Finite Group Databasebeta

← #139352

Subgroups — #139352

Order 512.

384 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 84 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#33486C211yesmaximal, characteristic
2#28554C211yesmaximal, characteristic
3#11892C212yesmaximal, complemented
4#8611C211yesmaximal, characteristic
5#7260C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2415—21no—
9D4×C16—24no—
10#2162—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#2086C411yescharacteristic
14#2075C411yescharacteristic
15#2056C411yescharacteristic
16#2001C411yescharacteristic
17#1933—21no—
18#1892—21no—
19#1873—21no—
20#1810C2212yes—
21#1735C2211yescharacteristic
22#1465—21no—
23#1350—24no—
24#1293—21no—
25C22×Q16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—21no—
30C2×C4×D4D411yescharacteristic
31C2×#117—21no—
32C22×C16—22no—
33C22×C16—22no—
34#467—22no—
35#467—22no—
36C8×D4—22no—
37C8×D4—22no—
38C2×#108—22no—
39C2×#107C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44#407—22no—
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#397—41no—
49#394—41no—
50#391—21no—
51#389—21no—
52#389—41no—
53#388—41no—
54#380—21no—
55C4⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58C22⋊C16D414yes—
59C4×C16—22no—
60#341—22no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#324—41no—
67#323—41no—
68C22×Q8—21no—
69C22×D4SD1611yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×Q16—22no—
73C2×Q16—81no—
74C2×M16—42no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22×C411yescharacteristic
82C22×C8C2×D412yes—
83C22×C8C2×D411yescharacteristic
84C22×C8#4112yes—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—81no—
89C4×D4—22no—
90C4×D4—22no—
91#117—42no—
92#117—81no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4D811yescharacteristic
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4D811yescharacteristic
105C2×C22⋊C4SD1611yescharacteristic
106C2×C42—41no—
107C2×C42C22⋊C411yescharacteristic
108C2×C16—24no—
109C2×C16—82no—
110C2×C16—82no—
111#108—82no—
112#107—22no—
113C4⋊C8—22no—
114#103—42no—
115#103—41no—
116#103—41no—
117#103—81no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C4×C8—21no—
123C4×C8—21no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128C24#10211yescharacteristic
129C24C2×SD1611yescharacteristic
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4#12112yes—
154C22×C4C2×D811yescharacteristic
155C22×C4#13512yes—
156Q16—82no—
157M16—82no—
158M16—82no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—81no—
169C2×C8—82no—
170C2×C8C4×D412yes—
171C2×C8#12011yescharacteristic
172C2×C8#12311yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—82no—
186C22⋊C4—82no—
187C22⋊C4—82no—
188C22⋊C4—82no—
189C42—21no—
190C42—21no—
191C42—82no—
192C16—44no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23#35912yes—
202C23C2×#10211yescharacteristic
203C23#46512yes—
204Q8—81no—
205Q8—81no—
206Q8—81no—
207Q8—81no—
208D4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4#35912yes—
237C2×C4#38611yescharacteristic
238C2×C4#41611yescharacteristic
239C2×C4#46612yes—
240C8—21no—
241C8—21no—
242C8—22no—
243C8—41no—
244C8—41no—
245C8—162no—
246C22—21no—
247C22—21no—
248C22—22no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#189311yescharacteristic
259C22C2×#35911yescharacteristic
260C22C2×#35911yescharacteristic
261C4—22no—
262C4—22no—
263C4—21no—
264C4—41no—
265C4—41no—
266C4—82no—
267C4—161no—
268C4—161no—
269C4—161no—
270C4#188411yescharacteristic
271C4#191311yescharacteristic
272C2—21no—
273C2—21no—
274C2—42no—
275C2—42no—
276C2#938411yescharacteristic
277C2#1306311yescharacteristic
278C2#1307811yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11892C2semidirect
5. #7260C2semidirect

Maximal subgroups

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

subgroupindex
#334862
#285542
#118922
#86112
#72602