Finite Group Databasebeta

← #10596100

Subgroups — #10596100

Order 576.

882 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 882 conjugacy classes of subgroups — too many to draw. Only its 191 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
1C2×#3832C211yesmaximal, characteristic, complemented
2C2×#3828C211yesmaximal, characteristic, complemented
3C32×C22⋊Q8C211yesmaximal, characteristic, complemented
4#64155C211yesmaximal, characteristic, complemented
5#64139C211yesmaximal, characteristic, complemented
6#64105C212yesmaximal, complemented
7#64104C211yesmaximal, characteristic, complemented
8#64104C212yesmaximal, complemented
9C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
10#64093C212yesmaximal, complemented
11#64093C212yesmaximal, complemented
12#5629—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×#638C2211yescharacteristic
17C2×#638C2212yes—
18#3832—24no—
19#3828—24no—
20C2×C4⋊(C3⋊D3)C2212yes—
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C4⋊(C3⋊D3)C2212yes—
24C2×C4×C3⋊D3C2212yes—
25C2×C4×C3⋊D3C2211yescharacteristic
26C2×C4×C3⋊D3C2212yes—
27C2×C4×C3⋊D3C2211yescharacteristic
28C2×#634C2211yescharacteristic
29C32×C4⋊C4C2211yescharacteristic
30C32×C4⋊C4C2212yes—
31C32×C22⋊C4C2212yes—
32#3752C2212yes—
33#3752C2212yes—
34#3752C2212yes—
35C4⋊(C3⋊Dic3)C2211yescharacteristic
36#3750C2212yes—
37#3750C2212yes—
38C2×#239—34no—
39C2×#235—34no—
40C3×C22⋊Q8D314yescomplemented
41#946—34no—
42#930—34no—
43#896—38no—
44#895—34no—
45#895—38no—
46D3×C4⋊C4—34no—
47#884—38no—
48#884—38no—
49C23×C32C2311yescharacteristic
50C22×C3⋊D3C2312yes—
51C22×C3⋊D3C2311yescharacteristic
52C22×C3⋊D3C2311yescharacteristic
53C32×Q8—22no—
54C2×C32×C4—22no—
55C2×C32×C4C2312yes—
56C2×C32×C4C2312yes—
57C2×C32×C4C2311yescharacteristic
58C2×C32×C4C2311yescharacteristic
59#638—22no—
60#638—22no—
61#638—42no—
62C2×C3⋊Dic3C2312yes—
63C2×C3⋊Dic3C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C4⋊(C3⋊D3)—22no—
66C4⋊(C3⋊D3)—24no—
67C4⋊(C3⋊D3)—22no—
68C4⋊(C3⋊D3)—42no—
69C4×C3⋊D3—22no—
70C4×C3⋊D3—24no—
71C4×C3⋊D3—42no—
72C4×C3⋊D3D414yescomplemented
73#634—22no—
74#537—91noSylow
75C2×C3×Q8C2×D314yes—
76C22×C3×C4C2×D314yes—
77C2×#67—34no—
78C2×#67—34no—
79C2×#67—38no—
80#239—616no—
81#235—616no—
82C2×D12—38no—
83C2×D12—34no—
84C2×D12—34no—
85C2×D12—38no—
86C2×C4×D3—34no—
87C2×C4×D3—38no—
88C2×C4×D3—34no—
89C2×C4×D3—38no—
90C2×Dic6—34no—
91C3×C4⋊C4C2×D314yes—
92C3×C4⋊C4C2×D318yes—
93C3×C22⋊C4C2×D318yes—
94(C2×D3)⋊C4—38no—
95(C2×D3)⋊C4—38no—
96(C2×D3)⋊C4—38no—
97C4⋊Dic3—34no—
98Dic3⋊C4—38no—
99Dic3⋊C4—38no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32C2411yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—22no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3—21no—
109C2×C3⋊D3—22no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3C2×D412yes—
113C32×C4—21no—
114C32×C4—22no—
115C32×C4—22no—
116C32×C4C2×D412yes—
117C3⋊Dic3—21no—
118C3⋊Dic3—22no—
119C3⋊Dic3C2×D412yes—
120C2×#41—91no—
121C2×#41—91no—
122C22⋊Q8—91no—
123C22⋊Q8—92no—
124C22⋊Q8C3⋊D311yescharacteristic, complemented
125#121—91no—
126#121—91no—
127#121—92no—
