Finite Group Databasebeta

← #150849

Subgroups — #150849

Order 512.

301 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 62 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#28727C211yesmaximal, characteristic
2#12798C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4#7412C214yesmaximal
5C2×#489—21no—
6#2862C411yescharacteristic, complemented
7C4×#124C2211yescharacteristic
8C2×#354—21no—
9#2074C412yescomplemented
10#2073C2212yes—
11#1993—21no—
12#1805—21no—
13#1804—21no—
14#1799—21no—
15#1767C411yescharacteristic, complemented
16#1461—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×#124D411yescharacteristic
22C2×C4×Q8D411yescharacteristic
23C2×C4×D4D411yescharacteristic
24#489—42no—
25#489—42no—
26#431—41no—
27C2×C4⋊C8—21no—
28C2×#103—22no—
29C2×#102—22no—
30C2×#99—22no—
31C2×#97—21no—
32#391—41no—
33#391—41no—
34#391C2×C411yescharacteristic
35#390C2311yescharacteristic
36#388—41no—
37#382—21no—
38#382C2×C412yes—
39#382D411yescharacteristic
40#380—41no—
41C4×C22⋊C4—41no—
42C43C2×C411yescharacteristic
43#354—42no—
44#342—41no—
45#340—22no—
46#340—22no—
47#336C2×C412yes—
48#324—22no—
49#324—41no—
50C22×Q8—41no—
51C22×Q8C22⋊C411yescharacteristic
52C22×D4C2×D411yescharacteristic
53C23×C4—21no—
54C22×C8—22no—
55C4⋊Q8—42no—
56C4⋊Q8—42no—
57#124—41no—
58#124D812yes—
59#124SD1612yes—
60C4×Q8—41no—
61C4×Q8—41no—
62C4×D4—41no—
63C4×D4—41no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—81no—
68C2×C4⋊C4C22⋊C411yescharacteristic
69C2×C4⋊C4C2×D411yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—41no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—42no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22×C411yescharacteristic
80C4⋊C8—41no—
81C4⋊C8—41no—
82C4⋊C8—81no—
83#103—44no—
84#102—42no—
85#102—42no—
86#99—42no—
87#99—42no—
88C22⋊C8—42no—
89#97—42no—
90C4×C8—81no—
91#95—41no—
92#95—41no—
93#95—42no—
94#95—41no—
95#95—41no—
96#95—81no—
97C24—21no—
98C2×Q8—21no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—82no—
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8#10212yes—
105C2×D4—21no—
106C2×D4—41no—
107C2×D4—41no—
108C2×D4C2×D811yescharacteristic
109C2×D4C2×SD1611yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—42no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4#9911yescharacteristic
121C22×C4#9911yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123C2×C8—42no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—82no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—81no—
132C4⋊C4—82no—
133C4⋊C4—162no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—81no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—81no—
147C42—81no—
148C42#10212yes—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23C2×#9911yescharacteristic
156Q8—42no—
157Q8—81no—
158Q8—164no—
159D4—41no—
160D4—41no—
161D4—81no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—82no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—161no—
185C2×C4—162no—
186C2×C4—162no—
187C2×C4—162no—
188C2×C4#35411yescharacteristic
189C2×C4#35411yescharacteristic
190C2×C4C2×#10211yescharacteristic
191C2×C4#42111yescharacteristic
192C8—82no—
193C8—162no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#151911yescharacteristic
204C22#212311yescharacteristic
205C22C2×#35411yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—42no—
213C4—81no—
214C4—81no—
215C4—81no—
216C4—162no—
217C4—322no—
218C2—21no—
219C2—21no—
220C2—81no—
221C2—81no—
222C2#1187811yescharacteristic
223C2#1359411yescharacteristic
224C2#1360311yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #2862C4semidirect
9. #2074C4semidirect
15. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#287272
#127982
#114762
#74122