Finite Group Databasebeta

← C3×#10593385 #13166719

Subgroups — C3×#10593385 #13166719

Order 1728.

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

normal subgroups only

This group has 382 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×#63919C211yesmaximal, characteristic
2C3×#63838C211yesmaximal, characteristic
3C3×#63779C211yesmaximal, characteristic
4C3×#63742C212yesmaximal
5C3×#63740C212yesmaximal
6#10593385C311yesmaximal, characteristic, direct product factor
7C2×C3×#627—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#622C2212yes—
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C32×C4⋊Dic3—21no—
12C32×Dic3⋊C4—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2212yes—
15C3×Dic32—21no—
16C3×#888—61no—
17C3×#887—61no—
18#63919C2×C311yescharacteristic
19#63838C2×C311yescharacteristic
20#63779C2×C311yescharacteristic
21#63742C2×C312yes—
22#63740C2×C312yes—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C2×C32×Dic3—21no—
27C2×C32×Dic3—22no—
28C2×C32×Dic3—21no—
29C3×#627—21no—
30C3×#627—42no—
31C3×#627Q812yescomplemented
32C3×#622—22no—
33C3×#475—91nomaximal
34C2×C3×Dic6—61no—
35C2×C3×Dic6—62no—
36C2×#627—21no—
37C2×#627C22×C311yescharacteristic
38C2×#622C22×C312yes—
39C32×C4⋊C4—61no—
40C32×C4⋊C4—61no—
41C4×C3⋊Dic3C22×C311yescharacteristic
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—41no—
45C3×Dic3⋊C4—21no—
46C3×Dic3⋊C4—21no—
47C3×Dic3⋊C4—41no—
48C3×Dic3⋊C4—61no—
49C3×Dic3⋊C4—62no—
50C3×C4×Dic3—61no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724C22×C311yescharacteristic
55#3724C22×C312yes—
56Dic32—21no—
57C22×C33C2×D411yescharacteristic
58C33×C4—21no—
59C3×C3⋊Dic3SD1612yes—
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#888—61no—
68#887—61no—
69C3×#106—92no—
70C3×C4⋊C8—91no—
71C3×#103—92no—
72C2×C32×C4—21no—
73C2×C32×C4—21no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C32×C4—61no—
77C2×C32×C4—61no—
78C2×C32×C4—62no—
79C2×C32×C4C3×D411yescharacteristic
80C2×C3⋊Dic3C3×D411yescharacteristic
81C2×C3⋊Dic3C23×C311yescharacteristic
82C2×C3×Dic3—21no—
83C2×C3×Dic3—21no—
84C2×C3×Dic3—21no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—41no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—42no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97C3×Dic6—124no—
98#627—21no—
99#627—42no—
100#627C3×Q812yescomplemented
101#622—22no—
102#475—91noSylow
103C2×C33C22⋊Q811yescharacteristic
104C2×C33C2×SD1611yescharacteristic
105C2×C33#13711yescharacteristic
106C2×C3×Q8—91no—
107C2×C3×Q8—181no—
108C2×Dic6—61no—
109C2×Dic6—62no—
110C2×C3×C8—92no—
111C3×C4⋊C4—61no—
112C3×C4⋊C4—61no—
113C3×C4⋊C4—61no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—92no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—121no—
118C3×C4⋊C4—121no—
119C3×C42—91no—
120C3×C42—181no—
121C4⋊Dic3—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—61no—
124Dic3⋊C4—62no—
125C4×Dic3—61no—
126C4×Dic3—61no—
127C4×Dic3—61no—
128C4×Dic3—61no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32C2×C3×D411yescharacteristic
134C32×C4—22no—
135C32×C4—21no—
136C32×C4—41no—
137C32×C4—42no—
138C32×C4—62no—
139C32×C4—81no—
140C32×C4—122no—
141C32×C4—121no—
142C3⋊Dic3C3×SD1612yes—
143C3⋊Dic3C2×C3×Q811yescharacteristic
144C3⋊Dic3C3×#4111yescharacteristic
145C3×Dic3—22no—
146C3×Dic3—22no—
147C3×Dic3—42no—
148C3×Dic3—42no—
149C3×Dic3—41no—
150C3×Dic3—42no—
151C3×Dic3—41no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—82no—
159C3×Dic3—81no—
160C4⋊Q8—91no—
161C4×Q8—91no—
162#106—92no—
163C4⋊C8—91no—
164#103—92no—
165C33#47511yescharacteristic, complemented, Sylow
166C3×Q8—92no—
167C3×Q8—181no—
168C3×Q8—362no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—62no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—121no—
182C2×C3×C4—121no—
183C2×C3×C4—122no—
184C2×C3×C4GO+(4,𝔽2)11yescharacteristic
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—22no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192Dic6—62no—
193Dic6—62no—
194Dic6—124no—
195C3×C8—182no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32C3×C22⋊Q811yescharacteristic
209C2×C32C2×C3×SD1611yescharacteristic
210C2×C32C3×#13711yescharacteristic
211C2×Q8—91no—
212C2×Q8—181no—
213C2×C8—92no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C4⋊C4—92no—
217C4⋊C4—91no—
218C42—91no—
219C42—181no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—42no—
229C3×C4—62no—
230C3×C4—62no—
231C3×C4—81no—
232C3×C4—91no—
233C3×C4—91no—
234C3×C4—92no—
235C3×C4—122no—
236C3×C4—122no—
237C3×C4—121no—
238C3×C4—122no—
239C3×C4—121no—
240C3×C4—241no—
241C3×C4—242no—
242Dic3—22no—
243Dic3—42no—
244Dic3—41no—
245Dic3—61no—
246Dic3—61no—
247Dic3—62no—
248Dic3—62no—
249Dic3—61no—
250Dic3—61no—
251C32—21no—
252C32—21no—
253C32—41no—
254C32—41no—
255C32C3×#47511yescharacteristic, complemented
256Q8—92no—
257Q8—181no—
258Q8—362no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4C3×GO+(4,𝔽2)11yescharacteristic
265C8—182no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3D32⋊Q811yescharacteristic
279C2×C3#6422711yescharacteristic
280C2×C3#6423911yescharacteristic
281C22C2×C3×GO+(4,𝔽2)11yescharacteristic
282C4—21no—
283C4—62no—
284C4—91no—
285C4—91no—
286C4—92no—
287C4—121no—
288C4—122no—
289C3—21no—
290C3—21no—
291C3—41no—
292C3—41no—
293C3#1059338511yescharacteristic, direct product factor
294C2C3×D32⋊Q811yescharacteristic
295C2C3×#6422711yescharacteristic
296C2C3×#6423911yescharacteristic
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×#475
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. #10593385C3direct (N × H)
31. C3×#627Q8semidirect
100. #627C3×Q8semidirect
165. C33#475semidirect
255. C32C3×#475semidirect
293. C3#10593385direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639192
C3×#638382
C3×#637792
C3×#637422
C3×#637402
#105933853
C3×#4759