Finite Group Databasebeta

← #138486

Subgroups — #138486

Order 512.

377 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 98 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#29878C211yesmaximal, characteristic
2#17814C211yesmaximal, characteristic
3#11972C212yesmaximal, complemented
4#11459C211yesmaximal, characteristic
5#7217C212yesmaximal, complemented
6C2×#500C2211yescharacteristic
7C2×#431C2211yescharacteristic
8C2×#423—21no—
9#2857—21no—
10#2162C2212yes—
11#2087—21no—
12#2087C412yes—
13#2076—21no—
14#2076C412yes—
15#1990—21no—
16#1834C2211yescharacteristic
17#1571—24no—
18#1567—21no—
19#1567C2212yes—
20#1365—24no—
21#1322—21no—
22#1314—21no—
23C22×M16—41no—
24C2×C4⋊Q8—21no—
25C2×C22⋊Q8—41no—
26C22×C42D411yescharacteristic
27#500—22no—
28#500—22no—
29#431—22no—
30#431—22no—
31C2×#107—21no—
32C2×#107C2×C412yes—
33#423—22no—
34#423—22no—
35#423—41no—
36#423—41no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—41no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#391—41no—
46#389—22no—
47#389—41no—
48#384—21no—
49#380—22no—
50#380—41no—
51C4⋊C16—22no—
52C22⋊C16—22no—
53C4×C16—22no—
54#340—22no—
55#340—22no—
56#340—21no—
57#340C2×C412yes—
58#335—24no—
59#335—22no—
60#335D414yes—
61C22×Q8—41no—
62C23×C4C22⋊C411yescharacteristic
63C23×C4D811yescharacteristic
64C23×C4SD1611yescharacteristic
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—82no—
68C2×M16—82no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4⋊Q8—41no—
77C4⋊Q8—41no—
78C22⋊Q8—84no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42D811yescharacteristic
92C2×C42SD1611yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C16—42no—
95#107—24no—
96#107—42no—
97C4⋊C8—22no—
98C4⋊C8—22no—
99C4⋊C8—24no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C4⋊C8—42no—
103C22⋊C8—22no—
104C22⋊C8—22no—
105C4×C8—21no—
106C4×C8—21no—
107C4×C8C2×D412yes—
108C4×C8#4112yes—
109#95—42no—
110#95—42no—
111C24#10211yescharacteristic
112C2×Q8—82no—
113C2×Q8—82no—
114C2×Q8—81no—
115C2×Q8—81no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4#10211yescharacteristic
130C22×C4#10211yescharacteristic
131C22×C4#10211yescharacteristic
132C22×C4SD3212yes—
133C22×C4Q3212yes—
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4C2×SD1611yescharacteristic
137M16—84no—
138M16—82no—
139M16—82no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—44no—
147C2×C8—84no—
148C2×C8C4×D412yes—
149C2×C8#12112yes—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—82no—
155C4⋊C4—84no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C22⋊C4—84no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42#12011yescharacteristic
168C42#12311yescharacteristic
169C42AGL(1,ℤ/8)11yescharacteristic
170C42#13711yescharacteristic
171C16—82no—
172C23—21no—
173C23—21no—
174C23—41no—
175C23—41no—
176C23#35812yes—
177C23C2×#10211yescharacteristic
178Q8—162no—
179Q8—162no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4#35812yes—
205C2×C4#38611yescharacteristic
206C2×C4#41711yescharacteristic
207C2×C4#44711yescharacteristic
208C2×C4#44811yescharacteristic
209C2×C4#48011yescharacteristic
210C2×C4#48111yescharacteristic
211C2×C4C2×SD3211yescharacteristic
212C2×C4C2×Q3211yescharacteristic
213C2×C4#50911yescharacteristic
214C2×C4#51011yescharacteristic
215C8—22no—
216C8—22no—
217C8—41no—
218C8—41no—
219C8—84no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22#187411yescharacteristic
228C22C2×#35811yescharacteristic
229C22#213911yescharacteristic
230C4—22no—
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—162no—
236C4—162no—
237C4#190911yescharacteristic
238C4#191511yescharacteristic
239C4#218711yescharacteristic
240C4#218811yescharacteristic
241C4#223511yescharacteristic
242C4#223611yescharacteristic
243C4#223812yes—
244C2—21no—
245C2—21no—
246C2—42no—
247C2#919511yescharacteristic
248C2#1299311yescharacteristic
249C2#1301011yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11972C2semidirect
5. #7217C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#178142
#119722
#114592
#72172