Finite Group Databasebeta

← #101850

Subgroups — #101850

Order 512.

403 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 102 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#28511C211yesmaximal, characteristic
2#23249C211yesmaximal, characteristic
3#9903C212yesmaximal
4#7653C211yesmaximal, characteristic
5#7022C212yesmaximal
6C2×#477—21no—
7C2×#477—21no—
8#2868C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#2095—21no—
11#2095—21no—
12#1985—21no—
13#1984—21no—
14#1956C412yes—
15#1940—22no—
16#1845C412yes—
17#1831C2212yes—
18#1771—22no—
19#1767C2212yes—
20C4×C4⋊C8C2211yescharacteristic
21#1296—22no—
22#1296—22no—
23#1272—21no—
24#1272—21no—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8C2×C411yescharacteristic
28C2×C4×Q8D411yescharacteristic
29C2×C4×Q8D411yescharacteristic
30#477—42no—
31#477—42no—
32#477—42no—
33#477—42no—
34C2×#107—22no—
35C2×#107—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D412yes—
41C2×C4⋊C8D412yes—
42C2×#103—22no—
43C2×#103—22no—
44C2×#97—22no—
45C2×#97—21no—
46C2×C4×C8—21no—
47C2×C4×C8C2×C412yes—
48#391D411yescharacteristic
49#391D411yescharacteristic
50#389—22no—
51#384—22no—
52#384—21no—
53#384C2×C412yes—
54#384C2×C411yescharacteristic
55C4×C4⋊C4—22no—
56C43C2311yescharacteristic
57#336—22no—
58#336—22no—
59#336—21no—
60#336—22no—
61#336—22no—
62#336—22no—
63C22×Q8—41no—
64C22×Q8C22⋊C411yescharacteristic
65C22×Q8C22⋊C411yescharacteristic
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—42no—
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76C4⋊Q8—84no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4C22⋊C411yescharacteristic
90C2×C4⋊C4C22⋊C411yescharacteristic
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C22×C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D412yes—
97C2×C42#4112yes—
98#107—44no—
99#107—44no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—81no—
108C4⋊C8—81no—
109C4⋊C8Q1614yes—
110#103—44no—
111#103—44no—
112#97—42no—
113#97—42no—
114#97—81no—
115C4×C8—42no—
116C4×C8—81no—
117#95—22no—
118#95—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—81no—
127C2×Q8—81no—
128C2×Q8—84no—
129C2×Q8#10112yes—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—42no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D412yes—
137C22×C4#12011yescharacteristic
138C22×C4#12112yes—
139C22×C4#12311yescharacteristic
140C2×C8—24no—
141C2×C8—24no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—82no—
152C4⋊C4—44no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—82no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—84no—
161C4⋊C4—84no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—42no—
170C42—41no—
171C42—41no—
172C42—41no—
173C42—41no—
174C42C2×Q1612yes—
175C42#13512yes—
176C42#13712yes—
177C42#13712yes—
178C23#38611yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—81no—
182Q8—81no—
183Q8—162no—
184Q8—162no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—82no—
204C2×C4—84no—
205C2×C4C2×#10111yescharacteristic
206C2×C4#41311yescharacteristic
207C2×C4C4×Q1612yes—
208C2×C4#44112yes—
209C2×C4#45112yes—
210C2×C4#45212yes—
211C2×C4#45311yescharacteristic
212C2×C4#45311yescharacteristic
213C2×C4#46212yes—
214C2×C4#46412yes—
215C8—44no—
216C8—44no—
217C8—82no—
218C8—162no—
219C22#187812yes—
220C22#188711yescharacteristic
221C22#190111yescharacteristic
222C22#190312yes—
223C22#190911yescharacteristic
224C4—22no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—42no—
232C4—81no—
233C4—81no—
234C4—81no—
235C4—162no—
236C4#163212yes—
237C4#163311yescharacteristic
238C4#163411yescharacteristic
239C2#906611yescharacteristic
240C2#925012yes—
241C2#928812yes—
242C2#942311yescharacteristic
243C2#945511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
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
#285112
#232492
#99032
#76532
#70222