Finite Group Databasebeta

← #511126

Subgroups — #511126

Order 512.

506 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 132 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#28704C211yesmaximal, characteristic
2#27661C211yesmaximal, characteristic
3#13494C212yesmaximal, complemented
4#13493C212yesmaximal, complemented
5#13451C212yesmaximal, complemented
6#13110C212yesmaximal, complemented
7#12977C211yesmaximal, characteristic
8#12919C212yesmaximal, complemented
9#12836C212yesmaximal, complemented
10C22×M32C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×C8×D4C2211yescharacteristic
13C22×#107C2211yescharacteristic
14#2547—21no—
15#2486—21no—
16C2×#367C2212yes—
17C2×#366C2212yes—
18C2×#366C2212yes—
19C2×C22⋊C16C2212yes—
20C2×C22⋊C16C2211yescharacteristic
21#2086—21no—
22#2075—21no—
23#2075—21no—
24#2075—21no—
25#1988—21no—
26#1942C2212yes—
27#1823—21no—
28#1808C2212yes—
29#1808C2211yescharacteristic
30#1808C2212yes—
31#1808C2212yes—
32C4×#107C2211yescharacteristic
33#1380C2212yes—
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—21no—
42C2×#124—41no—
43C2×#123—41no—
44C2×C4×D4D411yescharacteristic
45C22×C4⋊C4—21no—
46C2×M32—24no—
47C2×M32—41no—
48C22×C16C2312yes—
49C22×C16C2312yes—
50#500—41no—
51#500—41no—
52C8×D4—41no—
53C8×D4—41no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—41no—
57C2×#107—41no—
58C2×#107C2312yes—
59C2×#107C2312yes—
60C2×#107C2312yes—
61C2×#107C2312yes—
62C2×C4⋊C8D411yescharacteristic
63C2×C22⋊C8D411yescharacteristic
64C2×C22⋊C8D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#398—41no—
67#397—41no—
68#397—41no—
69#397—41no—
70#391—41no—
71#389—22no—
72#381—41no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—21no—
77C4×C4⋊C4—21no—
78#367—24no—
79#366—42no—
80#366—42no—
81C22⋊C16—24no—
82C22⋊C16—41no—
83#340—21no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—21no—
88#340—22no—
89C22×Q8—41no—
90C22×D4D811yescharacteristic
91C23×C4—41no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2411yescharacteristic
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108#124—84no—
109#123—84no—
110C4×D4—41no—
111C4×D4—41no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4D811yescharacteristic
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4D811yescharacteristic
127C2×C22⋊C4D811yescharacteristic
128C2×C42—41no—
129C2×C42C2×D411yescharacteristic
130M32—44no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16#4114yes—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—22no—
144#107—22no—
145#107—42no—
146#107—42no—
147C4⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C4×C8—21no—
151C4×C8—21no—
152#95—41no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—41no—
157#95—42no—
158C24C2×D811yescharacteristic
159C24C2×D811yescharacteristic
160C2×Q8—82no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4C2×D811yescharacteristic
182C22×C4C2×D811yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C2×D811yescharacteristic
185C22×C4C22×D411yescharacteristic
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8C2×#4112yes—
199C2×C8H(5,𝔽2)11yescharacteristic
200C2×C8H(5,𝔽2)11yescharacteristic
201C2×C82−1+412yes—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—82no—
217C22⋊C4—84no—
218C22⋊C4—84no—
219C22⋊C4—84no—
220C42—21no—
221C42—21no—
222C42—82no—
223C16—24no—
224C16—42no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23#50912yes—
235C23C22×D811yescharacteristic
236Q8—164no—
237D4—44no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—164no—
265C2×C4#50912yes—
266C2×C4#53411yescharacteristic
267C2×C4#54712yes—
268C2×C4#57511yescharacteristic
269C2×C4#57812yes—
270C8—41no—
271C8—41no—
272C8—41no—
273C8#54212yes—
274C8#55111yescharacteristic
275C8#56311yescharacteristic
276C22—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22#318611yescharacteristic
287C22#333412yes—
288C22C2×#50911yescharacteristic
289C22C2×#50911yescharacteristic
290C22#341712yes—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—162no—
296C4—162no—
297C4—162no—
298C4—162no—
299C4#271711yescharacteristic
300C4#320211yescharacteristic
301C4#334912yes—
302C2—42no—
303C2—81no—
304C2#3225811yescharacteristic
305C2#3375811yescharacteristic
306C2#3376811yescharacteristic
307C2#3391012yes—
308C2#3392012yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13494C2semidirect
4. #13493C2semidirect
5. #13451C2semidirect
6. #13110C2semidirect
8. #12919C2semidirect
9. #12836C2semidirect

Maximal subgroups

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

subgroupindex
#287042
#276612
#134942
#134932
#134512
#131102
#129772
#129192
#128362