Finite Group Databasebeta

← #145238

Subgroups — #145238

Order 512.

254 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 62 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#19195C211yesmaximal, characteristic
2#12032C211yesmaximal, characteristic, complemented
3#11976C211yesmaximal, characteristic, complemented
4#11968C211yesmaximal, characteristic, complemented
5#11633C211yesmaximal, characteristic
6#8640C211yesmaximal, characteristic
7#7292C211yesmaximal, characteristic, complemented
8#3246—21no—
9#3160C2211yescharacteristic
10#2931C2211yescharacteristic
11#2163C2211yescharacteristic
12#2162C2211yescharacteristic
13#1691—22no—
14#1675—21no—
15#1673—21no—
16#1567C2211yescharacteristic
17#1566C2211yescharacteristic
18C2×#335C2211yescharacteristic
19#1560—21no—
20#1550—21no—
21#1371—22no—
22#1365—21no—
23#1364—21no—
24#1363—22no—
25#1330—22no—
26#1309—21no—
27C2×#131—21no—
28#533—21no—
29#532D411yescharacteristic
30#500D411yescharacteristic
31#500D411yescharacteristic
32#499—21no—
33#499—21no—
34#498—21no—
35#476—22no—
36#475—41no—
37#465—42no—
38#461—22no—
39#460—41no—
40#436—21no—
41#436—21no—
42C8×D4—21no—
43C8×D4—21no—
44#427—21no—
45#427—21no—
46#423—21no—
47#423—21no—
48#423D411yescharacteristic
49C2×C4⋊C8—21no—
50#408—21no—
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16D412yes—
54C22⋊C16—21no—
55C22⋊C16—21no—
56#346—21no—
57C4×C16—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335D412yes—
62#334—21no—
63#334—21no—
64#333—22no—
65#333—41no—
66#332—22no—
67#331—41no—
68C2×#41—21no—
69C2×SD16—42no—
70#131—82no—
71C2×M16—21no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8#4111yescharacteristic
78C22×C8#4111yescharacteristic
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8D811yescharacteristic
82#127D811yescharacteristic
83#125—41no—
84#124—21no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87#121—21no—
88#121—21no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×D4—21no—
92C4×D4—21no—
93#117—41no—
94C2×C42C2×D411yescharacteristic
95C2×C16—22no—
96C2×C16—41no—
97#107—21no—
98#107—21no—
99#107—21no—
100#107—21no—
101#106—42no—
102#106—41no—
103#106—41no—
104C4⋊C8—21no—
105C4⋊C8—22no—
106C4⋊C8—21no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—42no—
113#102—42no—
114C22⋊C8—41no—
115C22⋊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—41no—
124C2×Q8—41no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C22×C4—41no—
130C22×C4#12311yescharacteristic
131C22×C4#13511yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133SD16—162no—
134M16—82no—
135M16—81no—
136M16—161no—
137M16—161no—
138C2×C8—22no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—82no—
143C2×C8—81no—
144C2×C8—81no—
145C2×C8—161no—
146C2×C8#12111yescharacteristic
147C2×C8#12111yescharacteristic
148C2×C8#12311yescharacteristic
149C2×C8#12411yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4—81no—
161C42—81no—
162C42#12011yescharacteristic
163C42#12111yescharacteristic
164C42C2×D811yescharacteristic
165C42#13711yescharacteristic
166C16—42no—
167C16—81no—
168C23—41no—
169C23#48211yescharacteristic
170Q8—81no—
171Q8—81no—
172Q8—81no—
173Q8—161no—
174D4—41no—
175D4—41no—
176D4—81no—
177D4—81no—
178D4—82no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4#39411yescharacteristic
190C2×C4#44711yescharacteristic
191C2×C4#45211yescharacteristic
192C2×C4#46611yescharacteristic
193C2×C4#46811yescharacteristic
194C2×C4#48011yescharacteristic
195C8—21no—
196C8—21no—
197C8—21no—
198C8—82no—
199C8—81no—
200C8—161no—
201C8—161no—
202C8#47112yes—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22#203311yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—81no—
215C4—161no—
216C4—161no—
217C4#204611yescharacteristic
218C4#204911yescharacteristic
219C4#219211yescharacteristic
220C4#219511yescharacteristic
221C2—41no—
222C2—161no—
223C2#1044411yescharacteristic
224C2#1338411yescharacteristic
225C2#1339111yescharacteristic
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. #12032C2semidirect
3. #11976C2semidirect
4. #11968C2semidirect
7. #7292C2semidirect

Maximal subgroups

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

subgroupindex
#191952
#120322
#119762
#119682
#116332
#86402
#72922