Finite Group Databasebeta

← #139387

Subgroups — #139387

Order 512.

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

normal subgroups only

This group has 291 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#33500C211yesmaximal, characteristic
2#28699C211yesmaximal, characteristic
3#11972C211yesmaximal, characteristic
4#11971C211yesmaximal, characteristic
5#8639C211yesmaximal, characteristic
6#7298C211yesmaximal, characteristic
7#7298C211yesmaximal, characteristic
8#3165—21no—
9C2×#500C2211yescharacteristic
10#2960C2211yescharacteristic
11#2183—21no—
12#2183—21no—
13Q8×C16—21no—
14Q8×C16—21no—
15#2164—22no—
16#2162C2211yescharacteristic
17#2162C2211yescharacteristic
18#2089C414yes—
19#1997—21no—
20#1952—22no—
21C4×#108C2211yescharacteristic
22#1567C2211yescharacteristic
23#1567C2211yescharacteristic
24#1561—21no—
25#1365—21no—
26#1365—21no—
27#1364—21no—
28#1364—21no—
29#1309—21no—
30#533D411yescharacteristic
31C2×C4⋊Q8—21no—
32#501—42no—
33#500—21no—
34#500—21no—
35#500—21no—
36#500—21no—
37#500—41no—
38#498—41no—
39#446—21no—
40#446—21no—
41C8×Q8—21no—
42C8×Q8—21no—
43C2×#108—21no—
44#428—24no—
45C2×#107—21no—
46C2×#107C2×C412yes—
47#424C2×C412yes—
48#423—21no—
49#423—21no—
50#423D411yescharacteristic
51#405D412yes—
52C4×M16—21no—
53C2×C4×C8C2311yescharacteristic
54C4⋊C16—22no—
55C4⋊C16—22no—
56C4⋊C16D411yescharacteristic
57C4⋊C16D411yescharacteristic
58C4⋊C16D411yescharacteristic
59C4⋊C16D411yescharacteristic
60C22⋊C16—21no—
61C22⋊C16—21no—
62#346—22no—
63C4×C16—22no—
64C4×C16—21no—
65C4×C16—21no—
66#343—24no—
67#341C2×C412yes—
68#335—21no—
69#335—21no—
70#335—21no—
71#335—21no—
72#332—41no—
73#332—41no—
74C22×Q8—41no—
75C2×M16—22no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16C22⋊C412yes—
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8D811yescharacteristic
86C4⋊Q8D811yescharacteristic
87#125SD1612yes—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92#117C22⋊C412yes—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C42C2×D411yescharacteristic
96C2×C16—21no—
97C2×C16—21no—
98C2×C16—21no—
99C2×C16—21no—
100#108—22no—
101#108—22no—
102#107—22no—
103#107—22no—
104#107—41no—
105#107—41no—
106#107—44no—
107#106—44no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116#97—22no—
117#97—41no—
118C4×C8—22no—
119C4×C8—41no—
120C4×C8C2×D411yescharacteristic
121C4×C8C2×D411yescharacteristic
122C4×C8#4111yescharacteristic
123C4×C8#4111yescharacteristic
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—81no—
127C2×Q8—81no—
128C2×Q8—161no—
129C2×Q8—161no—
130C22×C4—42no—
131C22×C4C2×C22⋊C411yescharacteristic
132C22×C4#13511yescharacteristic
133C22×C4#13511yescharacteristic
134M16—24no—
135M16—42no—
136M16—42no—
137M16—161no—
138M16—161no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—41no—
143C2×C8—42no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8C4×D411yescharacteristic
147C2×C8C4×D411yescharacteristic
148C2×C8#12011yescharacteristic
149C2×C8#12311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4#10212yes—
159C4⋊C4#10212yes—
160C22⋊C4—22no—
161C42—22no—
162C42#12111yescharacteristic
163C42#12111yescharacteristic
164C42C2×D811yescharacteristic
165C42C2×SD1611yescharacteristic
166C16—41no—
167C16—41no—
168C16—41no—
169C16—41no—
170C23#41611yescharacteristic
171Q8—41no—
172Q8—41no—
173Q8—41no—
174Q8—41no—
175Q8—162no—
176Q8—161no—
177Q8—161no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—162no—
188C2×C4#38611yescharacteristic
189C2×C4C2×#10211yescharacteristic
190C2×C4#46511yescharacteristic
191C2×C4#46511yescharacteristic
192C2×C4#46611yescharacteristic
193C2×C4#46611yescharacteristic
194C8—21no—
195C8—21no—
196C8—21no—
197C8—44no—
198C8—42no—
199C8—161no—
200C8—161no—
201C8#45611yescharacteristic
202C8#45611yescharacteristic
203C22—21no—
204C22—21no—
205C22—21no—
206C22#189311yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—162no—
213C4#188611yescharacteristic
214C4#191511yescharacteristic
215C4#222111yescharacteristic
216C4#222211yescharacteristic
217C4#222311yescharacteristic
218C4#222311yescharacteristic
219C2—41no—
220C2#939411yescharacteristic
221C2#1307211yescharacteristic
222C2#1307311yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#416
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#335002
#286992
#119722
#119712
#86392
#72982