Finite Group Databasebeta

← #10594180

Subgroups — #10594180

Order 576.

550 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#878C212yesmaximal, complemented
3#63974C211yesmaximal, characteristic
4#63964C212yesmaximal, complemented
5C2×#3722C211yesmaximal, characteristic
6#63895C212yesmaximal, complemented
7#63888C212yesmaximal, complemented
8#63871C212yesmaximal, complemented
9#63865C212yesmaximal, complemented
10#5634—32nomaximal
11C23×C3⋊D3C2211yescharacteristic
12C2×C4×C3⋊D3C2212yes—
13C22×C3×Dic3C2212yes—
14C2×C3×Dic6C2212yescomplemented
15C2×#627C2212yescomplemented
16C2×#624C2212yes—
17C32×C22⋊C4C2211yescharacteristic
18#3757C2211yescharacteristic
19#3752C2212yes—
20C3×#217C2212yes—
21C3×C4⋊Dic3C2212yescomplemented
22C3×Dic3⋊C4C2212yescomplemented
23C3×Dic3⋊C4C2212yescomplemented
24#3724C2212yescomplemented
25Dic3⋊Dic3C2212yescomplemented
26Dic3⋊Dic3C2212yescomplemented
27#3722C2211yescharacteristic
28#3722C2212yes—
29#3722C2211yescharacteristic
30#3722C2211yescharacteristic
31#3722C2211yescharacteristic
32C3×C22⋊Q8—32no—
33#946—32no—
34#946—32no—
35C2×(C2×D3)⋊C4—32no—
36#924—32no—
37#897—32no—
38#897—32no—
39#896—32no—
40#896—32no—
41#896—32no—
42#896—32no—
43D3×C22⋊C4—32no—
44D3×C22⋊C4—32no—
45D3×C22⋊C4—61no—
46#878—32no—
47#878D312yescomplemented
48C23×C32C2311yescharacteristic
49C22×C3⋊D3—42no—
50C22×C3⋊D3Q814yescomplemented
51C22×C3⋊D3C2311yescharacteristic
52C22×C3⋊D3C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C3⋊Dic3C2312yes—
55C4×C3⋊D3—42no—
56C2×C3×Dic3—42no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C3×Dic6—42no—
62#627—42no—
63#624—42no—
64#543—91noSylow
65C23×D3—32no—
66C23×D3—61no—
67C2×C3×Q8—32no—
68C22×C3×C4—32no—
69C22×Dic3C2×D312yes—
70C2×C4×D3—32no—
71C2×C4×D3—32no—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—62no—
75C2×Dic6—32no—
76C2×Dic6—32no—
77C2×Dic6C2×D312yescomplemented
78C3×C4⋊C4—32no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4—32no—
81C3×C22⋊C4—21no—
82C3×C22⋊C4—32no—
83C3×C22⋊C4C2×D312yes—
84#217—32no—
85#217—61no—
86#217C2×D312yes—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4—62no—
96C4⋊Dic3—32no—
97C4⋊Dic3—32no—
98C4⋊Dic3C2×D312yescomplemented
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4C2×D312yescomplemented
104Dic3⋊C4C2×D312yescomplemented
105C22×C32—22no—
106C22×C32C2×Q812yes—
107C22×C32C2411yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3—22no—
110C2×C3⋊D3—44no—
111C2×C3⋊D3C2×Q812yes—
112C2×C3⋊D3C2×Q812yes—
113C32×C4—22no—
114C3⋊Dic3—22no—
115C3×Dic3—22no—
116C3×Dic3—22no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119C22⋊Q8—32no—
120C22⋊Q8—92no—
121C22⋊Q8—92no—
122C22⋊Q8—92no—
123C22⋊Q8—92no—
124C22⋊Q8—92no—
125C2×C22⋊C4—91no—
126C2×C22⋊C4—91no—
127C2×C22⋊C4—91no—
128C23×C3—21no—
129C23×C3C22×D312yes—
130C22×D3—32no—
131C22×D3—32no—
132C22×D3—38no—
133C22×D3—61no—
134C22×D3—64no—
135C22×D3—61no—
136C22×D3—122no—
137C22×D3—122no—
138C22×D3—124no—
139C3×Q8—122no—
140C2×C3×C4—22no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—122no—
