Finite Group Databasebeta

← C3×#10593304 #13166673

Subgroups — C3×#10593304 #13166673

Order 1728.

362 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C3×#64226C212yesmaximal
2C3×#63803C211yesmaximal, characteristic
3C3×#63802C211yesmaximal, characteristic, complemented
4C3×#63771C211yesmaximal, characteristic, complemented
5C3×#63728C212yesmaximal, complemented
6#10593304C311yesmaximal, characteristic, direct product factor
7C3×#3798C2211yescharacteristic
8C3×#3797C2212yes—
9C3×#3788C2212yes—
10C3×#3776—22no—
11C3×#3775—22no—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×Dic12—21no—
14C32×D3×C8—21no—
15C3×#3718—21no—
16C3×#3712C2211yescharacteristic
17C3×#3711—21no—
18#64226C2×C312yes—
19C3×D3×Q16—61no—
20C3×#901—61no—
21#63803C2×C311yescharacteristic
22#63802C2×C311yescharacteristic, complemented
23#63771C2×C311yescharacteristic, complemented
24#63728C2×C312yescomplemented
25C3×C4×C3⋊D3C2311yescharacteristic
26C32×C4×D3—21no—
27C32×Dic6—22no—
28C3×#627—22no—
29C3×#627D411yescharacteristic
30C3×#626—21no—
31C3×#625D411yescharacteristic
32C3×#624—22no—
33C3×#622—22no—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36C32×C3⋊C8—21no—
37C3×#473—91nomaximal
38C3×D3×Q8—62no—
39C3×#235—61no—
40#3798C22×C311yescharacteristic
41#3797C22×C312yes—
42#3788C22×C312yes—
43#3776—22no—
44#3775—22no—
45C32×Q16—61no—
46C2×C32×C8—61no—
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#216—62no—
49C3×#208—61no—
50C3×Dic12—21no—
51C3×Dic12—21no—
52C3×Dic12—41no—
53C3×ΓL(1,𝔽25)—61no—
54C3×ΓL(1,𝔽25)—61no—
55C3×D3×C8—21no—
56C3×D3×C8—21no—
57C3×D3×C8—41no—
58C3×D3×C8—61no—
59C3×D3×C8—61no—
60#3718—21no—
61#3712C22×C311yescharacteristic
62#3711—21no—
63C2×C3×C3⋊D3#4111yescharacteristic
64C2×C32×D3—21no—
65C33×C4C2×D411yescharacteristic
66C3×C3⋊Dic3C2×D411yescharacteristic
67C32×Dic3—21no—
68C32×Dic3—42no—
69C3×#137—92no—
70C2×C3×Q16—91no—
71C3×#131—91no—
72D3×Q16—61no—
73#901—61no—
74C3×#108—91no—
75C3×#101—92no—
76C32×Q8—62no—
77C2×C32×C4—61no—
78C4×C3⋊D3C23×C311yescharacteristic
79C3×#67—61no—
80C3×#67—61no—
81C3×D12—61no—
82C3×C4×D3—21no—
83C3×C4×D3—21no—
84C3×C4×D3—41no—
85C3×C4×D3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—122no—
88C3×Dic6—22no—
89C3×Dic6—22no—
90C3×Dic6—42no—
91C3×Dic6—61no—
92C3×Dic6—122no—
93#627—22no—
94#627C3×D411yescharacteristic
95#626—21no—
96#625C3×D411yescharacteristic
97#624—22no—
98#622—22no—
99C32×C8—21no—
100C32×C8—21no—
101C32×C8—41no—
102C32×C8—41no—
103C32×C8—61no—
104C32×C8C3×D411yescharacteristic
105C3⋊(C3⋊C8)C3×D411yescharacteristic
106C3×C3⋊C8—21no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—41no—
109C3×C3⋊C8—61no—
110C3×C3⋊C8—61no—
111#473—91noSylow
112C2×C33#12111yescharacteristic
113C3×C3⋊D3—21no—
114C32×D3—41no—
115C3×#41—91no—
116C2×C3×Q8—92no—
117D3×Q8—62no—
118#235—61no—
119C3×Q16—61no—
120C3×Q16—61no—
121C3×Q16—121no—
122C3×Q16—181no—
123C3×Q16—182no—
124C3×SD16—182no—
125C3×M16—91no—
126C3×M16—92no—
127C3×M16—181no—
128C2×C3×C8—61no—
129C2×C3×C8—61no—
130C2×C3×C8—91no—
131C2×C3×C8—121no—
132#216—62no—
133#208—61no—
134Dic12—21no—
135ΓL(1,𝔽25)—61no—
136ΓL(1,𝔽25)—61no—
137D3×C8—21no—
138D3×C8—61no—
139D3×C8—61no—
