Finite Group Databasebeta

← #126313

Subgroups — #126313

Order 512.

316 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 66 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#32648C211yesmaximal, characteristic
2#13526C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic, complemented
4#10617C211yesmaximal, characteristic, complemented
5#7128C211yesmaximal, characteristic, complemented
6#7120C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2538—21no—
9C2×#366—21no—
10C2×#366—21no—
11C2×#357—21no—
12#2093—21no—
13#2083—21no—
14#2067—21no—
15#1984C2211yescharacteristic
16#1970C2211yescharacteristic
17#1942C2211yescharacteristic
18#1879—21no—
19#1862—21no—
20#1845C2211yescharacteristic
21#1803—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C2×C22⋊Q8—41no—
29C2×C4×Q8D411yescharacteristic
30C2×C4×D4—21no—
31C22×C16—21no—
32C22×C16—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×#107D411yescharacteristic
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×#103—41no—
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×C4×C8C2311yescharacteristic
46#396—41no—
47#396—41no—
48#390—21no—
49#390—21no—
50#389—21no—
51#389—21no—
52#384—21no—
53#384—21no—
54#380—41no—
55#366—42no—
56#366—42no—
57#357—42no—
58#340—21no—
59#340—21no—
60#340—41no—
61#336—21no—
62#336—21no—
63C22×Q8D811yescharacteristic
64C22×D4—21no—
65C23×C4—41no—
66C2×D8—21no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8Q1612yes—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22⋊Q8—82no—
75C22⋊Q8—82no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×D4—81no—
79C4×D4—81no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4D811yescharacteristic
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C42—21no—
98C2×C42C2×D411yescharacteristic
99C2×C16—22no—
100C2×C16—22no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16—41no—
104C2×C16—41no—
105#107—21no—
106#107—21no—
107#107—41no—
108#107—41no—
109#107—41no—
110#107—41no—
111#107—41no—
112#107—41no—
113#107—81no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117#103—82no—
118#102—41no—
119#102—41no—
120#102—81no—
121#102—81no—
122C4×C8—41no—
123#95—41no—
124#95—41no—
125C24—41no—
126C2×Q8—21no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8D1612yes—
130C2×D4—22no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4—81no—
145C22×C4#12011yescharacteristic
146C22×C4C2×D811yescharacteristic
147C22×C4AGL(1,ℤ/8)11yescharacteristic
148D8—42no—
149D8—81no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—81no—
158C2×C8—81no—
159C2×C8—81no—
160C2×C8C2×Q1611yescharacteristic
161C2×C8#13511yescharacteristic
162C2×C8AGL(1,ℤ/8)11yescharacteristic
163C2×C8#13711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—81no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4—161no—
181C4⋊C4—161no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—81no—
185C22⋊C4—81no—
186C22⋊C4—161no—
187C22⋊C4—161no—
188C42—21no—
189C42—21no—
190C42—41no—
191C42—41no—
192C16—42no—
193C16—42no—
194C23—42no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23—81no—
200C23—81no—
201C23#44711yescharacteristic
202Q8—42no—
203Q8—81no—
204D4—42no—
205D4—42no—
206D4—81no—
207D4—81no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—164no—
229C2×C4—161no—
230C2×C4—161no—
231C2×C4—161no—
232C2×C4#44911yescharacteristic
233C2×C4#45111yescharacteristic
234C2×C4#45311yescharacteristic
235C2×C4#45511yescharacteristic
236C2×C4#45611yescharacteristic
237C2×C4#45811yescharacteristic
238C2×C4C2×D1611yescharacteristic
239C2×C4#50911yescharacteristic
240C8—21no—
241C8—21no—
242C8—21no—
243C8—21no—
244C8—81no—
245C8—161no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—161no—
250C22—161no—
251C22—161no—
252C22—161no—
253C22#160711yescharacteristic
254C22#162111yescharacteristic
255C22#164911yescharacteristic
256C22#218411yescharacteristic
257C22#219111yescharacteristic
258C22#219211yescharacteristic
259C22#219511yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—41no—
264C4—81no—
265C4—161no—
266C4—161no—
267C4—161no—
268C4—321no—
269C4#160911yescharacteristic
270C4#161411yescharacteristic
271C4#164411yescharacteristic
272C4#166111yescharacteristic
273C2—162no—
274C2#778911yescharacteristic
275C2#1227811yescharacteristic
276C2#1228711yescharacteristic
277C2#1232811yescharacteristic
278C2#1233911yescharacteristic
279C2#1240411yescharacteristic
280C2#1241211yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13505C2semidirect
4. #10617C2semidirect
5. #7128C2semidirect
6. #7120C2semidirect

Maximal subgroups

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

subgroupindex
#326482
#135262
#135052
#106172
#71282
#71202
#71122