Finite Group Databasebeta

← #10596103

Subgroups — #10596103

Order 576.

840 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 211 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
2C22×#634C211yesmaximal, characteristic, complemented
3C32×C22⋊Q8C211yesmaximal, characteristic, complemented
4#64153C211yesmaximal, characteristic, complemented
5C4×#638C211yesmaximal, characteristic
6#64106C211yesmaximal, characteristic, complemented
7#64105C212yesmaximal, complemented
8#64097C212yesmaximal, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64090C212yesmaximal
11#64087C212yesmaximal, complemented
12#5632—34nomaximal
13C2×C32×Q8C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C22×C3⋊Dic3C2212yes—
17Q8×C3⋊D3—24no—
18C2×C4×C3⋊D3C2212yes—
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#634—24no—
21C2×#634C2212yes—
22C2×#634C2212yes—
23C2×#634C2211yescharacteristic
24C2×#634C2211yescharacteristic
25C32×C4⋊C4C2211yescharacteristic
26C32×C4⋊C4C2212yes—
27C32×C22⋊C4C2212yes—
28#3757C2211yescharacteristic
29#3752C2212yes—
30#3752C2211yescharacteristic
31C4⋊(C3⋊Dic3)C2212yes—
32#3750C2212yes—
33#3750C2212yes—
34#3750C2212yes—
35#3750C2211yescharacteristic
36C4×C3⋊Dic3C2212yes—
37C4×C3⋊Dic3C2211yescharacteristic
38C2×D3×Q8—34no—
39C22×Dic6—34no—
40C3×C22⋊Q8D314yescomplemented
41#944—34no—
42C4×#67—34no—
43#897—34no—
44#896—38no—
45#888—38no—
46#887—34no—
47#67⋊C4—38no—
48#878—38no—
49C23×C32C2311yescharacteristic
50C22×C3⋊D3C2311yescharacteristic
51C32×Q8—22no—
52C2×C32×C4—22no—
53C2×C32×C4C2312yes—
54C2×C32×C4C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57#638Q814yescomplemented
58C2×C3⋊Dic3—24no—
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2312yes—
61C2×C3⋊Dic3C2311yescharacteristic
62C2×C3⋊Dic3C2312yes—
63C2×C3⋊Dic3C2311yescharacteristic
64C4×C3⋊D3—22no—
65C4×C3⋊D3—24no—
66#634—22no—
67#634—24no—
68#634—24no—
69#634D414yescomplemented
70D4×Q8—91noSylow
71C2×C3×Q8C2×D314yes—
72C22×C3×C4C2×D314yes—
73C2×#67—34no—
74C22×Dic3—38no—
75D3×Q8—616no—
76C2×C4×D3—38no—
77C2×C4×D3—34no—
78C2×Dic6—34no—
79C2×Dic6—38no—
80C2×Dic6—38no—
81C2×Dic6—34no—
82C2×Dic6—616no—
83C3×C4⋊C4C2×D314yes—
84C3×C4⋊C4C2×D318yes—
85C3×C22⋊C4C2×D318yes—
86#217—34no—
87(C2×D3)⋊C4—38no—
88(C2×D3)⋊C4—34no—
89C4⋊Dic3—38no—
90Dic3⋊C4—38no—
91Dic3⋊C4—38no—
92Dic3⋊C4—38no—
93Dic3⋊C4—34no—
94C4×Dic3—38no—
95C4×Dic3—34no—
96C22×C32—21no—
97C22×C32—21no—
98C22×C32C2×Q812yescomplemented
99C22×C32C2411yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3C2×Q812yescomplemented
103C32×C4—21no—
104C32×C4—22no—
105C32×C4—22no—
106C32×C4C2×D412yes—
107C3⋊Dic3—21no—
108C3⋊Dic3—22no—
109C3⋊Dic3—21no—
110C3⋊Dic3C2×D414yes—
111C3⋊Dic3C2×Q812yescomplemented
112C22×Q8—91no—
113C22×Q8—91no—
114C4⋊Q8—91no—
115C4⋊Q8—92no—
116C22⋊Q8—92no—
117C22⋊Q8—91no—
118C22⋊Q8—92no—
119C22⋊Q8C3⋊D311yescharacteristic, complemented
120C4×Q8—91no—
121C4×D4—91no—
122C4×D4—92no—
123C23×C3C22×D314yes—
124C22×D3—34no—
