Finite Group Databasebeta

← #139113

Subgroups — #139113

Order 512.

357 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 74 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#33772C211yesmaximal, characteristic
2#22277C211yesmaximal, characteristic, complemented
3#11965C211yesmaximal, characteristic, complemented
4#11858C211yesmaximal, characteristic
5#9764C211yesmaximal, characteristic
6#7218C211yesmaximal, characteristic, complemented
7#7144C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#494C2211yescharacteristic
10#2327—21no—
11#2248C412yescomplemented
12#2241C412yescomplemented
13#2208—22no—
14#2185—22no—
15#2163C2211yescharacteristic
16#2162C2211yescharacteristic
17#1960—21no—
18#1878—22no—
19#1878—22no—
20#1835C2211yescharacteristic
21#1746C2211yescharacteristic
22#1568C2211yescharacteristic
23#1530—21no—
24#1505—21no—
25#1362—22no—
26#1347—22no—
27C22×Q16—21no—
28C22×Q16D412yes—
29#532—21no—
30C2×C4⋊Q8—21no—
31C2×C4×Q8—22no—
32C2×SD32—24no—
33#500C2×C411yescharacteristic
34#494—21no—
35#494—21no—
36#494C2×C412yes—
37#494D412yes—
38#493C2×C411yescharacteristic
39#486—21no—
40#466—21no—
41#466—21no—
42C4×D8—21no—
43C4×D8—21no—
44#432—21no—
45C2×#103—22no—
46C2×#103—22no—
47C2×#97D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#391—22no—
50#387—42no—
51#387—42no—
52#382—21no—
53#376D411yescharacteristic
54C4⋊C16—21no—
55C4⋊C16—21no—
56#358—24no—
57C22⋊C16—21no—
58C22⋊C16—21no—
59#346—21no—
60C4×C16C2×C412yes—
61#340—21no—
62#340D412yes—
63#336—21no—
64#336—21no—
65#335—21no—
66#334—21no—
67#329—42no—
68#327—42no—
69C2×#41—41no—
70C22×Q8—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—82no—
75C2×Q16—82no—
76C2×Q16C22⋊C414yes—
77C2×D8—21no—
78C2×D8—21no—
79C22×C8—21no—
80C22×C8C2×D411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—21no—
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87#127—21no—
88#124—41no—
89#121—41no—
90#121—41no—
91C4×Q8—82no—
92C4×Q8—82no—
93C4×D4—41no—
94C4×D4—41no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C42—21no—
100C2×C42—42no—
101C2×C42C2×D411yescharacteristic
102SD32—84no—
103C2×C16—22no—
104C2×C16—41no—
105#107—21no—
106#107—21no—
107C4⋊C8—41no—
108#103—82no—
109#103—82no—
110#102—41no—
111#102—41no—
112C22⋊C8—41no—
113#97—21no—
114#97—21no—
115#97—41no—
116C4×C8C22×C411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8#4111yescharacteristic
119C4×C8#4111yescharacteristic
120#95—21no—
121#95—44no—
122#95—41no—
123#95—41no—
124#95D812yes—
125#41—162no—
126C2×Q8—24no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—81no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4#12011yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4AGL(1,ℤ/8)11yescharacteristic
144Q16—44no—
145Q16—82no—
146Q16—82no—
147Q16—82no—
148D8—42no—
149D8—81no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—81no—
156C2×C8C2×C22⋊C411yescharacteristic
157C2×C8C4×D411yescharacteristic
158C2×C8#12111yescharacteristic
159C2×C8#12311yescharacteristic
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—84no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C42—41no—
172C42—41no—
173C42—84no—
174C42C4×D411yescharacteristic
175C42#12111yescharacteristic
176C42#13511yescharacteristic
177C42AGL(1,ℤ/8)11yescharacteristic
178C16—42no—
179C16—81no—
180C23—81no—
181C23#44711yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—82no—
185Q8—82no—
186Q8—161no—
187D4—81no—
188D4—81no—
189D4—161no—
190D4—161no—
191D4—162no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—82no—
206C2×C4—162no—
207C2×C4#38611yescharacteristic
208C2×C4C4×D811yescharacteristic
209C2×C4#44211yescharacteristic
210C2×C4#44911yescharacteristic
211C2×C4#45911yescharacteristic
212C2×C4#46311yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C8—161no—
219C22—21no—
220C22—21no—
221C22—161no—
222C22—161no—
223C22—161no—
224C22#187911yescharacteristic
225C22#219312yes—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—82no—
232C4—84no—
233C4—162no—
234C4—161no—
235C4#190411yescharacteristic
236C4#191111yescharacteristic
237C2—22no—
238C2—321no—
239C2#929911yescharacteristic
240C2#1301811yescharacteristic
241C2#1303411yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1746 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22277C2semidirect
3. #11965C2semidirect
6. #7218C2semidirect
7. #7144C2semidirect
11. #2248C4semidirect
12. #2241C4semidirect

Maximal subgroups

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

subgroupindex
#337722
#222772
#119652
#118582
#97642
#72182
#71442