Finite Group Databasebeta

← #145935

Subgroups — #145935

Order 512.

255 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 78 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#13212C211yesmaximal, characteristic
2#13163C211yesmaximal, characteristic
3#12009C211yesmaximal, characteristic
4#11932C211yesmaximal, characteristic
5#9733C211yesmaximal, characteristic
6#9627C211yesmaximal, characteristic
7#7281C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#364—21no—
10#2150C2211yescharacteristic
11C2×#362—21no—
12C2×#362—21no—
13C4×M32C2211yescharacteristic
14#1988—21no—
15#1848C2211yescharacteristic
16C8×C4⋊C4C2211yescharacteristic
17#1571—21no—
18#1564C2211yescharacteristic
19C2×#335C2211yescharacteristic
20C2×#332—21no—
21#1376—21no—
22#1372—21no—
23#1371—21no—
24#1362—21no—
25#1362—21no—
26#1303—21no—
27C22×Q16—21no—
28C22×Q16—21no—
29C2×C4⋊Q8—21no—
30C2×M32—21no—
31C2×M32D411yescharacteristic
32C2×M32D411yescharacteristic
33#494—21no—
34#494—21no—
35#494—21no—
36#494Q812yes—
37#431—21no—
38C2×#108—21no—
39C2×#108—21no—
40#423—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4×C8—21no—
44C2×C4×C8C2311yescharacteristic
45C4×C4⋊C4D411yescharacteristic
46#364—42no—
47C4⋊C16—21no—
48C4⋊C16—21no—
49#362—42no—
50#362—42no—
51#346—21no—
52C4×C16—21no—
53#340—42no—
54#336—21no—
55#335—21no—
56#335—21no—
57#335—42no—
58#335D411yescharacteristic
59#335D411yescharacteristic
60#334—21no—
61#332—42no—
62C22×Q8—41no—
63C2×Q16—22no—
64C2×Q16—22no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16—81no—
72C2×Q16—81no—
73C2×M16—41no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8C2×D411yescharacteristic
77C22×C8#4111yescharacteristic
78C4⋊Q8—22no—
79C4⋊Q8—41no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C42—21no—
86C2×C42C2×D411yescharacteristic
87M32—21no—
88M32—21no—
89M32—21no—
90M32—41no—
91M32—41no—
92M32D812yes—
93C2×C16—21no—
94C2×C16—21no—
95C2×C16—41no—
96#108—41no—
97#108—41no—
98#108—81no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8D812yes—
107C22⋊C8—41no—
108C4×C8—41no—
109C4×C8—41no—
110C4×C8—41no—
111C4×C8C2×D411yescharacteristic
112C4×C8C2×Q811yescharacteristic
113C4×C8#4111yescharacteristic
114C4×C8#4111yescharacteristic
115#95—21no—
116C2×Q8—42no—
117C2×Q8—81no—
118C2×Q8—81no—
119C2×Q8—81no—
120C2×Q8—81no—
121C2×Q8—81no—
122C22×C4—41no—
123C22×C4—81no—
124C22×C4#12111yescharacteristic
125C22×C4AGL(1,ℤ/8)11yescharacteristic
126C22×C4#13711yescharacteristic
127Q16—42no—
128Q16—42no—
129Q16—82no—
130Q16—82no—
131Q16—81no—
132Q16—81no—
133M16—81no—
134M16—81no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—82no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8#12111yescharacteristic
146C2×C8C22⋊Q811yescharacteristic
147C2×C8C22⋊Q811yescharacteristic
148C2×C8#12611yescharacteristic
149C2×C8C2×D811yescharacteristic
150C2×C8#13511yescharacteristic
151C2×C8AGL(1,ℤ/8)11yescharacteristic
152C2×C8#13711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C42—41no—
160C42—41no—
161C42#12111yescharacteristic
162C42C22⋊Q811yescharacteristic
163C42C2×D811yescharacteristic
164C42#13511yescharacteristic
165C16—22no—
166C16—41no—
167C16—81no—
168C23#46811yescharacteristic
169Q8—82no—
170Q8—82no—
171Q8—161no—
172Q8—161no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—82no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4#39611yescharacteristic
183C2×C4#45911yescharacteristic
184C2×C4#45911yescharacteristic
185C2×C4#46411yescharacteristic
186C2×C4#46411yescharacteristic
187C2×C4#46611yescharacteristic
188C2×C4#47111yescharacteristic
189C2×C4#47211yescharacteristic
190C2×C4#47411yescharacteristic
191C2×C4#47811yescharacteristic
192C8—21no—
193C8—21no—
194C8—42no—
195C8—81no—
196C8—81no—
197C8—161no—
198C8—161no—
199C8#46611yescharacteristic
200C8#46811yescharacteristic
201C8#47411yescharacteristic
202C8#47811yescharacteristic
203C22—21no—
204C22—21no—
205C22#168511yescharacteristic
206C22#169811yescharacteristic
207C22#206711yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—82no—
212C4—162no—
213C4#166711yescharacteristic
214C4#168211yescharacteristic
215C4#206311yescharacteristic
216C4#206711yescharacteristic
217C4#221211yescharacteristic
218C4#221411yescharacteristic
219C2—21no—
220C2—21no—
221C2#1059111yescharacteristic
222C2#1347311yescharacteristic
223C2#1347911yescharacteristic
subgroupG/H
center, socleC22#2067
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#468
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ 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
#132122
#131632
#120092
#119322
#97332
#96272
#72812