Finite Group Databasebeta

← #326646

Subgroups — #326646

Order 512.

436 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 122 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#21344C211yesmaximal, characteristic
3#10687C211yesmaximal, characteristic
4#10663C211yesmaximal, characteristic
5#10548C211yesmaximal, characteristic
6#10037C211yesmaximal, characteristic
7#9579C212yesmaximal
8#9236C212yesmaximal
9#9048C212yesmaximal
10#8881C212yesmaximal
11#8800C211yesmaximal, characteristic
12#3162C2211yescharacteristic
13#3104C2212yes—
14#2961C2212yes—
15#2295C2211yescharacteristic
16#1994C2212yes—
17#1985C2211yescharacteristic
18#1985C2211yescharacteristic
19#1946—21no—
20#1945—21no—
21#1940C2211yescharacteristic
22#1940C2211yescharacteristic
23#1916—21no—
24#1916—21no—
25#1849C2211yescharacteristic
26#1773—21no—
27#1772—21no—
28#1768—21no—
29C4×C4⋊C8C2211yescharacteristic
30#1746—21no—
31C4×#97C2211yescharacteristic
32#1570C2212yes—
33#1542C2212yes—
34#1503C2212yes—
35#1487C2212yes—
36#1481C2212yes—
37#1303C2212yes—
38#1301C2212yes—
39#1281C2212yes—
40#1276C2212yes—
41#1275C2211yescharacteristic
42#1275C2211yescharacteristic
43#533C2312yes—
44C2×C4×Q8C2311yescharacteristic
45#501—22no—
46#501—22no—
47#484—22no—
48#477—22no—
49#462—22no—
50#451—22no—
51#446—42no—
52#446—42no—
53C4×Q16—42no—
54#431C2312yes—
55#431C2312yes—
56C2×#108—21no—
57C2×#108—21no—
58#427—21no—
59#427—21no—
60#424—22no—
61#423C2312yes—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66#405—22no—
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×#97C2311yescharacteristic
70C2×#97C2311yescharacteristic
71C2×C4×C8—21no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74#391—21no—
75#389—21no—
76#387—22no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4C2311yescharacteristic
79#376—21no—
80C43C2311yescharacteristic
81#339—22no—
82#339—22no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#334—42no—
91#333—22no—
92#332—22no—
93#328—24no—
94Q8⋊C8—42no—
95Q8⋊C8—42no—
96C22×Q8#4111yescharacteristic
97C2×Q16—42no—
98C2×M16—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C4⋊Q8#4112yes—
106C4⋊Q8#4112yes—
107#125—22no—
108C22⋊Q8—22no—
109C22⋊Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114#117—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4#4111yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2411yescharacteristic
126#108—42no—
127#108—42no—
128#107—21no—
129#107—21no—
130#107—21no—
131#107—21no—
132#107—42no—
133#106—22no—
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—44no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#103—42no—
145#103—42no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148#97—22no—
149#97—22no—
150#97—22no—
151#97—22no—
152#97—42no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—42no—
158C4×C8—42no—
159#95—22no—
160#95—22no—
161#95—22no—
162#95#4112yes—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—81no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4C22×D411yescharacteristic
173C22×C4C2×#4111yescharacteristic
174C22×C4C2×#4111yescharacteristic
175Q16—84no—
176M16—42no—
177M16—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8#13512yes—
197C2×C8#13712yes—
198C2×C8#13712yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—81no—
209C22⋊C4—42no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—41no—
215C42—42no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42—81no—
220C42#13712yes—
221C42C2×#4112yes—
222C42H(5,𝔽2)12yes—
223C23#52511yescharacteristic
224Q8—42no—
225Q8—42no—
226Q8—82no—
227Q8—82no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4#52511yescharacteristic
246C2×C4#53511yescharacteristic
247C2×C4#55112yes—
248C2×C4#55312yes—
249C2×C4C2×#13511yescharacteristic
250C2×C4C2×#13711yescharacteristic
251C2×C4C2×#13711yescharacteristic
252C2×C4C2×#13711yescharacteristic
253C8—22no—
254C8—22no—
255C8—22no—
256C8—41no—
257C8—82no—
258C8—82no—
259C8—81no—
260C8—82no—
261C8—81no—
262C22—21no—
263C22—21no—
264C22#259811yescharacteristic
265C22#310711yescharacteristic
266C22#311311yescharacteristic
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—82no—
274C4—81no—
275C4—82no—
276C4#310811yescharacteristic
277C4#311311yescharacteristic
278C4#321011yescharacteristic
279C4#321811yescharacteristic
280C2—21no—
281C2—21no—
282C2#2937111yescharacteristic
283C2#2938511yescharacteristic
284C2#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

Maximal subgroups

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

subgroupindex
#217362
#213442
#106872
#106632
#105482
#100372
#95792
#92362
#90482
#88812
#88002