Finite Group Databasebeta

← #136308

Subgroups — #136308

Order 512.

271 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 76 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#17982C211yesmaximal, characteristic
2#12035C211yesmaximal, characteristic, complemented
3#11741C211yesmaximal, characteristic
4#11648C211yesmaximal, characteristic
5#9600C211yesmaximal, characteristic, complemented
6#7299C211yesmaximal, characteristic, complemented
7#7195C211yesmaximal, characteristic, complemented
8#3168C2211yescharacteristic
9#2966—21no—
10#2162C2211yescharacteristic
11#2162C2211yescharacteristic
12#1939—21no—
13C4×#106C2211yescharacteristic
14#1588—21no—
15#1587—21no—
16#1586—21no—
17#1585—21no—
18#1568C2211yescharacteristic
19C2×#335C2211yescharacteristic
20#1494—21no—
21C2×#322C2211yescharacteristic
22#1372—21no—
23#1371—21no—
24#1348C411yescharacteristic
25#1348C411yescharacteristic
26#1347C411yescharacteristic
27#1347C411yescharacteristic
28#1332—21no—
29#1331—21no—
30#1279—21no—
31#529—21no—
32#499Q811yescharacteristic
33#495Q811yescharacteristic
34#488—21no—
35#467C2×C411yescharacteristic
36#466C2×C411yescharacteristic
37#465—21no—
38#446—21no—
39C8×Q8—21no—
40C4×Q16D411yescharacteristic
41C4×SD16—21no—
42C4×D8D411yescharacteristic
43#436—21no—
44#435—41no—
45C8×D4—21no—
46C8×D4—41no—
47#433—41no—
48#432—21no—
49C2×#106—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—41no—
52#405—41no—
53C2×C4×C8D411yescharacteristic
54C2×C4×C8D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C4×C4⋊C4—21no—
57C4⋊C16—21no—
58C4⋊C16—21no—
59C22⋊C16—21no—
60C22⋊C16C2×C411yescharacteristic
61C22⋊C16C2×C411yescharacteristic
62C4×C16—21no—
63C4×C16—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#336—41no—
67#335—21no—
68#335—21no—
69#335D411yescharacteristic
70#335D411yescharacteristic
71#334—21no—
72Q8⋊C8—21no—
73Q8⋊C8—21no—
74D4⋊C8—21no—
75D4⋊C8—21no—
76#322—21no—
77#322—21no—
78C2×Q16C4⋊C411yescharacteristic
79C2×SD16—21no—
80C2×D8C4⋊C411yescharacteristic
81C2×M16—41no—
82C22×C8—21no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8D811yescharacteristic
86C22×C8D811yescharacteristic
87C22×C8C22×C411yescharacteristic
88C22×C8#4111yescharacteristic
89#126—41no—
90#125—21no—
91#124—21no—
92#123—41no—
93C22⋊Q8—21no—
94#121—21no—
95C4×Q8—21no—
96C4×D4—21no—
97C4×D4—41no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C42—41no—
102C2×C42C2×D411yescharacteristic
103C2×C16—21no—
104C2×C16—21no—
105C2×C16—41no—
106#107C4⋊C411yescharacteristic
107#107C4⋊C411yescharacteristic
108#106—21no—
109#106—42no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—82no—
117#103—21no—
118#103—21no—
119#102—21no—
120#102—21no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124#97—41no—
125#97—41no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—41no—
131C4×C8C2×D411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C4×C8C2×Q811yescharacteristic
134C4×C8#4111yescharacteristic
135#95—41no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4C4×D411yescharacteristic
143C22×C4C2×D811yescharacteristic
144C22×C4#13511yescharacteristic
145Q16—41no—
146SD16—81no—
147D8—41no—
148M16—82no—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—82no—
161C2×C8—81no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8C2×C4⋊C411yescharacteristic
165C2×C8C4×D411yescharacteristic
166C2×C8C22⋊Q811yescharacteristic
167C2×C8#12311yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C42—41no—
184C42—81no—
185C42—81no—
186C42#12011yescharacteristic
187C42C22⋊Q811yescharacteristic
188C42AGL(1,ℤ/8)11yescharacteristic
189C42#13711yescharacteristic
190C16—41no—
191C16—41no—
192C16—81no—
193C23—41no—
194C23C4×D811yescharacteristic
195Q8—41no—
196Q8—41no—
197D4—41no—
198D4—41no—
199D4—81no—
200D4—81no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4#38511yescharacteristic
219C2×C4#44011yescharacteristic
220C2×C4#44711yescharacteristic
221C2×C4#45211yescharacteristic
222C2×C4#47411yescharacteristic
223C2×C4#47811yescharacteristic
224C8—21no—
225C8—21no—
226C8—21no—
227C8—21no—
228C8—42no—
229C8—41no—
230C8—41no—
231C8—81no—
232C8—81no—
233C8—161no—
234C22—21no—
235C22—21no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22#186211yescharacteristic
240C22#217811yescharacteristic
241C22#217911yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—82no—
247C4—81no—
248C4—81no—
249C4—81no—
250C4—161no—
251C4#186111yescharacteristic
252C4#186811yescharacteristic
253C4#219011yescharacteristic
254C4#219111yescharacteristic
255C2—21no—
256C2—21no—
257C2—161no—
258C2#887811yescharacteristic
259C2#1293911yescharacteristic
260C2#1294311yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1862
prime radical#466C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12035C2semidirect
5. #9600C2semidirect
6. #7299C2semidirect
7. #7195C2semidirect

Maximal subgroups

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

subgroupindex
#179822
#120352
#117412
#116482
#96002
#72992
#71952