Finite Group Databasebeta

← #10596102

Subgroups — #10596102

Order 576.

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

normal subgroups only

This group has 876 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×#3832C211yesmaximal, characteristic, complemented
2C2×#3828C211yesmaximal, characteristic, complemented
3C32×C22⋊Q8C211yesmaximal, characteristic, complemented
4#64153C211yesmaximal, characteristic, complemented
5C4×#638C211yesmaximal, characteristic, complemented
6#64106C211yesmaximal, characteristic, complemented
7#64103C212yesmaximal, complemented
8#64102C211yesmaximal, characteristic, complemented
9C4⋊C4×C3⋊D3C212yesmaximal, complemented
10#64093C212yesmaximal, complemented
11#64092C212yesmaximal, complemented
12#5631—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×#638C2211yescharacteristic
17#3832—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⋊D3C2212yes—
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2211yescharacteristic
26C32×C4⋊C4C2212yes—
27C32×C4⋊C4C2211yescharacteristic
28C32×C22⋊C4C2212yes—
29#3757C2211yescharacteristic
30#3752C2212yes—
31#3752C2212yes—
32#3752C2211yescharacteristic
33C4⋊(C3⋊Dic3)C2212yes—
34#3750C2212yes—
35#3750C2212yes—
36#3750C2211yescharacteristic
37C4×C3⋊Dic3C2211yescharacteristic
38C2×#239—34no—
39C2×#235—34no—
40C3×C22⋊Q8D314yescomplemented
41#944—34no—
42C4×#67—34no—
43#897—34no—
44#894—38no—
45#893—34no—
46D3×C4⋊C4—38no—
47#884—38no—
48#883—38no—
49C23×C32C2311yescharacteristic
50C22×C3⋊D3C2311yescharacteristic
51C22×C3⋊D3C2312yes—
52C32×Q8—22no—
53C2×C32×C4—22no—
54C2×C32×C4C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2312yes—
57C2×C32×C4C2311yescharacteristic
58#638—24no—
59#638—22no—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62C2×C3⋊Dic3C2311yescharacteristic
63C2×C3⋊Dic3C2311yescharacteristic
64C4⋊(C3⋊D3)—24no—
65C4⋊(C3⋊D3)D414yescomplemented
66C4×C3⋊D3—24no—
67C4×C3⋊D3—24no—
68C4×C3⋊D3—22no—
69C4×C3⋊D3—24no—
70#634—22no—
71#547—91noSylow
72C2×C3×Q8C2×D314yes—
73C22×C3×C4C2×D314yes—
74C2×#67—38no—
75C2×#67—34no—
76#239—616no—
77#235—616no—
78C2×D12—34no—
79C2×D12—38no—
80C2×C4×D3—38no—
81C2×C4×D3—38no—
82C2×C4×D3—38no—
83C2×C4×D3—34no—
84C2×Dic6—34no—
85C3×C4⋊C4C2×D318yes—
86C3×C4⋊C4C2×D314yes—
87C3×C22⋊C4C2×D318yes—
88#217—34no—
89(C2×D3)⋊C4—38no—
90(C2×D3)⋊C4—38no—
91(C2×D3)⋊C4—34no—
92C4⋊Dic3—38no—
93Dic3⋊C4—38no—
94Dic3⋊C4—38no—
95Dic3⋊C4—34no—
96C4×Dic3—34no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32C2411yescharacteristic
101C2×C3⋊D3—22no—
102C2×C3⋊D3—22no—
103C2×C3⋊D3—21no—
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3C2×D414yes—
107C32×C4—21no—
108C32×C4—22no—
109C32×C4—22no—
110C32×C4C2×D412yes—
111C3⋊Dic3—22no—
112C3⋊Dic3—21no—
113C3⋊Dic3—21no—
114C3⋊Dic3#4112yescomplemented
115C2×#41—91no—
116C2×#41—91no—
117C4⋊Q8—91no—
118#123—92no—
