Finite Group Databasebeta

← #144067

Subgroups — #144067

Order 512.

387 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 84 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#33627C212yesmaximal
2#19618C211yesmaximal, characteristic, complemented
3#11936C212yesmaximal, complemented
4#9659C211yesmaximal, characteristic
5#7152C211yesmaximal, characteristic, complemented
6#3170—21no—
7#3170—21no—
8#3160C2212yes—
9C2×C4×D8C2211yescharacteristic
10#2228—24no—
11#2226—22no—
12#2224—22no—
13#2159—22no—
14#2150C2212yes—
15#2140—22no—
16#1954—21no—
17#1723—22no—
18#1723—22no—
19C2×#334C2211yescharacteristic
20C2×D4⋊C8—21no—
21C2×#322C2211yescharacteristic
22#1363—24no—
23C22×D8D411yescharacteristic
24#533—22no—
25C2×C4×D4—21no—
26C2×M32—22no—
27#500—21no—
28#500—21no—
29#500D412yes—
30#500D412yes—
31#499—22no—
32#499—22no—
33#498—22no—
34#494—21no—
35#494—21no—
36#487—22no—
37#487—22no—
38#467—42no—
39#467—42no—
40C4×Q16—42no—
41C4×Q16—42no—
42C4×D8—22no—
43C4×D8—22no—
44#427—22no—
45#427—22no—
46C2×#107D411yescharacteristic
47C2×C4⋊C8—21no—
48C2×#102—21no—
49C2×C22⋊C8—42no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#367—24no—
54#366—24no—
55C4⋊C16—22no—
56C4⋊C16D414yes—
57#357—24no—
58#357—24no—
59#334—22no—
60D4⋊C8—42no—
61#322—22no—
62C22×D4—21no—
63C23×C4—41no—
64C2×Q16—42no—
65C2×Q16—42no—
66C2×D8—22no—
67C2×D8—21no—
68C2×D8—22no—
69C2×D8Q1612yescomplemented
70C22×C8—21no—
71C22×C8—41no—
72C22×C8D812yes—
73C22×C8C2×D411yescharacteristic
74C22×C8#4111yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77#125—42no—
78C22⋊Q8—42no—
79C22⋊Q8—42no—
80C4×Q8—42no—
81C4×Q8—42no—
82C4×D4—41no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—41no—
86#117—42no—
87C2×C4⋊C4—21no—
88C2×C22⋊C4—41no—
89C2×C42C2×D411yescharacteristic
90M32—42no—
91M32—42no—
92C2×C16—24no—
93#107—22no—
94#107—22no—
95#107—22no—
96#107—22no—
97#107—22no—
98#107—42no—
99#107—42no—
100#106—42no—
101#106—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104#103—44no—
105#103—44no—
106#102—42no—
107C22⋊C8—84no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8D812yes—
112C4×C8C2×D412yes—
113C4×C8C2×D412yes—
114C24—41no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×D4—22no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C22×C4—41no—
124C22×C4—82no—
125C22×C4—82no—
126C22×C4—81no—
127C22×C4—81no—
128C22×C4#12111yescharacteristic
129C22×C4C2×D811yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131Q16—84no—
132Q16—84no—
133D8—24no—
134D8—42no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—42no—
140C2×C8—81no—
141C2×C8—82no—
142C2×C8—81no—
143C2×C8#12012yes—
144C2×C8#12112yes—
145C2×C8C2×Q1611yescharacteristic
146C2×C8#13511yescharacteristic
147C2×C8#13712yes—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—41no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C22⋊C4—82no—
157C22⋊C4—81no—
158C22⋊C4—81no—
159C42—82no—
160C42#12011yescharacteristic
161C42#12711yescharacteristic
162C42C2×D811yescharacteristic
163C42AGL(1,ℤ/8)11yescharacteristic
164C16—44no—
165C23—42no—
166C23—82no—
167C23—82no—
168C23—81no—
169C23—81no—
170C23#45911yescharacteristic
171Q8—84no—
172Q8—84no—
173D4—42no—
174D4—42no—
175D4—81no—
176D4—81no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—81no—
183C2×C4—161no—
184C2×C4—161no—
185C2×C4—162no—
186C2×C4#39211yescharacteristic
187C2×C4#44711yescharacteristic
188C2×C4#44711yescharacteristic
189C2×C4#45911yescharacteristic
190C2×C4#46711yescharacteristic
191C2×C4#47011yescharacteristic
192C2×C4#47311yescharacteristic
193C2×C4#47311yescharacteristic
194C2×C4#49311yescharacteristic
195C2×C4#49611yescharacteristic
196C8—22no—
197C8—42no—
198C8—42no—
199C8—161no—
200C8#45112yes—
201C8#45212yes—
202C22—21no—
203C22—21no—
204C22—84no—
205C22—82no—
206C22—162no—
207C22—161no—
208C22—161no—
209C22#166911yescharacteristic
210C22#168911yescharacteristic
211C22#201111yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—162no—
216C4—161no—
217C4—162no—
218C4#166911yescharacteristic
219C4#168911yescharacteristic
220C4#200111yescharacteristic
221C4#202011yescharacteristic
222C4#219012yes—
223C2—21no—
224C2—21no—
225C2—162no—
226C2#1027411yescharacteristic
227C2#1334011yescharacteristic
228C2#1334111yescharacteristic
subgroupG/H
center, socleC22#2011
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #19618C2semidirect
3. #11936C2semidirect
5. #7152C2semidirect
69. C2×D8Q16semidirect

Maximal subgroups

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

subgroupindex
#336272
#196182
#119362
#96592
#71522