Finite Group Databasebeta

← #140821

Subgroups — #140821

Order 512.

250 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 76 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#17931C211yesmaximal, characteristic
2#12014C211yesmaximal, characteristic
3#11644C211yesmaximal, characteristic
4#11640C211yesmaximal, characteristic
5#11497C211yesmaximal, characteristic
6#11495C211yesmaximal, characteristic
7#11416C211yesmaximal, characteristic
8#3104C411yescharacteristic
9#2960C2211yescharacteristic
10C2×#446C2211yescharacteristic
11#2163C2211yescharacteristic
12#2163C2211yescharacteristic
13#2163C2211yescharacteristic
14#2162C2211yescharacteristic
15#1591—21no—
16#1591—21no—
17#1584—21no—
18#1584—21no—
19#1571C411yescharacteristic
20#1487C411yescharacteristic
21#1481C411yescharacteristic
22#1450C2211yescharacteristic
23#1370—21no—
24#1367—21no—
25#1338—22no—
26Q8⋊C16—22no—
27#1323—22no—
28#1322—22no—
29#1313—21no—
30#1313—21no—
31#533C2×C411yescharacteristic
32C2×C4×Q8C2×C411yescharacteristic
33#484—21no—
34#477—21no—
35#462—21no—
36#451—21no—
37#446—21no—
38#446—21no—
39#446—21no—
40#446—21no—
41#446—21no—
42#446—21no—
43C8×Q8—21no—
44C8×Q8—21no—
45C4×Q16—21no—
46C4×Q16—21no—
47#431C2×C411yescharacteristic
48#431C2×C411yescharacteristic
49#424—21no—
50#423C2×C411yescharacteristic
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C411yescharacteristic
53#405—21no—
54C2×#97—21no—
55C2×C4×C8C2311yescharacteristic
56C4⋊C16—21no—
57C4⋊C16—21no—
58C4⋊C16—21no—
59C4⋊C16D412yes—
60C22⋊C16—21no—
61C22⋊C16—21no—
62C22⋊C16—21no—
63C22⋊C16D412yes—
64#346—21no—
65#346—21no—
66#346—21no—
67C4×C16—21no—
68#335—21no—
69#335—21no—
70Q8⋊C8—21no—
71Q8⋊C8—21no—
72Q8⋊C8—21no—
73Q8⋊C8—21no—
74#322D411yescharacteristic
75#322D411yescharacteristic
76C82D411yescharacteristic
77C82D411yescharacteristic
78C22×Q8M1611yescharacteristic
79C2×Q16—21no—
80C2×Q16—21no—
81C2×M16—21no—
82C22×C8—21no—
83C22×C8C2×D411yescharacteristic
84C22×C8#4111yescharacteristic
85C4⋊Q8M1611yescharacteristic
86C4⋊Q8M1611yescharacteristic
87#125—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96#117—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4M1611yescharacteristic
99C2×C42—21no—
100C2×C42C22×C411yescharacteristic
101C2×C16—22no—
102C2×C16—22no—
103C2×C16—41no—
104C2×C16—41no—
105#107—21no—
106#107—21no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—41no—
124#97—41no—
125#97—41no—
126C4×C8—41no—
127C4×C8C22⋊C411yescharacteristic
128C4×C8C22⋊C411yescharacteristic
129C4×C8C22⋊C411yescharacteristic
130C4×C8C22⋊C411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C4×C8C2×D411yescharacteristic
134C4×C8#4111yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—81no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4C4×D411yescharacteristic
143C22×C4C2×M1611yescharacteristic
144C22×C4#13111yescharacteristic
145Q16—81no—
146Q16—81no—
147M16—81no—
148C2×C8—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—81no—
157C2×C8#12111yescharacteristic
158C2×C8#12111yescharacteristic
159C2×C8#12411yescharacteristic
160C2×C8#12711yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C22⋊C4—41no—
172C42—41no—
173C42—41no—
174C42—41no—
175C42C2×C22⋊C411yescharacteristic
176C42C4×D411yescharacteristic
177C42C2×M1611yescharacteristic
178C42#13111yescharacteristic
179C16—42no—
180C16—42no—
181C16—81no—
182C16—81no—
183C23#43511yescharacteristic
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—41no—
189Q8—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4#39011yescharacteristic
201C2×C4#40711yescharacteristic
202C2×C4#40811yescharacteristic
203C2×C4#43511yescharacteristic
204C2×C4#43511yescharacteristic
205C2×C4#43511yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—41no—
210C8—41no—
211C8—41no—
212C8—41no—
213C8—81no—
214C8—81no—
215C8#47312yes—
216C22—21no—
217C22—21no—
218C22—21no—
219C22#195511yescharacteristic
220C4—21no—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—81no—
226C4—81no—
227C4—81no—
228C4—81no—
229C4#197611yescharacteristic
230C4#197911yescharacteristic
231C2—41no—
232C2#967111yescharacteristic
233C2#1319811yescharacteristic
234C2#1320111yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1955
prime radicalC23#435
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×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
#179312
#120142
#116442
#116402
#114972
#114952
#114162