Finite Group Databasebeta

← #10591022

Subgroups — #10591022

Order 576.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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#64297C211yesmaximal, characteristic, complemented
2#64117C211yesmaximal, characteristic, complemented
3#63864C211yesmaximal, characteristic, complemented
4#63825C211yesmaximal, characteristic, complemented
5C3×#825C211yesmaximal, characteristic, complemented
6C3×#824C211yesmaximal, characteristic, complemented
7#63570C211yesmaximal, characteristic, complemented
8#4827—31nomaximal
9#4753—31nomaximal
10#3828C2211yescharacteristic
11C2×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×#235C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14#3804—22no—
15#3799—21no—
16#3797—21no—
17#3796—21no—
18D3×Dic6—21no—
19C32×M16C2211yescharacteristic
20#3745—21no—
21#3744—21no—
22C3×C4×Dic3C2211yescharacteristic
23C3×#208C2211yescharacteristic
24#3722—22no—
25#3717—21no—
26#3714—21no—
27#1010—31no—
28#1007—31no—
29#949—31no—
30#947—31no—
31#908—31no—
32#908—31no—
33#908—31no—
34#908—61no—
35#875—31no—
36C3×#104—31no—
37C3×#101—31no—
38#825—31no—
39#825D311yescharacteristic, complemented
40#824D311yescharacteristic, complemented
41#805—31no—
42C22×C3⋊D3—21no—
43C2×C32×C4C2311yescharacteristic
44#638—21no—
45C4⋊(C3⋊D3)—21no—
46C4⋊(C3⋊D3)—21no—
47C4⋊(C3⋊D3)D411yescharacteristic
48C4×C3⋊D3—21no—
49#634D411yescharacteristic
50C3×#67—21no—
51C2×C3×Dic3—21no—
52C2×C3×Dic3D412yes—
53C3×D12D411yescharacteristic
54C3×C4×D3—21no—
55C3×Dic6—21no—
56C3×Dic6—21no—
57C3×Dic6D411yescharacteristic
58#627—22no—
59#626—22no—
60#624—22no—
61D3×Dic3—22no—
62C32×C8—21no—
63C3×C3⋊C8—21no—
64#454—91noSylow
65C3×#41—31no—
66C2×C3×Q8—31no—
67#239—61no—
68#239—61no—
69D3×Q8—61no—
70D3×Q8—61no—
71D3×Q8—61no—
72#237—61no—
73#237—62no—
74#235—31no—
75#235—31no—
76#235—61no—
77#235—61no—
78#235—62no—
79#235C2×D311yescharacteristic
80C2×D12—31no—
81C2×D12—31no—
82C2×D12—61no—
83C2×Dic6—61no—
84C2×Dic6C2×D311yescharacteristic
85C3×M16—21no—
86C3×M16—31no—
87C3×M16C2×D311yescharacteristic
88C3×M16C2×D311yescharacteristic
89C3×C42—31no—
90#215—61no—
91#213—61no—
92(C2×D3)⋊C4—62no—
93(C2×D3)⋊C4—62no—
94C4×Dic3C2×D311yescharacteristic
95#208C2×D311yescharacteristic
96D24—61no—
97D24—61no—
98D24—62no—
99D24—61no—
100#204—61no—
101#204—61no—
102#204—62no—
103#204—61no—
104C22×C32C2×D411yescharacteristic
105C2×C3⋊D3—21no—
106C2×C3⋊D3—41no—
107C2×C3⋊D3—42no—
108C2×C3×D3—21no—
109C32×C4C2×D411yescharacteristic
110C32×C4C2×D411yescharacteristic
111C3⋊Dic3—21no—
112C3×Dic3—22no—
113C3×Dic3—21no—
114C3×Dic3—41no—
1152−1+4—91no—
116AGL(1,ℤ/8)—91no—
117AGL(1,ℤ/8)—91no—
118#124—91no—
119#104—31no—
120#104—91no—
121#101—31no—
122C22×D3—61no—
123C22×D3—61no—
124C22×D3—121no—
125C3×Q8—31no—
126C3×Q8—61no—
127C3×Q8—61no—
128C3×D4—31no—
129C3×D4—61no—
130C2×C3×C4—21no—
131C2×C3×C4—32no—
132C2×C3×C4—61no—
133C2×C3×C4—61no—
134C2×C3×C4C22×D311yescharacteristic
135C2×C3×C4C22×D311yescharacteristic
136#67—21no—
137#67—62no—
138#67—61no—
