Finite Group Databasebeta

← C3×#10593507 #13166872

Subgroups — C3×#10593507 #13166872

Order 1728.

492 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3792C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C3×#63957C211yesmaximal, characteristic
4C3×#63743C212yesmaximal, complemented
5C3×#63742C212yesmaximal, complemented
6#10593507C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C3×#3792C414yescomplemented
10C2×C3×#622C2212yes—
11C32×#217—21no—
12C3×#3724—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2211yescharacteristic
15C3×#3724C2212yes—
16C3×Dic32—21no—
17C2×#3792C2×C311yescharacteristic
18C2×C3×Dic3⋊C4—61no—
19C3×#877—61no—
20C2×#3724C2×C311yescharacteristic
21#63957C2×C311yescharacteristic
22#63743C2×C312yescomplemented
23#63742C2×C312yescomplemented
24C23×C33D411yescharacteristic
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3Q812yes—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—21no—
30C2×C32×Dic3—21no—
31C2×C32×Dic3—22no—
32C2×C32×Dic3—42no—
33C3×#622C2×C414yescomplemented
34C3×#428—91nomaximal
35C22×C32×C4—61no—
36C22×C3⋊Dic3C22×C311yescharacteristic
37C22×C3×Dic3—21no—
38C22×C3×Dic3—21no—
39C22×C3×Dic3—41no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—61no—
42#3792C3×C414yescomplemented
43C2×#622C22×C312yes—
44C32×C22⋊C4—61no—
45C3×#217—21no—
46C3×#217—21no—
47C3×#217—41no—
48C3×Dic3⋊C4—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—62no—
51C3×Dic3⋊C4—62no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724—21no—
55#3724C22×C311yescharacteristic
56#3724C22×C311yescharacteristic
57#3724C22×C312yes—
58Dic32—21no—
59C22×C33—21no—
60C22×C33—21no—
61C22×C33C4⋊C412yes—
62C22×C33C2×D411yescharacteristic
63C3×C3⋊Dic3C4⋊C412yes—
64C3×C3⋊Dic3C22×C411yescharacteristic
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C32×Dic3—42no—
67C32×Dic3—41no—
68C32×Dic3—41no—
69C2×C3×M16—91no—
70C3×#117—91no—
71C2×C3×C4⋊C4—91no—
72C2×Dic3⋊C4—61no—
73#877—61no—
74C3×#107—92no—
75C3×#106—92no—
76C23×C32—21no—
77C23×C32—21no—
78C23×C32—41no—
79C23×C32—41no—
80C23×C32C3×D411yescharacteristic
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83C2×C32×C4—62no—
84C2×C32×C4—122no—
85C2×C3⋊Dic3C2×C3×C412yes—
86C2×C3⋊Dic3C3×D411yescharacteristic
87C2×C3⋊Dic3C3×Q812yes—
88C2×C3⋊Dic3C23×C311yescharacteristic
89C2×C3×Dic3—21no—
90C2×C3×Dic3—21no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—42no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—44no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109#622C2×C3×C414yescomplemented
110#428—91noSylow
111C2×C33—22no—
112C2×C33C2×C4⋊C411yescharacteristic
113C2×C33AGL(1,ℤ/8)11yescharacteristic
114C2×C33#13711yescharacteristic
115C22×C3×C4—61no—
116C22×C3×C4—61no—
117C22×C3×C4—91no—
118C22×C3×C4—121no—
119C22×Dic3—21no—
120C22×Dic3—61no—
121C22×Dic3—61no—
122C3×M16—94no—
123C2×C3×C8—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—92no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C22⋊C4—61no—
129C3×C22⋊C4—61no—
130C3×C22⋊C4—121no—
131C3×C42—181no—
132#217—21no—
133Dic3⋊C4—61no—
134Dic3⋊C4—61no—
135Dic3⋊C4—62no—
136Dic3⋊C4—62no—
137C4×Dic3—61no—
138C4×Dic3—61no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32—42no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—42no—
151C22×C32—41no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—81no—
