Finite Group Databasebeta

← #136383

Subgroups — #136383

Order 512.

351 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 90 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#22813C211yesmaximal, characteristic
2#12903C211yesmaximal, characteristic, complemented
3#12818C211yesmaximal, characteristic
4#11593C211yesmaximal, characteristic
5#9630C211yesmaximal, characteristic, complemented
6#7278C211yesmaximal, characteristic, complemented
7#7168C211yesmaximal, characteristic, complemented
8C2×C4×D8C2211yescharacteristic
9#2354—21no—
10C2×#365—22no—
11#2150C2211yescharacteristic
12C2×#361—21no—
13C4×M32C2211yescharacteristic
14#1972—21no—
15#1926—22no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18#1802—21no—
19#1776—21no—
20#1770C2211yescharacteristic
21#1571—21no—
22#1564C2211yescharacteristic
23#1387—21no—
24#1331C414yescomplemented
25#1303—21no—
26C22×D8D411yescharacteristic
27C2×C4×D4—21no—
28C2×#117—41no—
29C2×M32—21no—
30C2×M32—21no—
31C4×D8—21no—
32C4×D8—21no—
33C4×D8C2×C412yes—
34C4×D8Q812yes—
35#431—21no—
36C2×#108—21no—
37C2×#108—21no—
38C2×#107D411yescharacteristic
39C2×#106—21no—
40C2×#106D412yes—
41#423—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×#102—21no—
44C2×#102—22no—
45C2×#97—21no—
46C2×C4×C8C2311yescharacteristic
47#390—22no—
48#388—42no—
49#384—21no—
50#380—41no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4D411yescharacteristic
53#365—44no—
54C4⋊C16—21no—
55C4⋊C16—21no—
56#361—42no—
57#346—21no—
58C4×C16C2×C412yes—
59#340—22no—
60#336—21no—
61#335—42no—
62#335C2×C412yes—
63#334—21no—
64C22×D4—21no—
65C23×C4—41no—
66C2×D8—21no—
67C2×D8—21no—
68C2×D8—21no—
69C2×D8—21no—
70C2×D8—21no—
71C2×D8C4⋊C412yes—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—41no—
79C4×D4—41no—
80#117—84no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C42—21no—
90C2×C42—42no—
91C2×C42C2×D411yescharacteristic
92M32—22no—
93M32—22no—
94M32—41no—
95M32—41no—
96C2×C16—22no—
97C2×C16—41no—
98#108—42no—
99#108—41no—
100#108—41no—
101#107—21no—
102#107C4⋊C412yes—
103#106—41no—
104#106—41no—
105#106—42no—
106C4⋊C8—22no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#102—22no—
112#102—22no—
113#102—82no—
114C22⋊C8—41no—
115#97—41no—
116#97—41no—
117C4×C8—41no—
118C4×C8C22×C411yescharacteristic
119C4×C8C2×Q811yescharacteristic
120C4×C8#4111yescharacteristic
121C4×C8#4111yescharacteristic
122#95—42no—
123#95SD1612yes—
124C24—41no—
125C2×D4—22no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4#12011yescharacteristic
139C22×C4C2×SD1611yescharacteristic
140C22×C4AGL(1,ℤ/8)11yescharacteristic
141D8—22no—
142D8—22no—
143D8—41no—
144D8—41no—
145M16—81no—
146M16—81no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—42no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—81no—
154C2×C8C2×C4⋊C411yescharacteristic
155C2×C8C4×D411yescharacteristic
156C2×C8C22⋊Q811yescharacteristic
157C2×C8#12311yescharacteristic
158C2×C8#13511yescharacteristic
159C2×C8#13511yescharacteristic
160C2×C8AGL(1,ℤ/8)11yescharacteristic
161C2×C8#13711yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4—161no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—84no—
172C22⋊C4—162no—
173C42—41no—
174C42—41no—
175C42—84no—
176C42C4×D411yescharacteristic
177C42C22⋊Q811yescharacteristic
178C42#13511yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—24no—
181C16—81no—
182C23—42no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23#45011yescharacteristic
190D4—42no—
191D4—42no—
192D4—81no—
193D4—81no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—84no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4#38511yescharacteristic
208C2×C4C4×SD1611yescharacteristic
209C2×C4#44211yescharacteristic
210C2×C4#44911yescharacteristic
211C2×C4#44911yescharacteristic
212C2×C4#45211yescharacteristic
213C2×C4#45311yescharacteristic
214C2×C4#45611yescharacteristic
215C2×C4#47611yescharacteristic
216C2×C4#47811yescharacteristic
217C8—21no—
218C8—21no—
219C8—81no—
220C8—82no—
221C8—161no—
222C8#42711yescharacteristic
223C8#42811yescharacteristic
224C8#48211yescharacteristic
225C8#48511yescharacteristic
226C22—21no—
227C22—21no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22—161no—
235C22#163111yescharacteristic
236C22#165411yescharacteristic
237C22#186411yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—82no—
242C4—82no—
243C4—162no—
244C4#162411yescharacteristic
245C4#165211yescharacteristic
246C4#185411yescharacteristic
247C4#185911yescharacteristic
248C4AGL(1,ℤ/16)12yes—
249C2—21no—
250C2—21no—
251C2—162no—
252C2#888911yescharacteristic
253C2#1294911yescharacteristic
254C2#1295611yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12903C2semidirect
5. #9630C2semidirect
6. #7278C2semidirect
7. #7168C2semidirect
24. #1331C4semidirect

Maximal subgroups

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

subgroupindex
#228132
#129032
#128182
#115932
#96302
#72782
#71682