128#121—92no—
129#121—92no—
130C2×C4⋊C4—91no—
131C23×C3C22×D314yes—
132C22×D3—34no—
133C22×D3—38no—
134C22×D3—34no—
135C3×Q8—28no—
136C2×C3×C4—28no—
137C2×C3×C4C22×D318yes—
138C2×C3×C4C22×D318yes—
139C2×C3×C4C22×D314yes—
140C2×C3×C4C22×D314yes—
141#67—68no—
142#67—68no—
143#67—128no—
144C2×Dic3—34no—
145C2×Dic3—38no—
146C2×Dic3—34no—
147D12—68no—
148D12—616no—
149D12—68no—
150D12—128no—
151C4×D3—316no—
152C4×D3—616no—
153C4×D3—68no—
154C4×D3—128no—
155Dic6—68no—
156C2×C32—41no—
157C2×C32C22×D411yescharacteristic
158C2×C32H(5,𝔽2)11yescharacteristic
159C2×C322−1+411yescharacteristic
160C3⋊D3—22no—
161C3⋊D3—42no—
162C3⋊D3—41no—
163#41—184no—
164#41—184no—
165C2×Q8—91no—
166C2×Q8C2×C3⋊D311yescharacteristic
167C2×D4—91no—
168C2×D4—91no—
169C2×D4—92no—
170C2×D4—91no—
171C2×D4—92no—
172C2×D4—91no—
173C2×D4—92no—
174C22×C4—91no—
175C22×C4—92no—
176C22×C4—91no—
177C22×C4—92no—
178C22×C4C2×C3⋊D311yescharacteristic
179C4⋊C4—92no—
180C4⋊C4—91no—
181C4⋊C4—92no—
182C4⋊C4C2×C3⋊D311yescharacteristic
183C4⋊C4C2×C3⋊D312yes—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4C2×C3⋊D312yes—
188C22×C3—24no—
189C22×C3—24no—
190C22×C3—24no—
191C22×C3C23×D314yes—
192C2×D3—38no—
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—68no—
196C2×D3—64no—
197C2×D3—68no—
198C2×D3—64no—
199C2×D3—64no—
200C2×D3—68no—
201C3×C4—24no—
202C3×C4—28no—
203C3×C4—28no—
204C3×C4D3×D418yes—
205Dic3—38no—
206Dic3—64no—
207Dic3—68no—
208C32#53711yescharacteristic, complemented, Sylow
209C23—91no—
210C23—91no—
211C23—92no—
212C23C22×C3⋊D311yescharacteristic
213Q8—22no—
214Q8—182no—
215D4—182no—
216D4—184no—
217D4—182no—
218D4—182no—
219D4—182no—
220D4—362no—
221D4—362no—
222C2×C4—22no—
223C2×C4—91no—
224C2×C4—94no—
225C2×C4—92no—
226C2×C4—91no—
227C2×C4—184no—
228C2×C4—182no—
229C2×C4—362no—
230C2×C4C22×C3⋊D311yescharacteristic
231C2×C4C22×C3⋊D312yes—
232C2×C4C22×C3⋊D312yes—
233C2×C4C22×C3⋊D311yescharacteristic
234C2×C3—44no—
235C2×C3C2×D3×D414yes—
236C2×C3#100714yes—
237C2×C3#100914yes—
238D3—68no—
239D3—128no—
240D3—124no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—92no—
245C22—181no—
246C22—181no—
247C22—182no—
248C22—181no—
249C22—181no—
250C22—182no—
251C22—181no—
252C22—182no—
253C22C23×C3⋊D311yescharacteristic
254C4—21no—
255C4—22no—
256C4—22no—
257C4—92no—
258C4—181no—
259C4—182no—
260C4D4×C3⋊D312yes—
261C3#562914yescomplemented
262C2—41no—
263C2—182no—
264C2—362no—
265C2—361no—
266C2C2×D4×C3⋊D311yescharacteristic
267C2#6436411yescharacteristic
268C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#3832C2semidirect
2. C2×#3828C2semidirect
3. C32×C22⋊Q8C2semidirect
4. #64155C2semidirect
5. #64139C2semidirect
6. #64105C2semidirect
7. #64104C2semidirect
8. #64104C2semidirect
9. C4⋊C4×C3⋊D3C2semidirect
10. #64093C2semidirect
11. #64093C2semidirect
40. C3×C22⋊Q8D3semidirect
72. C4×C3⋊D3D4semidirect
124. C22⋊Q8C3⋊D3semidirect
208. C32#537semidirect
261. C3#5629semidirect

Maximal subgroups

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

subgroupindex
C2×#38322
C2×#38282
C32×C22⋊Q82
#641552
#641392
#641052
#641042
C4⋊C4×C3⋊D32
#640932
#56293