Finite Group Databasebeta

← #145197

Subgroups — #145197

Order 512.

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

normal subgroups only

This group has 306 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#28274C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11611C211yesmaximal, characteristic
5#11600C211yesmaximal, characteristic
6#8663C211yesmaximal, characteristic
7#7278C211yesmaximal, characteristic
8C2×#446C2211yescharacteristic
9C2×C4×Q16C2211yescharacteristic
10#2582—21no—
11#2150C2211yescharacteristic
12C2×#362—21no—
13C4×M32C2211yescharacteristic
14#2066—22no—
15#1971—21no—
16#1944—21no—
17#1848C2211yescharacteristic
18#1844C2211yescharacteristic
19#1803—21no—
20#1564C2211yescharacteristic
21#1487—21no—
22#1387—21no—
23#1383—22no—
24#1338—22no—
25#1334—22no—
26#1301—21no—
27C22×Q16D411yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8D411yescharacteristic
30C2×M32—21no—
31C2×M32—21no—
32#446—21no—
33#446—21no—
34#446—21no—
35#446—21no—
36C4×Q16—21no—
37C4×Q16—21no—
38C4×Q16—21no—
39C4×Q16—21no—
40#431—21no—
41C2×#108—21no—
42C2×#108—21no—
43C2×#107D411yescharacteristic
44C2×#106—21no—
45C2×#106D412yes—
46#423—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×#103—21no—
50C2×#103—22no—
51C2×#97—21no—
52C2×C4×C8C2311yescharacteristic
53#398—42no—
54#391—22no—
55#391—41no—
56#389—22no—
57#389—41no—
58#384—21no—
59C4⋊C16—21no—
60C4⋊C16—21no—
61#362—42no—
62#346D412yes—
63C4×C16—21no—
64#340—22no—
65#335—21no—
66#334—21no—
67Q8⋊C8—42no—
68C22×Q8—21no—
69C22×Q8SD1611yescharacteristic
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×Q16Q1612yes—
76C2×M16—41no—
77C22×C8—21no—
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×Q8—41no—
87C4×Q8—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4SD1611yescharacteristic
96C2×C42—21no—
97C2×C42—41no—
98C2×C42C2×D411yescharacteristic
99M32—41no—
100M32—41no—
101M32—41no—
102M32—41no—
103C2×C16—22no—
104C2×C16—41no—
105#108—42no—
106#108—41no—
107#108—41no—
108#107—21no—
109#107—21no—
110#106—41no—
111#106—41no—
112#106—42no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#103—41no—
121#103—41no—
122#103—82no—
123C22⋊C8—41no—
124#97—41no—
125#97—41no—
126C4×C8—41no—
127C4×C8C2×D411yescharacteristic
128C4×C8#4111yescharacteristic
129C4×C8#4111yescharacteristic
130C4×C8#4111yescharacteristic
131#95—42no—
132#95—42no—
133#95—42no—
134C2×Q8—22no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4#12011yescharacteristic
149C22×C4C2×SD1611yescharacteristic
150C22×C4AGL(1,ℤ/8)11yescharacteristic
151Q16—22no—
152Q16—22no—
153Q16—41no—
154Q16—41no—
155M16—81no—
156M16—81no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8#12111yescharacteristic
165C2×C8#12111yescharacteristic
166C2×C8#12311yescharacteristic
167C2×C8#12411yescharacteristic
168C2×C8C2×Q1611yescharacteristic
169C2×C8#13511yescharacteristic
170C2×C8AGL(1,ℤ/8)11yescharacteristic
171C2×C8#13711yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—161no—
184C4⋊C4—162no—
185C42—41no—
186C42—41no—
187C42—81no—
188C42—81no—
189C42#12111yescharacteristic
190C42#12311yescharacteristic
191C42#13511yescharacteristic
192C42AGL(1,ℤ/8)11yescharacteristic
193C16—42no—
194C16—81no—
195C23#45011yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—81no—
200Q8—81no—
201Q8—81no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4#39411yescharacteristic
218C2×C4#44911yescharacteristic
219C2×C4#44911yescharacteristic
220C2×C4#45111yescharacteristic
221C2×C4#45411yescharacteristic
222C2×C4#45511yescharacteristic
223C2×C4#46511yescharacteristic
224C2×C4#46911yescharacteristic
225C2×C4#48111yescharacteristic
226C2×C4#48211yescharacteristic
227C8—21no—
228C8—21no—
229C8—81no—
230C8—161no—
231C8—161no—
232C8#46711yescharacteristic
233C8#46811yescharacteristic
234C8#48411yescharacteristic
235C8#48511yescharacteristic
236C22—21no—
237C22—21no—
238C22#163711yescharacteristic
239C22#164211yescharacteristic
240C22#203411yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—81no—
245C4—81no—
246C4—82no—
247C4—162no—
248C4—161no—
249C4#162811yescharacteristic
250C4#164411yescharacteristic
251C4#203611yescharacteristic
252C4#204011yescharacteristic
253C4#223912yes—
254C2—21no—
255C2—21no—
256C2#1043611yescharacteristic
257C2#1339911yescharacteristic
258C2#1340011yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#282742
#128862
#128192
#116112
#116002
#86632
#72782