Finite Group Databasebeta

← #507767

Subgroups — #507767

Order 512.

415 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 124 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, complemented
2#21295C211yesmaximal, characteristic
3#13511C211yesmaximal, characteristic, complemented
4#13486C211yesmaximal, characteristic
5#13459C211yesmaximal, characteristic, complemented
6#13373C211yesmaximal, characteristic
7#13140C212yesmaximal
8#13117C212yesmaximal, complemented
9#13115C212yesmaximal, complemented
10#12951C212yesmaximal
11#12897C211yesmaximal, characteristic, complemented
12#3162C2211yescharacteristic
13#2960—21no—
14C4×#135C2211yescharacteristic
15#2923—21no—
16#2155C2212yes—
17#2155C2212yes—
18#2148C2212yes—
19#2145C2212yes—
20#1995—21no—
21#1975C2211yescharacteristic
22#1975C2211yescharacteristic
23#1946C2211yescharacteristic
24#1946C2211yescharacteristic
25#1945—21no—
26#1945C2212yes—
27#1937—21no—
28#1937—21no—
29#1849C2211yescharacteristic
30C4×#108C2211yescharacteristic
31#1773—21no—
32#1773C2212yes—
33#1759—21no—
34#1726C2211yescharacteristic
35#1389C2212yes—
36#1388C2212yes—
37#1385C2212yes—
38#1384C2212yes—
39#1377C2212yes—
40#1376C2212yes—
41#1375C2211yescharacteristic
42#1375C2211yescharacteristic
43C2×#135C2311yescharacteristic
44#533—21no—
45C4×#41—21no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#501—24no—
49#446—42no—
50C8×Q8—42no—
51C4×Q16—21no—
52C4×SD16—22no—
53C4×D8—21no—
54#433—42no—
55#432—42no—
56C2×#108—21no—
57C2×#108C2312yes—
58C2×#108C2311yescharacteristic
59C2×#108C2311yescharacteristic
60#427—21no—
61#427C2312yes—
62#427C2311yescharacteristic
63#424—21no—
64#424—21no—
65#424—22no—
66#423—21no—
67C2×#104—21no—
68C2×#104—21no—
69#416—21no—
70#408—21no—
71#408—21no—
72#405—22no—
73C4×M16—21no—
74C2×#97D411yescharacteristic
75C2×#97C2311yescharacteristic
76C2×#97C2311yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#376D411yescharacteristic
81#365—24no—
82#365—24no—
83#362—22no—
84#361—22no—
85#360—24no—
86#341—21no—
87#339—21no—
88#339—21no—
89#339—22no—
90#339—22no—
91C2×#41—21no—
92#135—22no—
93#135—22no—
94C2×Q16#4111yescharacteristic
95C2×SD16—21no—
96C2×D8#4111yescharacteristic
97C2×M16—22no—
98C2×M16—21no—
99C2×M16—21no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—22no—
105#125—22no—
106C22⋊Q8—42no—
107C4×Q8—22no—
108C4×Q8—42no—
109C4×D4—22no—
110C4×D4—42no—
111#117—21no—
112#117—22no—
113#117—41no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118M32—22no—
119M32—22no—
120M32—22no—
121M32—22no—
122C2×C16—22no—
123C2×C16—22no—
124#108—22no—
125#108—22no—
126#108—22no—
127#108—22no—
128#108—42no—
129#107—22no—
130#107—21no—
131#107—22no—
132#106—22no—
133#106#4114yes—
134#106#4112yes—
135C4⋊C8—44no—
136C4⋊C8—42no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139#104—42no—
140#104—42no—
141#103—22no—
142#102—22no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145#97—22no—
146#97—22no—
147#97—42no—
148#97—41no—
149#97SD1612yes—
150#97SD1612yes—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—22no—
154C4×C8—41no—
155C4×C8—42no—
156C4×C8—41no—
157#95—41no—
158#41—41no—
159#41—41no—
160#41—41no—
161#41—41no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×D4—21no—
165C2×D4—41no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4C22×D411yescharacteristic
172Q16—22no—
173SD16—42no—
174D8—22no—
175M16—41no—
176M16—42no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—42no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8C2×SD1612yes—
192C2×C8C2×SD1612yes—
193C2×C8C2×#4111yescharacteristic
194C2×C8C2×#4112yes—
195C2×C8C2×#4111yescharacteristic
196C2×C82−1+412yes—
197C4⋊C4—22no—
198C4⋊C4—24no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—41no—
211C42—42no—
212C42C2×SD1611yescharacteristic
213C42C2×SD1611yescharacteristic
214C42AGL(1,ℤ/8)12yes—
215C16—42no—
216C16—42no—
217C23—41no—
218C23#57511yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—84no—
222D4—41no—
223D4—41no—
224D4—81no—
225D4—81no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—44no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—162no—
239C2×C4#52511yescharacteristic
240C2×C4#52511yescharacteristic
241C2×C4#53611yescharacteristic
242C2×C4C22×SD1611yescharacteristic
243C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
244C2×C4#57511yescharacteristic
245C8—22no—
246C8—22no—
247C8—41no—
248C8—81no—
249C8—82no—
250C8—81no—
251C8#55212yes—
252C8#55612yes—
253C22—41no—
254C22—82no—
255C22—81no—
256C22#310111yescharacteristic
257C22#311411yescharacteristic
258C22#324011yescharacteristic
259C4—41no—
260C4—41no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—84no—
265C4—81no—
266C4#259911yescharacteristic
267C4#310311yescharacteristic
268C4#311511yescharacteristic
269C4#322711yescharacteristic
270C2—21no—
271C2—21no—
272C2—21no—
273C2—161no—
274C2#2939211yescharacteristic
subgroupG/H
centerC4#3227
commutator, Frattini, cosocleC22×C8C24
socleC2#29392
prime radicalC2×#135C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ 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. #21737C2semidirect
3. #13511C2semidirect
5. #13459C2semidirect
8. #13117C2semidirect
9. #13115C2semidirect
11. #12897C2semidirect

Maximal subgroups

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

subgroupindex
#217372
#212952
#135112
#134862
#134592
#133732
#131402
#131172
#131152
#129512
#128972