Finite Group Databasebeta

← #105213

Subgroups — #105213

Order 512.

258 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 82 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#9873C211yesmaximal, characteristic
2#8810C211yesmaximal, characteristic
3#8591C211yesmaximal, characteristic
4#8590C211yesmaximal, characteristic
5#7040C211yesmaximal, characteristic
6#7040C211yesmaximal, characteristic
7#7018C211yesmaximal, characteristic
8C4×C4⋊Q8C2211yescharacteristic
9#1842C2211yescharacteristic
10#1842C2211yescharacteristic
11#1831C2211yescharacteristic
12#1831C2211yescharacteristic
13#1831C2211yescharacteristic
14#1744C2211yescharacteristic
15C2×#335—21no—
16C2×#335—21no—
17C2×#333—21no—
18C2×#332—21no—
19#1296—21no—
20#1296—21no—
21#1279—21no—
22#1279—21no—
23C2×C4⋊Q8D411yescharacteristic
24C2×C4×Q8—21no—
25C2×C4⋊C8—21no—
26C2×C4⋊C8—21no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4×C8D411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#391—21no—
37#389—21no—
38#389—21no—
39#384D411yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4—21no—
42C43C2311yescharacteristic
43#336—21no—
44#336—21no—
45#336—21no—
46#336—21no—
47#336—21no—
48#336—21no—
49#336—21no—
50#336—21no—
51#335—42no—
52#335—42no—
53#333—42no—
54#332—42no—
55C22×Q8—21no—
56C22×C8—21no—
57C22×C8—21no—
58C22×C8—41no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8D812yes—
67C4×Q8—82no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—41no—
74C2×C42—41no—
75C2×C42C2×D411yescharacteristic
76C2×C42C2×D411yescharacteristic
77C2×C42#4111yescharacteristic
78C2×C42#4111yescharacteristic
79C2×C42#4111yescharacteristic
80C2×C42#4111yescharacteristic
81C2×C42#4111yescharacteristic
82C4⋊C8—41no—
83C4⋊C8—41no—
84C4⋊C8—41no—
85C4⋊C8—41no—
86C4⋊C8—41no—
87C4⋊C8—41no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90C4⋊C8—41no—
91C4⋊C8—41no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—41no—
95C4⋊C8—41no—
96C4⋊C8—81no—
97C4⋊C8—81no—
98C4×C8—21no—
99C4×C8—21no—
100C4×C8SD1612yes—
101C4×C8Q1612yes—
102#95—41no—
103#95—41no—
104C2×Q8—22no—
105C2×Q8—41no—
106C2×Q8—81no—
107C2×Q8—81no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4#12111yescharacteristic
111C22×C4#12311yescharacteristic
112C22×C4#12311yescharacteristic
113C22×C4#12311yescharacteristic
114C22×C4#12411yescharacteristic
115C22×C4#12611yescharacteristic
116C22×C4#12611yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—81no—
124C2×C8—81no—
125C2×C8—81no—
126C2×C8—81no—
127C2×C8—81no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8—81no—
135C2×C8—81no—
136C2×C8—81no—
137C2×C8—81no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—81no—
145C4⋊C4—81no—
146C4⋊C4—82no—
147C42—21no—
148C42—21no—
149C42—41no—
150C42—41no—
151C42—41no—
152C42—41no—
153C42—82no—
154C42C2×D811yescharacteristic
155C42C2×SD1611yescharacteristic
156C42C2×Q1611yescharacteristic
157C42#13511yescharacteristic
158C42AGL(1,ℤ/8)11yescharacteristic
159C42#13711yescharacteristic
160C42#13711yescharacteristic
161C42#13711yescharacteristic
162C23#39711yescharacteristic
163Q8—82no—
164Q8—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4#46011yescharacteristic
186C2×C4#46211yescharacteristic
187C2×C4#46611yescharacteristic
188C2×C4#47011yescharacteristic
189C2×C4#47211yescharacteristic
190C2×C4#47311yescharacteristic
191C2×C4#48011yescharacteristic
192C2×C4#48011yescharacteristic
193C2×C4#48311yescharacteristic
194C2×C4#48411yescharacteristic
195C2×C4#48511yescharacteristic
196C2×C4#48511yescharacteristic
197C2×C4#48711yescharacteristic
198C2×C4#48911yescharacteristic
199C8—42no—
200C8—42no—
201C8—161no—
202C8—161no—
203C8—161no—
204C8—161no—
205C8—161no—
206C22#207211yescharacteristic
207C22#207511yescharacteristic
208C22#207811yescharacteristic
209C22#208011yescharacteristic
210C22#208011yescharacteristic
211C22#208111yescharacteristic
212C22#208211yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—161no—
225C4#166811yescharacteristic
226C4#167411yescharacteristic
227C4#169011yescharacteristic
228C4#169611yescharacteristic
229C2#1061711yescharacteristic
230C2#1062811yescharacteristic
231C2#1063311yescharacteristic
232C2#1063711yescharacteristic
233C2#1063911yescharacteristic
234C2#1064511yescharacteristic
235C2#1064611yescharacteristic
subgroupG/H
center, socle, prime radicalC23#397
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#98732
#88102
#85912
#85902
#70402
#70182