Finite Group Databasebeta

← D23×#487 #15683705

Subgroups — D23×#487 #15683705

Order 2944.

424 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D23C211yesmaximal, characteristic
2C4⋊Q8×D23C211yesmaximal, characteristic
3#13092769C211yesmaximal, characteristic, complemented
4#13092768C211yesmaximal, characteristic, complemented
5#13092767C211yesmaximal, characteristic, complemented
6#13092766C214yesmaximal, complemented
7#103×D23C214yesmaximal
8C4×C8×D23C211yesmaximal, characteristic
9C23×#487C211yesmaximal, characteristic, complemented
10C2×Q8×D23—21no—
11C2×Q8×D23C2212yes—
12C23×C4⋊Q8C2211yescharacteristic
13C23×C4⋊Q8C2211yescharacteristic
14#10625115—21no—
15#10625115C2212yes—
16C2×C8×D23C2212yes—
17C4⋊C4×D23—21no—
18C4⋊C4×D23—21no—
19C4⋊C4×D23—21no—
20C4⋊C4×D23C2212yes—
21#10625059—21no—
22#10625059—21no—
23#10625059—21no—
24#10625059C2212yes—
25C42×D23C2211yescharacteristic
26#10625040C2211yescharacteristic
27#10625040C2211yescharacteristic
28C23×#103C2214yes—
29C4×C8×C23C2211yescharacteristic
30#10625017C2214yes—
31#10624998C2214yes—
32C8×Dic23C2212yes—
33#10624992C2214yes—
34C4×C23⋊C8C2211yescharacteristic
35C2×Q8×C23—21no—
36C2×Q8×C23C2312yes—
37Q8×D23—28no—
38Q8×D23—44no—
39C2×C4×D23—22no—
40C2×C4×D23—22no—
41C2×C4×D23D411yescharacteristic
42C2×C4×D23D411yescharacteristic
43C2×C4×D23C2311yescharacteristic
44C2×Dic46—21no—
45C2×Dic46—22no—
46C2×Dic46—22no—
47C2×Dic46C2312yes—
48C2×C8×C23C2312yes—
49C4⋊C4×C23—21no—
50C4⋊C4×C23—21no—
51C4⋊C4×C23—21no—
52C4⋊C4×C23C2312yes—
53C42×C23C2311yescharacteristic
54C4⋊Dic23—21no—
55C4⋊Dic23—21no—
56C4⋊Dic23—21no—
57C4⋊Dic23C2312yes—
58Dic23⋊C4—22no—
59Dic23⋊C4—22no—
60Dic23⋊C4—22no—
61Dic23⋊C4—22no—
62C4×Dic23D411yescharacteristic
63C4×Dic23D411yescharacteristic
64C4×Dic23C2311yescharacteristic
65C2×C23⋊C8C2312yes—
66C8×D23—24no—
67C22×D23C2×D411yescharacteristic
68Q8×C23—24no—
69Q8×C23—42no—
70C2×C4×C23—22no—
71C2×C4×C23—22no—
72C2×C4×C23C2×D411yescharacteristic
73C2×C4×C23C2×D411yescharacteristic
74C2×C4×C23C2411yescharacteristic
75C2×Dic23—22no—
76C2×Dic23—22no—
77C2×Dic23C2×D411yescharacteristic
78C2×Dic23C2×D411yescharacteristic
79C2×Dic23C2×D411yescharacteristic
80C4×D23—44no—
81C4×D23—44no—
82C4×D23SD1614yes—
83C4×D23Q1614yes—
84C4×D23#4114yes—
85Dic46—24no—
86Dic46—42no—
87Dic46—44no—
88Dic46—44no—
89C8×C23—22no—
90C23⋊C8—22no—
91C2×#487—231nomaximal, Sylow
92C22×C23C22×D411yescharacteristic
93C2×D23#12412yes—
94C2×D23C2×SD1612yes—
95C2×D23C2×Q1612yes—
96C4×C23—42no—
97C4×C23—42no—
98C4×C23C2×SD1612yes—
99C4×C23C2×Q1612yes—
100C4×C23C2×#4112yes—
101Dic23—42no—
102Dic23—42no—
103Dic23#12412yes—
104Dic23C2×SD1612yes—
105Dic23C2×Q1612yes—
106C2×C4⋊Q8—231no—
