Finite Group Databasebeta

← C3×#10593391 #13166740

Subgroups — C3×#10593391 #13166740

Order 1728.

394 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 70 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×#63873C211yesmaximal, characteristic
2C3×#63838C211yesmaximal, characteristic
3C3×#63779C211yesmaximal, characteristic
4C3×#63743C212yesmaximal
5C3×#63740C212yesmaximal
6#10593391C311yesmaximal, characteristic, direct product factor
7C2×C32×Dic6—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#622C2212yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×Dic3⋊C4—21no—
12C3×#3724—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2212yes—
15C3×Dic32—21no—
16C3×#944—61no—
17C3×#887—61no—
18#63873C2×C311yescharacteristic
19#63838C2×C311yescharacteristic
20#63779C2×C311yescharacteristic
21#63743C2×C312yes—
22#63740C2×C312yes—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—21no—
27C2×C32×Dic3—21no—
28C2×C32×Dic3—22no—
29C32×Dic6—42no—
30C3×#627—21no—
31C3×#627Q812yescomplemented
32C3×#622—22no—
33C3×#477—91nomaximal
34C2×C32×Q8—61no—
35C2×C3×Dic6—21no—
36C2×C3×Dic6—21no—
37C2×C3×Dic6—41no—
38C2×C3×Dic6—61no—
39C2×#627C22×C311yescharacteristic
40C2×#622C22×C312yes—
41C32×C4⋊C4—61no—
42C4×C3⋊Dic3C22×C311yescharacteristic
43C3×Dic3⋊C4—21no—
44C3×Dic3⋊C4—21no—
45C3×Dic3⋊C4—41no—
46C3×Dic3⋊C4—61no—
47C3×Dic3⋊C4—62no—
48C3×Dic3⋊C4—62no—
49C3×C4×Dic3—61no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—61no—
53#3724—21no—
54#3724C22×C311yescharacteristic
55#3724C22×C312yes—
56Dic32—21no—
57C22×C33C2×D411yescharacteristic
58C33×C4—21no—
59C3×C3⋊Dic3Q1612yes—
60C3×C3⋊Dic3C2×Q811yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—22no—
63C32×Dic3—42no—
64C32×Dic3—41no—
65C3×C4⋊Q8—91no—
66C3×C4×Q8—91no—
67#944—61no—
68#887—61no—
69C3×#107—92no—
70C3×C4⋊C8—91no—
71C3×#103—92no—
72C32×Q8—122no—
73C2×C32×C4—21no—
74C2×C32×C4—21no—
75C2×C32×C4—41no—
76C2×C32×C4—41no—
77C2×C32×C4—61no—
78C2×C32×C4—61no—
79C2×C32×C4—62no—
80C2×C32×C4C3×D411yescharacteristic
81C2×C3⋊Dic3C3×D411yescharacteristic
82C2×C3⋊Dic3C23×C311yescharacteristic
83C2×C3×Dic3—21no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—21no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C3×Dic6—42no—
97C3×Dic6—42no—
98C3×Dic6—44no—
99C3×Dic6—62no—
100C3×Dic6—62no—
101#627—21no—
102#627C3×Q812yescomplemented
103#622—22no—
104#477—91noSylow
105C2×C33C22⋊Q811yescharacteristic
106C2×C33C2×Q1611yescharacteristic
107C2×C33AGL(1,ℤ/8)11yescharacteristic
108C2×C3×Q8—61no—
109C2×C3×Q8—61no—
110C2×C3×Q8—91no—
111C2×C3×Q8—121no—
112C2×Dic6—21no—
113C2×Dic6—61no—
114C2×C3×C8—92no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—61no—
117C3×C4⋊C4—92no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4—121no—
120C3×C4⋊C4—181no—
121C3×C42—91no—
122C3×C42—181no—
123Dic3⋊C4—21no—
124Dic3⋊C4—61no—
125Dic3⋊C4—62no—
126Dic3⋊C4—62no—
127C4×Dic3—61no—
128C4×Dic3—61no—
129C4×Dic3—61no—
130C4×Dic3—61no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32C2×C3×D411yescharacteristic
136C32×C4—22no—
137C32×C4—21no—
138C32×C4—41no—
139C32×C4—42no—
140C32×C4—62no—
141C32×C4—81no—
142C32×C4—122no—
143C32×C4—121no—
144C3⋊Dic3C3×Q1612yes—
145C3⋊Dic3C2×C3×Q811yescharacteristic
146C3⋊Dic3C3×#4111yescharacteristic
147C3×Dic3—22no—
148C3×Dic3—22no—
149C3×Dic3—42no—
150C3×Dic3—42no—
151C3×Dic3—41no—
152C3×Dic3—44no—
153C3×Dic3—42no—
154C3×Dic3—41no—
155C3×Dic3—62no—
156C3×Dic3—62no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—81no—
162C4⋊Q8—91no—
163C4×Q8—91no—
164#107—92no—
165C4⋊C8—91no—
166#103—92no—
167C33#47711yescharacteristic, complemented, Sylow
168C3×Q8—92no—
169C3×Q8—122no—
170C3×Q8—122no—
171C3×Q8—124no—
172C3×Q8—181no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—62no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—121no—
186C2×C3×C4—122no—
187C2×C3×C4—121no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—22no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196Dic6—42no—
197Dic6—62no—
198Dic6—62no—
199C3×C8—182no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32C3×C22⋊Q811yescharacteristic
213C2×C32C2×C3×Q1611yescharacteristic
214C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
215C2×Q8—61no—
216C2×Q8—91no—
217C2×C8—92no—
218C4⋊C4—61no—
219C4⋊C4—92no—
220C4⋊C4—91no—
221C4⋊C4—181no—
222C42—91no—
223C42—181no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3C2×GO+(4,𝔽2)11yescharacteristic
229C3×C4—22no—
230C3×C4—21no—
231C3×C4—41no—
232C3×C4—42no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—81no—
236C3×C4—91no—
237C3×C4—91no—
238C3×C4—92no—
239C3×C4—122no—
240C3×C4—122no—
241C3×C4—121no—
242C3×C4—124no—
243C3×C4—122no—
244C3×C4—121no—
245C3×C4—241no—
246Dic3—22no—
247Dic3—42no—
248Dic3—41no—
249Dic3—61no—
250Dic3—61no—
251Dic3—62no—
252Dic3—62no—
253Dic3—61no—
254Dic3—61no—
255C32—21no—
256C32—21no—
257C32—41no—
258C32—41no—
259C32C3×#47711yescharacteristic, complemented
260Q8—92no—
261Q8—122no—
262Q8—181no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4C3×GO+(4,𝔽2)11yescharacteristic
269C8—182no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3D32⋊Q811yescharacteristic
283C2×C3#6422811yescharacteristic
284C2×C3#6423611yescharacteristic
285C22C2×C3×GO+(4,𝔽2)11yescharacteristic
286C4—21no—
287C4—62no—
288C4—91no—
289C4—91no—
290C4—92no—
291C4—122no—
292C4—121no—
293C3—21no—
294C3—21no—
295C3—41no—
296C3—41no—
297C3#1059339111yescharacteristic, direct product factor
298C2C3×D32⋊Q811yescharacteristic
299C2C3×#6422811yescharacteristic
300C2C3×#6423611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#477
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593391C3direct (N × H)
31. C3×#627Q8semidirect
102. #627C3×Q8semidirect
167. C33#477semidirect
259. C32C3×#477semidirect
297. C3#10593391direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#638732
C3×#638382
C3×#637792
C3×#637432
C3×#637402
#105933913
C3×#4779