Finite Group Databasebeta

← #119729

Subgroups — #119729

Order 512.

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

normal subgroups only

This group has 734 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#21829C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic
3#14155C211yesmaximal, characteristic
4#8098C212yesmaximal
5#8098C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#391—22no—
10C2×C4×C4⋊C4C2211yescharacteristic
11C2×C4×C4⋊C4C2211yescharacteristic
12#1997—24no—
13#1966—24no—
14#1821—22no—
15#1784—22no—
16#1445C414yes—
17C4⋊#95—22no—
18#1303—24no—
19#1303—24no—
20C23×Q8D412yes—
21C2×C4⋊Q8—22no—
22C2×C4⋊Q8—22no—
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4D412yes—
31C22×C42—21no—
32C22×C42—21no—
33C22×C42C2311yescharacteristic
34#432—22no—
35#432—22no—
36#432—22no—
37#432—22no—
38#431—22no—
39#431—22no—
40#431—22no—
41#431—22no—
42#423D414yes—
43#423D414yes—
44C2×#101—24no—
45C2×#101—24no—
46#407—22no—
47#407—22no—
48C4×M16—22no—
49C4×M16—22no—
50#391—48no—
51C4×C4⋊C4—22no—
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C4×C4⋊C4—22no—
55C2×#95—28no—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58#343—28no—
59C22×Q8—22no—
60C22×Q8—24no—
61C22×Q8—24no—
62C22×Q8—22no—
63C22×Q8—44no—
64C22×Q8—44no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4C22⋊C414yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—24no—
72C2×M16—24no—
73C4⋊Q8—42no—
74C4⋊Q8—42no—
75C4⋊Q8—42no—
76C4⋊Q8—42no—
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4⋊Q8—42no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C42—44no—
96C2×C42—44no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42C2×D412yes—
102C2×C42C2×D412yes—
103C2×C42#4112yes—
104C2×C42#4112yes—
105C2×C42#4112yes—
106C2×C42#4112yes—
107C4⋊C8—24no—
108C4⋊C8—24no—
109C4⋊C8—24no—
110C4⋊C8—24no—
111#101—84no—
112#101—84no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117#97—42no—
118#97—42no—
119C4×C8—42no—
120C4×C8—42no—
121#95—24no—
122#95—24no—
123#95—416no—
124C24C2×C22⋊C411yescharacteristic
125C2×Q8—416no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—82no—
131C2×Q8—82no—
132C2×Q8—88no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—42no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4#11711yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4C4×D412yes—
156C22×C4C4×D412yes—
157C22×C4#12011yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4#12112yes—
160C22×C4#12112yes—
161C22×C4#12312yes—
162C22×C4#12312yes—
163C22×C4#12311yescharacteristic
164C22×C4#12311yescharacteristic
165M16—44no—
166M16—44no—
167M16—44no—
168M16—44no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—48no—
172C4⋊C4—48no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C42—82no—
178C42—82no—
179C42—82no—
180C42—82no—
181C42#12112yes—
182C42#12112yes—
183C42#12312yes—
184C42#12312yes—
185C42#12412yes—
186C42#12412yes—
187C42#12612yes—
188C42#12612yes—
189C23—22no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23#38111yescharacteristic
195C23#38611yescharacteristic
196C23#38611yescharacteristic
197Q8—88no—
198Q8—88no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—48no—
214C2×C4—48no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—84no—
226C2×C4#38812yes—
227C2×C4#38812yes—
228C2×C4#39412yes—
229C2×C4#39412yes—
230C2×C4#39712yes—
231C2×C4#39712yes—
232C2×C4#45312yes—
233C2×C4#45412yes—
234C2×C4#45512yes—
235C2×C4#45612yes—
236C8—84no—
237C8—84no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22#144011yescharacteristic
249C22#183012yes—
250C22#188511yescharacteristic
251C22#188611yescharacteristic
252C22#188711yescharacteristic
253C22#188811yescharacteristic
254C4—22no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—84no—
259C4—84no—
260C4—84no—
261C4#203714yes—
262C4#203814yes—
263C2—21no—
264C2—21no—
265C2—22no—
266C2—21no—
267C2—21no—
268C2#1116111yescharacteristic
269C2#1116211yescharacteristic
270C2#1116811yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#218292
#217992
#141552
#80982