Finite Group Databasebeta

← #136405

Subgroups — #136405

Order 512.

340 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 96 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#22685C211yesmaximal, characteristic
2#12887C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11600C211yesmaximal, characteristic
5#9627C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic
7#7169C211yesmaximal, characteristic
8C2×C4×Q16—21no—
9C2×C4×Q16C2211yescharacteristic
10#2351—21no—
11#2150C2211yescharacteristic
12C2×#362—21no—
13C4×M32C2211yescharacteristic
14#1942—21no—
15#1865—22no—
16#1848C2211yescharacteristic
17#1846C2211yescharacteristic
18#1805—21no—
19C8×C4⋊C4C2211yescharacteristic
20#1571—21no—
21#1564C2211yescharacteristic
22#1387—21no—
23#1383—22no—
24#1334C414yescomplemented
25#1303—21no—
26C22×Q16D411yescharacteristic
27C2×C4×Q8—22no—
28C2×C4×Q8—21no—
29C2×M32—21no—
30C2×M32—21no—
31C4×Q16—21no—
32C4×Q16—21no—
33C4×Q16—42no—
34C4×Q16—42no—
35C4×Q16C2×C412yes—
36C4×Q16Q812yes—
37#431—21no—
38C2×#108—21no—
39C2×#108—21no—
40C2×#107—21no—
41C2×#107D411yescharacteristic
42C2×#106D412yes—
43#423—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—21no—
46C2×#103—22no—
47C2×C4×C8—21no—
48C2×C4×C8C2311yescharacteristic
49#389—22no—
50#389—42no—
51#382—21no—
52#382—41no—
53C4×C4⋊C4—41no—
54C4×C4⋊C4D411yescharacteristic
55C4⋊C16—21no—
56C4⋊C16—21no—
57#362—42no—
58#346C2×C412yes—
59C4×C16—21no—
60#340—22no—
61#336—21no—
62#335—21no—
63#335—42no—
64#334C2×C412yes—
65C22×Q8—21no—
66C2×Q16—21no—
67C2×Q16—21no—
68C2×Q16—21no—
69C2×Q16C4⋊C412yes—
70C2×Q16D812yes—
71C2×Q16Q1612yes—
72C2×M16—41no—
73C22×C8—21no—
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—44no—
81C4×Q8—82no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—42no—
91C2×C42C2×D411yescharacteristic
92M32—41no—
93M32—41no—
94M32—41no—
95M32—41no—
96C2×C16—22no—
97C2×C16—41no—
98#108—42no—
99#108—41no—
100#108—41no—
101#107—21no—
102#107—41no—
103#107—41no—
104#107C4⋊C412yes—
105#106—42no—
106C4⋊C8—22no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#103—22no—
112#103—22no—
113#103—44no—
114C22⋊C8—41no—
115C4×C8—41no—
116C4×C8—41no—
117C4×C8—41no—
118C4×C8C22×C411yescharacteristic
119C4×C8C2×Q811yescharacteristic
120C4×C8#4111yescharacteristic
121C4×C8#4111yescharacteristic
122#95—42no—
123#95—41no—
124#95—41no—
125#95SD1611yescharacteristic
126#95SD1611yescharacteristic
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4#12011yescharacteristic
139C22×C4C2×SD1611yescharacteristic
140C22×C4AGL(1,ℤ/8)11yescharacteristic
141Q16—22no—
142Q16—22no—
143Q16—22no—
144Q16#10714yes—
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×C8C2×D811yescharacteristic
159C2×C8C2×Q1611yescharacteristic
160C2×C8AGL(1,ℤ/8)11yescharacteristic
161C2×C8#13711yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C42—42no—
172C42—41no—
173C42—41no—
174C42—42no—
175C42—84no—
176C42C4×D411yescharacteristic
177C42C22⋊Q811yescharacteristic
178C42#13511yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—42no—
181C16—81no—
182C23#45011yescharacteristic
183Q8—24no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#38511yescharacteristic
204C2×C4C4×SD1611yescharacteristic
205C2×C4#44211yescharacteristic
206C2×C4#44711yescharacteristic
207C2×C4#44911yescharacteristic
208C2×C4#45111yescharacteristic
209C2×C4#45411yescharacteristic
210C2×C4#45511yescharacteristic
211C2×C4#47611yescharacteristic
212C2×C4#47811yescharacteristic
213C8—21no—
214C8—21no—
215C8—82no—
216C8—81no—
217C8—161no—
218C8C2×#10711yescharacteristic
219C8#42811yescharacteristic
220C8#48011yescharacteristic
221C8#48411yescharacteristic
222C22—21no—
223C22—21no—
224C22#163711yescharacteristic
225C22#163911yescharacteristic
226C22#186411yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—42no—
232C4—44no—
233C4—81no—
234C4—82no—
235C4—162no—
236C4#162511yescharacteristic
237C4#164411yescharacteristic
238C4#185211yescharacteristic
239C4#186011yescharacteristic
240C2—21no—
241C2—21no—
242C2#889011yescharacteristic
243C2#1295111yescharacteristic
244C2#1295411yescharacteristic
subgroupG/H
center, socleC22#1864
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1334C4semidirect

Maximal subgroups

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

subgroupindex
#226852
#128872
#128192
#116002
#96272
#72792
#71692