Finite Group Databasebeta

← #10596104

Subgroups — #10596104

Order 576.

870 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 193 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×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#3828C211yesmaximal, characteristic, complemented
3C32×C22⋊Q8C211yesmaximal, characteristic, complemented
4#64156C211yesmaximal, characteristic, complemented
5C4×#638C211yesmaximal, characteristic, complemented
6#64105C212yesmaximal, complemented
7#64104C211yesmaximal, characteristic, complemented
8#64102C212yesmaximal, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64094C212yesmaximal, complemented
11#64093C212yesmaximal, complemented
12#5633—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×#638C2212yes—
17Q8×C3⋊D3—24no—
18#3828—24no—
19C2×C4⋊(C3⋊D3)C2212yes—
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4×C3⋊D3C2212yes—
22C2×C4×C3⋊D3C2212yes—
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×#634C2212yes—
25C2×#634C2211yescharacteristic
26C32×C4⋊C4C2211yescharacteristic
27C32×C4⋊C4C2212yes—
28C32×C22⋊C4C2212yes—
29#3757C2211yescharacteristic
30#3752C2212yes—
31#3752C2212yes—
32#3752C2212yes—
33#3752C2211yescharacteristic
34#3750C2212yes—
35#3750C2211yescharacteristic
36C4×C3⋊Dic3C2212yes—
37C4×C3⋊Dic3C2211yescharacteristic
38C2×D3×Q8—34no—
39C2×#235—34no—
40C3×C22⋊Q8D314yescomplemented
41#947—34no—
42C4×#67—34no—
43#896—38no—
44#895—34no—
45#893—38no—
46#887—34no—
47#885—38no—
48#884—38no—
49C23×C32C2311yescharacteristic
50C22×C3⋊D3C2312yes—
51C22×C3⋊D3C2311yescharacteristic
52C32×Q8—22no—
53C2×C32×C4—22no—
54C2×C32×C4C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2312yes—
57C2×C32×C4C2311yescharacteristic
58#638—22no—
59#638—24no—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C2×C3⋊Dic3C2311yescharacteristic
63C2×C3⋊Dic3C2312yes—
64C4⋊(C3⋊D3)—24no—
65C4⋊(C3⋊D3)—22no—
66C4×C3⋊D3—22no—
67C4×C3⋊D3—24no—
68C4×C3⋊D3—24no—
69#634—24no—
70#634D414yescomplemented
71#548—91noSylow
72C2×C3×Q8C2×D314yes—
73C22×C3×C4C2×D314yes—
74C2×#67—34no—
75C2×#67—38no—
76D3×Q8—616no—
77#235—616no—
78C2×D12—38no—
79C2×D12—34no—
80C2×C4×D3—38no—
81C2×C4×D3—38no—
82C2×C4×D3—34no—
83C2×Dic6—34no—
84C2×Dic6—38no—
85C3×C4⋊C4C2×D314yes—
86C3×C4⋊C4C2×D318yes—
87C3×C22⋊C4C2×D318yes—
88#217—34no—
89(C2×D3)⋊C4—38no—
90(C2×D3)⋊C4—38no—
91(C2×D3)⋊C4—38no—
92(C2×D3)⋊C4—34no—
93Dic3⋊C4—38no—
94Dic3⋊C4—34no—
95C4×Dic3—38no—
96C4×Dic3—34no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32C2411yescharacteristic
101C2×C3⋊D3—21no—
102C2×C3⋊D3—21no—
103C2×C3⋊D3—22no—
104C2×C3⋊D3—22no—
105C2×C3⋊D3—22no—
106C2×C3⋊D3#4112yescomplemented
107C32×C4—22no—
108C32×C4—21no—
109C32×C4—22no—
110C32×C4C2×D412yes—
111C3⋊Dic3—21no—
112C3⋊Dic3—21no—
113C3⋊Dic3—21no—
114C3⋊Dic3C2×D414yes—
115C2×#41—91no—
116C22×Q8—91no—
117#124—91no—
118#124—92no—
119C22⋊Q8—92no—
120C22⋊Q8C3⋊D311yescharacteristic, complemented
121#121—91no—
122#121—92no—
