Finite Group Databasebeta

← #136566

Subgroups — #136566

Order 512.

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

normal subgroups only

This group has 249 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#18092C211yesmaximal, characteristic
2#12033C211yesmaximal, characteristic, complemented
3#11943C211yesmaximal, characteristic
4#11674C211yesmaximal, characteristic
5#9636C211yesmaximal, characteristic, complemented
6#7166C211yesmaximal, characteristic, complemented
7#7159C211yesmaximal, characteristic, complemented
8#3168C2211yescharacteristic
9#2994—21no—
10#2162C2211yescharacteristic
11C4×M32C2211yescharacteristic
12#1592—21no—
13#1584—21no—
14#1581—21no—
15SD16⋊C8—21no—
16#1574—21no—
17#1567C2211yescharacteristic
18#1564C2211yescharacteristic
19C2×#335C2211yescharacteristic
20#1483—21no—
21#1451C2211yescharacteristic
22#1372—21no—
23#1372—21no—
24#1363C411yescharacteristic
25#1363C411yescharacteristic
26#1360C411yescharacteristic
27#1360C411yescharacteristic
28#1332—21no—
29#1331—21no—
30#1299—21no—
31#529—21no—
32C2×M32—21no—
33#499C2×C411yescharacteristic
34#495C2×C411yescharacteristic
35#488—21no—
36#467Q811yescharacteristic
37#466Q811yescharacteristic
38#465—21no—
39#446—21no—
40C8×Q8—21no—
41C4×Q16D411yescharacteristic
42C4×SD16—21no—
43C4×D8D411yescharacteristic
44#436—21no—
45#435—41no—
46C8×D4—21no—
47C8×D4—41no—
48#433—41no—
49#431—21no—
50#431D411yescharacteristic
51#424—41no—
52#423—21no—
53C2×C4⋊C8—21no—
54C4×M16D411yescharacteristic
55C2×C4×C8C2311yescharacteristic
56C4⋊C16C2×C411yescharacteristic
57C4⋊C16C2×C411yescharacteristic
58C22⋊C16—21no—
59#346—21no—
60C4×C16—21no—
61C4×C16—21no—
62#335—21no—
63#335—21no—
64#335—21no—
65#335—21no—
66#335—41no—
67#335D411yescharacteristic
68#335D411yescharacteristic
69#334—41no—
70#334C2×C411yescharacteristic
71#334C2×C411yescharacteristic
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—21no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C2×Q811yescharacteristic
87C22×C8#4111yescharacteristic
88#126—41no—
89#125—21no—
90#124—21no—
91#123—41no—
92C22⋊Q8—21no—
93#121—21no—
94C4×Q8—21no—
95C4×D4—21no—
96C4×D4—41no—
97#117—41no—
98C2×C42C2×D411yescharacteristic
99M32—22no—
100M32—41no—
101C2×C16—21no—
102C2×C16—21no—
103C2×C16—41no—
104#107C4⋊C411yescharacteristic
105#107C4⋊C411yescharacteristic
106#106—21no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8D811yescharacteristic
117C4⋊C8D811yescharacteristic
118#103—21no—
119#103—21no—
120#102—21no—
121#102—21no—
122C22⋊C8—21no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#97—21no—
127#97—41no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—41no—
131C4×C8—41no—
132C4×C8C22×C411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8#4111yescharacteristic
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—41no—
139C22×C4—41no—
140C22×C4C22⋊Q811yescharacteristic
141C22×C4#13511yescharacteristic
142C22×C4AGL(1,ℤ/8)11yescharacteristic
143Q16—41no—
144SD16—41no—
145SD16—41no—
146D8—41no—
147M16—81no—
148M16—161no—
149M16—161no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—81no—
161C2×C8—161no—
162C2×C8—161no—
163C2×C8C2×C4⋊C411yescharacteristic
164C2×C8C4×D411yescharacteristic
165C2×C8C22⋊Q811yescharacteristic
166C2×C8#12311yescharacteristic
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C42—41no—
179C42C4×D411yescharacteristic
180C42#12011yescharacteristic
181C42C2×D811yescharacteristic
182C42#13711yescharacteristic
183C16—42no—
184C16—41no—
185C16—41no—
186C23—41no—
187C23#47811yescharacteristic
188Q8—41no—
189Q8—41no—
190D4—41no—
191D4—41no—
192D4—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4#38511yescharacteristic
204C2×C4C4×D811yescharacteristic
205C2×C4#44011yescharacteristic
206C2×C4#44711yescharacteristic
207C2×C4#45211yescharacteristic
208C2×C4#47411yescharacteristic
209C8—21no—
210C8—21no—
211C8—21no—
212C8—41no—
213C8—41no—
214C8—81no—
215C8—81no—
216C8—81no—
217C8—81no—
218C8—81no—
219C8—161no—
220C8#44311yescharacteristic
221C8#44411yescharacteristic
222C22—21no—
223C22—21no—
224C22—21no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22#186211yescharacteristic
229C4—21no—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4#185811yescharacteristic
237C4#186611yescharacteristic
238C4#219311yescharacteristic
239C4#219411yescharacteristic
240C2—41no—
241C2—161no—
242C2#893011yescharacteristic
243C2#1294411yescharacteristic
244C2#1294611yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1862
prime radical#466Q8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12033C2semidirect
5. #9636C2semidirect
6. #7166C2semidirect
7. #7159C2semidirect

Maximal subgroups

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

subgroupindex
#180922
#120332
#119432
#116742
#96362
#71662
#71592