Finite Group Databasebeta

← #134756

Subgroups — #134756

Order 512.

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

normal subgroups only

This group has 402 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#12798C212yesmaximal
3#8609C212yesmaximal
4#7237C212yesmaximal
5C2×C8×Q8C411yescharacteristic, complemented
6C2×C8×D4C2211yescharacteristic
7C4×#124C2211yescharacteristic
8C2×#354—22no—
9#1984C411yescharacteristic, complemented
10#1954C2211yescharacteristic
11#1923—22no—
12#1831C411yescharacteristic, complemented
13#1808—22no—
14#1799C2214yes—
15C8×C22⋊C4—22no—
16C42×C8C411yescharacteristic, complemented
17#1393—22no—
18#1289—22no—
19C23×C8D412yes—
20C2×#124D411yescharacteristic
21C2×C4×Q8C2×C411yescharacteristic
22C2×C4×D4C2311yescharacteristic
23C8×Q8—41no—
24C8×Q8—41no—
25C8×D4—41no—
26C8×D4—41no—
27C2×#107—42no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8C2×C411yescharacteristic
30C2×C4⋊C8C2×C411yescharacteristic
31C2×#99—24no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8D412yes—
34C2×C4×C8—21no—
35C2×C4×C8—21no—
36C2×C4×C8—21no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39#391Q811yescharacteristic
40#390D411yescharacteristic
41#388—22no—
42#382Q811yescharacteristic
43#380—24no—
44C4×C22⋊C4—22no—
45C43C2×C411yescharacteristic
46#354—44no—
47#342—24no—
48#340—42no—
49#336—22no—
50#336—22no—
51C22×Q8C4⋊C411yescharacteristic
52C22×D4C2×D411yescharacteristic
53C23×C4C2×D412yes—
54C22×C8—21no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—21no—
58C22×C8—44no—
59C22×C8C22⋊C411yescharacteristic
60C22×C8C22⋊C412yes—
61C22×C8C22⋊C411yescharacteristic
62#124—22no—
63#124D812yes—
64#124Q1612yes—
65C4×Q8—41no—
66C4×Q8—41no—
67C4×D4—41no—
68C4×D4—41no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4C4⋊C411yescharacteristic
72C2×C4⋊C4C2×Q811yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4#4112yes—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42C4⋊C411yescharacteristic
81C2×C42C4⋊C411yescharacteristic
82C2×C42C22×C411yescharacteristic
83#107—84no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—42no—
87#99—44no—
88#99—44no—
89C22⋊C8—42no—
90C22⋊C8—44no—
91C4×C8—21no—
92C4×C8—21no—
93C4×C8—41no—
94C4×C8—42no—
95C4×C8—42no—
96C4×C8—41no—
97C4×C8—41no—
98#95—22no—
99#95—22no—
100#95—44no—
101C24#12112yes—
102C2×Q8—21no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8#10712yescomplemented
106C2×D4—41no—
107C2×D4—41no—
108C2×D4C2×D811yescharacteristic
109C2×D4C2×Q1611yescharacteristic
110C2×D4#13512yes—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—84no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4C22⋊Q812yes—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—44no—
131C2×C8—41no—
132C2×C8—42no—
133C2×C8—42no—
134C2×C8—41no—
135C2×C8—82no—
136C2×C8#9911yescharacteristic
137C2×C8#9911yescharacteristic
138C4⋊C4—41no—
139C4⋊C4—41no—
140C4⋊C4—42no—
141C4⋊C4—84no—
142C4⋊C4—84no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C22⋊C4—84no—
148C22⋊C4—84no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—41no—
153C42—41no—
154C42—41no—
155C42—42no—
156C42—41no—
157C42—41no—
158C42#10712yescomplemented
159C23—22no—
160C23—22no—
161C23—42no—
162C23—42no—
163C23#38011yescharacteristic
164C23#46612yes—
165C23#46712yes—
166Q8—42no—
167Q8—42no—
168D4—41no—
169D4—41no—
170D4—42no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—164no—
194C2×C4—164no—
195C2×C4—164no—
196C2×C4C2×#9911yescharacteristic
197C2×C4#41011yescharacteristic
198C2×C4C2×#10711yescharacteristic
199C2×C4#42711yescharacteristic
200C8—21no—
201C8—42no—
202C8—42no—
203C8—41no—
204C8—81no—
205C8#35412yescomplemented
206C22—21no—
207C22—21no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—82no—
215C22#179911yescharacteristic
216C22#180811yescharacteristic
217C22#181211yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—41no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—324no—
228C4C2×#35411yescharacteristic
229C4#212611yescharacteristic
230C2—21no—
231C2—21no—
232C2—42no—
233C2—42no—
234C2#860911yescharacteristic
235C2#1279811yescharacteristic
236C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×C8×Q8C4semidirect
9. #1984C4semidirect
12. #1831C4semidirect
16. C42×C8C4semidirect
105. C2×Q8#107semidirect
158. C42#107semidirect
205. C8#354semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#127982
#86092
#72372