Finite Group Databasebeta

← #141574

Subgroups — #141574

Order 512.

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

normal subgroups only

This group has 317 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#28676C211yesmaximal, characteristic
2#11795C212yesmaximal
3#11574C212yesmaximal
4#8623C211yesmaximal, characteristic
5C8×#101C211yesmaximal, characteristic
6C2×#500C411yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2567C2212yes—
9#2087—21no—
10#2087—21no—
11#1987C412yes—
12#1942—21no—
13#1942—21no—
14C4×#107C411yescharacteristic
15C4×#101C2211yescharacteristic
16#1568—21no—
17#1467C2212yes—
18#1461C2211yescharacteristic
19C8×M16—21no—
20#1354—24no—
21#1325—24no—
22#1290—21no—
23#1283—21no—
24C2×C4⋊Q8C2×C412yes—
25C2×C4×Q8C2311yescharacteristic
26#500—41no—
27#500—41no—
28C8×Q8—41no—
29C8×Q8—41no—
30#432—22no—
31#431—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8D411yescharacteristic
37C2×C4⋊C8D411yescharacteristic
38C2×#101—21no—
39C4×M16—21no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43#395—22no—
44#395—22no—
45#391—22no—
46#389—22no—
47#389—22no—
48#384—22no—
49C4×C4⋊C4C2×C412yes—
50#343D412yes—
51#340—21no—
52#340—22no—
53#340—22no—
54#335—41no—
55#334—41no—
56#324—21no—
57#324—22no—
58#324—22no—
59#324D412yes—
60#322—41no—
61C82—41no—
62C22×Q8—22no—
63C22×Q8#4111yescharacteristic
64C2×M16—22no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C4⋊Q8—42no—
70C4⋊Q8—42no—
71C4×Q8—41no—
72C4×Q8—41no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4C2×D411yescharacteristic
79C2×C4⋊C4C2×D411yescharacteristic
80C2×C4⋊C4#4111yescharacteristic
81C2×C42—22no—
82C2×C42C22×C411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D411yescharacteristic
85#107—24no—
86#107—42no—
87#107—41no—
88#107—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—42no—
94#101—24no—
95C22⋊C8—22no—
96C22⋊C8—42no—
97#97—41no—
98#97—42no—
99C4×C8—21no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—21no—
103C4×C8—21no—
104C4×C8—21no—
105C4×C8—41no—
106C4×C8—41no—
107#95—22no—
108#95—42no—
109#95—42no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—41no—
113C2×Q8—82no—
114C2×Q8—162no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C4×D411yescharacteristic
119C22×C4C4×D411yescharacteristic
120C22×C4#12111yescharacteristic
121C22×C4#12111yescharacteristic
122C22×C4#12411yescharacteristic
123C22×C4#12711yescharacteristic
124M16—24no—
125M16—42no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—42no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—42no—
135C2×C8—82no—
136C2×C8#9911yescharacteristic
137C2×C8#9911yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C4⋊C4—162no—
147C4⋊C4—162no—
148C42—21no—
149C42—21no—
150C42—21no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—44no—
155C23#39011yescharacteristic
156Q8—44no—
157Q8—164no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—44no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4C2×#9911yescharacteristic
180C2×C4#41011yescharacteristic
181C2×C4#44312yes—
182C2×C4#47312yes—
183C8—21no—
184C8—44no—
185C8—41no—
186C8—41no—
187C8—41no—
188C8—82no—
189C8—162no—
190C8#35612yes—
191C22—21no—
192C22—21no—
193C22#196411yescharacteristic
194C22#197411yescharacteristic
195C22#197911yescharacteristic
196C4—22no—
197C4—22no—
198C4—21no—
199C4—41no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4—84no—
204C4—162no—
205C4C2×#35611yescharacteristic
206C4#213511yescharacteristic
207C2—21no—
208C2—21no—
209C2#981411yescharacteristic
210C2#1322711yescharacteristic
211C2#1323111yescharacteristic
subgroupG/H
center, socleC22#1964
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC23#390
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#286762
#117952
#115742
#86232
C8×#1012