Finite Group Databasebeta

← D23×#356 #15675076

Subgroups — D23×#356 #15675076

Order 2944.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
2#101×D23C212yesmaximal
3#13092216C211yesmaximal, characteristic, complemented
4#13092215C212yesmaximal, complemented
5C23×#356C211yesmaximal, characteristic, complemented
6C2×Q8×D23C411yescharacteristic
7C2×Q8×D23C2211yescharacteristic
8C23×C4⋊Q8C2211yescharacteristic
9#10625115C411yescharacteristic
10#10625115C2211yescharacteristic
11M16×D23—22no—
12C4⋊C4×D23—22no—
13#10625059—22no—
14C42×D23C411yescharacteristic
15#10625040C411yescharacteristic
16C23×#101C2212yes—
17#10625018C2212yes—
18#10625006—22no—
19C2×Q8×C23C2×C411yescharacteristic
20C2×Q8×C23C2311yescharacteristic
21Q8×D23—41no—
22Q8×D23—41no—
23Q8×D23—41no—
24Q8×D23—41no—
25C2×C4×D23—21no—
26C2×C4×D23—21no—
27C2×C4×D23C2×C411yescharacteristic
28C2×C4×D23D412yes—
29C2×Dic46—21no—
30C2×Dic46C2×C411yescharacteristic
31C2×Dic46C2×C411yescharacteristic
32C2×Dic46D412yes—
33M16×C23—22no—
34C4⋊C4×C23—22no—
35C42×C23C2×C411yescharacteristic
36C4⋊Dic23—22no—
37Dic23⋊C4—22no—
38Dic23⋊C4—22no—
39C4×Dic23—21no—
40C4×Dic23C2×C411yescharacteristic
41#67267—22no—
42#67262—42no—
43C8×D23—42no—
44C22×D23C22⋊C411yescharacteristic
45Q8×C23—41no—
46Q8×C23—41no—
47C2×C4×C23—21no—
48C2×C4×C23—21no—
49C2×C4×C23C22×C411yescharacteristic
50C2×C4×C23C2×D412yes—
51C2×Dic23—21no—
52C2×Dic23—21no—
53C2×Dic23C22⋊C411yescharacteristic
54C2×Dic23C22⋊C412yes—
55C4×D23—24no—
56C4×D23—22no—
57C4×D23—22no—
58C4×D23—21no—
59C4×D23—21no—
60C4×D23—41no—
61C4×D23—41no—
62Dic46—41no—
63Dic46—42no—
64Dic46—42no—
65Dic46—41no—
66C8×C23—42no—
67C23⋊C8—42no—
68C2×#356—231nomaximal, Sylow
69C22×C23C2×C22⋊C411yescharacteristic
70C2×D23—22no—
71C2×D23#9911yescharacteristic
72C2×D23#9911yescharacteristic
73C4×C23—22no—
74C4×C23—22no—
75C4×C23—21no—
76C4×C23—41no—
77Dic23—22no—
78Dic23—21no—
79Dic23—22no—
80Dic23—41no—
81C2×C4⋊Q8—231no—
82C2×#101—232no—
83#356—231no—
84#356—232no—
85#356D2311yescharacteristic, direct product factor
86C2×C23—21no—
87C2×C23C2×#9911yescharacteristic
88D23—21no—
89D23#35612yesdirect product factor
90C22×Q8—231no—
91C22×Q8—231no—
92C2×M16—462no—
93C4⋊Q8—231no—
94C4⋊Q8—231no—
95C4⋊Q8—231no—
96C4⋊Q8—462no—
97C4⋊Q8C2×D2311yescharacteristic
98C2×C4⋊C4—462no—
99C2×C42—231no—
100#101—232no—
101#101—462no—
102#101C2×D2312yes—
103C23C2×#35611yescharacteristic, complemented, Sylow
104C2×Q8—231no—
105C2×Q8—232no—
106C2×Q8—231no—
107C2×Q8—461no—
108C2×Q8—921no—
109C2×Q8—921no—
110C2×Q8—921no—
111C2×Q8—921no—
112C2×Q8C4×D2311yescharacteristic
113C2×Q8C22×D2311yescharacteristic
114C22×C4—231no—
115C22×C4—232no—
116C22×C4—461no—
117C22×C4—461no—
118M16—22no—
119M16—462no—
120M16—922no—
121C2×C8—922no—
122C4⋊C4—22no—
123C4⋊C4—462no—
124C4⋊C4—462no—
125C4⋊C4—462no—
126C42—231no—
127C42—461no—
128C42C4×D2311yescharacteristic
129C23—231no—
130Q8—41no—
131Q8—41no—
132Q8—922no—
133Q8—921no—
134Q8—921no—
135Q8—922no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—231no—
139C2×C4—232no—
140C2×C4—462no—
141C2×C4—462no—
142C2×C4—461no—
143C2×C4—461no—
144C2×C4—464no—
145C2×C4—461no—
146C2×C4—461no—
147C2×C4—921no—
148C2×C4—921no—
149C2×C4C2×C4×D2311yescharacteristic
150C2×C4D4×D2312yes—
151C8—42no—
152C8—922no—
153C22—231no—
154C22—231no—
155C22—462no—
156C22C22⋊C4×D2311yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—462no—
162C4—461no—
163C4—462no—
164C4—921no—
165C2—21no—
166C2—232no—
167C2—461no—
168C2#99×D2311yescharacteristic
subgroupG/H
centerC2#99×D23
commutatorC2×C4×C23C22×C4
FrattiniC2×Q8C22×D23
FittingC23×#356C2
socleC2×C23C2×#99
prime radicalC22×D23C22⋊C4
cosocleC2×Q8×C23C23
hypercenter#356D23
hypocenterC23C2×#356
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #13092216C2semidirect
4. #13092215C2semidirect
5. C23×#356C2semidirect
85. #356D23direct (N × H)
89. D23#356direct (N × H)
103. C23C2×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D232
#101×D232
#130922162
#130922152
C23×#3562
C2×#35623