139#67—61no—
140#67—121no—
141C2×Dic3—21no—
142C2×Dic3—62no—
143C2×Dic3D1212yes—
144D12—31no—
145D12—31no—
146D12—61no—
147D12—62no—
148D12—62no—
149D12—61no—
150D12—61no—
151D12—61no—
152D12—61no—
153D12—121no—
154D12D1211yescharacteristic
155C4×D3—21no—
156C4×D3—62no—
157C4×D3—62no—
158C4×D3—62no—
159C4×D3—61no—
160C4×D3—61no—
161C4×D3—121no—
162Dic6—21no—
163Dic6—21no—
164Dic6—31no—
165Dic6—31no—
166Dic6—61no—
167Dic6—62no—
168Dic6—62no—
169Dic6D1211yescharacteristic
170C3×C8—22no—
171C3×C8—21no—
172C3×C8—21no—
173C3×C8—61no—
174C3⋊C8—21no—
175C2×C32—21no—
176C2×C32#12011yescharacteristic
177C3⋊D3—41no—
178C3⋊D3—81no—
179C3×D3—41no—
180#41—31no—
181#41—91no—
182#41—182no—
183#41—181no—
184#41—181no—
185C2×Q8—31no—
186C2×Q8—181no—
187C2×Q8—181no—
188C2×D4—91no—
189SD16—181no—
190SD16—181no—
191D8—181no—
192D8—181no—
193M16—31no—
194M16D3211yescharacteristic
195C22⋊C4—182no—
196C42—31no—
197C22×C3—21no—
198C22×C3—61no—
199C22×C3C2×D1211yescharacteristic
200C22×C3D3×D411yescharacteristic
201C2×D3—21no—
202C2×D3—61no—
203C2×D3—61no—
204C2×D3—121no—
205C2×D3—122no—
206C2×D3—122no—
207C2×D3—122no—
208C2×D3—121no—
209C2×D3—121no—
210C2×D3—242no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—61no—
214C3×C4—62no—
215C3×C4—121no—
216C3×C4C2×D1211yescharacteristic
217C3×C4D3×D411yescharacteristic
218C3×C4D3×D411yescharacteristic
219C3×C4D3×D411yescharacteristic
220Dic3—22no—
221Dic3—21no—
222Dic3—41no—
223Dic3—61no—
224Dic3—61no—
225Dic3—121no—
226C32#45411yescharacteristic, complemented, Sylow
227C23—181no—
228Q8—31no—
229Q8—61no—
230Q8—61no—
231Q8—91no—
232Q8—182no—
233D4—31no—
234D4—61no—
235D4—91no—
236D4—182no—
237D4—181no—
238D4—181no—
239D4—181no—
240C2×C4—32no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4—181no—
246C2×C4C2×D3211yescharacteristic
247C8—21no—
248C8—61no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—121no—
254C2×C3#88211yescharacteristic
255C2×C3#88211yescharacteristic
256D3—41no—
257D3—121no—
258D3—121no—
259D3—241no—
260D3—242no—
261D3—241no—
262D3—241no—
263C22—61no—
264C22—181no—
265C22—362no—
266C22—361no—
267C22D3×D1211yescharacteristic
268C4—62no—
269C4—61no—
270C4—121no—
271C4—181no—
272C4D3×D1211yescharacteristic
273C4#381111yescharacteristic
274C3—21no—
275C3#475311yescharacteristic, complemented
276C3#482711yescharacteristic, complemented
277C2—21no—
278C2—121no—
279C2—361no—
280C2—721no—
281C2#6392311yescharacteristic
subgroupG/H
centerC2#63923
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×M16C22
socleC2×C32#120
hypercenterM16D32
hypocenterC32#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64297C2semidirect
2. #64117C2semidirect
3. #63864C2semidirect
4. #63825C2semidirect
5. C3×#825C2semidirect
6. C3×#824C2semidirect
7. #63570C2semidirect
39. #825D3semidirect
40. #824D3semidirect
226. C32#454semidirect
275. C3#4753semidirect
276. C3#4827semidirect

Maximal subgroups

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

subgroupindex
#642972
#641172
#638642
#638252
C3×#8252
C3×#8242
#635702
#48273
#47533