Finite Group Databasebeta

← #326609

Subgroups — #326609

Order 512.

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

normal subgroups only

This group has 461 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#21736C211yesmaximal, characteristic
2#21340C211yesmaximal, characteristic
3#10688C211yesmaximal, characteristic
4#10662C211yesmaximal, characteristic
5#10548C211yesmaximal, characteristic
6#10037C211yesmaximal, characteristic
7#9580C212yesmaximal
8#9232C212yesmaximal
9#9048C212yesmaximal
10#8879C212yesmaximal
11#8800C211yesmaximal, characteristic
12#3162C2211yescharacteristic
13#3104C2212yes—
14#2961C2212yes—
15C42×Q8C2211yescharacteristic
16#1994C2212yes—
17#1985C2211yescharacteristic
18#1985C2211yescharacteristic
19#1946—21no—
20#1945—21no—
21#1940C2211yescharacteristic
22#1940C2211yescharacteristic
23#1917—21no—
24#1917—21no—
25#1849C2211yescharacteristic
26#1773—21no—
27#1772—21no—
28C4×C4⋊C8—21no—
29C4×C4⋊C8C2211yescharacteristic
30#1744—21no—
31C4×#97C2211yescharacteristic
32#1571C2212yes—
33#1542C2212yes—
34#1503C2212yes—
35#1487C2212yes—
36#1481C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1281C2212yes—
40#1276C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43#533C2312yes—
44C2×C4×Q8—21no—
45C2×C4×Q8C2311yescharacteristic
46#501—22no—
47#501—22no—
48#484—22no—
49#477—22no—
50#462—22no—
51#451—22no—
52#446—42no—
53#446—42no—
54C4×Q16—42no—
55#431C2312yes—
56#431C2312yes—
57C2×#108—21no—
58C2×#108—21no—
59#427—21no—
60#427—21no—
61#424—22no—
62#423C2312yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67#405—22no—
68C2×#97D411yescharacteristic
69C2×#97D411yescharacteristic
70C2×#97C2311yescharacteristic
71C2×#97C2311yescharacteristic
72C2×C4×C8—21no—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4C2311yescharacteristic
80C43—21no—
81C43C2311yescharacteristic
82#339—22no—
83#339—22no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#335—42no—
92#333—22no—
93#332—22no—
94#328—22no—
95#328—22no—
96Q8⋊C8—42no—
97Q8⋊C8—42no—
98C22×Q8#4111yescharacteristic
99C2×Q16—42no—
100C2×M16—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8#4112yes—
108C4⋊Q8#4112yes—
109#125—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—44no—
117#117—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4#4111yescharacteristic
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2411yescharacteristic
134#108—42no—
135#108—42no—
136#107—21no—
137#107—21no—
138#107—21no—
139#107—21no—
140#107—42no—
141#106—22no—
142#106—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—42no—
150C4⋊C8—44no—
151C4⋊C8—42no—
152#103—42no—
153#103—42no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156#97—22no—
157#97—22no—
158#97—22no—
159#97—41no—
160#97—42no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—41no—
164C4×C8—42no—
165C4×C8—42no—
166C4×C8Q1614yescomplemented
167#95—21no—
168#95—21no—
169#95#4111yescharacteristic
170#95#4111yescharacteristic
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—42no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4C22×D411yescharacteristic
182C22×C4C2×#4111yescharacteristic
183C22×C4C2×#4111yescharacteristic
184Q16—84no—
185M16—42no—
186M16—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8C2×Q1612yes—
206C2×C8C2×Q1612yes—
207C2×C8#13712yes—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C22⋊C4—42no—
220C42—24no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—42no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—42no—
229C42—41no—
230C42—42no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42C2×Q1612yes—
235C42C2×#4112yes—
236C42H(5,𝔽2)12yes—
237C23#52511yescharacteristic
238Q8—24no—
239Q8—42no—
240Q8—82no—
241Q8—82no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4#52511yescharacteristic
265C2×C4#53511yescharacteristic
266C2×C4#55112yes—
267C2×C4#55312yes—
268C2×C4C22×Q1611yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4#57511yescharacteristic
271C2×C4#57511yescharacteristic
272C8—22no—
273C8—22no—
274C8—22no—
275C8—41no—
276C8—82no—
277C8—82no—
278C8—81no—
279C8—81no—
280C8—82no—
281C22—21no—
282C22—21no—
283C22#259811yescharacteristic
284C22#310411yescharacteristic
285C22#311611yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—42no—
290C4—42no—
291C4—81no—
292C4—81no—
293C4—82no—
294C4—81no—
295C4—82no—
296C4#310411yescharacteristic
297C4#311711yescharacteristic
298C4#320911yescharacteristic
299C4#321911yescharacteristic
300C2—21no—
301C2—21no—
302C2#2937011yescharacteristic
303C2#2938411yescharacteristic
304C2#2938611yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2598
prime radicalC23#525
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
166. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#217362
#213402
#106882
#106622
#105482
#100372
#95802
#92322
#90482
#88792
#88002