Finite Group Databasebeta

← #139172

Subgroups — #139172

Order 512.

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

normal subgroups only

This group has 294 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#33496C211yesmaximal, characteristic
2#19265C211yesmaximal, characteristic
3#11863C211yesmaximal, characteristic
4#11650C211yesmaximal, characteristic
5#8661C211yesmaximal, characteristic
6#7289C211yesmaximal, characteristic
7#7220C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3113—21no—
10C2×C8×Q8C2211yescharacteristic
11Q8×C16—22no—
12Q8×C16—22no—
13#2162—22no—
14#2162C2211yescharacteristic
15#2162C2211yescharacteristic
16#1837C2211yescharacteristic
17#1680—22no—
18#1674C412yes—
19#1670C412yes—
20#1568C2211yescharacteristic
21#1564C2211yescharacteristic
22#1552—21no—
23#1487—21no—
24#1350—22no—
25#1332—22no—
26#1301—21no—
27#533—21no—
28C2×C4×Q8D411yescharacteristic
29#500C2×C411yescharacteristic
30#494C2×C411yescharacteristic
31#487—21no—
32#483—41no—
33#475—22no—
34#467—21no—
35#467—21no—
36#462—22no—
37#451—41no—
38C8×Q8—21no—
39C8×Q8—21no—
40C8×Q8C2×C412yes—
41C8×Q8D412yes—
42#441—42no—
43C4×Q16—21no—
44C4×Q16—21no—
45#432—21no—
46#431—21no—
47#423—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—24no—
53C4⋊C16—21no—
54C4⋊C16—21no—
55C22⋊C16—21no—
56C22⋊C16D412yes—
57C4×C16—24no—
58C4×C16—21no—
59C4×C16D412yes—
60#341—21no—
61#341—21no—
62#335—21no—
63#335C2×C412yes—
64#334—21no—
65#334—21no—
66#332—22no—
67#332—22no—
68Q8⋊C8—42no—
69C22×Q8SD1611yescharacteristic
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—42no—
73C2×M16—41no—
74C22×C8—21no—
75C22×C8C2×D411yescharacteristic
76C22×C8#4111yescharacteristic
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79#125—41no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×Q8C22⋊C412yes—
87C4×Q8D812yes—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4SD1611yescharacteristic
91C2×C42—21no—
92C2×C42C2×D411yescharacteristic
93C2×C16—22no—
94C2×C16—22no—
95#107—21no—
96#107—21no—
97#106—42no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—22no—
101C4⋊C8—21no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8C22⋊C412yes—
106#103—42no—
107#103—41no—
108#103—41no—
109#103—42no—
110C22⋊C8—41no—
111C22⋊C8—41no—
112#97—41no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—41no—
116C4×C8C22×C411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—41no—
123C2×Q8—41no—
124C2×Q8#10212yes—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4#12111yescharacteristic
128C22×C4C2×SD1611yescharacteristic
129C22×C4#13511yescharacteristic
130Q16—42no—
131Q16—81no—
132Q16—84no—
133M16—81no—
134M16—81no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—42no—
139C2×C8—81no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8C4×D411yescharacteristic
143C2×C8C4×D411yescharacteristic
144C2×C8#12011yescharacteristic
145C2×C8#12311yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4#10212yes—
156C22⋊C4—81no—
157C42—22no—
158C42—22no—
159C42—81no—
160C42C2×C22⋊C411yescharacteristic
161C42#12111yescharacteristic
162C42C2×D811yescharacteristic
163C42#13511yescharacteristic
164C16—42no—
165C16—42no—
166C23#46511yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8—81no—
170Q8—82no—
171Q8—81no—
172Q8#35914yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4#38611yescharacteristic
184C2×C4C2×#10211yescharacteristic
185C2×C4#41611yescharacteristic
186C2×C4#46511yescharacteristic
187C2×C4#46611yescharacteristic
188C2×C4#46611yescharacteristic
189C8—21no—
190C8—21no—
191C8—21no—
192C8—21no—
193C8—42no—
194C8—82no—
195C8—161no—
196C8—161no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22#189311yescharacteristic
201C4—24no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—42no—
207C4—161no—
208C4—161no—
209C4#188111yescharacteristic
210C4#190511yescharacteristic
211C4C2×#35912yes—
212C2—41no—
213C2#930611yescharacteristic
214C2#1306311yescharacteristic
215C2#1306911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ 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
#334962
#192652
#118632
#116502
#86612
#72892
#72202