123C4×Q8—91no—
124C4×D4—91no—
125C4×D4—92no—
126C23×C3C22×D314yes—
127C22×D3—34no—
128C22×D3—38no—
129C3×Q8—28no—
130C2×C3×C4—28no—
131C2×C3×C4C22×D318yes—
132C2×C3×C4C22×D314yes—
133C2×C3×C4C22×D318yes—
134C2×C3×C4C22×D314yes—
135#67—616no—
136#67—68no—
137C2×Dic3—38no—
138C2×Dic3—34no—
139C2×Dic3—34no—
140C2×Dic3—34no—
141D12—616no—
142D12—68no—
143C4×D3—616no—
144C4×D3—616no—
145C4×D3—68no—
146Dic6—316no—
147Dic6—616no—
148C2×C32—41no—
149C2×C32C22×D411yescharacteristic
150C2×C32C2×#4111yescharacteristic
151C2×C322−1+411yescharacteristic
152C3⋊D3—22no—
153C3⋊D3—42no—
154#41—184no—
155C2×Q8—91no—
156C2×Q8—92no—
157C2×Q8—184no—
158C2×Q8C2×C3⋊D311yescharacteristic
159C2×D4—91no—
160C2×D4—92no—
161C2×D4—92no—
162C2×D4—91no—
163C22×C4—92no—
164C22×C4—91no—
165C22×C4—92no—
166C22×C4C2×C3⋊D311yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—92no—
169C4⋊C4C2×C3⋊D312yes—
170C4⋊C4C2×C3⋊D311yescharacteristic
171C22⋊C4—91no—
172C22⋊C4—91no—
173C22⋊C4—92no—
174C22⋊C4—92no—
175C22⋊C4—92no—
176C22⋊C4C2×C3⋊D312yes—
177C42—92no—
178C42—91no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3—24no—
182C22×C3C23×D314yes—
183C2×D3—38no—
184C2×D3—64no—
185C2×D3—64no—
186C2×D3—68no—
187C2×D3—68no—
188C2×D3—68no—
189C3×C4—28no—
190C3×C4—24no—
191C3×C4—28no—
192C3×C4D3×D418yes—
193Dic3—316no—
194Dic3—64no—
195Dic3—64no—
196Dic3—64no—
197C32#54811yescharacteristic, complemented, Sylow
198C23—91no—
199C23—92no—
200C23C22×C3⋊D311yescharacteristic
201Q8—22no—
202Q8—94no—
203Q8—184no—
204D4—184no—
205D4—184no—
206D4—182no—
207D4—182no—
208C2×C4—22no—
209C2×C4—92no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—184no—
214C2×C4—184no—
215C2×C4—182no—
216C2×C4C22×C3⋊D312yes—
217C2×C4C22×C3⋊D312yes—
218C2×C4C22×C3⋊D311yescharacteristic
219C2×C4C22×C3⋊D311yescharacteristic
220C2×C3—44no—
221C2×C3C2×D3×D414yes—
222C2×C3#100714yes—
223C2×C3D3×#4114yes—
224D3—68no—
225D3—128no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22—182no—
233C22—182no—
234C22—182no—
235C22C23×C3⋊D311yescharacteristic
236C4—21no—
237C4—22no—
238C4—22no—
239C4—94no—
240C4—181no—
241C4—181no—
242C4—181no—
243C4D4×C3⋊D312yes—
244C3#563314yescomplemented
245C2—41no—
246C2—182no—
247C2—362no—
248C2C2×D4×C3⋊D311yescharacteristic
249C2#6436411yescharacteristic
250C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
hypercenterC22⋊Q8C3⋊D3
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C3⋊D3C2semidirect
2. C2×#3828C2semidirect
3. C32×C22⋊Q8C2semidirect
4. #64156C2semidirect
5. C4×#638C2semidirect
6. #64105C2semidirect
7. #64104C2semidirect
8. #64102C2semidirect
9. #64096C2semidirect
10. #64094C2semidirect
11. #64093C2semidirect
40. C3×C22⋊Q8D3semidirect
70. #634D4semidirect
106. C2×C3⋊D3#41semidirect
120. C22⋊Q8C3⋊D3semidirect
197. C32#548semidirect
244. C3#5633semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D32
C2×#38282
C32×C22⋊Q82
#641562
C4×#6382
#641052
#641042
#641022
#640962
#640942
#640932
#56333