Finite Group Databasebeta

← #244145

Subgroups — #244145

Order 512.

579 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 162 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#21737C211yesmaximal, characteristic
2#21718C211yesmaximal, characteristic
3#17931C212yesmaximal
4#17903C212yesmaximal
5#10668C212yesmaximal
6#10662C211yesmaximal, characteristic
7#10662C211yesmaximal, characteristic
8#10037C211yesmaximal, characteristic
9#10037C211yesmaximal, characteristic
10#10017C212yesmaximal
11#8138C211yesmaximal, characteristic
12#3162C2211yescharacteristic
13#3104C2212yes—
14#3104C2212yes—
15#2960C2212yes—
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18#2399C2211yescharacteristic
19#1995C2212yes—
20#1985C2211yescharacteristic
21#1985C2211yescharacteristic
22#1945—22no—
23#1939—22no—
24#1849C2211yescharacteristic
25#1843C2211yescharacteristic
26#1773—22no—
27#1771—22no—
28#1726C2211yescharacteristic
29#1591—24no—
30#1584—24no—
31#1584—24no—
32C8×Q16—24no—
33#1571C2212yes—
34#1571C2212yes—
35#1487C2212yes—
36#1487C2212yes—
37#1481C2212yes—
38#1481C2212yes—
39#1450C2212yes—
40#1275C2211yescharacteristic
41#1275C2211yescharacteristic
42#1274C2212yes—
43#533C2312yes—
44C2×C4×Q8C2311yescharacteristic
45C2×C4×Q8C2311yescharacteristic
46#501—22no—
47#501—22no—
48#484—22no—
49#484—22no—
50#477—22no—
51#477—22no—
52#462—22no—
53#462—22no—
54#451—22no—
55#451—22no—
56#446—22no—
57#446—22no—
58#446—22no—
59#446—22no—
60C8×Q8—22no—
61C8×Q8—22no—
62C8×Q8—22no—
63C8×Q8—22no—
64C4×Q16—24no—
65C4×Q16—24no—
66#431C2312yes—
67#431C2312yes—
68C2×#108—22no—
69#427—22no—
70#424—22no—
71#423C2312yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77#405—22no—
78C2×#97—21no—
79C2×#97D412yes—
80C2×#97C2311yescharacteristic
81C2×C4×C8—21no—
82C2×C4×C8D412yes—
83C2×C4×C8C2311yescharacteristic
84#398—21no—
85#398—21no—
86#395—21no—
87#395—21no—
88C4×C4⋊C4—22no—
89#376C2311yescharacteristic
90#339—24no—
91#336—22no—
92#336—22no—
93#335—24no—
94#335—24no—
95Q8⋊C8—24no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98Q8⋊C8—24no—
99#322D414yes—
100C82D414yes—
101C22×Q8#4111yescharacteristic
102C22×Q8#4111yescharacteristic
103C2×Q16—24no—
104C2×Q16—24no—
105C2×M16—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D412yes—
110C22×C8C2×D411yescharacteristic
111C4⋊Q8#4112yes—
112C4⋊Q8#4112yes—
113#125—22no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122#117—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4#4111yescharacteristic
128C2×C4⋊C4#4111yescharacteristic
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2×D412yes—
132C2×C42C2411yescharacteristic
133#108—44no—
134#107—22no—
135#107—22no—
136#107—24no—
137#107—24no—
138#106—24no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—44no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152#103—24no—
153#103—24no—
154#103—24no—
155#103—24no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158#97—22no—
159#97—42no—
160#97—42no—
161#97Q1618yescomplemented
162C4×C8—42no—
163C4×C8—42no—
164C4×C8—42no—
165C4×C8C2×D414yes—
166C4×C8C2×D414yes—
167C4×C8C2×D412yes—
168C4×C8C2×D412yes—
169#95—22no—
170#95—22no—
171#95—21no—
172#95—21no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—81no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4C22×D411yescharacteristic
185C22×C4C2×#4111yescharacteristic
186C22×C4C2×#4111yescharacteristic
187Q16—48no—
188Q16—84no—
189M16—44no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8#12711yescharacteristic
201C2×C8#12711yescharacteristic
202C2×C8#12712yes—
203C2×C8C2×Q1614yes—
204C2×C8C2×Q1614yes—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—82no—
217C22⋊C4—42no—
218C42—22no—
219C42—41no—
220C42—41no—
221C42—41no—
222C42—42no—
223C42—41no—
224C42C2×Q1614yes—
225C42C22×D412yes—
226C42C2×#4112yes—
227C23#52511yescharacteristic
228Q8—24no—
229Q8—42no—
230Q8—44no—
231Q8—42no—
232Q8—44no—
233Q8—42no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4#52512yes—
252C2×C4#52512yes—
253C2×C4#52511yescharacteristic
254C2×C4C2×#12711yescharacteristic
255C2×C4C22×Q1612yes—
256C2×C4#57512yes—
257C8—24no—
258C8—21no—
259C8—21no—
260C8—44no—
261C8—44no—
262C8—81no—
263C8—81no—
264C8—82no—
265C8#49414yes—
266C22—21no—
267C22—21no—
268C22#239711yescharacteristic
269C22#310412yes—
270C4—22no—
271C4—22no—
272C4—44no—
273C4—42no—
274C4—81no—
275C4—82no—
276C4—81no—
277C4—81no—
278C4—82no—
279C4#310412yes—
280C4C2×#49411yescharacteristic
281C4#315011yescharacteristic
282C2—22no—
283C2#2379411yescharacteristic
284C2#2379511yescharacteristic
285C2#2380111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutator, Frattini, cosocleC2×C42C24
socleC22#2397
prime radicalC23#525
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
161. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#217372
#217182
#179312
#179032
#106682
#106622
#100372
#100172
#81382