Finite Group Databasebeta

← #506322

Subgroups — #506322

Order 512.

478 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 144 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#33474C211yesmaximal, characteristic
2C2×C4×Q32C211yesmaximal, characteristic
3#28297C212yesmaximal
4#13517C212yesmaximal
5#13441C212yesmaximal
6#13145C214yesmaximal
7#12891C211yesmaximal, characteristic
8#12819C212yesmaximal
9C22×Q32C2211yescharacteristic
10#3078—22no—
11#2960C2211yescharacteristic
12C2×C4×Q16C2212yes—
13#2183—21no—
14#2183—21no—
15Q8×C16—21no—
16Q8×C16—21no—
17C4×Q32—21no—
18C4×Q32—21no—
19C4×Q32—21no—
20C4×Q32—21no—
21C2×#366C2211yescharacteristic
22#2151C2212yes—
23#2150C2211yescharacteristic
24C2×#362C2214yes—
25C2×#358C2212yes—
26#2108C2212yes—
27C2×C4×C16C2211yescharacteristic
28#2064—24no—
29#1979C2214yes—
30#1952C2214yes—
31#1848C2212yes—
32#1801—22no—
33#1387C2214yes—
34#1381C2212yes—
35#1381C2212yes—
36C22×Q16C2312yes—
37#533D411yescharacteristic
38C2×C4×Q8—22no—
39C2×Q32—24no—
40C2×Q32—21no—
41C2×Q32—21no—
42C2×M32C2312yes—
43C22×C16C2311yescharacteristic
44C22×C16C2311yescharacteristic
45#479—44no—
46#477—42no—
47#475—42no—
48#446—21no—
49#446—21no—
50C8×Q8—21no—
51C8×Q8—21no—
52C4×Q16—22no—
53C4×Q16—22no—
54C4×Q16—22no—
55C4×Q16—22no—
56C2×#108—22no—
57C2×#108C2314yes—
58#428—24no—
59C2×#107C2312yes—
60#424D412yes—
61#423—22no—
62#423D411yescharacteristic
63#419—24no—
64C2×#103—22no—
65C2×#101—24no—
66#405C2312yes—
67C2×C4×C8C2311yescharacteristic
68#366—21no—
69#366—21no—
70C4⋊C16—22no—
71C4⋊C16—22no—
72C4⋊C16—21no—
73C4⋊C16—21no—
74#362—28no—
75#358—22no—
76#358—22no—
77#346—22no—
78C4×C16—22no—
79C4×C16D411yescharacteristic
80C4×C16D411yescharacteristic
81C4×C16D411yescharacteristic
82C4×C16D411yescharacteristic
83#343—24no—
84C22×Q8—22no—
85C2×Q16—22no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16#4114yes—
91C2×M16—24no—
92C2×M16C2×D412yes—
93C22×C8C2×D411yescharacteristic
94C22×C8C2411yescharacteristic
95C4⋊Q8D811yescharacteristic
96C4⋊Q8D811yescharacteristic
97#125D812yes—
98C22⋊Q8—21no—
99C22⋊Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—42no—
103C4×Q8—42no—
104C4×Q8—42no—
105C4×Q8—42no—
106#117C2×D412yes—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C42—42no—
110C2×C42C2×D411yescharacteristic
111Q32—24no—
112Q32—24no—
113M32—24no—
114C2×C16—22no—
115C2×C16—22no—
116C2×C16—22no—
117C2×C16C2×D411yescharacteristic
118C2×C16C2×D411yescharacteristic
119C2×C16C2×D411yescharacteristic
120C2×C16C2×D411yescharacteristic
121#108—24no—
122#108—24no—
123#108—82no—
124#107—22no—
125#107—22no—
126#107—44no—
127#106—44no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134#103—42no—
135#103—42no—
136#103—44no—
137#103—44no—
138#101—84no—
139#97—22no—
140C4×C8—22no—
141C4×C8C2×D411yescharacteristic
142C4×C8C2×D411yescharacteristic
143C4×C8#4111yescharacteristic
144C4×C8#4111yescharacteristic
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4#13511yescharacteristic
156C22×C4#13511yescharacteristic
157C22×C4C22×D411yescharacteristic
158Q16—24no—
159Q16—24no—
160Q16—42no—
161Q16—42no—
162M16—24no—
163M16—44no—
164M16—44no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—44no—
168C2×C8#12111yescharacteristic
169C2×C8#12111yescharacteristic
170C2×C8C22×D411yescharacteristic
171C2×C8C2×#4112yes—
172C2×C8C2×#4111yescharacteristic
173C2×C82−1+412yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4C2×D812yes—
183C4⋊C4C2×D812yes—
184C22⋊C4—22no—
185C42—22no—
186C42—82no—
187C42—82no—
188C42#12111yescharacteristic
189C42#12111yescharacteristic
190C42C2×D811yescharacteristic
191C42C2×D811yescharacteristic
192C16—21no—
193C16—21no—
194C16—21no—
195C16—21no—
196C16—42no—
197C23C2×#13511yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—44no—
201Q8—44no—
202Q8—82no—
203Q8—82no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—162no—
213C2×C4#46611yescharacteristic
214C2×C4#46611yescharacteristic
215C2×C4#46611yescharacteristic
216C2×C4#46611yescharacteristic
217C2×C4C2×#12111yescharacteristic
218C2×C4#54812yes—
219C2×C4C22×D811yescharacteristic
220C2×C4#57712yes—
221C8—21no—
222C8—21no—
223C8—42no—
224C8—84no—
225C8#52511yescharacteristic
226C8#53611yescharacteristic
227C8#55212yes—
228C22—21no—
229C22—21no—
230C22C2×#46611yescharacteristic
231C22#336012yes—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—84no—
238C4—162no—
239C4#222212yes—
240C4#253611yescharacteristic
241C4#306011yescharacteristic
242C4#335812yes—
243C2—22no—
244C2#2818511yescharacteristic
245C2C2×#222211yescharacteristic
246C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC22×C8C24
prime radicalC23C2×#135
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334742
C2×C4×Q322
#282972
#135172
#134412
#131452
#128912
#128192