Finite Group Databasebeta

← #10591474

Subgroups — #10591474

Order 576.

298 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 66 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#64145C211yesmaximal, characteristic, complemented
2C2×C3×#214C211yesmaximal, characteristic, complemented
3C2×C3×#213C211yesmaximal, characteristic
4#63863C211yesmaximal, characteristic, complemented
5C2×#3719C211yesmaximal, characteristic, complemented
6C2×#3716C211yesmaximal, characteristic, complemented
7#63553C211yesmaximal, characteristic, complemented
8#5168—31nomaximal
9#5155—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C3×D12C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14#3757—22no—
15C4×C3⋊Dic3C2211yescharacteristic
16C3×#214—22no—
17C3×#213—22no—
18C2×C3×C3⋊C8C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic3—22no—
21#3719—22no—
22#3716—22no—
23C2×C3×SD16—31no—
24C2×C3×D8—31no—
25#947—31no—
26C2×#216—31no—
27#936—31no—
28#936—31no—
29#936—31no—
30#936—61no—
31C2×#214—31no—
32C2×#214D311yescharacteristic, complemented
33C2×#213D311yescharacteristic
34C2×Dic12—31no—
35C2×#204—31no—
36C8×Dic3—31no—
37C8×Dic3—31no—
38C23×C32—21no—
39C22×C3×D3—21no—
40C32×D4—21no—
41C32×D4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C2×C3⋊Dic3D412yes—
45#634—21no—
46#634—21no—
47C2×C3×Dic3—21no—
48C3×D12—21no—
49C3×D12—21no—
50C3×Dic6—21no—
51C3×Dic6—21no—
52C3×C3⋊C8D412yes—
53C3×C3⋊C8D412yes—
54#495—91noSylow
55C2×C3×Q8—31no—
56C2×C3×D4—21no—
57C2×C3×D4—31no—
58C2×C3×D4C2×D311yescharacteristic
59C2×C3×D4C2×D311yescharacteristic
60C2×D12C2×D311yescharacteristic
61C2×Dic6—31no—
62C2×Dic6—31no—
63C2×Dic6—61no—
64C2×Dic6C2×D311yescharacteristic
65C3×SD16—62no—
66C3×D8—62no—
67C2×C3×C8—31no—
68C2×C3×C8—31no—
69#217—62no—
70#217—62no—
71#217—62no—
72#217—64no—
73#216—62no—
74#214—22no—
75#214—62no—
76#213—22no—
77(C2×D3)⋊C4—62no—
78C4×Dic3—31no—
79C4×Dic3—31no—
80C4×Dic3—61no—
81C2×C3⋊C8C2×D311yescharacteristic
82C2×C3⋊C8C2×D311yescharacteristic
83Dic12—62no—
84#204—62no—
85C22×C32—41no—
86C22×C32—42no—
87C22×C32C2×D411yescharacteristic
88C2×C3×D3—41no—
89C2×C3×D3—42no—
90C32×C4C2×D411yescharacteristic
91C32×C4C2×D411yescharacteristic
92C3⋊Dic3—22no—
93C3⋊Dic3—41no—
94C3×Dic3—41no—
95C2×Q16—91no—
96C2×SD16—31no—
97C2×SD16—91no—
98C2×D8—31no—
99#124—91no—
100#124—91no—
101C4×C8—91no—
102C23×C3—22no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3—61no—
106C22×D3—21no—
107C3×Q8—61no—
108C3×Q8—61no—
109C3×D4—21no—
110C3×D4—21no—
111C3×D4—21no—
112C3×D4—21no—
113C3×D4—41no—
114C3×D4—41no—
115C3×D4—61no—
116C3×D4—61no—
117C2×C3×C4—21no—
118C2×C3×C4—61no—
119C2×C3×C4C22×D311yescharacteristic
120C2×C3×C4C22×D311yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—32no—
123C2×Dic3—32no—
124C2×Dic3—62no—
125C2×Dic3—62no—
126C2×Dic3—61no—
127C2×Dic3—61no—
128D12—21no—
129D12—21no—
130Dic6—21no—
131Dic6—21no—
132Dic6—61no—
133Dic6—61no—
134Dic6—61no—
135Dic6—61no—
136Dic6—121no—
137Dic6—121no—
138C3×C8—32no—
139C3×C8—32no—
140C3⋊C8#6712yes—
141C3⋊C8#6712yes—
142C2×C32—81no—
143C2×C32#12711yescharacteristic
144C2×C32#13512yes—
145C3×D3—81no—
146C2×Q8—31no—
147C2×Q8—91no—
148C2×D4—31no—
149C2×D4D3211yescharacteristic
150Q16—182no—
151SD16—62no—
152SD16—182no—
153D8—62no—
154C2×C8—31no—
155C2×C8—31no—
156C22⋊C4—182no—
157C22⋊C4—182no—
158C42—91no—
159C22×C3—21no—
160C22×C3—42no—
161C22×C3—44no—
162C22×C3—42no—
163C22×C3—42no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—122no—
167C22×C3—121no—
168C22×C3D3×D411yescharacteristic
169C22×C3D3×D411yescharacteristic
170C2×D3—42no—
171C2×D3—41no—
172C3×C4—21no—
173C3×C4—21no—
174C3×C4—121no—
175C3×C4D3×D411yescharacteristic
176C3×C4D3×D411yescharacteristic
177C3×C4C2×#6711yescharacteristic
178C3×C4C2×#6711yescharacteristic
179Dic3—41no—
180Dic3—62no—
181Dic3—62no—
182Dic3—122no—
183Dic3—122no—
184Dic3—121no—
185Dic3—121no—
186C32#49511yescharacteristic, complemented, Sylow
187C23—21no—
188C23—61no—
189Q8—61no—
190Q8—61no—
191Q8—181no—
192Q8—181no—
193D4—21no—
194D4—21no—
195D4—61no—
196D4—61no—
197C2×C4—61no—
198C2×C4—92no—
199C2×C4—181no—
200C2×C4C2×D3211yescharacteristic
201C8—32no—
202C8—32no—
203C2×C3—22no—
204C2×C3—21no—
205C2×C3—82no—
206C2×C3—81no—
207C2×C3—81no—
208C2×C3—241no—
209C2×C3#91212yes—
210C2×C3#91612yes—
211C2×C3#94011yescharacteristic
212C2×C3#94011yescharacteristic
213D3—81no—
214C22—42no—
215C22—41no—
216C22—122no—
217C22—121no—
218C22#381111yescharacteristic
219C4—121no—
220C4—182no—
221C4—361no—
222C4D3×#6711yescharacteristic
223C4D3×#6711yescharacteristic
224C3—21no—
225C3#515511yescharacteristic, complemented
226C3#516811yescharacteristic, complemented
227C2—81no—
228C2—241no—
229C2#6393612yes—
230C2#6397911yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC2×C3×#213C2
hypercenterC2×D4D32
hypocenterC32#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64145C2semidirect
2. C2×C3×#214C2semidirect
4. #63863C2semidirect
5. C2×#3719C2semidirect
6. C2×#3716C2semidirect
7. #63553C2semidirect
32. C2×#214D3semidirect
186. C32#495semidirect
225. C3#5155semidirect
226. C3#5168semidirect

Maximal subgroups

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

subgroupindex
#641452
C2×C3×#2142
C2×C3×#2132
#638632
C2×#37192
C2×#37162
#635532
#51683
#51553