Finite Group Databasebeta

← #137891

Subgroups — #137891

Order 512.

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

normal subgroups only

This group has 358 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
1#33487C211yesmaximal, characteristic
2#22212C211yesmaximal, characteristic
3#11644C212yesmaximal, complemented
4#11488C211yesmaximal, characteristic
5#7143C212yesmaximal, complemented
6C2×#446C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#124C411yescharacteristic
9#2168—24no—
10#2163C2212yes—
11#2163C2212yes—
12#2112—21no—
13C2×C22⋊C16—21no—
14#1984C411yescharacteristic
15#1954C411yescharacteristic
16#1924—21no—
17#1924—21no—
18#1850—21no—
19#1850—21no—
20#1842C411yescharacteristic
21#1744C2211yescharacteristic
22#1338—24no—
23#1314—21no—
24#1314—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#124C811yescharacteristic
28C2×C4×Q8C2×C411yescharacteristic
29C2×C4×D4C2×C411yescharacteristic
30C2×M32—22no—
31C22×C16—22no—
32#446—22no—
33#446—22no—
34C8×D4—22no—
35C8×D4—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8D411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C2×#97—21no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8C2311yescharacteristic
48#391C811yescharacteristic
49#390C811yescharacteristic
50#388—21no—
51#388—21no—
52#382C811yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C43C2×C411yescharacteristic
56C4⋊C16—22no—
57C4⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61C22⋊C16—22no—
62#346—22no—
63#346D414yes—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68C22×Q8C2×C811yescharacteristic
69C22×D4C2×C811yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—81no—
79C22×C8—81no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84#124—42no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×D4—22no—
88C4×D4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4C2×C811yescharacteristic
91C2×C4⋊C4C2×C811yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4M1611yescharacteristic
97C2×C22⋊C4M1611yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2×C811yescharacteristic
102C2×C42C2×C811yescharacteristic
103C2×C42C22×C411yescharacteristic
104M32—82no—
105C2×C16—24no—
106C2×C16—42no—
107C2×C16—82no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—81no—
113C4⋊C8—81no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116C22⋊C8—81no—
117C22⋊C8—81no—
118#97—42no—
119C4×C8—22no—
120C4×C8C2×D412yes—
121C4×C8#4112yes—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24C22⋊C811yescharacteristic
127C24C22⋊C811yescharacteristic
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C22⋊C811yescharacteristic
147C22×C4C22⋊C811yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C22×C811yescharacteristic
150C22×C4C2×M1611yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8#12011yescharacteristic
168C2×C8#12112yes—
169C2×C8#12311yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—41no—
180C42—41no—
181C42—41no—
182C42—41no—
183C42—42no—
184C42—41no—
185C42—41no—
186C42C4×D412yes—
187C42#13112yes—
188C16—44no—
189C16—82no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23#34912yes—
200C23C2×C22⋊C811yescharacteristic
201Q8—82no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4#38611yescharacteristic
231C2×C4#40811yescharacteristic
232C2×C4C8×D412yes—
233C2×C4#43512yes—
234C8—22no—
235C8—21no—
236C8—21no—
237C8—41no—
238C8—41no—
239C8—81no—
240C8—81no—
241C22—21no—
242C22—21no—
243C22—22no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—81no—
252C22—81no—
253C22#186911yescharacteristic
254C22C2×#34911yescharacteristic
255C22#211511yescharacteristic
256C4—21no—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—41no—
261C4—41no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4#190911yescharacteristic
266C4#191111yescharacteristic
267C2—21no—
268C2—21no—
269C2—42no—
270C2—81no—
271C2#909911yescharacteristic
272C2#1298211yescharacteristic
273C2#1298511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11644C2semidirect
5. #7143C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#222122
#116442
#114882
#71432