Finite Group Databasebeta

← C3×#10593359 #13166702

Subgroups — C3×#10593359 #13166702

Order 1728.

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

normal subgroups only

This group has 458 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
1C3×#63919C211yesmaximal, characteristic, complemented
2C3×C4×#625C211yesmaximal, characteristic
3C3×C4⋊#622C211yesmaximal, characteristic, complemented
4C3×#63742C212yesmaximal, complemented
5C3×#63739C212yesmaximal
6#10593359C311yesmaximal, characteristic, direct product factor
7C2×C32×C4×D3—21no—
8C2×C3×#627—21no—
9C2×C3×#625C2211yescharacteristic
10C2×C3×#622C2212yes—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2212yes—
15C3×(C2×D3)⋊Dic3—21no—
16C3×C4×#67—61no—
17C3×#888—61no—
18#63919C2×C311yescharacteristic, complemented
19C4×#625C2×C311yescharacteristic
20C4⋊#622C2×C311yescharacteristic, complemented
21#63742C2×C312yescomplemented
22#63739C2×C312yes—
23C22×C32×D3—21no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—22no—
28C2×C32×Dic3—21no—
29C32×C4×D3—42no—
30C3×#627—42no—
31C3×#625—21no—
32C3×#625Q812yescomplemented
33C3×#622—22no—
34C3×#476—91nomaximal
35C22×C32×C4—61no—
36C2×C3×#67—61no—
37C2×C3×C4×D3—21no—
38C2×C3×C4×D3—21no—
39C2×C3×C4×D3—41no—
40C2×C3×Dic6—62no—
41C2×#627—21no—
42C2×#625C22×C311yescharacteristic
43C2×#622C22×C312yes—
44C32×C4⋊C4—61no—
45C4×C3⋊Dic3C22×C311yescharacteristic
46C3×#217—61no—
47C3×(C2×D3)⋊C4—61no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—41no—
51C3×Dic3⋊C4—62no—
52C3×Dic3⋊C4—61no—
53C3×C4×Dic3—61no—
54C3×C4×Dic3—61no—
55#3724C22×C311yescharacteristic
56#3724C22×C312yes—
57(C2×D3)⋊Dic3—21no—
58C22×C33C2×D411yescharacteristic
59C2×C32×D3—22no—
60C2×C32×D3—41no—
61C2×C32×D3—41no—
62C33×C4SD1612yes—
63C3×C3⋊Dic3—21no—
64C3×C3⋊Dic3C2×Q811yescharacteristic
65C3×C3⋊Dic3#4111yescharacteristic
66C32×Dic3—41no—
67C32×Dic3—42no—
68C3×C4⋊Q8—91no—
69C3×C4×D4—91no—
70C4×#67—61no—
71#888—61no—
72C3×#106—92no—
73C3×C4⋊C8—91no—
74C3×#102—92no—
75C23×C32—61no—
76C22×C3×D3—21no—
77C22×C3×D3—21no—
78C22×C3×D3—41no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—61no—
84C2×C32×C4—62no—
85C2×C32×C4—122no—
86C2×C32×C4C3×D411yescharacteristic
87C2×C3⋊Dic3C3×D411yescharacteristic
88C2×C3⋊Dic3C23×C311yescharacteristic
89C3×#67—62no—
90C3×#67—62no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—42no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C3×C4×D3—42no—
102C3×C4×D3—44no—
103C3×C4×D3—42no—
104C3×Dic6—124no—
105#627—42no—
106#625—21no—
107#625C3×Q812yescomplemented
108#622—22no—
109#476—91noSylow
110C2×C33C22⋊Q811yescharacteristic
111C2×C33C2×SD1611yescharacteristic
112C2×C33AGL(1,ℤ/8)11yescharacteristic
113C32×D3—42no—
114C2×C3×Q8—181no—
115C2×C3×D4—91no—
116C22×C3×C4—61no—
117C22×C3×C4—61no—
118C22×C3×C4—121no—
119C2×#67—61no—
120C2×C4×D3—21no—
121C2×Dic6—62no—
122C2×C3×C8—92no—
123C3×C4⋊C4—61no—
124C3×C4⋊C4—61no—
125C3×C4⋊C4—92no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—121no—
128C3×C22⋊C4—181no—
129C3×C42—91no—
130#217—61no—
131(C2×D3)⋊C4—61no—
132C4⋊Dic3—21no—
133Dic3⋊C4—62no—
134Dic3⋊C4—61no—
135C4×Dic3—61no—
136C4×Dic3—61no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32—62no—
142C22×C32—121no—
143C22×C32—121no—
144C22×C32C2×C3×D411yescharacteristic
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3—42no—
148C2×C3×D3—41no—
149C2×C3×D3—41no—
150C2×C3×D3—42no—
151C2×C3×D3—42no—
152C2×C3×D3—41no—
153C2×C3×D3—41no—
