Finite Group Databasebeta

← #139097

Subgroups — #139097

Order 512.

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

normal subgroups only

This group has 301 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#33775C211yesmaximal, characteristic
2#17991C211yesmaximal, characteristic
3#11965C211yesmaximal, characteristic
4#11865C211yesmaximal, characteristic
5#9779C211yesmaximal, characteristic
6#7220C211yesmaximal, characteristic
7#7147C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9C2×#494C2211yescharacteristic
10#2985—21no—
11#2247C412yes—
12#2241C412yes—
13#2209—22no—
14#2188—22no—
15#2163C2211yescharacteristic
16#2162C2211yescharacteristic
17#1837C2211yescharacteristic
18#1587—22no—
19#1584—22no—
20#1568C2211yescharacteristic
21#1546—21no—
22#1546—21no—
23#1497—21no—
24#1452C2211yescharacteristic
25#1362—22no—
26#1350—22no—
27C22×Q16—21no—
28C22×Q16—21no—
29#533—21no—
30C2×C4⋊Q8—21no—
31C2×Q32—24no—
32#500C2×C411yescharacteristic
33#494—21no—
34#494—21no—
35#494—21no—
36#494C2×C412yes—
37#494C2×C411yescharacteristic
38#487—21no—
39#467—21no—
40#467D412yes—
41#446—22no—
42#446—42no—
43C8×Q8—22no—
44C4×Q16—21no—
45C4×Q16D412yes—
46#433—42no—
47#432—21no—
48#431D411yescharacteristic
49#431D411yescharacteristic
50#423—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54#358—24no—
55C22⋊C16—21no—
56C22⋊C16—21no—
57#346—21no—
58C4×C16C2×C412yes—
59#341—21no—
60#341—21no—
61#335D412yes—
62#334—21no—
63#332—42no—
64#332—42no—
65Q8⋊C8—24no—
66#322—21no—
67#322—21no—
68C22×Q8—41no—
69C2×Q16—22no—
70C2×Q16—21no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—82no—
75C2×Q16—82no—
76C2×Q16C22⋊C412yes—
77C2×M16—41no—
78C22×C8C2×D411yescharacteristic
79C22×C8#4111yescharacteristic
80C4⋊Q8—21no—
81C4⋊Q8—21no—
82C4⋊Q8—22no—
83C4⋊Q8—41no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86#125—41no—
87C22⋊Q8—22no—
88C22⋊Q8—41no—
89C4×Q8—22no—
90C4×Q8—41no—
91#117—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C42C2×D411yescharacteristic
95Q32—84no—
96C2×C16—22no—
97C2×C16—41no—
98#107—21no—
99#107C22⋊C412yes—
100C4⋊C8—22no—
101C4⋊C8—21no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8D812yes—
107#103—22no—
108#103—22no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—41no—
112#97—42no—
113#97—41no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—42no—
117C4×C8C22×C411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8C2×D411yescharacteristic
120C4×C8#4111yescharacteristic
121C2×Q8—22no—
122C2×Q8—41no—
123C2×Q8—42no—
124C2×Q8—82no—
125C2×Q8—82no—
126C2×Q8—81no—
127C22×C4—81no—
128C22×C4#12111yescharacteristic
129C22×C4#13511yescharacteristic
130C22×C4AGL(1,ℤ/8)11yescharacteristic
131Q16—42no—
132Q16—44no—
133Q16—81no—
134Q16—82no—
135Q16—82no—
136Q16—82no—
137M16—81no—
138M16—81no—
139C2×C8—22no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—81no—
145C2×C8C2×C22⋊C411yescharacteristic
146C2×C8C4×D411yescharacteristic
147C2×C8#12111yescharacteristic
148C2×C8#12311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C22⋊C4—42no—
160C42—42no—
161C42C4×D411yescharacteristic
162C42#12011yescharacteristic
163C42C2×D811yescharacteristic
164C42AGL(1,ℤ/8)11yescharacteristic
165C16—42no—
166C16—81no—
167C23#46311yescharacteristic
168Q8—42no—
169Q8—42no—
170Q8—82no—
171Q8—82no—
172Q8—161no—
173Q8—162no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—82no—
181C2×C4—162no—
182C2×C4#38611yescharacteristic
183C2×C4C4×D811yescharacteristic
184C2×C4#44211yescharacteristic
185C2×C4#44711yescharacteristic
186C2×C4#44911yescharacteristic
187C2×C4#45911yescharacteristic
188C8—21no—
189C8—21no—
190C8—21no—
191C8—21no—
192C8—42no—
193C8—81no—
194C8—82no—
195C8—161no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22#187911yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—82no—
205C4—82no—
206C4—162no—
207C4#190311yescharacteristic
208C4#191011yescharacteristic
209C4#219512yes—
210C2—41no—
211C2#929311yescharacteristic
212C2#1302011yescharacteristic
213C2#1303511yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#463
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452 ◁ 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
#337752
#179912
#119652
#118652
#97792
#72202
#71472