Finite Group Databasebeta

← #99709

Subgroups — #99709

Order 512.

515 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 98 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#23551C211yesmaximal, characteristic
2#21957C211yesmaximal, characteristic
3#8314C212yesmaximal
4#7564C211yesmaximal, characteristic
5#7029C212yesmaximal
6C2×#533C2211yescharacteristic
7C2×#433—21no—
8C2×#427—21no—
9#2296C2211yescharacteristic
10#2059—22no—
11#2058C412yes—
12#2057C412yes—
13#1940—22no—
14#1939—22no—
15#1798—21no—
16#1797C2212yes—
17C4×C22⋊C8C2212yes—
18#1735—21no—
19C2×#336C2211yescharacteristic
20C2×Q8⋊C8—22no—
21C2×Q8⋊C8—22no—
22#1271—22no—
23C23×C8—21no—
24#533—44no—
25C2×C4⋊Q8C2×C412yes—
26C2×#125—21no—
27C2×C22⋊Q8—22no—
28C2×C4×Q8—22no—
29C2×#117—21no—
30C2×#117D411yescharacteristic
31C2×#117D411yescharacteristic
32C22×C42C2311yescharacteristic
33#433—42no—
34#433—42no—
35#427—44no—
36C2×#107—22no—
37C2×#106—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×#97—21no—
45C2×C4×C8—21no—
46C2×C4×C8—22no—
47#389C2×C412yes—
48#389D412yes—
49#385—22no—
50C4×C4⋊C4—22no—
51#340—22no—
52#340—22no—
53#336—21no—
54#336—21no—
55#336—22no—
56#336—21no—
57#336—22no—
58#336C2×C411yescharacteristic
59#336C2×C411yescharacteristic
60Q8⋊C8—44no—
61Q8⋊C8—44no—
62C22×Q8—22no—
63C23×C4C2×D411yescharacteristic
64C23×C4#4112yes—
65C22×C8—24no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—42no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—42no—
74C4⋊Q8—44no—
75#125—44no—
76C22⋊Q8—44no—
77C22⋊Q8—44no—
78C4×Q8—24no—
79C4×Q8—24no—
80C4×Q8—42no—
81C4×Q8—42no—
82#117—21no—
83#117—21no—
84#117—24no—
85#117—42no—
86#117—42no—
87#117SD1612yes—
88#117Q1612yes—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4C22⋊C412yes—
95C2×C4⋊C4C22⋊C412yes—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—41no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—42no—
102C2×C42—41no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D412yes—
105C2×C42C2×D411yescharacteristic
106#107—44no—
107#106—44no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110C4⋊C8—44no—
111C4⋊C8—82no—
112C22⋊C8—24no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#97—42no—
117C4×C8—42no—
118C4×C8—42no—
119C4×C8—42no—
120#95—22no—
121#95—22no—
122C24#12311yescharacteristic
123C2×Q8—24no—
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C4×D412yes—
140C22×C4#12011yescharacteristic
141C22×C4#12112yes—
142C22×C4C2×SD1611yescharacteristic
143C22×C4C2×Q1611yescharacteristic
144C22×C4AGL(1,ℤ/8)11yescharacteristic
145C22×C4#13711yescharacteristic
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—82no—
162C2×C8—82no—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—82no—
174C4⋊C4#10314yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—44no—
178C22⋊C4—81no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—41no—
184C42—41no—
185C42—42no—
186C42—44no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23#38611yescharacteristic
192C23#48311yescharacteristic
193C23#48411yescharacteristic
194Q8—44no—
195Q8—44no—
196Q8—82no—
197Q8—82no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4C2×#10311yescharacteristic
221C2×C4#41711yescharacteristic
222C2×C4#44312yes—
223C2×C4#44412yes—
224C2×C4#45011yescharacteristic
225C2×C4#45111yescharacteristic
226C2×C4#45612yes—
227C2×C4B(4,𝔽2)12yes—
228C8—42no—
229C8—42no—
230C8—84no—
231C8—82no—
232C8—82no—
233C22—22no—
234C22—22no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22#187211yescharacteristic
241C22#187612yes—
242C22#188112yes—
243C22#188611yescharacteristic
244C22#188911yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—44no—
250C4—42no—
251C4—82no—
252C4—84no—
253C4#160412yes—
254C4#160512yes—
255C2—41no—
256C2—41no—
257C2#899312yes—
258C2#900611yescharacteristic
259C2#901511yescharacteristic
260C2#903712yes—
261C2#906811yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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
#235512
#219572
#83142
#75642
#70292