140C22×C32—61no—
141C2×C3⋊D3C3×#4111yescharacteristic
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—41no—
145C2×C3×D3—61no—
146C2×C3×D3—61no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—61no—
152C32×C4—122no—
153C32×C4C2×C3×D411yescharacteristic
154C3⋊Dic3C2×C3×D411yescharacteristic
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3—41no—
158C3×Dic3—42no—
159C3×Dic3—42no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—82no—
163#137—92no—
164C2×Q16—91no—
165#131—91no—
166#108—91no—
167#101—92no—
168C33#47311yescharacteristic, complemented, Sylow
169C3×Q8—62no—
170C3×Q8—62no—
171C3×Q8—91no—
172C3×Q8—122no—
173C3×Q8—182no—
174C3×D4—91no—
175C3×D4—181no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—91no—
179C2×C3×C4—121no—
180C2×C3×C4—182no—
181#67—61no—
182#67—61no—
183D12—61no—
184C4×D3—21no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—122no—
188Dic6—22no—
189Dic6—61no—
190Dic6—122no—
191C3×C8—21no—
192C3×C8—21no—
193C3×C8—41no—
194C3×C8—41no—
195C3×C8—61no—
196C3×C8—61no—
197C3×C8—91no—
198C3×C8—121no—
199C3×C8—182no—
200C3×C8GO+(4,𝔽2)11yescharacteristic
201C3⋊C8—21no—
202C3⋊C8—61no—
203C3⋊C8—61no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—121no—
209C2×C32C3×#12111yescharacteristic
210C3⋊D3—21no—
211C3×D3—41no—
212C3×D3—42no—
213C3×D3—41no—
214C3×D3—62no—
215C3×D3—121no—
216#41—91no—
217C2×Q8—92no—
218Q16—61no—
219Q16—181no—
220Q16—182no—
221SD16—182no—
222M16—91no—
223M16—92no—
224M16—181no—
225C2×C8—61no—
226C2×C8—91no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—91no—
230C22×C3—121no—
231C2×D3—21no—
232C2×D3—61no—
233C2×D3—61no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—91no—
241C3×C4—121no—
242C3×C4—122no—
243C3×C4—122no—
244C3×C4—242no—
245C3×C4C2×GO+(4,𝔽2)11yescharacteristic
246Dic3—21no—
247Dic3—42no—
248Dic3—61no—
249Dic3—61no—
250C32—21no—
251C32—21no—
252C32—41no—
253C32—41no—
254C32C3×#47311yescharacteristic, complemented
255Q8—62no—
256Q8—91no—
257Q8—182no—
258D4—91no—
259D4—181no—
260C2×C4—61no—
261C2×C4—91no—
262C2×C4—182no—
263C8—61no—
264C8—91no—
265C8—182no—
266C8C3×GO+(4,𝔽2)11yescharacteristic
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—121no—
272C2×C3—122no—
273C2×C3—121no—
274C2×C3—181no—
275C2×C3#6423011yescharacteristic
276D3—41no—
277D3—62no—
278D3—121no—
279C22—61no—
280C22—91no—
281C4—61no—
282C4—91no—
283C4—122no—
284C4C2×C3×GO+(4,𝔽2)11yescharacteristic
285C3—21no—
286C3—21no—
287C3—41no—
288C3—41no—
289C3#1059330411yescharacteristic, direct product factor
290C2—121no—
291C2—181no—
292C2C3×#6423011yescharacteristic
subgroupG/H
centerC2×C3#64230
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#121
prime radicalC3×#3798C22
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#473
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63802C2semidirect
4. C3×#63771C2semidirect
5. C3×#63728C2semidirect
6. #10593304C3direct (N × H)
22. #63802C2×C3semidirect
23. #63771C2×C3semidirect
24. #63728C2×C3semidirect
168. C33#473semidirect
254. C32C3×#473semidirect
289. C3#10593304direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642262
C3×#638032
C3×#638022
C3×#637712
C3×#637282
#105933043
C3×#4739