Finite Group Databasebeta

← #144068

Subgroups — #144068

Order 512.

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

normal subgroups only

This group has 361 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#33628C212yesmaximal
2#19618C211yesmaximal, characteristic
3#11936C212yesmaximal
4#9658C211yesmaximal, characteristic
5#7152C211yesmaximal, characteristic
6#3170—21no—
7#3170—21no—
8#3160C2212yes—
9C2×C4×Q16C2211yescharacteristic
10#2229—24no—
11#2227—22no—
12#2225—22no—
13#2159—22no—
14#2150C2212yes—
15#2141—22no—
16#1984—21no—
17#1723—22no—
18#1723—22no—
19C2×#334C2211yescharacteristic
20C2×Q8⋊C8—21no—
21C2×#322C2211yescharacteristic
22#1363—24no—
23C22×Q16D411yescharacteristic
24#533—22no—
25C2×C4×Q8—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—22no—
41C4×Q16—22no—
42C4×Q16—42no—
43C4×Q16—42no—
44#427—22no—
45#427—22no—
46C2×#107D411yescharacteristic
47C2×C4⋊C8—21no—
48C2×#103—21no—
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52#367—24no—
53#366—24no—
54C4⋊C16—22no—
55C4⋊C16D414yes—
56#358—24no—
57#358—24no—
58#336—42no—
59#334—22no—
60Q8⋊C8—42no—
61#322—22no—
62C22×Q8—21no—
63C2×Q16—22no—
64C2×Q16—21no—
65C2×Q16—22no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16Q1612yescomplemented
69C22×C8—21no—
70C22×C8—41no—
71C22×C8D812yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4111yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—22no—
76#125—42no—
77C22⋊Q8—42no—
78C22⋊Q8—42no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×Q8—41no—
82C4×Q8—42no—
83C4×Q8—41no—
84C4×Q8—42no—
85#117—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C42—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#103—42no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8D812yes—
111C4×C8C2×D412yes—
112C4×C8C2×D412yes—
113C2×Q8—22no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4#12111yescharacteristic
124C22×C4C2×D811yescharacteristic
125C22×C4AGL(1,ℤ/8)11yescharacteristic
126Q16—24no—
127Q16—42no—
128Q16—84no—
129Q16—84no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—81no—
136C2×C8—82no—
137C2×C8—81no—
138C2×C8#12012yes—
139C2×C8#12112yes—
140C2×C8C2×Q1611yescharacteristic
141C2×C8#13511yescharacteristic
142C2×C8#13712yes—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—81no—
150C4⋊C4—82no—
151C4⋊C4—81no—
152C4⋊C4—82no—
153C22⋊C4—82no—
154C42—81no—
155C42—82no—
156C42—81no—
157C42#12011yescharacteristic
158C42#12711yescharacteristic
159C42C2×D811yescharacteristic
160C42AGL(1,ℤ/8)11yescharacteristic
161C16—44no—
162C23#45911yescharacteristic
163Q8—42no—
164Q8—42no—
165Q8—84no—
166Q8—81no—
167Q8—81no—
168Q8—84no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4—82no—
175C2×C4—81no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4#39211yescharacteristic
179C2×C4#44711yescharacteristic
180C2×C4#44711yescharacteristic
181C2×C4#45911yescharacteristic
182C2×C4#46711yescharacteristic
183C2×C4#47011yescharacteristic
184C2×C4#47311yescharacteristic
185C2×C4#47311yescharacteristic
186C2×C4#49311yescharacteristic
187C2×C4#49611yescharacteristic
188C8—22no—
189C8—42no—
190C8—42no—
191C8—161no—
192C8#45112yes—
193C8#45212yes—
194C22—21no—
195C22—21no—
196C22#166911yescharacteristic
197C22#168911yescharacteristic
198C22#201111yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—82no—
203C4—161no—
204C4—162no—
205C4—162no—
206C4#166911yescharacteristic
207C4#168911yescharacteristic
208C4#200111yescharacteristic
209C4#202011yescharacteristic
210C4#219112yes—
211C2—21no—
212C2—21no—
213C2#1027411yescharacteristic
214C2#1334011yescharacteristic
215C2#1334211yescharacteristic
subgroupG/H
center, socleC22#2011
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#459
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
68. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#336282
#196182
#119362
#96582
#71522