Finite Group Databasebeta

← #28400086

Subgroups — #28400086

Order 14464.

494 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#16813648C211yesmaximal, characteristic
2C22×#14268635C211yesmaximal, characteristic, complemented
3#20299174C211yesmaximal, characteristic, complemented
4#20298813C214yesmaximal, complemented
5#20298803C214yesmaximal, complemented
6#20298625C211yesmaximal, characteristic, complemented
7#20298606C211yesmaximal, characteristic, complemented
8#20298592C211yesmaximal, characteristic, complemented
9#518×C113C211yesmaximal, characteristic, complemented
10C2×#14268647C2211yescharacteristic
11C22×Dic226C2211yescharacteristic
12#16813648—24no—
13C2×#14268635—24no—
14C2×#14268635C2212yes—
15C2×#14268635C2214yes—
16C2×#14268632C2212yes—
17C2×#14268631C2211yescharacteristic
18C22×#11744751C2211yescharacteristic
19#16813622C2212yescomplemented
20#16813610C2212yescomplemented
21#16813606—22no—
22#16813602—22no—
23C4⋊#11744751C2214yescomplemented
24#16813545C2212yescomplemented
25#16813545C2214yescomplemented
26#16813542C2212yescomplemented
27C22×D4×C113C2211yescharacteristic
28C4×D4×C113—22no—
29C4×D4×C113C2214yes—
30#117×C113C2211yescharacteristic
31#117×C113C2211yescharacteristic
32C2×C22⋊C4×C113—22no—
33#14268647—22no—
34#14268647—22no—
35C2×Dic226—22no—
36C2×Dic226—22no—
37C2×Dic226—21no—
38C2×Dic226C2312yes—
39C2×Dic226C2311yescharacteristic
40C2×D452C2311yescharacteristic
41C2×C22⋊D113—21no—
42C22×Dic113—21no—
43C2×C4×D113—21no—
44#14268635—28no—
45#14268635—24no—
46#14268632—24no—
47#14268631—22no—
48C2×#11744751—22no—
49C2×#11744751C2312yes—
50C2×#11744751C2312yes—
51C4⋊Dic113—24no—
52#14268624—24no—
53C24×C113—21no—
54C2×D4×C113—21no—
55C2×D4×C113—22no—
56C2×D4×C113D414yes—
57C2×D4×C113C2312yes—
58C22×C4×C113—24no—
59C22×C4×C113C2311yescharacteristic
60C4⋊C4×C113D412yes—
61C4⋊C4×C113C2312yes—
62C22⋊C4×C113—21no—
63C22⋊C4×C113—24no—
64C22⋊C4×C113D412yes—
65C42×C113—21no—
66C42×C113C2312yes—
67C23×C113—22no—
68C23×C113—24no—
69C23×C113—82no—
70C23×C113C2×D411yescharacteristic
71C22×D113—21no—
72D4×C113—24no—
73D4×C113—22no—
74D4×C113C2×D414yes—
75C2×C4×C113—22no—
76C2×C4×C113—84no—
77C2×C4×C113C2×D414yes—
78C2×C4×C113C2×D411yescharacteristic
79C2×C4×C113C2×D412yes—
80C2×C4×C113#4112yes—
81C2×C4×C113C2411yescharacteristic
82C22⋊D113—42no—
83C2×Dic113—22no—
84C2×Dic113—21no—
85C2×Dic113—42no—
86D452—21no—
87D452—21no—
88C4×D113—42no—
89Dic226—22no—
90Dic226—21no—
91Dic226—22no—
92Dic226—21no—
93Dic226—42no—
94#11744751—22no—
95#11744751—22no—
96C22×C113—22no—
97C22×C113—24no—
98C22×C113—44no—
99C22×C113—84no—
100C22×C113#12112yes—
101C22×C113C22×D411yescharacteristic
102C2×D113—41no—
103C2×D113—42no—
104C4×C113—24no—
105C4×C113—42no—
106C4×C113#12112yes—
107C4×C113C22×D411yescharacteristic
108C4×C113C2×#4111yescharacteristic
109Dic113—42no—
110Dic113—41no—
111C2×C113—22no—
112C2×C113—44no—
113C2×C113C2×#12111yescharacteristic
114C2×C113#57712yes—
115D113—81no—
116#3042—1131nomaximal, Sylow
