Finite Group Databasebeta

← #137260

Subgroups — #137260

Order 512.

372 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 82 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#17815C211yesmaximal, characteristic
3#11664C212yesmaximal, complemented
4#11460C211yesmaximal, characteristic
5#7302C212yesmaximal, complemented
6C2×C4×Q16C2211yescharacteristic
7#2920—21no—
8C2×#431C2211yescharacteristic
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#1572—24no—
20#1567—21no—
21#1566C2212yes—
22#1334—24no—
23#1322—21no—
24#1314—21no—
25C22×Q16C2×C411yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C4×Q8—21no—
28C22×C42D411yescharacteristic
29C4×Q16—22no—
30C4×Q16—22no—
31#432—22no—
32#432—41no—
33#431—22no—
34#431—22no—
35#431—22no—
36#431—41no—
37C2×#107C2×C411yescharacteristic
38C2×#106—21no—
39#423—22no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45#407—41no—
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8D411yescharacteristic
48C4×M16—41no—
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#335—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—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22⋊Q8—42no—
81C22⋊Q8—42no—
82C4×Q8—41no—
83C4×Q8—41no—
84C4×Q8—41no—
85C4×Q8—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42SD1611yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C16—42no—
105#107—22no—
106#106—42no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—24no—
110C4⋊C8—42no—
111#103—42no—
112#103—42no—
113#103—42no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118#97—42no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—42no—
122C4×C8C2×D412yes—
123C4×C8#4112yes—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129C24#10211yescharacteristic
130C2×Q8—22no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C2×D811yescharacteristic
156C22×C4C2×SD1611yescharacteristic
157Q16—24no—
158Q16—42no—
159M16—82no—
160M16—82no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—44no—
168C2×C8C4×D412yes—
169C2×C8#12112yes—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—44no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—44no—
181C22⋊C4—82no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—82no—
189C42—81no—
190C42—81no—
191C42#12011yescharacteristic
192C42#12311yescharacteristic
193C42AGL(1,ℤ/8)11yescharacteristic
194C42#13711yescharacteristic
195C16—82no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23C2×#10211yescharacteristic
202Q8—42no—
203Q8—42no—
204Q8—81no—
205Q8—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
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#38611yescharacteristic
238C2×C4#41711yescharacteristic
239C2×C4#44711yescharacteristic
240C2×C4#44811yescharacteristic
241C2×C4#48011yescharacteristic
242C2×C4#48111yescharacteristic
243C8—22no—
244C8—22no—
245C8—41no—
246C8—41no—
247C8—84no—
248C22—21no—
249C22—21no—
250C22—42no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22#187411yescharacteristic
256C22#214511yescharacteristic
257C22#214511yescharacteristic
258C4—21no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—82no—
266C4—82no—
267C4—161no—
268C4—161no—
269C4#188411yescharacteristic
270C4#189111yescharacteristic
271C4#223912yes—
272C2—21no—
273C2—21no—
274C2—81no—
275C2#903511yescharacteristic
276C2#1300211yescharacteristic
277C2#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
#178152
#116642
#114602
#73022