Finite Group Databasebeta

← #120529

Subgroups — #120529

Order 512.

471 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 126 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#21976C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic
3#8002C211yesmaximal, characteristic
4#7954C212yesmaximal
5#7941C212yesmaximal
6#2920C2212yes—
7C2×#432C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2283C2212yes—
10C4×#117C2211yescharacteristic
11#1949—24no—
12#1940—22no—
13#1850—21no—
14#1850—21no—
15#1842C412yescomplemented
16#1831C412yescomplemented
17#1808—21no—
18#1807—21no—
19#1292—22no—
20#1285—24no—
21#1284—22no—
22#1284—22no—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#117—21no—
26C22×C42C2311yescharacteristic
27#432—22no—
28#432—22no—
29#432—22no—
30#432Q814yes—
31#431—22no—
32#431—22no—
33C2×#107—22no—
34C2×#106—22no—
35#423—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C412yes—
43C2×C4⋊C8D412yes—
44#407—22no—
45C2×C22⋊C8—21no—
46C4×M16—22no—
47C2×#97—21no—
48#383—22no—
49#382—22no—
50#381—22no—
51#380—22no—
52C4×C4⋊C4—21no—
53C4×C4⋊C4D414yes—
54C4×C4⋊C4Q812yes—
55C4×C22⋊C4—21no—
56C4×C22⋊C4—22no—
57C4×C22⋊C4D411yescharacteristic
58C4×C22⋊C4D411yescharacteristic
59#376—21no—
60#376—22no—
61C43C2×C412yes—
62#340—22no—
63#340—22no—
64#336—22no—
65#336—22no—
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4#4111yescharacteristic
69C2×M16—42no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—81no—
77C22×C8—81no—
78C22×C8C22⋊C412yes—
79C22×C8C22⋊C412yes—
80#117—44no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—24no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—42no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—24no—
89C2×C42C4⋊C412yes—
90C2×C42C4⋊C412yes—
91C2×C42C22×C411yescharacteristic
92C2×C42C2×D412yes—
93C2×C42C2×D411yescharacteristic
94C2×C42C2×Q812yes—
95C2×C42C2×Q811yescharacteristic
96C2×C42#4112yes—
97C2×C42#4112yes—
98C2×C42#4111yescharacteristic
99#107—82no—
100#106—82no—
101C4⋊C8—24no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—24no—
105C4⋊C8—24no—
106C4⋊C8—42no—
107C4⋊C8—82no—
108C4⋊C8—82no—
109C22⋊C8—24no—
110C22⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—41no—
113#97—24no—
114#97—42no—
115C4×C8—42no—
116#95—21no—
117#95—21no—
118#95—22no—
119#95—24no—
120#95—42no—
121#95—42no—
122#95—42no—
123C24#12111yescharacteristic
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4C2×C4⋊C411yescharacteristic
133C22×C4#11711yescharacteristic
134C22×C4#12011yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12112yes—
137C22×C4C22⋊Q812yes—
138C22×C4C22⋊Q811yescharacteristic
139C22×C4C22⋊Q811yescharacteristic
140C22×C4#12311yescharacteristic
141C22×C4#12312yes—
142C22×C4#12411yescharacteristic
143C22×C4#12512yes—
144C22×C4C4⋊Q811yescharacteristic
145M16—84no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—44no—
157C2×C8—81no—
158C2×C8—81no—
159C4⋊C4—44no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—84no—
165C42—42no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—44no—
170C42—84no—
171C42C4×D412yes—
172C42C4×Q812yes—
173C42#12112yes—
174C42C22⋊Q812yes—
175C42C22⋊Q812yes—
176C42#12512yes—
177C42#12612yes—
178C42C4⋊Q812yes—
179C23—22no—
180C23—22no—
181C23—41no—
182C23#38011yescharacteristic
183C23#39411yescharacteristic
184C23#39811yescharacteristic
185C2×C4—24no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4#38611yescharacteristic
212C2×C4#38712yes—
213C2×C4#38912yes—
214C2×C4#39111yescharacteristic
215C2×C4#39511yescharacteristic
216C2×C4#39612yes—
217C2×C4#39812yes—
218C2×C4#40011yescharacteristic
219C8—42no—
220C8—42no—
221C8—84no—
222C8—81no—
223C8—81no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—44no—
232C22#144111yescharacteristic
233C22#181212yes—
234C22#205012yes—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—22no—
242C4—44no—
243C4—82no—
244C4—164no—
245C4#191111yescharacteristic
246C4#191211yescharacteristic
247C2—22no—
248C2—22no—
249C2—41no—
250C2#1122411yescharacteristic
251C2#1122911yescharacteristic
252C2#1124311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1842C4semidirect
16. #1831C4semidirect

Maximal subgroups

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

subgroupindex
#219762
#219602
#80022
#79542
#79412