Finite Group Databasebeta

← #150922

Subgroups — #150922

Order 512.

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

normal subgroups only

This group has 331 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#12801C211yesmaximal, characteristic
3#11477C211yesmaximal, characteristic
4#7437C214yesmaximal, complemented
5C2×#477—21no—
6C2×#476—21no—
7#2868C411yescharacteristic
8C4×#124C2211yescharacteristic
9C2×#355—21no—
10#2096C2212yes—
11#2093C412yes—
12#1959—21no—
13#1847—21no—
14#1801—21no—
15#1767C411yescharacteristic
16#1459—21no—
17#1394C2212yes—
18#1324—21no—
19#1319C2212yes—
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—
29C2×#107—22no—
30C2×#106—22no—
31#423—41no—
32C2×C4⋊C8—21no—
33C2×#103—22no—
34C2×#102—22no—
35C2×#101—22no—
36C2×C22⋊C8—41no—
37C2×#97—21no—
38#391C2311yescharacteristic
39#390C2×C411yescharacteristic
40#388—41no—
41#384—21no—
42#384—41no—
43#384C2×C412yes—
44#382D411yescharacteristic
45C4×C22⋊C4—41no—
46C43C2×C411yescharacteristic
47#355—42no—
48#343—41no—
49#336C2×C412yes—
50#324—22no—
51#323—41no—
52C22×Q8—41no—
53C22×Q8C2×D411yescharacteristic
54C22×D4C22⋊C411yescharacteristic
55C23×C4—21no—
56C2×M16—42no—
57C22×C8—22no—
58C22×C8—81no—
59C4⋊Q8—41no—
60C4⋊Q8—41no—
61C4⋊Q8—41no—
62C4⋊Q8—41no—
63C4⋊Q8—84no—
64#124—41no—
65#124SD1612yes—
66#124Q1612yes—
67C4×Q8—41no—
68C4×Q8—41no—
69C4×D4—41no—
70C4×D4—41no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
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—
92C4⋊C8—81no—
93#103—44no—
94#102—44no—
95#101—42no—
96#101—42no—
97C22⋊C8—42no—
98C22⋊C8—81no—
99#97—42no—
100#95—41no—
101#95—41no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—81no—
107C2×Q8—81no—
108C2×Q8—84no—
109C2×Q8C2×SD1611yescharacteristic
110C2×Q8C2×Q1611yescharacteristic
111C2×D4—21no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4#10312yes—
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#9911yescharacteristic
125C22×C4#9911yescharacteristic
126C22×C4C2×C22⋊C411yescharacteristic
127M16—82no—
128M16—82no—
129C2×C8—24no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—82no—
133C2×C8—82no—
134C2×C8—162no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—81no—
140C4⋊C4—84no—
141C4⋊C4—84no—
142C4⋊C4—81no—
143C4⋊C4—81no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—81no—
147C22⋊C4—82no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—41no—
152C42—42no—
153C42—42no—
154C42—41no—
155C42—41no—
156C42#10312yes—
157C23—22no—
158C23—41no—
159C23—41no—
160C23—81no—
161C23—81no—
162C23C2×#9911yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—81no—
166Q8—162no—
167Q8—162no—
168D4—42no—
169D4—81no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
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#35311yescharacteristic
194C2×C4#35311yescharacteristic
195C2×C4C2×#10311yescharacteristic
196C2×C4#42111yescharacteristic
197C8—44no—
198C8—162no—
199C8—162no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—161no—
207C22#152011yescharacteristic
208C22#212311yescharacteristic
209C22C2×#35311yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—42no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—81no—
220C4—162no—
221C4#140712yes—
222C2—21no—
223C2—21no—
224C2—82no—
225C2#1190111yescharacteristic
226C2C2×#140711yescharacteristic
227C2#1359111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect

Maximal subgroups

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

subgroupindex
#285192
#128012
#114772
#74372