146C2×C3×C4C22×D312yes—
147C2×C3×C4C22×D312yes—
148C2×Dic3—32no—
149C2×Dic3—32no—
150C2×Dic3—42no—
151C2×Dic3—62no—
152C2×Dic3C22×D312yes—
153C2×Dic3C22×D312yes—
154C2×Dic3C22×D312yes—
155C2×Dic3C22×D312yes—
156C4×D3—122no—
157C4×D3—122no—
158C4×D3—122no—
159C4×D3—122no—
160C4×D3—124no—
161Dic6—42no—
162Dic6—122no—
163Dic6—122no—
164C2×C32—22no—
165C2×C32C22×Q811yescharacteristic
166C2×C32H(5,𝔽2)12yes—
167C3⋊D3—22no—
168C3⋊D3—22no—
169C24—91no—
170C2×Q8—32no—
171C2×Q8—92no—
172C22×C4—32no—
173C22×C4—92no—
174C22×C4—92no—
175C4⋊C4—32no—
176C4⋊C4—32no—
177C4⋊C4—32no—
178C4⋊C4—92no—
179C4⋊C4—92no—
180C4⋊C4—92no—
181C22⋊C4—32no—
182C22⋊C4—91no—
183C22⋊C4—91no—
184C22⋊C4—91no—
185C22⋊C4—92no—
186C22⋊C4—91no—
187C22⋊C4—91no—
188C22⋊C4—92no—
189C22⋊C4D3211yescharacteristic
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3D3×Q814yes—
196C22×C3C23×D312yes—
197C2×D3—34no—
198C2×D3—34no—
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—64no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—62no—
206C2×D3—122no—
207C2×D3—128no—
208C2×D3—128no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—62no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—62no—
216Dic3—22no—
217Dic3—22no—
218Dic3—22no—
219Dic3—22no—
220Dic3—62no—
221Dic3—62no—
222Dic3—122no—
223C32#54311yescharacteristic, complemented, Sylow
224C23—91no—
225C23—91no—
226C23—94no—
227C23—362no—
228C23C2×D3211yescharacteristic
229Q8—122no—
230Q8—362no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—92no—
236C2×C4—122no—
237C2×C4—362no—
238C2×C4—362no—
239C2×C4C2×D3212yes—
240C2×C3—24no—
241C2×C3—21no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3#100412yes—
245C2×C3C2×D3×Q812yes—
246C2×C3#100912yes—
247D3—64no—
248D3—64no—
249D3—64no—
250D3—122no—
251C22—22no—
252C22—92no—
253C22—92no—
254C22—182no—
255C22—182no—
256C22—364no—
257C22#379712yes—
258C22C22×D3211yescharacteristic
259C4—22no—
260C4—62no—
261C4—62no—
262C4—62no—
263C4—62no—
264C4—182no—
265C3—21no—
266C3#563412yescomplemented
267C2—22no—
268C2—182no—
269C2—182no—
270C2C2×#379711yescharacteristic
271C2#6431412yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC23×C3⋊D3C22
hypercenterC22⋊C4D32
hypocenterC32#543
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#878C2semidirect
4. #63964C2semidirect
6. #63895C2semidirect
7. #63888C2semidirect
8. #63871C2semidirect
9. #63865C2semidirect
14. C2×C3×Dic6C22semidirect
15. C2×#627C22semidirect
21. C3×C4⋊Dic3C22semidirect
22. C3×Dic3⋊C4C22semidirect
23. C3×Dic3⋊C4C22semidirect
24. #3724C22semidirect
25. Dic3⋊Dic3C22semidirect
26. Dic3⋊Dic3C22semidirect
47. #878D3semidirect
50. C22×C3⋊D3Q8semidirect
77. C2×Dic6C2×D3semidirect
98. C4⋊Dic3C2×D3semidirect
103. Dic3⋊C4C2×D3semidirect
104. Dic3⋊C4C2×D3semidirect
223. C32#543semidirect
266. C3#5634semidirect

Maximal subgroups

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

subgroupindex
C22⋊C4×C3⋊D32
C3×#8782
#639742
#639642
C2×#37222
#638952
#638882
#638712
#638652
#56343