Finite Group Databasebeta

← #134759

Subgroups — #134759

Order 512.

370 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C8×#124C211yesmaximal, characteristic
2#12801C212yesmaximal
3#8617C212yesmaximal, complemented
4#7238C212yesmaximal, complemented
5C2×C8×Q8C2211yescharacteristic
6C2×C8×D4C411yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#355—22no—
9#1984C2211yescharacteristic
10#1954C411yescharacteristic
11#1923—22no—
12#1831C411yescharacteristic
13#1801C2214yes—
14C8×C22⋊C4—22no—
15C42×C8C411yescharacteristic
16#1567—22no—
17#1394—22no—
18#1292—22no—
19C23×C8—21no—
20C2×#124D411yescharacteristic
21C2×C4×Q8C2311yescharacteristic
22C2×C4×D4C2×C411yescharacteristic
23C8×Q8—41no—
24C8×Q8—41no—
25C8×D4—41no—
26C8×D4—41no—
27#423—24no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8D412yes—
31C2×#101—24no—
32C2×C22⋊C8—21no—
33C2×C22⋊C8—22no—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D412yes—
39#391D411yescharacteristic
40#390Q811yescharacteristic
41#388—22no—
42#382Q811yescharacteristic
43C4×C22⋊C4—22no—
44C43C2×C411yescharacteristic
45#355—44no—
46#343—24no—
47#336—22no—
48#336—22no—
49#335—42no—
50#335—42no—
51C22×Q8C2×D411yescharacteristic
52C22×D4C4⋊C411yescharacteristic
53C23×C4—21no—
54C2×M16—44no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—44no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8C22⋊C411yescharacteristic
61C22×C8C22⋊C412yes—
62C22×C8C22⋊C411yescharacteristic
63#124—22no—
64#124D812yes—
65#124Q1612yes—
66C4×Q8—41no—
67C4×Q8—41no—
68C4×D4—41no—
69C4×D4—41no—
70C2×C4⋊C4C4⋊C411yescharacteristic
71C2×C4⋊C4C2×Q811yescharacteristic
72C2×C4⋊C4#4112yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—22no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C4⋊C411yescharacteristic
79C2×C42C4⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D412yes—
82C4⋊C8—41no—
83C4⋊C8—42no—
84C4⋊C8—41no—
85C4⋊C8—44no—
86C4⋊C8—44no—
87#101—44no—
88#101—44no—
89C22⋊C8—44no—
90C22⋊C8—42no—
91C4×C8—22no—
92C4×C8—22no—
93C4×C8—21no—
94C4×C8—21no—
95C4×C8—41no—
96C4×C8—42no—
97C4×C8—42no—
98#95—22no—
99#95—22no—
100C24—21no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8C2×D811yescharacteristic
104C2×Q8C2×Q1611yescharacteristic
105C2×Q8#13512yes—
106C2×D4—21no—
107C2×D4—41no—
108C2×D4—41no—
109C2×D4#10712yes—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C2×C4⋊C411yescharacteristic
119C22×C4#11711yescharacteristic
120C22×C4#12112yes—
121C22×C4C22⋊Q812yes—
122M16—84no—
123M16—84no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8—41no—
134C2×C8—82no—
135C2×C8—82no—
136C2×C8—84no—
137C2×C8#9911yescharacteristic
138C2×C8#9911yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—41no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—44no—
145C22⋊C4—42no—
146C42—22no—
147C42—21no—
148C42—21no—
149C42—22no—
150C42—21no—
151C42—41no—
152C42—41no—
153C42—41no—
154C42—42no—
155C42#10712yes—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—41no—
160C23—41no—
161C23#38011yescharacteristic
162Q8—41no—
163Q8—41no—
164Q8—42no—
165D4—42no—
166D4—42no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4C2×#9911yescharacteristic
189C2×C4#41011yescharacteristic
190C2×C4C2×#10711yescharacteristic
191C2×C4#42711yescharacteristic
192C2×C4#46612yes—
193C2×C4#46712yes—
194C8—21no—
195C8—42no—
196C8—41no—
197C8—42no—
198C8—81no—
199C8—164no—
200C8#35512yescomplemented
201C22—21no—
202C22—21no—
203C22—44no—
204C22—42no—
205C22—82no—
206C22—81no—
207C22—81no—
208C22#179911yescharacteristic
209C22#180811yescharacteristic
210C22#181211yescharacteristic
211C4—21no—
212C4—22no—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C4—42no—
220C4—81no—
221C4C2×#35511yescharacteristic
222C4#213211yescharacteristic
223C2—21no—
224C2—21no—
225C2—82no—
226C2#860911yescharacteristic
227C2#1280111yescharacteristic
228C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8617C2semidirect
4. #7238C2semidirect
200. C8#355semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#128012
#86172
#72382