125C3×Q8—28no—
126C2×C3×C4—28no—
127C2×C3×C4C22×D318yes—
128C2×C3×C4C22×D318yes—
129C2×C3×C4C22×D314yes—
130C2×C3×C4C22×D314yes—
131#67—316no—
132C2×Dic3—34no—
133C2×Dic3—38no—
134C2×Dic3—34no—
135C2×Dic3—38no—
136C2×Dic3—34no—
137C2×Dic3—616no—
138C4×D3—616no—
139C4×D3—68no—
140Dic6—316no—
141Dic6—616no—
142Dic6—616no—
143Dic6—68no—
144C2×C32—22no—
145C2×C32C22×D411yescharacteristic
146C2×C32C22×Q811yescharacteristic
147C2×C322−1+411yescharacteristic
148C3⋊D3—22no—
149C2×Q8—91no—
150C2×Q8—92no—
151C2×Q8—92no—
152C2×Q8—91no—
153C2×Q8—184no—
154C2×Q8—184no—
155C2×Q8C2×C3⋊D311yescharacteristic
156C2×D4—91no—
157C22×C4—92no—
158C22×C4—92no—
159C22×C4—91no—
160C22×C4C2×C3⋊D311yescharacteristic
161C4⋊C4—91no—
162C4⋊C4—92no—
163C4⋊C4—92no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4C2×C3⋊D311yescharacteristic
167C4⋊C4C2×C3⋊D312yes—
168C22⋊C4—91no—
169C22⋊C4—91no—
170C22⋊C4—92no—
171C22⋊C4C2×C3⋊D312yes—
172C42—92no—
173C42—91no—
174C22×C3—24no—
175C22×C3—24no—
176C22×C3D3×Q818yescomplemented
177C22×C3C23×D314yes—
178C2×D3—38no—
179C2×D3—64no—
180C2×D3—64no—
181C3×C4—24no—
182C3×C4—28no—
183C3×C4—28no—
184C3×C4D3×D418yes—
185Dic3—316no—
186Dic3—38no—
187Dic3—64no—
188Dic3—68no—
189Dic3—64no—
190C32D4×Q811yescharacteristic, complemented, Sylow
191C23—91no—
192C23C22×C3⋊D311yescharacteristic
193Q8—22no—
194Q8—94no—
195Q8—184no—
196Q8—184no—
197Q8—182no—
198D4—94no—
199C2×C4—22no—
200C2×C4—91no—
201C2×C4—92no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—92no—
205C2×C4—184no—
206C2×C4—184no—
207C2×C4—182no—
208C2×C4C22×C3⋊D312yes—
209C2×C4C22×C3⋊D311yescharacteristic
210C2×C4C22×C3⋊D312yes—
211C2×C4C22×C3⋊D311yescharacteristic
212C2×C3—28no—
213C2×C3C2×D3×D414yes—
214C2×C3C2×D3×Q814yes—
215C2×C3#101014yes—
216D3—68no—
217C22—21no—
218C22—21no—
219C22—92no—
220C22—181no—
221C22—181no—
222C22Q8×C3⋊D312yescomplemented
223C22C23×C3⋊D311yescharacteristic
224C4—21no—
225C4—22no—
226C4—22no—
227C4—94no—
228C4—92no—
229C4—182no—
230C4—181no—
231C4—181no—
232C4D4×C3⋊D312yes—
233C3#563214yescomplemented
234C2—22no—
235C2—182no—
236C2C2×D4×C3⋊D311yescharacteristic
237C2C2×Q8×C3⋊D311yescharacteristic
238C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊Q8C2
prime radicalC2×#638C22
hypercenterC22⋊Q8C3⋊D3
hypocenterC32D4×Q8
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. C22×#634C2semidirect
3. C32×C22⋊Q8C2semidirect
4. #64153C2semidirect
6. #64106C2semidirect
7. #64105C2semidirect
8. #64097C2semidirect
9. #64096C2semidirect
11. #64087C2semidirect
40. C3×C22⋊Q8D3semidirect
57. #638Q8semidirect
69. #634D4semidirect
98. C22×C32C2×Q8semidirect
102. C2×C3⋊D3C2×Q8semidirect
111. C3⋊Dic3C2×Q8semidirect
119. C22⋊Q8C3⋊D3semidirect
176. C22×C3D3×Q8semidirect
190. C32D4×Q8semidirect
222. C22Q8×C3⋊D3semidirect
233. C3#5632semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D32
C22×#6342
C32×C22⋊Q82
#641532
C4×#6382
#641062
#641052
#640972
#640962
#640902
#640872
#56323