Finite Group Databasebeta

← #137259

Subgroups — #137259

Order 512.

395 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 86 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#23699C211yesmaximal, characteristic
2#17813C211yesmaximal, characteristic
3#11664C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#7302C212yesmaximal, complemented
6C2×C4×Q16C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9C4×C22⋊Q8—21no—
10#2163C2212yes—
11#2086C411yescharacteristic
12#2075C411yescharacteristic
13#2066—21no—
14#2056C411yescharacteristic
15#2035—21no—
16#2001C411yescharacteristic
17C4×#106C2211yescharacteristic
18C4×#103—21no—
19#1570—24no—
20#1566—21no—
21#1566C2212yes—
22#1334—24no—
23#1322—21no—
24#1314—21no—
25C22×Q16C2×C411yescharacteristic
26C22×M16—41no—
27C2×C22⋊Q8—21no—
28C2×C4×Q8—21no—
29C22×C42D411yescharacteristic
30C4×Q16—22no—
31C4×Q16—22no—
32#431—22no—
33#431—22no—
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36#423—22no—
37#423—22no—
38#423—41no—
39#423—41no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#391—21no—
52#389—21no—
53#387—41no—
54#385—41no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4—41no—
58C4×C22⋊C4—41no—
59C4⋊C16—22no—
60C22⋊C16—22no—
61#346—22no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#334—24no—
67#334—22no—
68#334D414yes—
69C22×Q8—21no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4D811yescharacteristic
72C23×C4SD1611yescharacteristic
73C2×Q16—22no—
74C2×Q16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C2×M16—82no—
78C2×M16—82no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C4×Q8—41no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×Q8—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42SD1611yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C16—42no—
109#107—22no—
110#106—42no—
111C4⋊C8—22no—
112C4⋊C8—22no—
113C4⋊C8—24no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#103—42no—
118#103—42no—
119#103—42no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8C2×D412yes—
125C4×C8#4112yes—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131C24#10211yescharacteristic
132C2×Q8—22no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159Q16—24no—
160Q16—42no—
161M16—84no—
162M16—82no—
163M16—82no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—44no—
171C2×C8—84no—
172C2×C8C4×D412yes—
173C2×C8#12112yes—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—82no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C22⋊C4—44no—
185C22⋊C4—82no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—82no—
193C42—81no—
194C42—81no—
195C42#12011yescharacteristic
196C42#12311yescharacteristic
197C42AGL(1,ℤ/8)11yescharacteristic
198C42#13711yescharacteristic
199C16—82no—
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23#36012yes—
205C23C2×#10211yescharacteristic
206Q8—42no—
207Q8—42no—
208Q8—81no—
209Q8—81no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4#36012yes—
242C2×C4#38611yescharacteristic
243C2×C4#41711yescharacteristic
244C2×C4#44711yescharacteristic
245C2×C4#44811yescharacteristic
246C2×C4#48011yescharacteristic
247C2×C4#48111yescharacteristic
248C8—22no—
249C8—22no—
250C8—41no—
251C8—41no—
252C8—84no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—42no—
257C22—41no—
258C22—41no—
259C22—81no—
260C22#187411yescharacteristic
261C22C2×#36011yescharacteristic
262C22C2×#36011yescharacteristic
263C4—21no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4—161no—
273C4—161no—
274C4#188411yescharacteristic
275C4#189111yescharacteristic
276C4#223912yes—
277C2—21no—
278C2—21no—
279C2—42no—
280C2#903511yescharacteristic
281C2#1300111yescharacteristic
282C2#1300911yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11664C2semidirect
5. #7302C2semidirect

Maximal subgroups

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

subgroupindex
#236992
#178132
#116642
#114602
#73022