Finite Group Databasebeta

← #145046

Subgroups — #145046

Order 512.

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

normal subgroups only

This group has 321 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#28744C211yesmaximal, characteristic
2#13212C211yesmaximal, characteristic
3#13185C211yesmaximal, characteristic
4#11932C211yesmaximal, characteristic
5#11611C211yesmaximal, characteristic
6#8663C211yesmaximal, characteristic
7#7174C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#446C2211yescharacteristic
10#2568—21no—
11C2×#365—21no—
12#2150C2211yescharacteristic
13C2×#362—22no—
14C4×M32C2211yescharacteristic
15#2056—22no—
16#1981—21no—
17#1971—21no—
18#1971—21no—
19#1848C2211yescharacteristic
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1487—21no—
23#1383—21no—
24#1362—22no—
25#1338—22no—
26#1301—21no—
27C22×Q16—21no—
28C22×Q16D412yes—
29C2×C4⋊Q8—21no—
30C2×C4×Q8D411yescharacteristic
31C2×M32—21no—
32C2×M32—21no—
33#494—21no—
34#494—21no—
35#494—21no—
36#494—21no—
37#446—21no—
38#446—21no—
39#446—21no—
40#446—21no—
41#431—21no—
42C2×#108—21no—
43C2×#108—21no—
44C2×#106D411yescharacteristic
45C2×#106D411yescharacteristic
46#423—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×#103—22no—
50C2×#103—22no—
51C2×#97—21no—
52C2×C4×C8C2311yescharacteristic
53#400—42no—
54#391—22no—
55#391—22no—
56#391—41no—
57#382—41no—
58C4×C4⋊C4—21no—
59#365—42no—
60C4⋊C16—21no—
61C4⋊C16D412yes—
62#362—44no—
63#346—21no—
64C4×C16—21no—
65#340—21no—
66#335—21no—
67#334—21no—
68Q8⋊C8—42no—
69C22×Q8—22no—
70C22×Q8SD1611yescharacteristic
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—42no—
74C2×Q16—42no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×M16—41no—
82C22×C8—21no—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4SD1611yescharacteristic
99C2×C42—21no—
100C2×C42—41no—
101C2×C42C2×D411yescharacteristic
102M32—22no—
103M32—41no—
104M32—41no—
105M32—41no—
106C2×C16—22no—
107C2×C16—41no—
108#108—42no—
109#108—41no—
110#108—41no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122#103—82no—
123#103—82no—
124C22⋊C8—41no—
125#97—41no—
126#97—41no—
127C4×C8—41no—
128C4×C8C2×D411yescharacteristic
129C4×C8#4111yescharacteristic
130C4×C8#4111yescharacteristic
131C4×C8#4111yescharacteristic
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—41no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4#12011yescharacteristic
152C22×C4C2×SD1611yescharacteristic
153C22×C4AGL(1,ℤ/8)11yescharacteristic
154Q16—44no—
155Q16—82no—
156Q16—81no—
157Q16—81no—
158Q16—81no—
159Q16—81no—
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×C8#13511yescharacteristic
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—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C42—41no—
190C42—41no—
191C42—82no—
192C42#12111yescharacteristic
193C42#12311yescharacteristic
194C42#13511yescharacteristic
195C42AGL(1,ℤ/8)11yescharacteristic
196C16—42no—
197C16—42no—
198C23#45011yescharacteristic
199Q8—42no—
200Q8—81no—
201Q8—82no—
202Q8—82no—
203Q8—82no—
204Q8—82no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4#39411yescharacteristic
221C2×C4#44911yescharacteristic
222C2×C4#44911yescharacteristic
223C2×C4#45211yescharacteristic
224C2×C4#45311yescharacteristic
225C2×C4#45511yescharacteristic
226C2×C4#46511yescharacteristic
227C2×C4#46911yescharacteristic
228C2×C4#48111yescharacteristic
229C2×C4#48211yescharacteristic
230C8—21no—
231C8—21no—
232C8—81no—
233C8—161no—
234C8—161no—
235C8#46311yescharacteristic
236C8#46411yescharacteristic
237C8#48811yescharacteristic
238C8#48911yescharacteristic
239C22—21no—
240C22—21no—
241C22#163111yescharacteristic
242C22#164211yescharacteristic
243C22#203411yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—81no—
248C4—81no—
249C4—161no—
250C4—161no—
251C4—162no—
252C4#162011yescharacteristic
253C4#164011yescharacteristic
254C4#203711yescharacteristic
255C4#204211yescharacteristic
256C4#223912yes—
257C2—21no—
258C2—21no—
259C2#1041011yescharacteristic
260C2#1339711yescharacteristic
261C2#1339911yescharacteristic
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
#287442
#132122
#131852
#119322
#116112
#86632
#71742