Finite Group Databasebeta

← #102934

Subgroups — #102934

Order 512.

813 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C2×#2057C212yesmaximal
2#17075C211yesmaximal, characteristic
3#11010C211yesmaximal, characteristic
4C2×#1271C212yesmaximal
5#7025C211yesmaximal, characteristic
6C22×C4⋊Q8C2211yescharacteristic
7C22×#107—22no—
8C22×#103—22no—
9C2×#398—22no—
10C2×#389C2212yes—
11#2057—24no—
12#2057—24no—
13C2×#340—22no—
14C2×#336C2211yescharacteristic
15C2×#336C2212yes—
16#1445—22no—
17#1444—22no—
18C4⋊#95C2211yescharacteristic
19#1271—24no—
20C23×Q8—21no—
21C23×C8—22no—
22C23×C8—21no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8—44no—
26C2×C4⋊Q8D412yes—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4D412yes—
33C22×C42C2311yescharacteristic
34C2×#107—28no—
35C2×#107—48no—
36C2×#103—44no—
37C2×#103—44no—
38C2×#103—44no—
39#398—44no—
40#398—44no—
41#389—22no—
42#389—22no—
43#389—22no—
44#389—22no—
45C2×#95—24no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49#340—28no—
50#336—22no—
51#336—22no—
52#336D414yes—
53C22×Q8—24no—
54C22×Q8—22no—
55C22×Q8—42no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—41no—
59C22×Q8—42no—
60C22×Q8—42no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—41no—
65C23×C4C2×D412yes—
66C23×C4C2×D411yescharacteristic
67C22×C8—28no—
68C22×C8—24no—
69C22×C8—22no—
70C22×C8—44no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8—41no—
76C22×C8—42no—
77C4⋊Q8—44no—
78C4⋊Q8—48no—
79C4⋊Q8—44no—
80C2×C4⋊C4—28no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—44no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—82no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4D814yes—
103C2×C4⋊C4Q1614yes—
104C2×C42—42no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42#4111yescharacteristic
108#107—416no—
109#103—416no—
110#95—44no—
111#95—48no—
112#95—42no—
113#95—42no—
114#95—42no—
115#95—42no—
116C24#12011yescharacteristic
117C2×Q8—24no—
118C2×Q8—44no—
119C2×Q8—42no—
120C2×Q8—44no—
121C2×Q8—44no—
122C2×Q8—44no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—82no—
127C2×Q8—82no—
128C2×Q8—82no—
129C2×Q8—82no—
130C2×Q8—82no—
131C2×Q8—81no—
132C2×Q8—81no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—82no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4#12012yes—
159C22×C4#12111yescharacteristic
160C22×C4#12112yes—
161C22×C4#12711yescharacteristic
162C22×C4C2×D812yes—
163C22×C4C2×Q1612yes—
164C22×C4#13512yes—
165C22×C4AGL(1,ℤ/8)12yes—
166C22×C4#13712yes—
167C22×C4#13712yes—
168C2×C8—28no—
169C2×C8—44no—
170C2×C8—48no—
171C2×C8—48no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—42no—
175C2×C8—82no—
176C2×C8—81no—
177C2×C8—81no—
178C4⋊C4—28no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—88no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C42—41no—
194C42—42no—
195C42—41no—
196C23#39211yescharacteristic
197C23#44712yes—
198C23#45112yes—
199C23#45212yes—
200C23#45311yescharacteristic
201C23#45411yescharacteristic
202C23#45614yes—
203C23B(4,𝔽2)12yes—
204Q8—48no—
205Q8—84no—
206Q8—84no—
207Q8—84no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—48no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4—84no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#45112yes—
245C2×C4#45212yes—
246C2×C4#45912yes—
247C2×C4#46712yes—
248C2×C4#47012yes—
249C2×C4#49311yescharacteristic
250C2×C4#49611yescharacteristic
251C8—48no—
252C8—82no—
253C22#159512yes—
254C22#159812yes—
255C22#160314yes—
256C22#160514yes—
257C22#163011yescharacteristic
258C22#163311yescharacteristic
259C22#163612yes—
260C22#164714yes—
261C22#164914yes—
262C22#166012yes—
263C22#166312yes—
264C22#200112yes—
265C22#200512yes—
266C22#200611yescharacteristic
267C22#201111yescharacteristic
268C22#202111yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—21no—
274C4—22no—
275C4—21no—
276C4—84no—
277C4—84no—
278C4—84no—
279C4—162no—
280C2#775112yes—
281C2#775712yes—
282C2#776314yes—
283C2#992512yes—
284C2#996211yescharacteristic
285C2#1007612yes—
286C2#1008711yescharacteristic
287C2#1023711yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC22×C42C23
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20572
#170752
#110102
C2×#12712
#70252