Finite Group Databasebeta

← #145869

Subgroups — #145869

Order 512.

305 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 100 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#28814C211yesmaximal, characteristic
2#13177C211yesmaximal, characteristic
3#12888C211yesmaximal, characteristic
4#9615C211yesmaximal, characteristic
5#7274C211yesmaximal, characteristic
6#7171C211yesmaximal, characteristic
7#7167C211yesmaximal, characteristic
8C2×#499—21no—
9#2552—21no—
10C2×#367—21no—
11C2×C4⋊C16C2211yescharacteristic
12C2×C4×C16C2211yescharacteristic
13#2059—22no—
14#1948—21no—
15#1948—21no—
16#1942—21no—
17#1845C2211yescharacteristic
18#1845C2211yescharacteristic
19C4×#107C2211yescharacteristic
20C8×C4⋊C4C2211yescharacteristic
21C2×#335C2211yescharacteristic
22#1380—22no—
23#1380—21no—
24#1296—21no—
25C2×#125—22no—
26C22×C16—21no—
27C22×C16—21no—
28#499—42no—
29#499—42no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#107D412yes—
36C2×#106—22no—
37C2×#106—41no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4×C8—21no—
41C2×C4×C8C2311yescharacteristic
42#398—42no—
43#389—22no—
44#389—41no—
45#384—21no—
46#384—21no—
47#384—41no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4D411yescharacteristic
50#367—42no—
51C4⋊C16—21no—
52C4⋊C16Q812yes—
53C4×C16—21no—
54C4×C16Q812yes—
55#340—22no—
56#340—21no—
57#340—41no—
58#336—21no—
59#335—21no—
60#335Q812yes—
61C22×C8—21no—
62C22×C8—41no—
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65#125—44no—
66#125—82no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4D811yescharacteristic
79C2×C4⋊C4D811yescharacteristic
80C2×C42—21no—
81C2×C42—41no—
82C2×C42C2×D411yescharacteristic
83C2×C16—22no—
84C2×C16—22no—
85C2×C16—41no—
86C2×C16—41no—
87C2×C16—41no—
88C2×C16—41no—
89#107—21no—
90#107—21no—
91#107—41no—
92#107—41no—
93#107—42no—
94#107—42no—
95#107—41no—
96#107—41no—
97#107SD1612yes—
98#107SD1612yes—
99#106—44no—
100#106—82no—
101C4⋊C8—22no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8Q1612yes—
105C4×C8—41no—
106C4×C8—41no—
107C4×C8C2×Q811yescharacteristic
108C4×C8C2×Q811yescharacteristic
109C4×C8C2×Q811yescharacteristic
110C4×C8#4111yescharacteristic
111#95—21no—
112#95—42no—
113#95—41no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—81no—
120C22×C4#12011yescharacteristic
121C22×C4C2×D811yescharacteristic
122C22×C4AGL(1,ℤ/8)11yescharacteristic
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—42no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8C22⋊Q811yescharacteristic
132C2×C8C22⋊Q811yescharacteristic
133C2×C8C4⋊Q811yescharacteristic
134C2×C8C4⋊Q811yescharacteristic
135C2×C8C2×SD1611yescharacteristic
136C2×C8C2×SD1611yescharacteristic
137C2×C8AGL(1,ℤ/8)11yescharacteristic
138C2×C8#13711yescharacteristic
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—84no—
150C4⋊C4—84no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—161no—
158C4⋊C4—161no—
159C4⋊C4—162no—
160C42—41no—
161C42—41no—
162C42—81no—
163C42—81no—
164C42C22⋊Q811yescharacteristic
165C42C4⋊Q811yescharacteristic
166C42C2×Q1611yescharacteristic
167C42#13711yescharacteristic
168C16—42no—
169C16—42no—
170C23#44711yescharacteristic
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—161no—
189C2×C4—161no—
190C2×C4—161no—
191C2×C4#39511yescharacteristic
192C2×C4#44811yescharacteristic
193C2×C4#45011yescharacteristic
194C2×C4#45111yescharacteristic
195C2×C4#45411yescharacteristic
196C2×C4#45511yescharacteristic
197C2×C4#47411yescharacteristic
198C2×C4#47711yescharacteristic
199C2×C4#50011yescharacteristic
200C2×C4#50111yescharacteristic
201C2×C4#50811yescharacteristic
202C2×C4#50811yescharacteristic
203C2×C4#50911yescharacteristic
204C2×C4#51011yescharacteristic
205C8—21no—
206C8—21no—
207C8—42no—
208C8—82no—
209C8#47511yescharacteristic
210C8#47611yescharacteristic
211C8#49811yescharacteristic
212C8#50111yescharacteristic
213C22#162711yescharacteristic
214C22#163911yescharacteristic
215C22#205711yescharacteristic
216C22#218611yescharacteristic
217C22#218911yescharacteristic
218C22#219311yescharacteristic
219C22#219411yescharacteristic
220C4—21no—
221C4—21no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—82no—
226C4—162no—
227C4—321no—
228C4#163711yescharacteristic
229C4#164511yescharacteristic
230C4#205811yescharacteristic
231C4#206111yescharacteristic
232C4#222811yescharacteristic
233C4#222911yescharacteristic
234C4#225311yescharacteristic
235C4#225511yescharacteristic
236C2#1057511yescharacteristic
237C2#1235311yescharacteristic
238C2#1235911yescharacteristic
239C2#1236911yescharacteristic
240C2#1237511yescharacteristic
241C2#1345311yescharacteristic
242C2#1345611yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#288142
#131772
#128882
#96152
#72742
#71712
#71672