Finite Group Databasebeta

← C3×#10593289 #13166658

Subgroups — C3×#10593289 #13166658

Order 1728.

386 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 62 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×D32⋊Q8C212yesmaximal
2C3×#63802C211yesmaximal, characteristic, complemented
3C3×#63791C211yesmaximal, characteristic
4C3×#63768C211yesmaximal, characteristic, complemented
5C3×#63730C212yesmaximal, complemented
6#10593289C311yesmaximal, characteristic, direct product factor
7C3×C4×D32C2211yescharacteristic
8C3×#3797C2212yes—
9C3×C4⋊#157C2212yes—
10C3×#3773—22no—
11C3×D32⋊C4—22no—
12C3×C8×C3⋊D3C2211yescharacteristic
13C32×Dic12—21no—
14C32×ΓL(1,𝔽25)—21no—
15C3×#3718—21no—
16C3×#3711—21no—
17C3×#3710C2211yescharacteristic
18D32⋊Q8C2×C312yes—
19C3×D3×Q16—61no—
20C3×D3×M16—61no—
21#63802C2×C311yescharacteristic, complemented
22#63791C2×C311yescharacteristic
23#63768C2×C311yescharacteristic, complemented
24#63730C2×C312yescomplemented
25C2×C3×D32D411yescharacteristic
26C2×C3×#157—22no—
27C3×C4×C3⋊D3C2311yescharacteristic
28C32×C4×D3—21no—
29C32×Dic6—22no—
30C3×#627—22no—
31C3×#624—22no—
32C3×#624D411yescharacteristic
33C3×D3×Dic3—21no—
34C33×C8D411yescharacteristic
35C3×C3⋊(C3⋊C8)D411yescharacteristic
36C32×C3⋊C8—21no—
37C3×#470—91nomaximal
38C3×D3×Q8—62no—
39C2×C3×C4×D3—61no—
40C4×D32C22×C311yescharacteristic
41#3797C22×C312yes—
42C4⋊#157C22×C312yes—
43#3773—22no—
44D32⋊C4—22no—
45C32×Q16—61no—
46C32×M16—61no—
47C8×C3⋊D3C22×C311yescharacteristic
48C3×#216—62no—
49C3×#208—61no—
50C3×Dic12—21no—
51C3×Dic12—21no—
52C3×Dic12—41no—
53C3×ΓL(1,𝔽25)—21no—
54C3×ΓL(1,𝔽25)—21no—
55C3×ΓL(1,𝔽25)—41no—
56C3×ΓL(1,𝔽25)—61no—
57C3×D3×C8—61no—
58C3×D3×C8—61no—
59C3×D3×C8—61no—
60#3718—21no—
61#3711—21no—
62#3710C22×C311yescharacteristic
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C32×D3—21no—
65C3×D32—22no—
66C3×#157—42no—
67C33×C4C2×D411yescharacteristic
68C3×C3⋊Dic3#4111yescharacteristic
69C32×Dic3—21no—
70C32×Dic3—42no—
71C2×C3×Q16—91no—
72C2×C3×M16—91no—
73C3×C22⋊Q8—92no—
74D3×Q16—61no—
75D3×M16—61no—
76C3×#106—91no—
77C3×#103—92no—
78C22×C3×D3—61no—
79C2×D32C3×D411yescharacteristic
80C2×#157—22no—
81C32×Q8—62no—
82C2×C32×C4—61no—
83C4×C3⋊D3C23×C311yescharacteristic
84C2×C3×Dic3—61no—
85C3×C4×D3—21no—
86C3×C4×D3—21no—
87C3×C4×D3—41no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—122no—
93C3×Dic6—22no—
94C3×Dic6—22no—
95C3×Dic6—42no—
96C3×Dic6—122no—
97#627—22no—
98#624—22no—
99#624C3×D411yescharacteristic
100D3×Dic3—21no—
101C32×C8—21no—
102C32×C8—21no—
103C32×C8—41no—
104C32×C8—41no—
105C32×C8—61no—
106C32×C8C3×D411yescharacteristic
107C3⋊(C3⋊C8)C3×D411yescharacteristic
108C3×C3⋊C8—21no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—61no—
113#470—91noSylow
114C2×C33#12111yescharacteristic
115C3×C3⋊D3#13712yes—
116C32×D3—41no—
117C2×C3×Q8—92no—
118C22×C3×C4—91no—
119D3×Q8—62no—
120C2×C4×D3—61no—
121C3×Q16—61no—
122C3×Q16—61no—
123C3×Q16—121no—
124C3×Q16—181no—
125C3×M16—61no—
126C3×M16—61no—
127C3×M16—121no—
128C3×M16—181no—
129C2×C3×C8—91no—
130C2×C3×C8—91no—
131C3×C4⋊C4—92no—
132C3×C4⋊C4—182no—
133C3×C22⋊C4—182no—
134#216—62no—
135#208—61no—
136Dic12—21no—
137ΓL(1,𝔽25)—21no—
138ΓL(1,𝔽25)—61no—
139D3×C8—61no—
140D3×C8—61no—
141D3×C8—61no—
142C22×C32—61no—
