Finite Group Databasebeta

← #120216

Subgroups — #120216

Order 512.

599 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 136 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#21948C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#8008C211yesmaximal, characteristic
4C2×#1293C212yesmaximal
5#7929C212yesmaximal
6C2×#407C2212yes—
7C2×#407C2212yes—
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1855—24no—
12#1850—22no—
13#1850C414yes—
14#1810—22no—
15#1797—22no—
16#1293—24no—
17#1293—22no—
18#1282—24no—
19C24×C4C2311yescharacteristic
20C22×M16—22no—
21C22×M16—22no—
22C23×C8C2×C412yes—
23C23×C8D412yes—
24C22×C22⋊C4Q811yescharacteristic
25C22×C22⋊C4Q811yescharacteristic
26C22×C42—21no—
27C2×#108—24no—
28C2×C4⋊C8—24no—
29#407—22no—
30#407—22no—
31#407—22no—
32#407—22no—
33#407—22no—
34#407—22no—
35#407—22no—
36#407D414yes—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8C2×C412yes—
47C2×C22⋊C8D412yes—
48C2×C22⋊C8Q814yes—
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4C2×C412yes—
52C4×C22⋊C4D412yes—
53C2×#95—21no—
54#341—24no—
55#336—24no—
56C25C2×Q811yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—82no—
60C23×C4C22×C411yescharacteristic
61C23×C4C2×D411yescharacteristic
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D412yes—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×Q812yes—
66C23×C4#4112yes—
67C23×C4#4111yescharacteristic
68C23×C4#4111yescharacteristic
69C23×C4#4112yes—
70C2×M16—24no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—84no—
75C2×M16—84no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—42no—
82C22×C8—82no—
83C22×C8—82no—
84C22×C8C22⋊C414yes—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4C4⋊C412yes—
90C2×C22⋊C4C4⋊C412yes—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—82no—
94#108—84no—
95C4⋊C8—84no—
96C22⋊C8—24no—
97C22⋊C8—24no—
98C22⋊C8—24no—
99C22⋊C8—24no—
100C22⋊C8—24no—
101C22⋊C8—24no—
102C22⋊C8—41no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—41no—
108C22⋊C8—42no—
109#95—22no—
110#95—22no—
111C24—22no—
112C24—22no—
113C24—81no—
114C24—81no—
115C24C2×C4⋊C411yescharacteristic
116C24C22⋊Q811yescharacteristic
117C24C22⋊Q811yescharacteristic
118C24#12512yes—
119C24C4⋊Q812yes—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—81no—
130C22×C4—82no—
131C22×C4—81no—
132C22×C4—82no—
133C22×C4—81no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#11711yescharacteristic
140C22×C4C4×D412yes—
141C22×C4C4×Q812yes—
142C22×C4#12011yescharacteristic
143C22×C4#12112yes—
144C22×C4#12112yes—
145C22×C4#12111yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4C22⋊Q812yes—
148C22×C4C22⋊Q812yes—
149C22×C4C22⋊Q812yes—
150C22×C4#12311yescharacteristic
151C22×C4#12312yes—
152C22×C4#12411yescharacteristic
153C22×C4#12512yes—
154C22×C4#12612yes—
155C22×C4C4⋊Q811yescharacteristic
156M16—48no—
157M16—84no—
158M16—84no—
159M16—84no—
160C2×C8—24no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8—44no—
167C2×C8—82no—
168C2×C8—84no—
169C2×C8—84no—
170C2×C8—84no—
171C22⋊C4—24no—
172C22⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—42no—
175C22⋊C4#10814yescomplemented
176C42—44no—
177C42—82no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—24no—
183C23—24no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—42no—
194C23—84no—
195C23—84no—
196C23#38011yescharacteristic
197C23#38912yes—
198C23#39511yescharacteristic
199C23#39812yes—
200C23#40011yescharacteristic
201C23C2×#10812yes—
202C23#43012yes—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4#38611yescharacteristic
230C2×C4#38712yes—
231C2×C4#39111yescharacteristic
232C2×C4#39411yescharacteristic
233C2×C4#39612yes—
234C2×C4#39811yescharacteristic
235C8—44no—
236C8—84no—
237C8—84no—
238C8—82no—
239C22—21no—
240C22—22no—
241C22—21no—
242C22—22no—
243C22—21no—
244C22—22no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—21no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—84no—
259C22—84no—
260C22#144111yescharacteristic
261C22#181011yescharacteristic
262C22#181111yescharacteristic
263C22#195014yes—
264C4—21no—
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—44no—
270C4—82no—
271C4#191312yes—
272C2—21no—
273C2—21no—
274C2—24no—
275C2—41no—
276C2—42no—
277C2—41no—
278C2#1120511yescharacteristic
279C2#1123311yescharacteristic
280C2#1123511yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
175. C22⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
#219482
#219302
#80082
C2×#12932
#79292