Finite Group Databasebeta

← #507765

Subgroups — #507765

Order 512.

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

normal subgroups only

This group has 435 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#21723C211yesmaximal, characteristic, complemented
2#21296C211yesmaximal, characteristic
3#13508C211yesmaximal, characteristic, complemented
4#13484C211yesmaximal, characteristic
5#13458C211yesmaximal, characteristic, complemented
6#13372C211yesmaximal, characteristic
7#13143C212yesmaximal
8#13119C212yesmaximal, complemented
9#13114C212yesmaximal, complemented
10#12943C212yesmaximal
11#12897C211yesmaximal, characteristic, complemented
12C2×#499C2211yescharacteristic
13#2943C2211yescharacteristic
14#2922—21no—
15#2330—21no—
16#2159C2212yes—
17C2×#365C2212yes—
18C2×#360C2212yes—
19#2139C2212yes—
20#1992—21no—
21#1975C2211yescharacteristic
22#1974C2211yescharacteristic
23#1946C2211yescharacteristic
24#1945C2211yescharacteristic
25#1944C2212yes—
26#1942—21no—
27#1938—21no—
28#1937—21no—
29#1845C2211yescharacteristic
30#1778C2211yescharacteristic
31#1776C2212yes—
32C4×#107—21no—
33#1759—21no—
34#1726C2211yescharacteristic
35#1387C2212yes—
36#1387C2212yes—
37#1386C2212yes—
38#1383C2212yes—
39#1377C2212yes—
40#1376C2211yescharacteristic
41#1376C2211yescharacteristic
42#1375C2212yes—
43C2×#135C2311yescharacteristic
44C2×#125—21no—
45C4×#41—21no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#499—24no—
49#442—21no—
50#441—21no—
51#440—21no—
52#440—21no—
53#433—42no—
54#431—42no—
55C2×#108C2311yescharacteristic
56C2×#108C2311yescharacteristic
57#427C2311yescharacteristic
58C2×#107—21no—
59C2×#107—21no—
60C2×#107C2312yes—
61C2×#106C2312yes—
62#424—21no—
63#424—21no—
64C2×#104—21no—
65C2×#104—21no—
66#416—21no—
67#408—21no—
68#408—21no—
69C4×M16—21no—
70C2×#97D411yescharacteristic
71C2×#97C2311yescharacteristic
72C2×#97C2311yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#389—22no—
77#389—42no—
78#387—42no—
79#384—21no—
80C4×C4⋊C4—22no—
81#376D411yescharacteristic
82#367—22no—
83#366—22no—
84#365—24no—
85#360—24no—
86#358—22no—
87#357—22no—
88#341—21no—
89#340—22no—
90#340—22no—
91#339—21no—
92#339—21no—
93C2×#41—21no—
94#135—22no—
95#135—22no—
96C2×Q16#4111yescharacteristic
97C2×SD16—21no—
98C2×D8#4111yescharacteristic
99C2×M16—21no—
100C2×M16—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2411yescharacteristic
105#125—24no—
106#125—42no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4—41no—
112C4×D4—41no—
113#117—21no—
114#117—41no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125M32—22no—
126M32—22no—
127M32—22no—
128M32—22no—
129C2×C16—22no—
130C2×C16—22no—
131#108—22no—
132#108—22no—
133#107—24no—
134#107—22no—
135#107—22no—
136#107—42no—
137#107#4111yescharacteristic
138#107#4111yescharacteristic
139#106—22no—
140#106—21no—
141#106#4114yes—
142C4⋊C8—42no—
143C4⋊C8—42no—
144#104—42no—
145#104—42no—
146#103—21no—
147#103—21no—
148#102—21no—
149#102—21no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152#97—21no—
153#97—21no—
154#97—22no—
155#97—22no—
156#97—41no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—41no—
161#95—21no—
162#95—21no—
163#95—42no—
164#95—42no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169C2×Q8—21no—
170C2×D4—21no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4AGL(1,ℤ/8)11yescharacteristic
180C22×C4AGL(1,ℤ/8)11yescharacteristic
181C22×C4C22×D411yescharacteristic
182Q16—22no—
183SD16—42no—
184D8—22no—
185M16—41no—
186M16—41no—
187M16—41no—
188M16—41no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8AGL(1,ℤ/8)12yes—
200C2×C8C2×#4111yescharacteristic
201C2×C8C2×#4112yes—
202C2×C8C2×#4111yescharacteristic
203C2×C82−1+412yes—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—24no—
207C4⋊C4—48no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—162no—
214C4⋊C4—162no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—41no—
223C42—41no—
224C42—41no—
225C42—162no—
226C42#13512yes—
227C42#13712yes—
228C16—42no—
229C16—42no—
230C23—41no—
231C23C2×AGL(1,ℤ/8)11yescharacteristic
232Q8—41no—
233Q8—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—81no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—44no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#52511yescharacteristic
262C2×C4#52511yescharacteristic
263C2×C4#53611yescharacteristic
264C2×C4C2×#13511yescharacteristic
265C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C4C2×#13711yescharacteristic
267C8—22no—
268C8—41no—
269C8—41no—
270C8—81no—
271C8—81no—
272C8#55212yes—
273C8#55612yes—
274C22—21no—
275C22—21no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22#310511yescharacteristic
280C22#310911yescharacteristic
281C22#323911yescharacteristic
282C4—42no—
283C4—42no—
284C4—41no—
285C4—41no—
286C4—84no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—162no—
291C4#259911yescharacteristic
292C4#310711yescharacteristic
293C4#311211yescharacteristic
294C4#322811yescharacteristic
295C2—21no—
296C2—21no—
297C2—161no—
298C2#2938911yescharacteristic
299C2#3369211yescharacteristic
300C2#3369311yescharacteristic
subgroupG/H
center, socleC22#3109
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#135C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21723C2semidirect
3. #13508C2semidirect
5. #13458C2semidirect
8. #13119C2semidirect
9. #13114C2semidirect
11. #12897C2semidirect

Maximal subgroups

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

subgroupindex
#217232
#212962
#135082
#134842
#134582
#133722
#131432
#131192
#131142
#129432
#128972