Finite Group Databasebeta

← #150916

Subgroups — #150916

Order 512.

337 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 64 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#28519C211yesmaximal, characteristic
2#12798C211yesmaximal, characteristic
3#11476C211yesmaximal, characteristic
4#7434C214yesmaximal
5C2×#477—21no—
6C2×#476—21no—
7#2868C411yescharacteristic, complemented
8C4×#124C2211yescharacteristic
9C2×#354—21no—
10#2096C412yescomplemented
11#2093C2212yes—
12#1959—21no—
13#1847—21no—
14#1799—21no—
15#1767C411yescharacteristic, complemented
16#1461—21no—
17#1393C2212yes—
18#1324—21no—
19#1318C2212yes—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8—41no—
22C2×#124D411yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#477—42no—
26#477—42no—
27#476—42no—
28#476—42no—
29#431—41no—
30C2×#107—22no—
31C2×#106—22no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#99—22no—
36C2×#97—21no—
37#391C2×C411yescharacteristic
38#390C2311yescharacteristic
39#388—41no—
40#384—21no—
41#384—41no—
42#384C2×C412yes—
43#382D411yescharacteristic
44#380—41no—
45C4×C22⋊C4—41no—
46C43C2×C411yescharacteristic
47#354—42no—
48#342—41no—
49#336C2×C412yes—
50#324—22no—
51#324—41no—
52C22×Q8—41no—
53C22×Q8C22⋊C411yescharacteristic
54C22×D4C2×D411yescharacteristic
55C23×C4—21no—
56C22×C8—22no—
57C4⋊Q8—41no—
58C4⋊Q8—41no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8—84no—
62#124—41no—
63#124SD1612yes—
64#124Q1612yes—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×D4—41no—
68C4×D4—41no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—81no—
75C2×C4⋊C4C22⋊C411yescharacteristic
76C2×C4⋊C4C2×D411yescharacteristic
77C2×C22⋊C4—21no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—42no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22⋊C411yescharacteristic
86C2×C42C22×C411yescharacteristic
87#107—44no—
88#106—44no—
89C4⋊C8—22no—
90C4⋊C8—22no—
91C4⋊C8—81no—
92#103—44no—
93#102—44no—
94#99—42no—
95#99—42no—
96C22⋊C8—42no—
97#97—42no—
98C4×C8—81no—
99#95—41no—
100#95—41no—
101#95—81no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—81no—
107C2×Q8—81no—
108C2×Q8—84no—
109C2×Q8#10312yes—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—41no—
113C2×D4C2×SD1611yescharacteristic
114C2×D4C2×Q1611yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—42no—
123C22×C4—82no—
124C22×C4—82no—
125C22×C4#9911yescharacteristic
126C22×C4#9911yescharacteristic
127C22×C4C2×C22⋊C411yescharacteristic
128C2×C8—24no—
129C2×C8—42no—
130C2×C8—42no—
131C2×C8—82no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—81no—
137C4⋊C4—84no—
138C4⋊C4—84no—
139C4⋊C4—81no—
140C4⋊C4—81no—
141C4⋊C4—162no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—81no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C22⋊C4—82no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—41no—
152C42—42no—
153C42—42no—
154C42—41no—
155C42—41no—
156C42#10312yes—
157C23—21no—
158C23—21no—
159C23—41no—
160C23—41no—
161C23—81no—
162C23—81no—
163C23C2×#9911yescharacteristic
164Q8—42no—
165Q8—81no—
166Q8—162no—
167Q8—162no—
168D4—41no—
169D4—41no—
170D4—81no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—82no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—84no—
191C2×C4—82no—
192C2×C4—161no—
193C2×C4—162no—
194C2×C4—162no—
195C2×C4—162no—
196C2×C4#35311yescharacteristic
197C2×C4#35311yescharacteristic
198C2×C4C2×#10311yescharacteristic
199C2×C4#42111yescharacteristic
200C8—44no—
201C8—162no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—161no—
211C22#152011yescharacteristic
212C22#212311yescharacteristic
213C22C2×#35311yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—42no—
218C4—42no—
219C4—41no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—162no—
224C4—322no—
225C4#140712yes—
226C2—21no—
227C2—21no—
228C2—81no—
229C2—81no—
230C2#1190111yescharacteristic
231C2C2×#140711yescharacteristic
232C2#1358511yescharacteristic
subgroupG/H
center, socleC22C2×#353
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
7. #2868C4semidirect
10. #2096C4semidirect
15. #1767C4semidirect

Maximal subgroups

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

subgroupindex
#285192
#127982
#114762
#74342