Finite Group Databasebeta

← #101265

Subgroups — #101265

Order 512.

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

normal subgroups only

This group has 569 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#23507C211yesmaximal, characteristic
2#21957C211yesmaximal, characteristic
3#8340C212yesmaximal
4#7598C211yesmaximal, characteristic
5#7045C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C2×#433—21no—
8C2×#419—21no—
9C2×#419—21no—
10#2296C2211yescharacteristic
11#2043—22no—
12#2043C414yes—
13#1940—22no—
14#1939—22no—
15#1798—21no—
16#1797C2212yes—
17#1797C2212yes—
18#1728C2211yescharacteristic
19C2×#332—22no—
20C2×#332—22no—
21#1271—22no—
22C23×Q8—21no—
23C22×M16—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—44no—
27C2×C4⋊Q8C2×C412yes—
28C2×#117D411yescharacteristic
29C2×#117D411yescharacteristic
30C22×C4⋊C4—22no—
31C22×C42C2311yescharacteristic
32#433—42no—
33#433—42no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8D412yes—
38C2×C4⋊C8D412yes—
39#419—44no—
40#419—42no—
41#419—42no—
42C2×#103—24no—
43C2×#103—24no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×#97—21no—
47C2×C4×C8—21no—
48#389C2×C412yes—
49#389D412yes—
50#385—22no—
51C4×C4⋊C4—22no—
52#336—21no—
53#336—21no—
54#336—22no—
55#336—21no—
56#336—22no—
57#336C2×C412yes—
58#332—44no—
59#332—44no—
60C22×Q8—24no—
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C23×C4—41no—
71C23×C4C2×D411yescharacteristic
72C23×C4#4112yes—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—42no—
81C4⋊Q8—24no—
82C4⋊Q8—24no—
83C4⋊Q8—42no—
84C4⋊Q8—42no—
85C4⋊Q8—44no—
86C4⋊Q8—44no—
87#117—21no—
88#117—21no—
89#117—42no—
90#117D812yes—
91#117SD1612yes—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—48no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C22⋊C412yes—
98C2×C4⋊C4C22⋊C412yes—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—44no—
110C4⋊C8—42no—
111C4⋊C8—42no—
112C4⋊C8—82no—
113#103—48no—
114#103—48no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117C22⋊C8—82no—
118#97—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—44no—
128C2×Q8—44no—
129C2×Q8—44no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—82no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—82no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8—82no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—82no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C4×D412yes—
156C22×C4#12011yescharacteristic
157C22×C4#12112yes—
158C22×C4C2×D811yescharacteristic
159C22×C4C2×SD1611yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161C22×C4#13711yescharacteristic
162M16—82no—
163M16—82no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8—82no—
176C4⋊C4—22no—
177C4⋊C4—28no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—82no—
185C4⋊C4#10214yes—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—81no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—41no—
194C42—41no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23#38611yescharacteristic
201C23#48011yescharacteristic
202C23#48111yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—82no—
206Q8—82no—
207Q8—84no—
208Q8—84no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4—84no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C2×#10211yescharacteristic
234C2×C4#41711yescharacteristic
235C2×C4#44711yescharacteristic
236C2×C4#44811yescharacteristic
237C2×C4#45412yes—
238C2×C4B(4,𝔽2)12yes—
239C2×C4#47112yes—
240C2×C4#47312yes—
241C8—82no—
242C8—82no—
243C8—82no—
244C8—162no—
245C22—21no—
246C22—21no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—41no—
251C22—41no—
252C22#187411yescharacteristic
253C22#188811yescharacteristic
254C22#188911yescharacteristic
255C22#190212yes—
256C22#190812yes—
257C4—22no—
258C4—22no—
259C4—21no—
260C4—21no—
261C4—42no—
262C4—44no—
263C4—82no—
264C4—84no—
265C4#159812yes—
266C4#159912yes—
267C2—41no—
268C2—41no—
269C2#901911yescharacteristic
270C2#917112yes—
271C2#942211yescharacteristic
272C2#943611yescharacteristic
273C2#951912yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ 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
#235072
#219572
#83402
#75982
#70452