Finite Group Databasebeta

← #145196

Subgroups — #145196

Order 512.

340 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 78 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#28273C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic, complemented
3#12818C211yesmaximal, characteristic
4#11611C211yesmaximal, characteristic, complemented
5#11599C211yesmaximal, characteristic
6#8663C211yesmaximal, characteristic, complemented
7#7278C211yesmaximal, characteristic, complemented
8C2×#446C2211yescharacteristic
9C2×C4×D8C2211yescharacteristic
10#2538—21no—
11#2150C2211yescharacteristic
12C2×#361—21no—
13C4×M32C2211yescharacteristic
14#2065—22no—
15#1972—21no—
16#1944—21no—
17#1848C2211yescharacteristic
18#1844C2211yescharacteristic
19#1802—21no—
20#1564C2211yescharacteristic
21#1487—21no—
22#1387—21no—
23#1383—22no—
24#1338—22no—
25#1333—22no—
26#1301—21no—
27C22×D8D411yescharacteristic
28C2×C22⋊Q8—41no—
29C2×C4×Q8D411yescharacteristic
30C2×C4×D4—21no—
31C2×M32—21no—
32C2×M32—21no—
33#446—21no—
34#446—21no—
35#446—21no—
36#446—21no—
37C4×D8—21no—
38C4×D8—21no—
39C4×D8—21no—
40C4×D8—21no—
41#431—21no—
42C2×#108—21no—
43C2×#108—21no—
44C2×#107D411yescharacteristic
45C2×#106—21no—
46C2×#106D412yes—
47#423—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×#102—21no—
51C2×#102—22no—
52C2×#97—21no—
53C2×C4×C8C2311yescharacteristic
54#396—42no—
55#390—22no—
56#389—22no—
57#384—21no—
58#380—41no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61#361—42no—
62#346D412yes—
63C4×C16—21no—
64#340—22no—
65#335—21no—
66#334—21no—
67Q8⋊C8—42no—
68C22×Q8SD1611yescharacteristic
69C22×D4—21no—
70C23×C4—41no—
71C2×D8—21no—
72C2×D8—21no—
73C2×D8—21no—
74C2×D8—21no—
75C2×D8—21no—
76C2×D8Q1612yes—
77C2×M16—41no—
78C22×C8—21no—
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C22⋊Q8—84no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4SD1611yescharacteristic
97C2×C22⋊C4—42no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C42—21no—
101C2×C42C2×D411yescharacteristic
102M32—41no—
103M32—41no—
104M32—41no—
105M32—41no—
106C2×C16—22no—
107C2×C16—41no—
108#108—42no—
109#108—41no—
110#108—41no—
111#107—21no—
112#107—21no—
113#106—41no—
114#106—41no—
115#106—42no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123#102—41no—
124#102—41no—
125#102—82no—
126C22⋊C8—41no—
127#97—41no—
128#97—41no—
129C4×C8—41no—
130C4×C8C2×D411yescharacteristic
131C4×C8#4111yescharacteristic
132C4×C8#4111yescharacteristic
133C4×C8#4111yescharacteristic
134#95—42no—
135C24—41no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—42no—
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4#12011yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155C22×C4AGL(1,ℤ/8)11yescharacteristic
156D8—22no—
157D8—22no—
158D8—41no—
159D8—41no—
160M16—81no—
161M16—81no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8#12111yescharacteristic
170C2×C8#12111yescharacteristic
171C2×C8#12311yescharacteristic
172C2×C8#12411yescharacteristic
173C2×C8C2×Q1611yescharacteristic
174C2×C8#13511yescharacteristic
175C2×C8AGL(1,ℤ/8)11yescharacteristic
176C2×C8#13711yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—82no—
186C4⋊C4—84no—
187C4⋊C4—161no—
188C22⋊C4—84no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—162no—
192C42—41no—
193C42—41no—
194C42#12111yescharacteristic
195C42#12311yescharacteristic
196C42#13511yescharacteristic
197C42AGL(1,ℤ/8)11yescharacteristic
198C16—42no—
199C16—81no—
200C23—42no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23#45011yescharacteristic
208Q8—42no—
209Q8—81no—
210D4—42no—
211D4—42no—
212D4—81no—
213D4—81no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—162no—
227C2×C4—162no—
228C2×C4#39411yescharacteristic
229C2×C4#44911yescharacteristic
230C2×C4#44911yescharacteristic
231C2×C4#45111yescharacteristic
232C2×C4#45411yescharacteristic
233C2×C4#45511yescharacteristic
234C2×C4#46511yescharacteristic
235C2×C4#46911yescharacteristic
236C2×C4#48111yescharacteristic
237C2×C4#48211yescharacteristic
238C8—21no—
239C8—21no—
240C8—81no—
241C8—161no—
242C8—161no—
243C8#46711yescharacteristic
244C8#46811yescharacteristic
245C8#48411yescharacteristic
246C8#48511yescharacteristic
247C22—21no—
248C22—21no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22—161no—
253C22—161no—
254C22—161no—
255C22—161no—
256C22#163711yescharacteristic
257C22#164211yescharacteristic
258C22#203411yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—81no—
263C4—81no—
264C4—162no—
265C4—161no—
266C4#162811yescharacteristic
267C4#164411yescharacteristic
268C4#203611yescharacteristic
269C4#204011yescharacteristic
270C4#223812yes—
271C2—21no—
272C2—21no—
273C2—162no—
274C2#1043611yescharacteristic
275C2#1339811yescharacteristic
276C2#1340011yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12886C2semidirect
4. #11611C2semidirect
6. #8663C2semidirect
7. #7278C2semidirect

Maximal subgroups

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

subgroupindex
#282732
#128862
#128182
#116112
#115992
#86632
#72782