Finite Group Databasebeta

← #139124

Subgroups — #139124

Order 512.

294 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 76 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#33797C211yesmaximal, characteristic
2#22256C211yesmaximal, characteristic
3#11968C211yesmaximal, characteristic
4#11863C211yesmaximal, characteristic
5#9853C211yesmaximal, characteristic
6#7219C211yesmaximal, characteristic
7#7142C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3160C2211yescharacteristic
10#2960—21no—
11#2254C411yescharacteristic, complemented
12#2252C411yescharacteristic, complemented
13#2241C411yescharacteristic, complemented
14#2241C411yescharacteristic, complemented
15#2237—22no—
16#2229—22no—
17#2162C2211yescharacteristic
18#2162C2211yescharacteristic
19#1994—21no—
20#1994—21no—
21#1922—22no—
22#1922—22no—
23#1837C2211yescharacteristic
24#1744C2211yescharacteristic
25#1566C2211yescharacteristic
26#1493—21no—
27#1363—22no—
28#1350—22no—
29#533—21no—
30#533—21no—
31#500C2×C411yescharacteristic
32#500C2×C411yescharacteristic
33#500C2×C411yescharacteristic
34#499—21no—
35#499D412yes—
36#498—21no—
37#494C2×C411yescharacteristic
38#487—21no—
39#467—21no—
40#467—21no—
41#446—41no—
42#446—41no—
43C8×Q8—41no—
44C8×Q8—41no—
45C4×Q16—21no—
46C4×Q16—21no—
47#427—21no—
48#427D412yes—
49#424—22no—
50#423—21no—
51#423—21no—
52#405—22no—
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55C43D411yescharacteristic
56#367—22no—
57#366—22no—
58C4⋊C16—21no—
59C4⋊C16—21no—
60#358—22no—
61#358—22no—
62C22⋊C16—21no—
63C22⋊C16—21no—
64C4×C16—21no—
65C4×C16C2×C411yescharacteristic
66C4×C16C2×C411yescharacteristic
67#341—21no—
68#341D412yes—
69#339—24no—
70#339—42no—
71#336—21no—
72#336—21no—
73#334—21no—
74#334—21no—
75Q8⋊C8—41no—
76Q8⋊C8—41no—
77C2×Q16—21no—
78C2×Q16—21no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8—21no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88#125—22no—
89#125—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98#117—22no—
99#117—41no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2×D411yescharacteristic
104C2×C16—21no—
105C2×C16—21no—
106C2×C16—41no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107C22⋊C412yes—
112#107C22⋊C412yes—
113#106—22no—
114#106—22no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121#103—41no—
122#103—41no—
123#97—42no—
124C4×C8—21no—
125C4×C8—42no—
126C4×C8D811yescharacteristic
127C4×C8D811yescharacteristic
128C4×C8C22×C411yescharacteristic
129C4×C8C2×D411yescharacteristic
130C4×C8#4111yescharacteristic
131C4×C8#4111yescharacteristic
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C22×C4—41no—
137C22×C4#12011yescharacteristic
138C22×C4#13511yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140Q16—42no—
141Q16—81no—
142M16—42no—
143M16—42no—
144M16—82no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—81no—
153C2×C8C2×C22⋊C411yescharacteristic
154C2×C8C4×D411yescharacteristic
155C2×C8#12111yescharacteristic
156C2×C8#12311yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C22⋊C4—42no—
170C22⋊C4—81no—
171C42—22no—
172C42—41no—
173C42—41no—
174C42—41no—
175C42—42no—
176C42—42no—
177C42—81no—
178C42C4×D411yescharacteristic
179C42#12111yescharacteristic
180C42C2×D811yescharacteristic
181C42AGL(1,ℤ/8)11yescharacteristic
182C16—41no—
183C16—41no—
184C16—81no—
185C23#44911yescharacteristic
186Q8—81no—
187Q8—81no—
188Q8—82no—
189Q8—81no—
190Q8—81no—
191Q8—81no—
192Q8—81no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4#38611yescharacteristic
206C2×C4C4×D811yescharacteristic
207C2×C4#44211yescharacteristic
208C2×C4#44711yescharacteristic
209C2×C4#45911yescharacteristic
210C2×C4#46311yescharacteristic
211C8—21no—
212C8—21no—
213C8—21no—
214C8—21no—
215C8—41no—
216C8—41no—
217C8—82no—
218C8—161no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22#187911yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—82no—
227C4—82no—
228C4—82no—
229C4—161no—
230C4—161no—
231C4#190511yescharacteristic
232C4#191211yescharacteristic
233C4#217811yescharacteristic
234C4#217911yescharacteristic
235C4#218012yes—
236C2—41no—
237C2#930011yescharacteristic
238C2#1301111yescharacteristic
239C2#1303911yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#449
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2254C4semidirect
12. #2252C4semidirect
13. #2241C4semidirect
14. #2241C4semidirect

Maximal subgroups

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

subgroupindex
#337972
#222562
#119682
#118632
#98532
#72192
#71422