Finite Group Databasebeta

← #136376

Subgroups — #136376

Order 512.

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

normal subgroups only

This group has 348 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#12904C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11600C211yesmaximal, characteristic
5#9628C211yesmaximal, characteristic
6#7279C211yesmaximal, characteristic
7#7169C211yesmaximal, characteristic
8C2×C4×Q16—21no—
9C2×C4×Q16C2211yescharacteristic
10#2351—21no—
11C2×#365—22no—
12#2150C2211yescharacteristic
13C2×#362—21no—
14C4×M32C2211yescharacteristic
15#1942—21no—
16#1865—22no—
17#1848C2211yescharacteristic
18#1846C2211yescharacteristic
19#1805—21no—
20C8×C4⋊C4C2211yescharacteristic
21#1570—21no—
22#1564C2211yescharacteristic
23#1387—21no—
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
55#365—44no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58#362—42no—
59#346C2×C412yes—
60C4×C16—21no—
61#340—22no—
62#336—21no—
63#335—21no—
64#334—42no—
65#334C2×C412yes—
66C22×Q8—21no—
67C2×Q16—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16C4⋊C412yes—
71C2×Q16SD1612yes—
72C2×Q16SD1612yes—
73C2×M16—41no—
74C22×C8—21no—
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D411yescharacteristic
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×Q8—44no—
82C4×Q8—82no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C42—21no—
90C2×C42—41no—
91C2×C42—42no—
92C2×C42C2×D411yescharacteristic
93M32—22no—
94M32—22no—
95M32—41no—
96M32—41no—
97C2×C16—22no—
98C2×C16—41no—
99#108—42no—
100#108—41no—
101#108—41no—
102#107—21no—
103#107—41no—
104#107—41no—
105#107C4⋊C412yes—
106#106—42no—
107C4⋊C8—22no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—22no—
113#103—22no—
114#103—44no—
115C22⋊C8—41no—
116C4×C8—41no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8C22×C411yescharacteristic
120C4×C8C2×Q811yescharacteristic
121C4×C8#4111yescharacteristic
122C4×C8#4111yescharacteristic
123#95—42no—
124#95—41no—
125#95—41no—
126#95SD1611yescharacteristic
127#95SD1611yescharacteristic
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—42no—
139C22×C4#12011yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141C22×C4AGL(1,ℤ/8)11yescharacteristic
142Q16—22no—
143Q16—22no—
144Q16—22no—
145Q16#10614yes—
146M16—81no—
147M16—81no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—81no—
155C2×C8C2×C4⋊C411yescharacteristic
156C2×C8C4×D411yescharacteristic
157C2×C8C22⋊Q811yescharacteristic
158C2×C8#12311yescharacteristic
159C2×C8C2×SD1611yescharacteristic
160C2×C8C2×SD1611yescharacteristic
161C2×C8AGL(1,ℤ/8)11yescharacteristic
162C2×C8#13711yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—84no—
171C4⋊C4—84no—
172C42—42no—
173C42—41no—
174C42—41no—
175C42—42no—
176C42—84no—
177C42C4×D411yescharacteristic
178C42C22⋊Q811yescharacteristic
179C42#13511yescharacteristic
180C42AGL(1,ℤ/8)11yescharacteristic
181C16—24no—
182C16—81no—
183C23#45011yescharacteristic
184Q8—24no—
185Q8—42no—
186Q8—42no—
187Q8—42no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#38511yescharacteristic
205C2×C4C4×SD1611yescharacteristic
206C2×C4#44211yescharacteristic
207C2×C4#44811yescharacteristic
208C2×C4#44911yescharacteristic
209C2×C4#45011yescharacteristic
210C2×C4#45311yescharacteristic
211C2×C4#45611yescharacteristic
212C2×C4#47611yescharacteristic
213C2×C4#47811yescharacteristic
214C8—21no—
215C8—21no—
216C8—82no—
217C8—81no—
218C8—161no—
219C8C2×#10611yescharacteristic
220C8#42811yescharacteristic
221C8#48111yescharacteristic
222C8#48311yescharacteristic
223C22—21no—
224C22—21no—
225C22#162911yescharacteristic
226C22#165111yescharacteristic
227C22#186411yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—44no—
234C4—81no—
235C4—82no—
236C4—162no—
237C4#162211yescharacteristic
238C4#165411yescharacteristic
239C4#185311yescharacteristic
240C4#185911yescharacteristic
241C2—21no—
242C2—21no—
243C2#888611yescharacteristic
244C2#1295111yescharacteristic
245C2#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
#129042
#128192
#116002
#96282
#72792
#71692