Finite Group Databasebeta

← #28406012

Subgroups — #28406012

Order 14464.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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×#16813679C211yesmaximal, characteristic
2#20299540C211yesmaximal, characteristic
3#20299452C212yesmaximal, complemented
4#20299451C212yesmaximal, complemented
5C22⋊Q8×D113C211yesmaximal, characteristic
6C22×#11744753—21no—
7C2×C4⋊#90673—21no—
8C2×C22⋊#90673—21no—
9#16813679—21no—
10#16813679C2212yes—
11(C2×#90673)⋊C4C2212yes—
12C2×Q8×D113C411yescharacteristic, complemented
13C22×C4×D113C2211yescharacteristic
14#16813602C2211yescharacteristic
15#16813601C411yescharacteristic, complemented
16#16813600—21no—
17#16813599—21no—
18#16813598C2211yescharacteristic
19C4⋊C4×D113—21no—
20C4⋊C4×D113C411yescharacteristic, complemented
21C22⋊C4×D113—21no—
22C22⋊Q8×C113C411yescharacteristic, complemented
23C22×#90673—22no—
24C2×#11744753—22no—
25C2×#11744753—42no—
26C4⋊#90673—42no—
27C22⋊#90673—42no—
28C23×D113D411yescharacteristic
29C2×Dic226—21no—
30C2×Dic226C2×C411yescharacteristic
31Q8×D113—42no—
32C22×Dic113D411yescharacteristic
33C2×C4×D113—21no—
34C2×C4×D113—21no—
35C2×C4×D113—42no—
36C2×C4×D113C2×C411yescharacteristic
37C2×C4×D113D411yescharacteristic
38C2×C4×D113D411yescharacteristic
39C2×C4×D113C2311yescharacteristic
40C4⋊Dic113—21no—
41C4⋊Dic113C2×C411yescharacteristic
42#14268626—21no—
43#14268626—21no—
44#14268626—21no—
45C22⋊Dic113—21no—
46#14268624—21no—
47#14268624—21no—
48C2×Q8×C113C2×C411yescharacteristic
49C22×C4×C113C2×C411yescharacteristic
50C4⋊C4×C113—21no—
51C4⋊C4×C113C2×C411yescharacteristic
52C22⋊C4×C113—21no—
53C23×C113C22⋊C411yescharacteristic
54C22×D113—41no—
55C22×D113—41no—
56C22×D113C22⋊C411yescharacteristic
57C22×D113SD1612yes—
58C22×D113Q1612yes—
59C22×D113C2×D411yescharacteristic
60C2×#90673—42no—
61C2×#90673—42no—
62C2×#90673—42no—
63Q8×C113—41no—
64C2×C4×C113—21no—
65C2×C4×C113—21no—
66C2×C4×C113—41no—
67C2×C4×C113C22⋊C411yescharacteristic
68C2×C4×C113C22×C411yescharacteristic
69C2×Dic113—22no—
70C2×Dic113—21no—
71C2×Dic113—21no—
72C2×Dic113C22⋊C411yescharacteristic
73C2×Dic113C2×D411yescharacteristic
74C4×D113—22no—
75C4×D113—22no—
76C4×D113—41no—
77C4×D113—41no—
78C4×D113—42no—
79C4×D113—42no—
80Dic226—42no—
81Dic226—41no—
82#11744753—42no—
83C22×C113—21no—
84C22×C113—21no—
85C22×C113#10312yes—
86C22×C113C2×C22⋊C411yescharacteristic
87C2×D113—21no—
88C2×D113—21no—
89C2×D113—21no—
90C2×D113—21no—
91C2×D113—42no—
92C2×D113—42no—
93C2×D113#10312yes—
94C2×D113C2×SD1611yescharacteristic
95C2×D113C2×Q1611yescharacteristic
96#90673—82no—
97C4×C113—21no—
98C4×C113—21no—
99C4×C113—41no—
100C4×C113—41no—
101Dic113—21no—
102Dic113—41no—
103Dic113—41no—
104Dic113#9912yes—
105C2×C113—22no—
106C2×C113C2×#9911yescharacteristic
107C2×C113C2×#10311yescharacteristic
108C2×C113#42111yescharacteristic
