Finite Group Databasebeta

← #136563

Subgroups — #136563

Order 512.

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

normal subgroups only

This group has 255 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#18074C211yesmaximal, characteristic
2#12036C211yesmaximal, characteristic, complemented
3#11944C211yesmaximal, characteristic
4#11674C211yesmaximal, characteristic
5#9626C211yesmaximal, characteristic, complemented
6#7166C211yesmaximal, characteristic, complemented
7#7158C211yesmaximal, characteristic, complemented
8#3168C2211yescharacteristic
9#2969—21no—
10#2162C2211yescharacteristic
11C4×M32C2211yescharacteristic
12#1950—21no—
13#1592—21no—
14#1587—21no—
15#1586—21no—
16SD16⋊C8—21no—
17#1567C2211yescharacteristic
18#1564C2211yescharacteristic
19C2×#334C2211yescharacteristic
20#1482—21no—
21#1451C2211yescharacteristic
22#1372—21no—
23#1371—21no—
24#1363C411yescharacteristic
25#1363C411yescharacteristic
26#1360C411yescharacteristic
27#1360C411yescharacteristic
28#1332—21no—
29#1331—21no—
30#1301—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#436—41no—
46#435—41no—
47C8×D4—21no—
48#433—41no—
49#431—21no—
50#431D411yescharacteristic
51C2×#108—41no—
52#423—21no—
53C2×C4⋊C8—21no—
54#405—41no—
55C4×M16D411yescharacteristic
56C2×C4×C8C2311yescharacteristic
57C4⋊C16C2×C411yescharacteristic
58C4⋊C16C2×C411yescharacteristic
59C22⋊C16—21no—
60#346—21no—
61C4×C16—21no—
62C4×C16—21no—
63#341—41no—
64#335—21no—
65#335—21no—
66#335D411yescharacteristic
67#335D411yescharacteristic
68#334—21no—
69#334—21no—
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—41no—
82C2×M16—41no—
83C2×M16D811yescharacteristic
84C2×M16D811yescharacteristic
85C22×C8—41no—
86C22×C8—41no—
87C22×C8C2×Q811yescharacteristic
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×C42C2×D411yescharacteristic
100M32—22no—
101M32—41no—
102C2×C16—21no—
103C2×C16—21no—
104C2×C16—41no—
105#108—82no—
106#107C4⋊C411yescharacteristic
107#107C4⋊C411yescharacteristic
108#106—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—21no—
120#103—21no—
121#102—21no—
122#102—21no—
123C22⋊C8—21no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#97—21no—
128#97—41no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—41no—
132C4×C8—41no—
133C4×C8C22×C411yescharacteristic
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8#4111yescharacteristic
137C2×Q8—21no—
138C2×D4—21no—
139C2×D4—41no—
140C22×C4—41no—
141C22×C4C22⋊Q811yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4#13711yescharacteristic
144Q16—41no—
145SD16—41no—
146SD16—41no—
147D8—41no—
148M16—41no—
149M16—41no—
150M16—82no—
151M16—81no—
152M16—81no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—82no—
164C2×C8—81no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8C2×C4⋊C411yescharacteristic
168C2×C8C4×D411yescharacteristic
169C2×C8C22⋊Q811yescharacteristic
170C2×C8#12311yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—41no—
183C42C4×D411yescharacteristic
184C42#12011yescharacteristic
185C42#13511yescharacteristic
186C42AGL(1,ℤ/8)11yescharacteristic
187C16—42no—
188C16—41no—
189C16—41no—
190C23—41no—
191C23#47411yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—161no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4#38511yescharacteristic
208C2×C4C4×D811yescharacteristic
209C2×C4#44011yescharacteristic
210C2×C4#44711yescharacteristic
211C2×C4#45211yescharacteristic
212C2×C4#47811yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—41no—
217C8—41no—
218C8—81no—
219C8—81no—
220C8—81no—
221C8—81no—
222C8—81no—
223C8—161no—
224C8#44311yescharacteristic
225C8#44411yescharacteristic
226C22—21no—
227C22—21no—
228C22—21no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22#186211yescharacteristic
233C4—21no—
234C4—21no—
235C4—21no—
236C4—21no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4#185811yescharacteristic
241C4#186611yescharacteristic
242C4#219011yescharacteristic
243C4#219111yescharacteristic
244C2—41no—
245C2—161no—
246C2#893011yescharacteristic
247C2#1294311yescharacteristic
248C2#1294711yescharacteristic
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. #12036C2semidirect
5. #9626C2semidirect
6. #7166C2semidirect
7. #7158C2semidirect

Maximal subgroups

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

subgroupindex
#180742
#120362
#119442
#116742
#96262
#71662
#71582