Finite Group Databasebeta

← #136713

Subgroups — #136713

Order 512.

343 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 104 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#33475C211yesmaximal, characteristic, complemented
2#17902C211yesmaximal, characteristic
3#13190C211yesmaximal, characteristic
4#11610C211yesmaximal, characteristic, complemented
5#11583C211yesmaximal, characteristic
6#7153C211yesmaximal, characteristic, complemented
7#7140C211yesmaximal, characteristic, complemented
8#3101C411yescharacteristic
9#2960C2211yescharacteristic
10C2×C8×D4C2211yescharacteristic
11#2183—22no—
12#2183—22no—
13#2151—21no—
14#2150C2211yescharacteristic
15C2×C4⋊C16C2211yescharacteristic
16C2×C22⋊C16—21no—
17C2×C22⋊C16—21no—
18#2108—21no—
19C4×M32C2211yescharacteristic
20C2×#346C2211yescharacteristic
21#1582—22no—
22C8×SD16—22no—
23#1570C411yescharacteristic
24#1486C411yescharacteristic
25#1481C411yescharacteristic
26#1450C2211yescharacteristic
27#1338—22no—
28D4⋊C16—22no—
29C23×C8—21no—
30#533C2×C411yescharacteristic
31C2×C4×D4C2×C411yescharacteristic
32C2×M32—21no—
33C2×M32—21no—
34C22×C16—21no—
35C22×C16D411yescharacteristic
36C22×C16D411yescharacteristic
37#483C812yes—
38#476—21no—
39#461—21no—
40#450C812yes—
41#446—21no—
42#446D412yes—
43C8×Q8—21no—
44C8×Q8D412yes—
45C4×SD16—22no—
46C8×D4—21no—
47C8×D4—21no—
48C8×D4—21no—
49C8×D4—21no—
50#431C2×C411yescharacteristic
51#431C2×C411yescharacteristic
52#424—21no—
53#423C2×C411yescharacteristic
54C2×C4⋊C8C2×C411yescharacteristic
55C2×C22⋊C8—21no—
56#405—21no—
57C2×C4×C8C2311yescharacteristic
58C4⋊C16—24no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62C4⋊C16D412yes—
63C22⋊C16—42no—
64C22⋊C16—42no—
65#346—22no—
66#346—21no—
67#346—21no—
68#346—21no—
69C4×C16—22no—
70C4×C16—21no—
71#334—22no—
72Q8⋊C8—22no—
73D4⋊C8—22no—
74#322—21no—
75C82—21no—
76C22×D4C2×C811yescharacteristic
77C23×C4—21no—
78C2×SD16—22no—
79C2×M16—21no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8C2×D411yescharacteristic
88C22×C8#4111yescharacteristic
89C4⋊Q8M1611yescharacteristic
90C4⋊Q8M1611yescharacteristic
91#125—21no—
92C22⋊Q8—21no—
93C22⋊Q8C2×C812yes—
94C4×Q8—21no—
95C4×Q8C22⋊C412yes—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100#117—21no—
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C22⋊C4—21no—
103C2×C42C22×C411yescharacteristic
104M32—22no—
105M32—41no—
106M32—41no—
107M32—41no—
108C2×C16—21no—
109C2×C16—21no—
110C2×C16—22no—
111C2×C16—21no—
112C2×C16—21no—
113C2×C16—21no—
114C2×C16—41no—
115C2×C16—41no—
116C2×C16—41no—
117C2×C16—41no—
118C2×C16SD1612yescomplemented
119#106—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8C22⋊C412yes—
128#103—22no—
129#102—22no—
130C22⋊C8—21no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8C2×C812yes—
134#97—22no—
135C4×C8—22no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8C2×D411yescharacteristic
139C4×C8C2×D411yescharacteristic
140C4×C8C2×D411yescharacteristic
141C4×C8#4111yescharacteristic
142C24—21no—
143C2×Q8—21no—
144C2×Q8C22⋊C812yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—42no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4C4×D411yescharacteristic
156C22×C4C22×C811yescharacteristic
157C22×C4#13111yescharacteristic
158SD16—44no—
159M16—41no—
160M16—41no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8—81no—
173C2×C8#12011yescharacteristic
174C2×C8#12111yescharacteristic
175C2×C8#12111yescharacteristic
176C2×C8#12311yescharacteristic
177C2×C8C2×SD1611yescharacteristic
178C2×C8#13511yescharacteristic
179C2×C8AGL(1,ℤ/8)11yescharacteristic
180C2×C8#13711yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4C22⋊C812yes—
189C22⋊C4—22no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C42—22no—
193C42C2×C22⋊C411yescharacteristic
194C42C4×D411yescharacteristic
195C42C2×M1611yescharacteristic
196C42#13111yescharacteristic
197C16—22no—
198C16—42no—
199C16—41no—
200C16—42no—
201C16—81no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C8×D411yescharacteristic
210Q8—42no—
211Q8—81no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4#38611yescharacteristic
228C2×C4C2×C22⋊C811yescharacteristic
229C2×C4#40811yescharacteristic
230C2×C4C8×D411yescharacteristic
231C2×C4#43511yescharacteristic
232C2×C4#43511yescharacteristic
233C2×C4C4×SD1611yescharacteristic
234C2×C4#44011yescharacteristic
235C2×C4#44311yescharacteristic
236C2×C4#44411yescharacteristic
237C8—21no—
238C8—21no—
239C8—42no—
240C8—42no—
241C8—81no—
242C8—81no—
243C8#44811yescharacteristic
244C8#44911yescharacteristic
245C8#46011yescharacteristic
246C8#46411yescharacteristic
247C22—21no—
248C22—21no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22C8×SD1611yescharacteristic
257C22SD16⋊C811yescharacteristic
258C22#186911yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—81no—
266C4#158311yescharacteristic
267C4#158811yescharacteristic
268C4#187611yescharacteristic
269C4#188011yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2#894811yescharacteristic
274C2#1297911yescharacteristic
275C2#1298111yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33475C2semidirect
4. #11610C2semidirect
6. #7153C2semidirect
7. #7140C2semidirect
118. C2×C16SD16semidirect

Maximal subgroups

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

subgroupindex
#334752
#179022
#131902
#116102
#115832
#71532
#71402