107C2×C4⋊Q8—231no—
108#487—231no—
109#487—231no—
110#487—234no—
111#487—231no—
112#487D2311yescharacteristic, direct product factor
113C2×#103—234no—
114C2×C4×C8—231no—
115C2×C23C2×#12411yescharacteristic
116C2×C23C22×SD1611yescharacteristic
117C2×C23C22×Q1611yescharacteristic
118D23#48714yesdirect product factor
119C22×Q8—232no—
120C22×Q8—461no—
121C22×C8—232no—
122C4⋊Q8—232no—
123C4⋊Q8—232no—
124C4⋊Q8—231no—
125C4⋊Q8—231no—
126C4⋊Q8—461no—
127C4⋊Q8—461no—
128C4⋊Q8—461no—
129C4⋊Q8—461no—
130C4⋊Q8C2×D2311yescharacteristic
131C4⋊Q8C2×D2311yescharacteristic
132C2×C4⋊C4—232no—
133C2×C4⋊C4—461no—
134C2×C4⋊C4—461no—
135C2×C4⋊C4—461no—
136C2×C42—231no—
137#103—234no—
138#103—234no—
139#103—234no—
140#103C2×D2314yes—
141C4×C8—232no—
142C4×C8—231no—
143C4×C8C2×D2311yescharacteristic
144C23C2×#48711yescharacteristic, complemented, Sylow
145C2×Q8—21no—
146C2×Q8—232no—
147C2×Q8—462no—
148C2×Q8—462no—
149C2×Q8—468no—
150C2×Q8—461no—
151C2×Q8—924no—
152C2×Q8C22×D2312yes—
153C22×C4—231no—
154C22×C4—231no—
155C22×C4—231no—
156C22×C4—462no—
157C22×C4—462no—
158C2×C8—232no—
159C2×C8—464no—
160C2×C8C22×D2312yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—232no—
165C4⋊C4—462no—
166C4⋊C4—462no—
167C4⋊C4—462no—
168C4⋊C4—462no—
169C4⋊C4—461no—
170C4⋊C4—461no—
171C4⋊C4—461no—
172C4⋊C4C22×D2312yes—
173C42—231no—
174C42—231no—
175C42—231no—
176C42C22×D2311yescharacteristic
177C23—231no—
178Q8—24no—
179Q8—42no—
180Q8—464no—
181Q8—924no—
182Q8—924no—
183Q8—922no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—231no—
187C2×C4—231no—
188C2×C4—231no—
189C2×C4—234no—
190C2×C4—234no—
191C2×C4—234no—
192C2×C4—462no—
193C2×C4—462no—
194C2×C4—924no—
195C2×C4—924no—
196C2×C4D4×D2311yescharacteristic
197C2×C4D4×D2311yescharacteristic
198C2×C4C23×D2311yescharacteristic
199C8—22no—
200C8—462no—
201C22—232no—
202C22—232no—
203C22—232no—
204C22C2×D4×D2311yescharacteristic
205C4—42no—
206C4—42no—
207C4—232no—
208C4—232no—
209C4—232no—
210C4—922no—
211C4—922no—
212C4SD16×D2312yes—
213C4Q16×D2312yes—
214C4#41×D2312yes—
215C2—234no—
216C2#124×D2311yescharacteristic
217C2C2×SD16×D2311yescharacteristic
218C2C2×Q16×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#487C2
socleC22×C23C22×D4
prime radicalC22×D23C2×D4
hypercenter#487D23
hypocenterC23C2×#487
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #13092769C2semidirect
4. #13092768C2semidirect
5. #13092767C2semidirect
6. #13092766C2semidirect
9. C23×#487C2semidirect
112. #487D23direct (N × H)
118. D23#487direct (N × H)
144. C23C2×#487semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D232
#130927692
#130927682
#130927672
#130927662
#103×D232
C4×C8×D232
C23×#4872
C2×#48723