Finite Group Databasebeta

← #145212

Subgroups — #145212

Order 512.

275 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 64 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#19188C211yesmaximal, characteristic
2#12036C211yesmaximal, characteristic, complemented
3#11962C211yesmaximal, characteristic, complemented
4#11962C211yesmaximal, characteristic, complemented
5#11630C211yesmaximal, characteristic
6#8638C211yesmaximal, characteristic
7#7291C211yesmaximal, characteristic, complemented
8#3226—21no—
9C2×#494C2211yescharacteristic
10#2931C2211yescharacteristic
11#2162C2211yescharacteristic
12#2162C2211yescharacteristic
13#1689—22no—
14#1669—21no—
15#1669—21no—
16#1567C2211yescharacteristic
17#1567C2211yescharacteristic
18#1564C2211yescharacteristic
19#1560—21no—
20#1546—21no—
21#1372—21no—
22#1371—21no—
23#1362—22no—
24#1362—22no—
25D4⋊C16—22no—
26#1309—21no—
27C22×Q16—21no—
28C22×Q16—21no—
29C2×#131—21no—
30#532D411yescharacteristic
31C2×C4⋊Q8—21no—
32#500—41no—
33#494—21no—
34#494—21no—
35#494—21no—
36#494D412yes—
37#486—41no—
38#480—42no—
39#474—22no—
40#474—22no—
41#436—21no—
42#436—21no—
43C8×D4—21no—
44C8×D4—21no—
45#431—21no—
46#423—21no—
47#423—21no—
48#423D411yescharacteristic
49#408—21no—
50C2×C4×C8C2311yescharacteristic
51C4⋊C16—21no—
52C4⋊C16D412yes—
53C22⋊C16—21no—
54C22⋊C16—21no—
55C4×C16—21no—
56C4×C16—21no—
57#335—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335D411yescharacteristic
62#335D411yescharacteristic
63#334—21no—
64#333—41no—
65#332—22no—
66#332—22no—
67#331—41no—
68C2×#41—21no—
69C22×Q8—41no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—82no—
75C2×Q16—82no—
76#131—82no—
77C2×M16—21no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—41no—
82C22×C8#4111yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8D811yescharacteristic
89#127D811yescharacteristic
90#124—21no—
91#121—21no—
92#121—21no—
93C4×D4—21no—
94C4×D4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C42C2×D411yescharacteristic
98C2×C16—22no—
99C2×C16—41no—
100#107—42no—
101#107—42no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C4⋊C8—21no—
105C4⋊C8—21no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110#102—42no—
111#102—42no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C4×C8—41no—
116C4×C8C2×D411yescharacteristic
117C4×C8C2×D411yescharacteristic
118C4×C8C2×D411yescharacteristic
119C4×C8#4111yescharacteristic
120#41—81no—
121#41—81no—
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8—82no—
125C2×Q8—82no—
126C2×Q8—81no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—41no—
131C22×C4—41no—
132C22×C4—81no—
133C22×C4#12311yescharacteristic
134C22×C4#13511yescharacteristic
135C22×C4AGL(1,ℤ/8)11yescharacteristic
136Q16—82no—
137Q16—82no—
138Q16—82no—
139Q16—82no—
140M16—82no—
141M16—81no—
142M16—161no—
143M16—161no—
144C2×C8—22no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—82no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8#12111yescharacteristic
153C2×C8#12111yescharacteristic
154C2×C8#12311yescharacteristic
155C2×C8#12411yescharacteristic
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C22⋊C4—41no—
163C22⋊C4—41no—
164C42#12011yescharacteristic
165C42#12111yescharacteristic
166C42C2×D811yescharacteristic
167C42#13711yescharacteristic
168C16—42no—
169C16—81no—
170C23—41no—
171C23#48211yescharacteristic
172Q8—81no—
173Q8—82no—
174Q8—82no—
175Q8—162no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—82no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—162no—
193C2×C4#39411yescharacteristic
194C2×C4#44711yescharacteristic
195C2×C4#45211yescharacteristic
196C2×C4#46611yescharacteristic
197C2×C4#46811yescharacteristic
198C2×C4#48011yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—81no—
203C8—81no—
204C8—81no—
205C8—161no—
206C8—161no—
207C8#47212yes—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22#203311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—81no—
219C4—162no—
220C4#204511yescharacteristic
221C4#204811yescharacteristic
222C4#219311yescharacteristic
223C4#219511yescharacteristic
224C4#222211yescharacteristic
225C4#222311yescharacteristic
226C2—41no—
227C2—161no—
228C2#1044011yescharacteristic
229C2#1338811yescharacteristic
230C2#1339411yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#532D4
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #12036C2semidirect
3. #11962C2semidirect
4. #11962C2semidirect
7. #7291C2semidirect

Maximal subgroups

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

subgroupindex
#191882
#120362
#119622
#116302
#86382
#72912