Finite Group Databasebeta

← C3×#10593447 #13166815

Subgroups — C3×#10593447 #13166815

Order 1728.

410 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×C3×#3776C212yesmaximal
2C3×#63919C211yesmaximal, characteristic
3C3×C4×#627C211yesmaximal, characteristic
4C3×C4⋊#622C211yesmaximal, characteristic
5C3×#63740C212yesmaximal
6#10593447C311yesmaximal, characteristic, direct product factor
7C2×C3×#627C2212yes—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#622C2212yes—
10C3×#3776—24no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3—21no—
13C32×C4×Dic3—21no—
14C3×#3724—21no—
15C3×#3724C2211yescharacteristic
16C3×Dic3⋊Dic3—21no—
17C2×#3776C2×C312yes—
18C3×#888—61no—
19C3×C4×Dic6—61no—
20#63919C2×C311yescharacteristic
21C4×#627C2×C311yescharacteristic
22C4⋊#622C2×C311yescharacteristic
23#63740C2×C312yes—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—21no—
28C2×C32×Dic3—22no—
29C2×C32×Dic3—21no—
30C3×#627—21no—
31C3×#627—22no—
32C3×#627—22no—
33C3×#627D412yes—
34C3×#622—22no—
35C3×#462—91nomaximal
36C2×C3×Dic6—61no—
37C2×C3×Dic6—62no—
38C2×#627C22×C312yes—
39C2×#627C22×C311yescharacteristic
40C2×#622C22×C312yes—
41#3776—24no—
42C32×C4⋊C4—61no—
43C32×C42—61no—
44C4×C3⋊Dic3C22×C311yescharacteristic
45C3×C4⋊Dic3—21no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—41no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—62no—
51C3×Dic3⋊C4—61no—
52C3×C4×Dic3—21no—
53C3×C4×Dic3—21no—
54C3×C4×Dic3—41no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57#3724—21no—
58#3724C22×C311yescharacteristic
59Dic3⋊Dic3—21no—
60C22×C33C2×D411yescharacteristic
61C33×C4Q1612yescomplemented
62C3×C3⋊Dic3—21no—
63C3×C3⋊Dic3C2×D411yescharacteristic
64C3×C3⋊Dic3#4111yescharacteristic
65C32×Dic3—22no—
66C32×Dic3—42no—
67C32×Dic3—41no—
68C2×C3×Q16—92no—
69C3×C4⋊Q8—91no—
70C3×C4×Q8—91no—
71#888—61no—
72C4×Dic6—61no—
73C3×C4⋊C8—91no—
74C3×#103—92no—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—41no—
78C2×C32×C4—41no—
79C2×C32×C4—61no—
80C2×C32×C4—61no—
81C2×C32×C4—62no—
82C2×C32×C4C3×D411yescharacteristic
83C2×C3⋊Dic3C3×D411yescharacteristic
84C2×C3⋊Dic3C23×C311yescharacteristic
85C2×C3×Dic3—21no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—21no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C3×Dic6—62no—
99C3×Dic6—62no—
100C3×Dic6—122no—
101C3×Dic6—122no—
102#627—21no—
103#627—22no—
104#627—22no—
105#627C3×D412yes—
106#622—22no—
107#462—91noSylow
108C2×C33#12111yescharacteristic
109C2×C33C2×Q1611yescharacteristic
110C2×C33#13711yescharacteristic
111C2×C3×Q8—91no—
112C2×C3×Q8—92no—
113C2×Dic6—61no—
114C2×Dic6—62no—
115C3×Q16—184no—
116C2×C3×C8—92no—
117C3×C4⋊C4—61no—
118C3×C4⋊C4—61no—
119C3×C4⋊C4—91no—
120C3×C4⋊C4—121no—
121C3×C4⋊C4—181no—
122C3×C4⋊C4—181no—
123C3×C42—61no—
124C3×C42—61no—
125C3×C42—91no—
126C3×C42—121no—
127C4⋊Dic3—21no—
128C4⋊Dic3—61no—
129Dic3⋊C4—61no—
130Dic3⋊C4—61no—
131Dic3⋊C4—62no—
132C4×Dic3—21no—
133C4×Dic3—61no—
134C4×Dic3—61no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32C2×C3×D411yescharacteristic
