Finite Group Databasebeta

← #120632

Subgroups — #120632

Order 512.

492 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 134 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#21969C211yesmaximal, characteristic
2#10834C211yesmaximal, characteristic
3#8098C212yesmaximal
4#8002C211yesmaximal, characteristic
5#7885C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C22×C4⋊C8C2211yescharacteristic
9C22×C4×C8C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1950—22no—
12#1950—22no—
13#1810—22no—
14C2×#336—22no—
15#1303—22no—
16#1303—22no—
17#1292C414yescomplemented
18#1278—24no—
19C23×C8—21no—
20C23×C8D412yes—
21C22×C4⋊C4—21no—
22C22×C42—21no—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32C2×#108—22no—
33C2×#108—22no—
34#423C2×C414yes—
35#423D412yes—
36#423D412yes—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8C2×C412yes—
40C2×C4⋊C8Q812yes—
41#407—22no—
42#407—22no—
43C4×M16—22no—
44C4×M16—22no—
45C2×C4×C8—22no—
46C2×C4×C8—22no—
47C2×C4×C8—21no—
48C2×C4×C8D412yes—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51C2×#95Q812yes—
52#341—24no—
53#336—44no—
54C23×C4—22no—
55C23×C4C2×D411yescharacteristic
56C23×C4C2×Q811yescharacteristic
57C23×C4#4111yescharacteristic
58C2×M16—24no—
59C2×M16—22no—
60C2×M16—22no—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8C4⋊C414yes—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—42no—
75C2×C4⋊C4—42no—
76C2×C42—42no—
77C2×C42—42no—
78C2×C42—41no—
79C2×C42—41no—
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D412yes—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×Q811yescharacteristic
84C2×C42#4112yes—
85C2×C42#4112yes—
86C2×C42#4112yes—
87C2×C42#4111yescharacteristic
88#108—82no—
89#108—82no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—22no—
94C4⋊C8—22no—
95C4⋊C8—24no—
96C4⋊C8—22no—
97C4⋊C8—22no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102#97—42no—
103#97—42no—
104C4×C8—24no—
105C4×C8—42no—
106C4×C8—42no—
107C4×C8—42no—
108#95—24no—
109C24C22⋊Q811yescharacteristic
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—24no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4C2×C4⋊C411yescharacteristic
123C22×C4C4×D411yescharacteristic
124C22×C4C4×Q811yescharacteristic
125C22×C4#12011yescharacteristic
126C22×C4#12111yescharacteristic
127C22×C4C22⋊Q812yes—
128C22×C4#12311yescharacteristic
129C22×C4#12411yescharacteristic
130C22×C4#12411yescharacteristic
131C22×C4#12412yes—
132C22×C4#12512yes—
133C22×C4#12511yescharacteristic
134C22×C4#12612yes—
135C22×C4C4⋊Q811yescharacteristic
136M16—44no—
137M16—44no—
138M16—42no—
139M16—42no—
140M16—42no—
141M16—42no—
142C2×C8—24no—
143C2×C8—24no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—44no—
153C2×C8—44no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—82no—
157C4⋊C4—48no—
158C42—82no—
159C42—81no—
160C42—81no—
161C42#11712yes—
162C42C4×D412yes—
163C42#12112yes—
164C42C22⋊Q812yes—
165C42#12312yes—
166C42#12312yes—
167C42#12512yes—
168C42#12612yes—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#38911yescharacteristic
175C23#39311yescharacteristic
176C23#40011yescharacteristic
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—44no—
196C2×C4—48no—
197C2×C4—84no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#38211yescharacteristic
201C2×C4#38511yescharacteristic
202C2×C4#38712yes—
203C2×C4#38812yes—
204C2×C4#39411yescharacteristic
205C2×C4#39711yescharacteristic
206C2×C4#39812yes—
207C2×C4#40012yes—
208C2×C4#44312yes—
209C2×C4#44412yes—
210C2×C4#45311yescharacteristic
211C2×C4#45411yescharacteristic
212C2×C4#45511yescharacteristic
213C2×C4#45611yescharacteristic
214C8—44no—
215C8—44no—
216C8—84no—
217C8—82no—
218C8—82no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22#144211yescharacteristic
230C22#194511yescharacteristic
231C22#194611yescharacteristic
232C22#202711yescharacteristic
233C22#202811yescharacteristic
234C22#210312yes—
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—84no—
241C4#185611yescharacteristic
242C4#185711yescharacteristic
243C4#192914yes—
244C4#203911yescharacteristic
245C4#204011yescharacteristic
246C2—21no—
247C2—22no—
248C2—21no—
249C2—21no—
250C2—21no—
251C2#1125811yescharacteristic
252C2#1129511yescharacteristic
253C2#1129611yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1442
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1292C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#108342
#80982
#80022
#78852