143C2×C3⋊D3C2×C3×D411yescharacteristic
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—41no—
147C2×C3×D3—61no—
148C2×C3×D3—61no—
149C2×C3×D3—122no—
150D32—22no—
151#157—42no—
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—61no—
157C32×C4—122no—
158C32×C4C2×C3×D411yescharacteristic
159C3⋊Dic3C3×#4111yescharacteristic
160C3×Dic3—21no—
161C3×Dic3—21no—
162C3×Dic3—41no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—82no—
168C2×Q16—91no—
169C2×M16—91no—
170C22⋊Q8—92no—
171#106—91no—
172#103—92no—
173C33#47011yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C22×D3—61no—
176C3×Q8—62no—
177C3×Q8—62no—
178C3×Q8—122no—
179C3×Q8—182no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—121no—
185C2×C3×C4—181no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×Dic3—61no—
189C4×D3—21no—
190C4×D3—61no—
191C4×D3—61no—
192C4×D3—61no—
193C4×D3—61no—
194C4×D3—122no—
195Dic6—22no—
196Dic6—122no—
197C3×C8—21no—
198C3×C8—21no—
199C3×C8—41no—
200C3×C8—41no—
201C3×C8—61no—
202C3×C8—61no—
203C3×C8—91no—
204C3×C8—121no—
205C3×C8GO+(4,𝔽2)11yescharacteristic
206C3⋊C8—21no—
207C3⋊C8—61no—
208C3⋊C8—61no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—121no—
214C2×C32C3×#12111yescharacteristic
215C3⋊D3C3×#13712yes—
216C3×D3—41no—
217C3×D3—41no—
218C3×D3—62no—
219C3×D3—62no—
220C3×D3—81no—
221C2×Q8—92no—
222C22×C4—91no—
223Q16—61no—
224Q16—181no—
225M16—61no—
226M16—181no—
227C2×C8—91no—
228C2×C8—91no—
229C4⋊C4—92no—
230C4⋊C4—182no—
231C22⋊C4—182no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—91no—
235C22×C3—121no—
236C22×C3—182no—
237C2×D3—21no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—122no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—91no—
248C3×C4—121no—
249C3×C4—122no—
250C3×C4—122no—
251C3×C4—242no—
252C3×C4—362no—
253C3×C4C2×GO+(4,𝔽2)11yescharacteristic
254Dic3—21no—
255Dic3—42no—
256Dic3—61no—
257Dic3—61no—
258C32—21no—
259C32—21no—
260C32—41no—
261C32—41no—
262C32C3×#47011yescharacteristic, complemented
263C23—91no—
264Q8—62no—
265Q8—182no—
266C2×C4—61no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—181no—
270C2×C4—182no—
271C2×C4—182no—
272C8—61no—
273C8—91no—
274C8C3×GO+(4,𝔽2)11yescharacteristic
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—92no—
280C2×C3—121no—
281C2×C3—121no—
282C2×C3—241no—
283C2×C3#6423011yescharacteristic
284D3—41no—
285D3—62no—
286D3—62no—
287C22—61no—
288C22—91no—
289C22—182no—
290C4—61no—
291C4—91no—
292C4—122no—
293C4—362no—
294C4C2×C3×GO+(4,𝔽2)11yescharacteristic
295C3—21no—
296C3—21no—
297C3—41no—
298C3—41no—
299C3#1059328911yescharacteristic, direct product factor
300C2—92no—
301C2—121no—
302C2C3×#6423011yescharacteristic
subgroupG/H
centerC2×C3#64230
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33#121
prime radicalC2×C3×D32D4
hypercenterC3×C8GO+(4,𝔽2)
hypocenterC32C3×#470
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
2. C3×#63802C2semidirect
4. C3×#63768C2semidirect
5. C3×#63730C2semidirect
6. #10593289C3direct (N × H)
21. #63802C2×C3semidirect
23. #63768C2×C3semidirect
24. #63730C2×C3semidirect
173. C33#470semidirect
262. C32C3×#470semidirect
299. C3#10593289direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D32⋊Q82
C3×#638022
C3×#637912
C3×#637682
C3×#637302
#105932893
C3×#4709