Finite Group Databasebeta

← #145113

Subgroups — #145113

Order 512.

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

normal subgroups only

This group has 330 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#28538C211yesmaximal, characteristic
2#13284C211yesmaximal, characteristic
3#12812C211yesmaximal, characteristic
4C2×#1363C211yesmaximal, characteristic
5C2×#1334C211yesmaximal, characteristic
6#9620C211yesmaximal, characteristic
7#7170C211yesmaximal, characteristic
8C2×#499C2211yescharacteristic
9C2×#479—21no—
10C2×C4×Q16C2211yescharacteristic
11#2565—21no—
12C2×C4⋊C16C2211yescharacteristic
13C2×#358—21no—
14C2×#346C2211yescharacteristic
15#2097—21no—
16#2095—21no—
17#1971—21no—
18#1947—21no—
19#1847—21no—
20#1845C2211yescharacteristic
21#1770C2211yescharacteristic
22C2×#334C2211yescharacteristic
23#1380—21no—
24#1380—21no—
25#1380—21no—
26#1363—22no—
27#1334—22no—
28#1296—21no—
29C22×Q16D411yescharacteristic
30C2×#125—21no—
31C2×C4×Q8—21no—
32C22×C16—21no—
33C22×C16—21no—
34#499—21no—
35#499—21no—
36#499—21no—
37#499—21no—
38#479—42no—
39#479—42no—
40C4×Q16—21no—
41C4×Q16—21no—
42C4×Q16—21no—
43C4×Q16—21no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#107—21no—
47C2×#107—41no—
48C2×#107D411yescharacteristic
49C2×#107D411yescharacteristic
50C2×#107D411yescharacteristic
51C2×#106—21no—
52C2×#106—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8D411yescharacteristic
55C2×#103—21no—
56C2×#103—21no—
57C2×#103—21no—
58C2×#97—21no—
59C2×C4×C8C2311yescharacteristic
60#398—41no—
61#398—41no—
62#391—21no—
63#391—21no—
64#387—41no—
65#387—41no—
66#384—21no—
67#384—21no—
68#382—21no—
69C4×C4⋊C4D411yescharacteristic
70C4⋊C16—21no—
71C4⋊C16—21no—
72#358—42no—
73#346—21no—
74#346—21no—
75#340—41no—
76#336—21no—
77#334—21no—
78#334D412yes—
79C22×Q8—21no—
80C2×Q16—21no—
81C2×Q16—21no—
82C2×Q16—21no—
83C2×Q16—21no—
84C2×Q16—21no—
85C2×Q16Q1612yes—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90#125—41no—
91#125—41no—
92#125—41no—
93#125—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4D811yescharacteristic
110C2×C4⋊C4D811yescharacteristic
111C2×C42—21no—
112C2×C42—41no—
113C2×C42C2×D411yescharacteristic
114C2×C16—22no—
115C2×C16—41no—
116C2×C16—41no—
117C2×C16—41no—
118C2×C16—41no—
119C2×C16—41no—
120#107—21no—
121#107—21no—
122#107—21no—
123#107—21no—
124#107—21no—
125#107—21no—
126#107—42no—
127#107—42no—
128#107—41no—
129#107—41no—
130#107—82no—
131#106—42no—
132#106—41no—
133#106—41no—
134C4⋊C8—22no—
135C4⋊C8—21no—
136C4⋊C8—41no—
137C4⋊C8SD1612yes—
138#103—42no—
139#103—42no—
140#103—41no—
141#103—41no—
142#97—41no—
143#97—41no—
144C4×C8C2×D411yescharacteristic
145C4×C8#4111yescharacteristic
146C4×C8#4111yescharacteristic
147C4×C8#4111yescharacteristic
148#95—21no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155C2×Q8—22no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—81no—
168C22×C4#12011yescharacteristic
169C22×C4C2×D811yescharacteristic
170C22×C4AGL(1,ℤ/8)11yescharacteristic
171Q16—22no—
172Q16—22no—
173Q16—41no—
174Q16—41no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—42no—
181C2×C8—81no—
182C2×C8—81no—
183C2×C8#12111yescharacteristic
184C2×C8#12111yescharacteristic
185C2×C8#12311yescharacteristic
186C2×C8#12311yescharacteristic
187C2×C8C2×Q1611yescharacteristic
188C2×C8#13511yescharacteristic
189C2×C8AGL(1,ℤ/8)11yescharacteristic
190C2×C8#13711yescharacteristic
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—161no—
212C42—41no—
213C42—41no—
214C42—81no—
215C42—81no—
216C42#12111yescharacteristic
217C42#12411yescharacteristic
218C42C2×SD1611yescharacteristic
219C42AGL(1,ℤ/8)11yescharacteristic
220C16—42no—
221C16—81no—
222C23#44711yescharacteristic
223Q8—42no—
224Q8—42no—
225Q8—81no—
226Q8—81no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—161no—
250C2×C4—161no—
251C2×C4—161no—
252C2×C4#39411yescharacteristic
253C2×C4#44911yescharacteristic
254C2×C4#45011yescharacteristic
255C2×C4#45111yescharacteristic
256C2×C4#45411yescharacteristic
257C2×C4#45611yescharacteristic
258C2×C4#45911yescharacteristic
259C2×C4#46011yescharacteristic
260C2×C4#48611yescharacteristic
261C2×C4#49011yescharacteristic
262C8—21no—
263C8—21no—
264C8—42no—
265C8—82no—
266C8#46711yescharacteristic
267C8#46811yescharacteristic
268C8#48411yescharacteristic
269C8#48511yescharacteristic
270C22#162511yescharacteristic
271C22#164911yescharacteristic
272C22#204111yescharacteristic
273C22#219011yescharacteristic
274C22#219111yescharacteristic
275C22#219311yescharacteristic
276C22#219511yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—81no—
282C4—81no—
283C4—82no—
284C4—161no—
285C4—161no—
286C4—161no—
287C4—321no—
288C4#163711yescharacteristic
289C4#165411yescharacteristic
290C4#203611yescharacteristic
291C4#203811yescharacteristic
292C2#1042111yescharacteristic
293C2#1234911yescharacteristic
294C2#1235111yescharacteristic
295C2#1241311yescharacteristic
296C2#1241511yescharacteristic
297C2#1341011yescharacteristic
298C2#1341211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#285382
#132842
#128122
C2×#13632
C2×#13342
#96202
#71702