Finite Group Databasebeta

← #43252720

Subgroups — #43252720

Order 28544.

294 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 37 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#28383435C211yesmaximal, characteristic
2#28383304C211yesmaximal, characteristic, complemented
3#323×C223C211yesmaximal, characteristic, complemented
4C24×D223C411yescharacteristic
5C2×C22⋊Dic223C411yescharacteristic
6C2×(C2×D223)⋊C4—21no—
7C2×(C2×D223)⋊C4—21no—
8C22⋊(C223⋊C8)—21no—
9C2×C22⋊C4×C223C2211yescharacteristic
10C22⋊C8×C223—21no—
11C23×D223—21no—
12C23×D223—21no—
13C23×D223—41no—
14C23×D223—41no—
15C23×D223—42no—
16C23×D223—42no—
17C23×D223C812yescomplemented
18C22×Dic223—21no—
19C2×C223⋊C8—41no—
20C22⋊Dic223—41no—
21(C2×D223)⋊C4—42no—
22(C2×D223)⋊C4—42no—
23C24×C223C2×C411yescharacteristic
24C22×C4×C223D411yescharacteristic
25C22×C4×C223D411yescharacteristic
26C2×C8×C223—41no—
27C22⋊C4×C223—41no—
28C22×D223—22no—
29C22×D223—22no—
30C22×D223—22no—
31C22×D223—44no—
32C22×D223—44no—
33C22×D223—41no—
34C22×D223—41no—
35C22×D223—42no—
36C22×D223—48no—
37C22×D223—42no—
38C22×D223—42no—
39C22×D223—41no—
40C22×D223—41no—
41C22×D223—42no—
42C22×D223—42no—
43C22×D223—42no—
44C2×Dic223—41no—
45C2×Dic223—41no—
46C2×Dic223—41no—
47C223⋊C8—81no—
48C23×C223—21no—
49C23×C223—21no—
50C23×C223—41no—
51C23×C223—41no—
52C23×C223C22⋊C411yescharacteristic
53C23×C223C2×C811yescharacteristic
54C23×C223M1611yescharacteristic
55C2×C4×C223—21no—
56C2×C4×C223—21no—
57C2×C4×C223—41no—
58C2×C4×C223—41no—
59C8×C223—81no—
60C22×C223—21no—
61C22×C223—21no—
62C22×C223—21no—
63C22×C223—21no—
64C22×C223—41no—
65C22×C223—41no—
66C22×C223—42no—
67C22×C223—41no—
68C22×C223—41no—
69C22×C223C22⋊C811yescharacteristic
70C22×C223#9911yescharacteristic
71C22×C223#10011yescharacteristic
72C2×D223—24no—
73C2×D223—42no—
74C2×D223—42no—
75C2×D223—42no—
76C2×D223—44no—
77C2×D223—44no—
78C2×D223—44no—
79C2×D223—42no—
80C2×D223—42no—
81C2×D223—42no—
82C2×D223—44no—
83C4×C223—41no—
84C4×C223—41no—
85Dic223—81no—
86C2×C223—21no—
87C2×C223—21no—
88C2×C223—41no—
89C2×C223—41no—
90C2×C223#32311yescharacteristic
91C2×C223AΣL(1,𝔽16)11yescharacteristic
92C2×C223AΣL(1,𝔽16)11yescharacteristic
93D223—44no—
94C223#131611yescharacteristic, complemented, Sylow
95#1316—2231nomaximal, Sylow
96#379—2231no—
97#323—2231no—
98#323D22311yescharacteristic, complemented
99C25—2231no—
100C2×C22⋊C4—4461no—
101C2×C22⋊C4—4461no—
102C2×C22⋊C4—2231no—
103C2×C22⋊C4C2×D22311yescharacteristic
104C22⋊C8—21no—
105C22⋊C8—4461no—
106C24—8921no—
107C24—8921no—
108C24—8922no—
109C24—8922no—
110C24—4461no—
111C24—4461no—
112C24—2232no—
113C24C4×D22311yescharacteristic
114C22×C4—4461no—
115C22×C4C22⋊D22311yescharacteristic
116C22×C4D89211yescharacteristic
117C2×C8—41no—
118C2×C8—8921no—
119C22⋊C4—41no—
120C22⋊C4—8922no—
121C22⋊C4—8922no—
122C22⋊C4—8921no—
123C23—21no—
124C23—21no—
125C23—41no—
126C23—41no—
127C23—8922no—
128C23—8921no—
129C23—8921no—
130C23—8922no—
131C23—8922no—
132C23—8922no—
133C23—8928no—
134C23—8922no—
135C23—8922no—
136C23—8924no—
137C23—8924no—
138C23—8921no—
139C23—8921no—
140C23—4462no—
141C23—4462no—
142C23—4462no—
143C23(C2×D223)⋊C411yescharacteristic
144C23C8×D22311yescharacteristic
145C23#1681315311yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—8921no—
151C2×C4—8921no—
152C2×C4—8921no—
153C8—81no—
154C8—17841no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22—8922no—
165C22—8922no—
166C22—8922no—
167C22—8924no—
168C22—8924no—
169C22—8924no—
170C22—8924no—
171C22—8922no—
172C22—8922no—
173C22—8922no—
174C22—4464no—
175C22(C2×D223)⋊C811yescharacteristic
176C22#2029721811yescharacteristic
177C22#2029722011yescharacteristic
178C4—41no—
179C4—41no—
180C4—17841no—
181C2—21no—
182C2—21no—
183C2—41no—
184C2—41no—
185C2—8924no—
186C2#2838330311yescharacteristic
187C2#2838336611yescharacteristic
188C2#2838336811yescharacteristic
subgroupG/H
centerC22#20297218
commutatorC23×C223C2×C8
FrattiniC2×C22⋊C4C2×D223
Fitting#323×C223C2
socleC22×C223#99
prime radicalC24×D223C4
cosocleC2×C22⋊C4×C223C22
hypercenter#323D223
hypocenterC223#1316
LCSG ▷ C23×C223 ▷ C22×C223 ▷ C2×C223 ▷ C223
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #28383304C2semidirect
3. #323×C223C2semidirect
17. C23×D223C8semidirect
94. C223#1316semidirect
98. #323D223semidirect

Maximal subgroups

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

subgroupindex
#283834352
#283833042
#323×C2232
#1316223