Finite Group Databasebeta

← #102287

Subgroups — #102287

Order 512.

394 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 134 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#33289C211yesmaximal, characteristic
2#9898C211yesmaximal, characteristic
3#9896C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic
5#9694C211yesmaximal, characteristic
6#8784C211yesmaximal, characteristic
7#8260C211yesmaximal, characteristic
8#2868C2211yescharacteristic
9#2868C2211yescharacteristic
10#1990—21no—
11#1990—21no—
12#1990—22no—
13#1842C2211yescharacteristic
14#1842C2211yescharacteristic
15#1834C414yes—
16C4×C4⋊C8C2211yescharacteristic
17C4×C4⋊C8C2211yescharacteristic
18C42×C8C2211yescharacteristic
19C2×#332—22no—
20C2×#322—21no—
21#1296—22no—
22#1296—22no—
23#1296—22no—
24#1277—22no—
25#1273—21no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8D412yes—
39C2×C4⋊C8D412yes—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8C2×C412yes—
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D412yes—
46#384—21no—
47#384—21no—
48#384—21no—
49#384—21no—
50#384—21no—
51#384—21no—
52#384C2×C412yes—
53#384C2×C412yes—
54C43C2311yescharacteristic
55#340—24no—
56#340—24no—
57#336—21no—
58#336—21no—
59#332—44no—
60#322—42no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C22⋊C412yes—
68C22×C8C22⋊C412yes—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4⋊Q8—42no—
72C4⋊Q8—42no—
73C4⋊Q8—41no—
74C4⋊Q8—42no—
75C4⋊Q8—41no—
76C4⋊Q8—42no—
77C4⋊Q8—82no—
78C4⋊Q8—82no—
79C4⋊Q8—82no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42#4111yescharacteristic
98C2×C42#4111yescharacteristic
99C4⋊C8—22no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4×C8—24no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—41no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8—41no—
120C2×Q8—82no—
121C2×Q8—82no—
122C2×Q8—81no—
123C2×Q8—81no—
124C2×Q8—82no—
125C2×Q8—82no—
126C2×Q8—82no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C4×D411yescharacteristic
131C22×C4C4×D411yescharacteristic
132C22×C4#12111yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12411yescharacteristic
135C22×C4#12711yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—81no—
153C2×C8#10114yes—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42#13511yescharacteristic
174C42#13511yescharacteristic
175C42#13512yes—
176C42#13512yes—
177C42AGL(1,ℤ/8)11yescharacteristic
178C42AGL(1,ℤ/8)12yes—
179C42#13711yescharacteristic
180C42#13712yes—
181C42#13712yes—
182C42#13712yes—
183C23#39011yescharacteristic
184Q8—164no—
185Q8—162no—
186Q8—162no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4C2×#10112yes—
200C2×C4#41611yescharacteristic
201C2×C4#41612yes—
202C2×C4#41711yescharacteristic
203C2×C4#44012yes—
204C2×C4#44112yes—
205C2×C4#46311yescharacteristic
206C2×C4#46311yescharacteristic
207C2×C4#46411yescharacteristic
208C2×C4#46411yescharacteristic
209C2×C4#46812yes—
210C2×C4#47012yes—
211C2×C4#47112yes—
212C2×C4#47212yes—
213C2×C4#48812yes—
214C2×C4#49511yescharacteristic
215C2×C4#49511yescharacteristic
216C2×C4#49611yescharacteristic
217C2×C4#49711yescharacteristic
218C2×C4#49711yescharacteristic
219C2×C4#49711yescharacteristic
220C8—24no—
221C8—42no—
222C8—82no—
223C8—82no—
224C8—161no—
225C8—161no—
226C22#195811yescharacteristic
227C22#196611yescharacteristic
228C22#197111yescharacteristic
229C22#197311yescharacteristic
230C22#197711yescharacteristic
231C22#197811yescharacteristic
232C22#198311yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—162no—
237C4—162no—
238C4#154412yes—
239C4#155212yes—
240C4#169712yes—
241C4#169812yes—
242C4#170312yes—
243C4#170412yes—
244C4#171811yescharacteristic
245C4#171811yescharacteristic
246C4#171911yescharacteristic
247C4#172011yescharacteristic
248C2#970011yescharacteristic
249C2#974511yescharacteristic
250C2#974711yescharacteristic
251C2#976911yescharacteristic
252C2#980611yescharacteristic
253C2#981011yescharacteristic
254C2#984011yescharacteristic
subgroupG/H
center, socle, prime radicalC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#332892
#98982
#98962
#96942
#87842
#82602