109D113—21no—
110D113—22no—
111D113—21no—
112#1520—1131nomaximal, Sylow
113C113#152011yescharacteristic, complemented, Sylow
114C2×C22⋊Q8—1131no—
115C2×C22⋊C8—1131no—
116#380—1131no—
117#328—1132no—
118#328—1132no—
119C22×Q8—1131no—
120C23×C4—1131no—
121C22×C8—2261no—
122C22⋊Q8—1131no—
123C22⋊Q8—1131no—
124C22⋊Q8—1131no—
125C22⋊Q8—2261no—
126C22⋊Q8—2261no—
127C22⋊Q8#9067311yescharacteristic, complemented
128C2×C4⋊C4—1131no—
129C2×C4⋊C4—2261no—
130C2×C4⋊C4—2261no—
131C2×C22⋊C4—2261no—
132C2×C22⋊C4—2261no—
133C22⋊C8—1132no—
134C22⋊C8—2261no—
135#95—1132no—
136C24—1131no—
137C2×Q8—1131no—
138C2×Q8—4522no—
139C2×Q8—2261no—
140C2×Q8C2×#9067311yescharacteristic
141C22×C4—1131no—
142C22×C4—1131no—
143C22×C4—1131no—
144C22×C4—1131no—
145C22×C4—1131no—
146C22×C4—4522no—
147C22×C4—2261no—
148C22×C4—2261no—
149C22×C4—2262no—
150C22×C4C2×#9067311yescharacteristic
151C2×C8—4522no—
152C2×C8—2262no—
153C4⋊C4—21no—
154C4⋊C4—1131no—
155C4⋊C4—4522no—
156C4⋊C4—2261no—
157C4⋊C4—2261no—
158C4⋊C4—2261no—
159C4⋊C4—2261no—
160C4⋊C4C2×#9067311yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—4522no—
163C22⋊C4—2261no—
164C22⋊C4—2261no—
165C22⋊C4—2261no—
166C23—1132no—
167C23—1132no—
168C23—1131no—
169C23—1131no—
170C23—4521no—
171C23—4521no—
172C23C22⋊#9067311yescharacteristic
173Q8—41no—
174Q8—4521no—
175Q8—4522no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—1131no—
180C2×C4—1131no—
181C2×C4—4521no—
182C2×C4—4521no—
183C2×C4—4522no—
184C2×C4—4522no—
185C2×C4—4522no—
186C2×C4—4522no—
187C2×C4—4522no—
188C2×C4—2262no—
189C2×C4—2262no—
190C2×C4—2262no—
191C2×C4—2261no—
192C2×C4—2261no—
193C2×C4C22⋊#9067311yescharacteristic
194C2×C4C22×#9067311yescharacteristic
195C8—4522no—
196C22—21no—
197C22—21no—
198C22—1131no—
199C22—1132no—
200C22—1131no—
201C22—4522no—
202C22—4522no—
203C22—2261no—
204C22—2261no—
205C22—2261no—
206C22—2261no—
207C22#1681368512yes—
208C22C2×C22⋊#9067311yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—1132no—
214C4—9042no—
215C4—4521no—
216C4—4521no—
217C4—2261no—
218C2—22no—
219C2—2261no—
220C2—2262no—
221C2—2261no—
222C2D113⋊#9911yescharacteristic
223C2C2×#1681368511yescharacteristic
224C2#2029959511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#90673
commutatorC2×C4×C113C22×C4
FrattiniC2×C4C22×#90673
FittingC22⋊Q8×C113C4
socleC22×C113C2×C22⋊C4
prime radicalC23×D113D4
cosocleC2×C4×D113C23
hypercenterC22⋊Q8#90673
hypocenterC113#1520
LCSG ▷ C2×C4×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20299452C2semidirect
4. #20299451C2semidirect
12. C2×Q8×D113C4semidirect
15. #16813601C4semidirect
20. C4⋊C4×D113C4semidirect
22. C22⋊Q8×C113C4semidirect
113. C113#1520semidirect
127. C22⋊Q8#90673semidirect

Maximal subgroups

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

subgroupindex
C2×#168136792
#202995402
#202994522
#202994512
C22⋊Q8×D1132
#1520113