140C32×C4—22no—
141C32×C4—22no—
142C32×C4—42no—
143C32×C4—42no—
144C32×C4—62no—
145C32×C4—121no—
146C32×C4—122no—
147C32×C4C3×Q1612yescomplemented
148C3⋊Dic3—21no—
149C3⋊Dic3C2×C3×D411yescharacteristic
150C3⋊Dic3C3×#4111yescharacteristic
151C3×Dic3—22no—
152C3×Dic3—22no—
153C3×Dic3—42no—
154C3×Dic3—42no—
155C3×Dic3—41no—
156C3×Dic3—42no—
157C3×Dic3—42no—
158C3×Dic3—41no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—62no—
164C3×Dic3—82no—
165C3×Dic3—121no—
166C2×Q16—92no—
167C4⋊Q8—91no—
168C4×Q8—91no—
169C4⋊C8—91no—
170#103—92no—
171C33#46211yescharacteristic, complemented, Sylow
172C3×Q8—92no—
173C3×Q8—181no—
174C3×Q8—182no—
175C3×Q8—182no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—121no—
189C2×C3×C4—121no—
190C2×C3×C4—122no—
191C2×C3×C4GO+(4,𝔽2)11yescharacteristic
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—22no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199Dic6—62no—
200Dic6—62no—
201Dic6—122no—
202Dic6—122no—
203C3×C8—182no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32C3×#12111yescharacteristic
217C2×C32C2×C3×Q1611yescharacteristic
218C2×C32C3×#13711yescharacteristic
219C2×Q8—91no—
220C2×Q8—92no—
221Q16—184no—
222C2×C8—92no—
223C4⋊C4—61no—
224C4⋊C4—91no—
225C4⋊C4—181no—
226C4⋊C4—181no—
227C42—61no—
228C42—91no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3C2×GO+(4,𝔽2)11yescharacteristic
234C3×C4—22no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—42no—
238C3×C4—62no—
239C3×C4—62no—
240C3×C4—91no—
241C3×C4—91no—
242C3×C4—122no—
243C3×C4—122no—
244C3×C4—121no—
245C3×C4—122no—
246C3×C4—121no—
247C3×C4—122no—
248C3×C4—181no—
249C3×C4—242no—
250C3×C4#377612yescomplemented
251Dic3—22no—
252Dic3—42no—
253Dic3—41no—
254Dic3—61no—
255Dic3—61no—
256Dic3—61no—
257Dic3—61no—
258Dic3—62no—
259Dic3—121no—
260C32—21no—
261C32—21no—
262C32—41no—
263C32—41no—
264C32C3×#46211yescharacteristic, complemented
265Q8—92no—
266Q8—181no—
267Q8—182no—
268Q8—182no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4C3×GO+(4,𝔽2)11yescharacteristic
275C8—182no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3#6422611yescharacteristic
289C2×C3#6423011yescharacteristic
290C2×C3C2×#377611yescharacteristic
291C22C2×C3×GO+(4,𝔽2)11yescharacteristic
292C4—62no—
293C4—91no—
294C4—91no—
295C4—121no—
296C4—122no—
297C4—181no—
298C4C3×#377612yescomplemented
299C3—21no—
300C3—21no—
301C3—41no—
302C3—41no—
303C3#1059344711yescharacteristic, direct product factor
304C2C3×#6422611yescharacteristic
305C2C3×#6423011yescharacteristic
306C2C2×C3×#377611yescharacteristic
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×#462
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. #10593447C3direct (N × H)
61. C33×C4Q16semidirect
147. C32×C4C3×Q16semidirect
171. C33#462semidirect
250. C3×C4#3776semidirect
264. C32C3×#462semidirect
298. C4C3×#3776semidirect
303. C3#10593447direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C3×#639192
C3×C4×#6272
C3×C4⋊#6222
C3×#637402
#105934473
C3×#4629