#1281711yescharacteristic
222C2#1282311yescharacteristic
subgroupG/H
center, socleC22#1802
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#135D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11996C2semidirect
4. #8814C2semidirect
5. #7278C2semidirect
13. #1591C4semidirect
15. #1589C4semidirect

Maximal subgroups

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

subgroupindex
#178862
#128162
#119962
#88142
#72782