Finite Group Databasebeta

← #134618

Subgroups — #134618

Order 512.

314 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4×#499C211yesmaximal, characteristic
2#12879C211yesmaximal, characteristic
3C2×#1363C212yesmaximal
4#8807C211yesmaximal, characteristic
5#7275C212yesmaximal
6C2×#499C2211yescharacteristic
7C4×#125—21no—
8C2×C4⋊C16C2212yes—
9#1991—21no—
10#1942C411yescharacteristic
11#1942C411yescharacteristic
12#1941—21no—
13#1846C2211yescharacteristic
14#1831—21no—
15C4×#107C411yescharacteristic
16C4×#107C411yescharacteristic
17C4×#106—21no—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334C2212yes—
21#1374—21no—
22#1374—21no—
23#1363—24no—
24C2×#125—21no—
25C22×C16—22no—
26#499—21no—
27#499—21no—
28#499D412yes—
29#499Q812yes—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35C2×#106—21no—
36C2×C4⋊C8—22no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8C2311yescharacteristic
43#389—21no—
44#389—21no—
45#387—41no—
46#387—41no—
47#382—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—21no—
50C4×C4⋊C4—41no—
51C4×C4⋊C4—41no—
52C43D411yescharacteristic
53C4⋊C16—22no—
54C4⋊C16—22no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#336—41no—
60#336—41no—
61#335—42no—
62#334—22no—
63#334—22no—
64C22×C8—42no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69#125—22no—
70#125—22no—
71#125—41no—
72#125—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C42—41no—
82C2×C42—41no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42D811yescharacteristic
86C2×C42D811yescharacteristic
87C2×C42SD1611yescharacteristic
88C2×C42SD1611yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C16—42no—
91C2×C16—42no—
92C2×C16—42no—
93#107—21no—
94#107—21no—
95#107—41no—
96#107—41no—
97#107C4⋊C412yes—
98#107C4⋊C412yes—
99#106—22no—
100#106—22no—
101#106—42no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—41no—
110C4×C8C2×D411yescharacteristic
111C4×C8C2×Q811yescharacteristic
112C4×C8#4111yescharacteristic
113C4×C8#4111yescharacteristic
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4#10211yescharacteristic
125C22×C4#10211yescharacteristic
126C22×C4#10211yescharacteristic
127C22×C4#10211yescharacteristic
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4C2×SD1611yescharacteristic
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8—82no—
141C2×C8#10112yes—
142C2×C8C2×C4⋊C411yescharacteristic
143C2×C8#11711yescharacteristic
144C2×C8#12111yescharacteristic
145C2×C8C22⋊Q811yescharacteristic
146C2×C8#13512yes—
147C2×C8#13712yes—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C42—22no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—82no—
171C42—82no—
172C42#12111yescharacteristic
173C42C22⋊Q811yescharacteristic
174C42AGL(1,ℤ/8)11yescharacteristic
175C42#13711yescharacteristic
176C16—82no—
177C23C2×#10211yescharacteristic
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4#35912yes—
206C2×C4#38011yescharacteristic
207C2×C4C2×#10111yescharacteristic
208C2×C4#41311yescharacteristic
209C2×C4#41611yescharacteristic
210C2×C4#41711yescharacteristic
211C2×C4#41911yescharacteristic
212C2×C4#45911yescharacteristic
213C2×C4#46111yescharacteristic
214C2×C4#47411yescharacteristic
215C2×C4#47611yescharacteristic
216C8—21no—
217C8—21no—
218C8—22no—
219C8—41no—
220C8—162no—
221C8#46412yes—
222C8#47912yes—
223C22#154011yescharacteristic
224C22#154611yescharacteristic
225C22#180211yescharacteristic
226C22C2×#35911yescharacteristic
227C22#214311yescharacteristic
228C22#214511yescharacteristic
229C22#214511yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—22no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—82no—
239C4—82no—
240C4—161no—
241C4—161no—
242C4#155211yescharacteristic
243C4#155911yescharacteristic
244C4#180111yescharacteristic
245C4#180511yescharacteristic
246C2#858911yescharacteristic
247C2#1194311yescharacteristic
248C2#1194611yescharacteristic
249C2#1196711yescharacteristic
250C2#1197011yescharacteristic
251C2#1281511yescharacteristic
252C2#1281711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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
C4×#4992
#128792
C2×#13632
#88072
#72752