Finite Group Databasebeta

← #139407

Subgroups — #139407

Order 512.

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

normal subgroups only

This group has 420 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#33487C211yesmaximal, characteristic
2#28486C211yesmaximal, characteristic
3#11965C212yesmaximal, complemented
4#8622C211yesmaximal, characteristic
5#7301C212yesmaximal, complemented
6C2×#494C2211yescharacteristic
7C2×#467—21no—
8C2×#467—21no—
9C2×C8×D4C2211yescharacteristic
10#2542—21no—
11#2168—24no—
12#2163C2212yes—
13#2112—21no—
14C2×C22⋊C16—21no—
15#2001C412yes—
16#2001C412yes—
17#1981—21no—
18C4×#107C2211yescharacteristic
19#1568C2212yes—
20#1466—21no—
21#1362—24no—
22#1290—21no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8—21no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C4×D4D411yescharacteristic
31C2×M32—22no—
32C22×C16—22no—
33#494—22no—
34#494—22no—
35#467—44no—
36#467—44no—
37C8×D4—22no—
38C8×D4—22no—
39#432—22no—
40C2×#107—21no—
41C2×C4⋊C8D411yescharacteristic
42C2×#103—22no—
43C2×#103—22no—
44C2×C22⋊C8C2×C411yescharacteristic
45C2×C22⋊C8C2×C411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#399—41no—
48#399—41no—
49#390—41no—
50#384—41no—
51C4×C4⋊C4—21no—
52C4⋊C16D414yes—
53C22⋊C16—22no—
54C22⋊C16—22no—
55C22⋊C16—22no—
56#346—22no—
57#340C2×C412yes—
58#335—22no—
59#334—22no—
60#323—41no—
61#323—41no—
62C22×Q8—22no—
63C22×D4SD1611yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C2×Q16—24no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16—82no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C22⋊Q8—44no—
85C22⋊Q8—44no—
86C22⋊Q8—82no—
87C22⋊Q8—82no—
88C4×D4—22no—
89C4×D4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4SD1611yescharacteristic
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4D811yescharacteristic
99C2×C22⋊C4D811yescharacteristic
100C2×C42—41no—
101C2×C42C2×D411yescharacteristic
102M32—82no—
103C2×C16—24no—
104C2×C16—82no—
105#107—24no—
106C4⋊C8—22no—
107C4⋊C8—42no—
108#103—44no—
109#103—44no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112C22⋊C8—42no—
113#97—42no—
114C4×C8C2×D412yes—
115C4×C8#4112yes—
116#95—41no—
117C24#10211yescharacteristic
118C24#10211yescharacteristic
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143Q16—84no—
144Q16—82no—
145Q16—82no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—81no—
155C2×C8—81no—
156C2×C8—82no—
157C2×C8C4×D412yes—
158C2×C8#12011yescharacteristic
159C2×C8#12311yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—24no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—84no—
170C22⋊C4—84no—
171C42—82no—
172C42#12112yes—
173C42#13512yes—
174C16—44no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23#36212yes—
185C23C2×#10211yescharacteristic
186Q8—82no—
187Q8—82no—
188Q8—82no—
189Q8—82no—
190D4—44no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—44no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4#38611yescharacteristic
213C2×C4#41611yescharacteristic
214C2×C4#46512yes—
215C2×C4#46612yes—
216C8—21no—
217C8—22no—
218C8—41no—
219C8—41no—
220C8—162no—
221C8#45312yes—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—81no—
233C22—81no—
234C22#189311yescharacteristic
235C22C2×#36211yescharacteristic
236C22#214811yescharacteristic
237C4—22no—
238C4—22no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—84no—
243C4—162no—
244C4#188711yescharacteristic
245C4#191411yescharacteristic
246C2—21no—
247C2—21no—
248C2—42no—
249C2—81no—
250C2#939711yescharacteristic
251C2#1306611yescharacteristic
252C2#1307511yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11965C2semidirect
5. #7301C2semidirect

Maximal subgroups

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

subgroupindex
#334872
#284862
#119652
#86222
#73012