Finite Group Databasebeta

← #10591600

Subgroups — #10591600

Order 576.

257 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 62 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#64155C211yesmaximal, characteristic
2C2×C3×#216C211yesmaximal, characteristic, complemented
3#63871C211yesmaximal, characteristic
4C2×#3712C211yesmaximal, characteristic
5C3×#835C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7#63566C211yesmaximal, characteristic, complemented
8#5189—31nomaximal
9#4816—31nomaximal
10C2×C32×Q8C2211yescharacteristic
11C2×C4×C3⋊D3C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13#3752—21no—
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15#3750—21no—
16C3×#216—22no—
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21#3722—21no—
22#3712—22no—
23C2×C3×Q16—31no—
24#946—31no—
25#946—31no—
26#946—31no—
27#946—61no—
28C2×#216D311yescharacteristic, complemented
29C2×ΓL(1,𝔽25)—31no—
30C2×ΓL(1,𝔽25)—31no—
31#897—31no—
32C3×#103—31no—
33#835—31no—
34#835D311yescharacteristic, complemented
35#817—31no—
36#816—31no—
37#808—31no—
38C22×C3⋊D3D411yescharacteristic
39C32×Q8—21no—
40C32×Q8—21no—
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44C2×C3⋊Dic3D411yescharacteristic
45C4×C3⋊D3—21no—
46C4×C3⋊D3—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3×C3⋊C8—21no—
52C3×C3⋊C8D412yes—
53#470—91noSylow
54C2×C3×Q8—21no—
55C2×C3×Q8—31no—
56C2×C3×Q8C2×D311yescharacteristic
57C2×C3×Q8C2×D311yescharacteristic
58C2×C4×D3—31no—
59C2×C4×D3—31no—
60C2×C4×D3—61no—
61C2×Dic6C2×D311yescharacteristic
62C3×Q16—62no—
63C2×C3×C8—31no—
64C2×C3×C8—31no—
65C3×C4⋊C4—31no—
66#216—22no—
67(C2×D3)⋊C4—61no—
68(C2×D3)⋊C4—61no—
69(C2×D3)⋊C4—61no—
70(C2×D3)⋊C4—61no—
71(C2×D3)⋊C4—62no—
72C4⋊Dic3—31no—
73C4⋊Dic3—31no—
74C4⋊Dic3—61no—
75C4⋊Dic3—61no—
76C4⋊Dic3C2×D311yescharacteristic
77Dic3⋊C4—61no—
78Dic3⋊C4—61no—
79Dic3⋊C4—61no—
80Dic3⋊C4—62no—
81C2×C3⋊C8C2×D311yescharacteristic
82C2×C3⋊C8C2×D311yescharacteristic
83ΓL(1,𝔽25)—62no—
84ΓL(1,𝔽25)—62no—
85C22×C32C2×D411yescharacteristic
86C2×C3⋊D3—21no—
87C2×C3⋊D3—21no—
88C2×C3⋊D3—21no—
89C32×C4—41no—
90C32×C4C2×D411yescharacteristic
91C32×C4#4111yescharacteristic
92C3⋊Dic3—21no—
93C3⋊Dic3—41no—
94C3×Dic3—41no—
95C3×Dic3—41no—
96C2×Q16—31no—
97C2×M16—91no—
98C22⋊Q8—91no—
99C22⋊Q8—91no—
100#106—91no—
101#103—31no—
102#103—91no—
103C22×D3—31no—
104C22×D3—31no—
105C22×D3—61no—
106C3×Q8—21no—
107C3×Q8—21no—
108C3×Q8—21no—
109C3×Q8—21no—
110C3×Q8—41no—
111C3×Q8—41no—
112C3×Q8—61no—
113C3×Q8—61no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4C22×D311yescharacteristic
121C2×C3×C4C22×D311yescharacteristic
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3—61no—
127C2×Dic3—62no—
128C2×Dic3—61no—
129C2×Dic3—61no—
130C4×D3—61no—
131C4×D3—61no—
132C4×D3—61no—
133C4×D3—61no—
134C4×D3—121no—
135C4×D3—121no—
136Dic6—21no—
137Dic6—21no—
138C3×C8—32no—
139C3×C8—61no—
140C3⋊C8—21no—
141C3⋊C8#6712yes—
142C2×C32#12111yescharacteristic
143C2×C32#13711yescharacteristic
144C2×C32#13711yescharacteristic
145C3⋊D3—41no—
146C2×Q8—31no—
147C2×Q8D3211yescharacteristic
148C22×C4—91no—
149Q16—62no—
150M16—182no—
151C2×C8—31no—
152C2×C8—31no—
153C4⋊C4—31no—
154C4⋊C4—91no—
155C4⋊C4—181no—
156C4⋊C4—181no—
157C22⋊C4—181no—
158C22⋊C4—181no—
159C22×C3—21no—
160C22×C3D3×D411yescharacteristic
161C22×C3D3×D411yescharacteristic
162C2×D3—62no—
163C2×D3—61no—
164C2×D3—61no—
165C2×D3—61no—
166C2×D3—61no—
167C2×D3—61no—
168C2×D3—61no—
169C2×D3—121no—
170C2×D3—121no—
171C3×C4—21no—
172C3×C4—21no—
173C3×C4—42no—
174C3×C4—41no—
175C3×C4—41no—
176C3×C4—121no—
177C3×C4—121no—
178C3×C4#23511yescharacteristic
179C3×C4D3×D411yescharacteristic
180C3×C4#23711yescharacteristic
181C3×C4C2×#6711yescharacteristic
182Dic3—41no—
183Dic3—41no—
184Dic3—61no—
185Dic3—61no—
186Dic3—121no—
187Dic3—122no—
188Dic3—121no—
189Dic3—121no—
190C32#47011yescharacteristic, complemented, Sylow
191C23—91no—
192Q8—21no—
193Q8—21no—
194Q8—61no—
195Q8—61no—
196C2×C4—21no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—91no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—181no—
203C2×C4C2×D3211yescharacteristic
204C8—32no—
205C8—61no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3#88411yescharacteristic
210C2×C3#91511yescharacteristic
211C2×C3#91811yescharacteristic
212C2×C3#91811yescharacteristic
213C2×C3#91811yescharacteristic
214C2×C3#93811yescharacteristic
215D3—122no—
216D3—121no—
217D3—121no—
218C22—181no—
219C22—181no—
220C22—181no—
221C22#381111yescharacteristic
222C4—41no—
223C4—121no—
224C4—121no—
225C4—181no—
226C4—361no—
227C4#380411yescharacteristic
228C4D3×#6711yescharacteristic
229C3—21no—
230C3#481611yescharacteristic, complemented
231C3#518911yescharacteristic, complemented
232C2—361no—
233C2#6394511yescharacteristic
234C2#6395111yescharacteristic
235C2#6395811yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×Q8C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC2×Q8D32
hypocenterC32#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#216C2semidirect
5. C3×#835C2semidirect
6. #63573C2semidirect
7. #63566C2semidirect
28. C2×#216D3semidirect
34. #835D3semidirect
190. C32#470semidirect
230. C3#4816semidirect
231. C3#5189semidirect

Maximal subgroups

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

subgroupindex
#641552
C2×C3×#2162
#638712
C2×#37122
C3×#8352
#635732
#635662
#51893
#48163