154C32×C4—22no—
155C32×C4—22no—
156C32×C4—42no—
157C32×C4—42no—
158C32×C4—121no—
159C32×C4—122no—
160C32×C4C3×SD1612yes—
161C3⋊Dic3—21no—
162C3⋊Dic3C2×C3×Q811yescharacteristic
163C3⋊Dic3C3×#4111yescharacteristic
164C3×Dic3—42no—
165C3×Dic3—41no—
166C3×Dic3—42no—
167C3×Dic3—41no—
168C3×Dic3—42no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—62no—
174C3×Dic3—82no—
175C3×Dic3—121no—
176C4⋊Q8—91no—
177C4×D4—91no—
178#106—92no—
179C4⋊C8—91no—
180#102—92no—
181C33#47611yescharacteristic, complemented, Sylow
182C23×C3—61no—
183C23×C3—61no—
184C23×C3—121no—
185C22×D3—21no—
186C3×Q8—362no—
187C3×D4—92no—
188C3×D4—181no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—61no—
194C2×C3×C4—62no—
195C2×C3×C4—61no—
196C2×C3×C4—62no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—121no—
200C2×C3×C4—122no—
201C2×C3×C4—122no—
202C2×C3×C4—124no—
203C2×C3×C4—122no—
204C2×C3×C4GO+(4,𝔽2)11yescharacteristic
205#67—62no—
206#67—62no—
207C2×Dic3—22no—
208C2×Dic3—21no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213C4×D3—42no—
214Dic6—124no—
215C3×C8—182no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—122no—
229C2×C32C3×C22⋊Q811yescharacteristic
230C2×C32C2×C3×SD1611yescharacteristic
231C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
232C3×D3—42no—
233C3×D3—44no—
234C3×D3—42no—
235C2×Q8—181no—
236C2×D4—91no—
237C22×C4—61no—
238C2×C8—92no—
239C4⋊C4—61no—
240C4⋊C4—92no—
241C4⋊C4—91no—
242C22⋊C4—181no—
243C42—91no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—122no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—122no—
254C22×C3—122no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3C2×GO+(4,𝔽2)11yescharacteristic
258C2×D3—22no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—42no—
264C3×C4—42no—
265C3×C4—91no—
266C3×C4—91no—
267C3×C4—122no—
268C3×C4—121no—
269C3×C4—122no—
270C3×C4—122no—
271C3×C4—121no—
272C3×C4—181no—
273C3×C4—242no—
274C3×C4#377512yes—
275Dic3—42no—
276Dic3—41no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—62no—
282Dic3—121no—
283C32—21no—
284C32—21no—
285C32—41no—
286C32—41no—
287C32C3×#47611yescharacteristic, complemented
288C23—61no—
289Q8—362no—
290D4—92no—
291D4—181no—
292C2×C4—61no—
293C2×C4—62no—
294C2×C4—91no—
295C2×C4—91no—
296C2×C4—122no—
297C2×C4C3×GO+(4,𝔽2)11yescharacteristic
298C8—182no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—122no—
312C2×C3—124no—
313C2×C3—122no—
314C2×C3D32⋊Q811yescharacteristic
315C2×C3#6422411yescharacteristic
316C2×C3C2×#377511yescharacteristic
317D3—42no—
318C22—62no—
319C22—121no—
320C22—121no—
321C22C2×C3×GO+(4,𝔽2)11yescharacteristic
322C4—91no—
323C4—91no—
324C4—121no—
325C4—122no—
326C4—181no—
327C4C3×#377512yes—
328C3—21no—
329C3—21no—
330C3—41no—
331C3—41no—
332C3#1059335911yescharacteristic, direct product factor
333C2—122no—
334C2C3×D32⋊Q811yescharacteristic
335C2C3×#6422411yescharacteristic
336C2C2×C3×#377511yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#476
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
1. C3×#63919C2semidirect
3. C3×C4⋊#622C2semidirect
4. C3×#63742C2semidirect
6. #10593359C3direct (N × H)
18. #63919C2×C3semidirect
20. C4⋊#622C2×C3semidirect
21. #63742C2×C3semidirect
32. C3×#625Q8semidirect
107. #625C3×Q8semidirect
181. C33#476semidirect
287. C32C3×#476semidirect
332. C3#10593359direct (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×C4×#6252
C3×C4⋊#6222
C3×#637422
C3×#637392
#105933593
C3×#4769