117C113#304211yescharacteristic, complemented, Sylow
118C2×AGL(1,ℤ/8)—1131no—
119C22×SD16—1131no—
120#536—1131no—
121#518D11311yescharacteristic, complemented
122#463—1134no—
123#461—1134no—
124#424—1131no—
125#419—1131no—
126#416—1131no—
127C2×#41—1131no—
128C22×Q8—1131no—
129C22×D4C2×D11311yescharacteristic
130AGL(1,ℤ/8)—2264no—
131C2×SD16—1134no—
132C2×SD16—1132no—
133C2×SD16—2264no—
134C2×D8—1132no—
135C2×M16—1131no—
136C22×C8—1131no—
137C4⋊Q8—1132no—
138#124—1132no—
139#123—2262no—
140C22⋊Q8—2262no—
141C4×D4—22no—
142C4×D4C2×D11314yes—
143#117C2×D11311yescharacteristic
144#117C2×D11311yescharacteristic
145C2×C22⋊C4—22no—
146C4⋊C8—1134no—
147#103—1132no—
148#103—1134no—
149#102—1132no—
150C24—21no—
151#41—2262no—
152#41—2262no—
153C2×Q8—1132no—
154C2×Q8—1131no—
155C2×Q8—2262no—
156C2×Q8—2262no—
157C2×Q8—2261no—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—1131no—
161C2×D4—2261no—
162C2×D4D45214yes—
163C2×D4C22×D11312yes—
164C22×C4—24no—
165C22×C4—2261no—
166C22×C4—2261no—
167C22×C4C22×D11311yescharacteristic
168SD16—2268no—
169SD16—2264no—
170D8—2264no—
171M16—2262no—
172C2×C8—1132no—
173C2×C8—1132no—
174C2×C8—2262no—
175C4⋊C4—2264no—
176C4⋊C4C22⋊D11312yes—
177C4⋊C4C22×D11312yes—
178C22⋊C4—21no—
179C22⋊C4—24no—
180C22⋊C4—2264no—
181C22⋊C4C22⋊D11312yes—
182C42—21no—
183C42C22×D11312yes—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—2261no—
188C23C2×C22⋊D11311yescharacteristic
189Q8—4522no—
190Q8—2262no—
191Q8—2261no—
192Q8—2262no—
193Q8—2261no—
194D4—24no—
195D4—22no—
196D4—4522no—
197D4—2261no—
198D4—2261no—
199D4C2×D45214yes—
200C2×C4—22no—
201C2×C4—84no—
202C2×C4—4522no—
203C2×C4—4522no—
204C2×C4—2262no—
205C2×C4—2261no—
206C2×C4C2×C22⋊D11314yes—
207C2×C4C2×C22⋊D11311yescharacteristic
208C2×C4C2×D45212yes—
209C2×C4#1426864712yes—
210C2×C4C23×D11311yescharacteristic
211C8—2262no—
212C8—2262no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—4521no—
218C22—4522no—
219C22#1681359612yes—
220C22C22×C22⋊D11311yescharacteristic
221C4—24no—
222C4—42no—
223C4—4521no—
224C4—4522no—
225C4#1681359612yes—
226C4C22×D45211yescharacteristic
227C4C2×#1426864711yescharacteristic
228C2—22no—
229C2—44no—
230C2—9041no—
231C2C2×#1681359611yescharacteristic
232C2#2029940912yes—
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#518×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813648C2
hypercenter#518D113
hypocenterC113#3042
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C22×#14268635C2semidirect
3. #20299174C2semidirect
4. #20298813C2semidirect
5. #20298803C2semidirect
6. #20298625C2semidirect
7. #20298606C2semidirect
8. #20298592C2semidirect
9. #518×C113C2semidirect
19. #16813622C22semidirect
20. #16813610C22semidirect
23. C4⋊#11744751C22semidirect
24. #16813545C22semidirect
25. #16813545C22semidirect
26. #16813542C22semidirect
117. C113#3042semidirect
121. #518D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136482
C22×#142686352
#202991742
#202988132
#202988032
#202986252
#202986062
#202985922
#518×C1132
#3042113