119#123—92no—
120C22⋊Q8—91no—
121C22⋊Q8C3⋊D311yescharacteristic, complemented
122#121—92no—
123C4×D4—91no—
124C4×D4—91no—
125C2×C4⋊C4—92no—
126C23×C3C22×D314yes—
127C22×D3—38no—
128C22×D3—34no—
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—68no—
136#67—616no—
137C2×Dic3—34no—
138C2×Dic3—34no—
139C2×Dic3—38no—
140C2×Dic3—34no—
141D12—316no—
142D12—616no—
143C4×D3—616no—
144C4×D3—616no—
145C4×D3—616no—
146C4×D3—68no—
147Dic6—68no—
148C2×C32—41no—
149C2×C32C22×D411yescharacteristic
150C2×C32C2×#4111yescharacteristic
151C2×C322−1+411yescharacteristic
152C3⋊D3—24no—
153C3⋊D3—41no—
154#41—184no—
155#41—184no—
156C2×Q8—91no—
157C2×Q8C2×C3⋊D311yescharacteristic
158C2×D4—91no—
159C2×D4—92no—
160C2×D4—91no—
161C2×D4—92no—
162C22×C4—92no—
163C22×C4—92no—
164C22×C4—92no—
165C22×C4—91no—
166C22×C4C2×C3⋊D311yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—92no—
169C4⋊C4—92no—
170C4⋊C4—92no—
171C4⋊C4C2×C3⋊D312yes—
172C4⋊C4C2×C3⋊D311yescharacteristic
173C22⋊C4—91no—
174C22⋊C4—92no—
175C22⋊C4—91no—
176C22⋊C4—92no—
177C22⋊C4C2×C3⋊D312yes—
178C42—91no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3—24no—
182C22×C3C23×D314yes—
183C2×D3—316no—
184C2×D3—68no—
185C2×D3—68no—
186C2×D3—64no—
187C2×D3—64no—
188C2×D3—64no—
189C3×C4—24no—
190C3×C4—28no—
191C3×C4—28no—
192C3×C4D3×D418yes—
193Dic3—38no—
194Dic3—68no—
195Dic3—64no—
196Dic3—64no—
197C32#54711yescharacteristic, complemented, Sylow
198C23—92no—
199C23—91no—
200C23C22×C3⋊D311yescharacteristic
201Q8—22no—
202Q8—182no—
203D4—94no—
204D4—184no—
205D4—182no—
206D4—184no—
207C2×C4—22no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—92no—
212C2×C4—184no—
213C2×C4—184no—
214C2×C4—184no—
215C2×C4—182no—
216C2×C4C22×C3⋊D312yes—
217C2×C4C22×C3⋊D311yescharacteristic
218C2×C4C22×C3⋊D312yes—
219C2×C4C22×C3⋊D311yescharacteristic
220C2×C3—44no—
221C2×C3C2×D3×D414yes—
222C2×C3#100714yes—
223C2×C3D3×#4114yes—
224D3—616no—
225D3—124no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—94no—
230C22—182no—
231C22—182no—
232C22—181no—
233C22—181no—
234C22—181no—
235C22C23×C3⋊D311yescharacteristic
236C4—21no—
237C4—22no—
238C4—22no—
239C4—92no—
240C4—181no—
241C4—182no—
242C4—181no—
243C4D4×C3⋊D312yes—
244C3#563114yescomplemented
245C2—41no—
246C2—184no—
247C2—361no—
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#547
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. #64153C2semidirect
5. C4×#638C2semidirect
6. #64106C2semidirect
7. #64103C2semidirect
8. #64102C2semidirect
9. C4⋊C4×C3⋊D3C2semidirect
10. #64093C2semidirect
11. #64092C2semidirect
40. C3×C22⋊Q8D3semidirect
65. C4⋊(C3⋊D3)D4semidirect
114. C3⋊Dic3#41semidirect
121. C22⋊Q8C3⋊D3semidirect
197. C32#547semidirect
244. C3#5631semidirect

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
#641532
C4×#6382
#641062
#641032
#641022
C4⋊C4×C3⋊D32
#640932
#640922
#56313