Finite Group Databasebeta

← #141589

Subgroups — #141589

Order 512.

365 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 72 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#27661C211yesmaximal, characteristic
2#11780C212yesmaximal
3#11562C212yesmaximal
4#8612C211yesmaximal, characteristic
5C8×#100C211yesmaximal, characteristic
6C2×C8×D4C2211yescharacteristic
7C22×#107C411yescharacteristic
8#2486C2212yes—
9#2086—21no—
10#2075—21no—
11#1942—21no—
12#1823C412yes—
13#1810—21no—
14#1808—21no—
15#1808C411yescharacteristic
16C4×#100C2211yescharacteristic
17#1465C2212yes—
18#1460C2211yescharacteristic
19C8×M16—21no—
20#1346—24no—
21#1317—24no—
22#1293—21no—
23#1283—21no—
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—22no—
27C2×C4×D4C2311yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C8×D4—41no—
30C8×D4—41no—
31#431—21no—
32C2×#108—41no—
33C2×#107—21no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—41no—
37C2×#107—41no—
38C2×C4⋊C8D411yescharacteristic
39C2×#100—21no—
40#407—22no—
41C2×C22⋊C8C2×C411yescharacteristic
42C2×C22⋊C8D411yescharacteristic
43C4×M16—21no—
44C2×C4×C8D411yescharacteristic
45#398—22no—
46#397—22no—
47#389—22no—
48#381—22no—
49#380—22no—
50#380C2×C412yes—
51#343D412yes—
52#341—41no—
53#340—22no—
54#340—21no—
55#340—22no—
56#324—22no—
57#323—21no—
58#323—22no—
59#323D412yes—
60#322—41no—
61C82—41no—
62C22×D4#4111yescharacteristic
63C23×C4—22no—
64C23×C4C22×C411yescharacteristic
65C23×C4C2×D411yescharacteristic
66C2×M16—22no—
67C2×M16—42no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78#123—84no—
79C4×D4—41no—
80C4×D4—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C22⋊C4—22no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4C2×D411yescharacteristic
91C2×C22⋊C4#4111yescharacteristic
92C2×C42C2×D411yescharacteristic
93#108—82no—
94#107—42no—
95#107—42no—
96#107—41no—
97#107—41no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100#100—24no—
101C22⋊C8—22no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—41no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—41no—
110C4×C8—41no—
111#95—22no—
112#95—42no—
113#95—42no—
114C24C4×D411yescharacteristic
115C24#12411yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—41no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—82no—
131C22×C4—82no—
132C22×C4C2×C22⋊C411yescharacteristic
133C22×C4C4×D411yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12111yescharacteristic
136C22×C4#12711yescharacteristic
137M16—24no—
138M16—42no—
139M16—82no—
140M16—82no—
141C2×C8—22no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—42no—
153C2×C8—81no—
154C2×C8—82no—
155C2×C8#9911yescharacteristic
156C2×C8#9911yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4—84no—
164C22⋊C4—41no—
165C22⋊C4—41no—
166C22⋊C4—84no—
167C22⋊C4—82no—
168C42—21no—
169C42—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#39011yescharacteristic
180C23#44312yes—
181D4—44no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—164no—
205C2×C4C2×#9911yescharacteristic
206C2×C4#41011yescharacteristic
207C2×C4#47212yes—
208C8—21no—
209C8—44no—
210C8—41no—
211C8—41no—
212C8—41no—
213C8—82no—
214C8—162no—
215C8#35212yes—
216C22—21no—
217C22—21no—
218C22—22no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—81no—
228C22#196411yescharacteristic
229C22#197411yescharacteristic
230C22#197811yescharacteristic
231C4—22no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—162no—
237C4—162no—
238C4C2×#35211yescharacteristic
239C4#212311yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2—81no—
244C2#981511yescharacteristic
245C2#1321711yescharacteristic
246C2#1323311yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4#41
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#276612
#117802
#115622
#86122
C8×#1002