156C22×C32—81no—
157C22×C32C3×C4⋊C412yes—
158C22×C32C2×C3×D411yescharacteristic
159C32×C4—122no—
160C32×C4—121no—
161C32×C4—121no—
162C3⋊Dic3C3×C4⋊C412yes—
163C3⋊Dic3C22×C3×C411yescharacteristic
164C3⋊Dic3C2×C3×Q811yescharacteristic
165C3×Dic3—41no—
166C3×Dic3—41no—
167C3×Dic3—42no—
168C3×Dic3—44no—
169C3×Dic3—41no—
170C3×Dic3—42no—
171C3×Dic3—41no—
172C3×Dic3—62no—
173C3×Dic3—62no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—81no—
179C3×Dic3—81no—
180C2×M16—91no—
181#117—91no—
182C2×C4⋊C4—91no—
183#107—92no—
184#106—92no—
185C33#42811yescharacteristic, complemented, Sylow
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3GO+(4,𝔽2)11yescharacteristic
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—62no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—62no—
197C2×C3×C4—91no—
198C2×C3×C4—91no—
199C2×C3×C4—92no—
200C2×C3×C4—92no—
201C2×C3×C4—122no—
202C2×C3×C4—121no—
203C2×C3×C4—121no—
204C2×C3×C4—122no—
205C2×C3×C4—122no—
206C2×C3×C4—124no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—22no—
210C2×Dic3—42no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C3×C8—94no—
220C2×C32—24no—
221C2×C32—22no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—42no—
235C2×C32—44no—
236C2×C32—82no—
237C2×C32C2×C3×C4⋊C411yescharacteristic
238C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
239C2×C32C3×#13711yescharacteristic
240C22×C4—61no—
241C22×C4—91no—
242M16—94no—
243C2×C8—92no—
244C4⋊C4—91no—
245C4⋊C4—92no—
246C4⋊C4—91no—
247C4⋊C4—181no—
248C22⋊C4—61no—
249C42—181no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#377312yes—
269C22×C3C2×GO+(4,𝔽2)11yescharacteristic
270C3×C4—91no—
271C3×C4—92no—
272C3×C4—91no—
273C3×C4—122no—
274C3×C4—121no—
275C3×C4—121no—
276C3×C4—124no—
277C3×C4—122no—
278C3×C4—121no—
279C3×C4—121no—
280C3×C4—241no—
281C3×C4—241no—
282Dic3—42no—
283Dic3—41no—
284Dic3—41no—
285Dic3—62no—
286Dic3—62no—
287Dic3—61no—
288Dic3—61no—
289Dic3—61no—
290Dic3—61no—
291C32—21no—
292C32—21no—
293C32—41no—
294C32—41no—
295C32C3×#42811yescharacteristic, complemented
296C23C3×GO+(4,𝔽2)11yescharacteristic
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—62no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—122no—
305C8—94no—
306C2×C3—24no—
307C2×C3—22no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—42no—
315C2×C3—44no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—82no—
323C2×C3C2×#377311yescharacteristic
324C2×C3#6423611yescharacteristic
325C2×C3#6423911yescharacteristic
326C22—21no—
327C22—21no—
328C22C3×#377312yes—
329C22C2×C3×GO+(4,𝔽2)11yescharacteristic
330C4—91no—
331C4—92no—
332C4—91no—
333C4—122no—
334C4—121no—
335C4—121no—
336C3—21no—
337C3—21no—
338C3—41no—
339C3—41no—
340C3#1059350711yescharacteristic, direct product factor
341C2—22no—
342C2C2×C3×#377311yescharacteristic
343C2C3×#6423611yescharacteristic
344C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#428
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
6. #10593507C3direct (N × H)
9. C3×#3792C4semidirect
22. #63743C2×C3semidirect
23. #63742C2×C3semidirect
33. C3×#622C2×C4semidirect
42. #3792C3×C4semidirect
109. #622C2×C3×C4semidirect
185. C33#428semidirect
295. C32C3×#428semidirect
340. C3#10593507direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×C3×#37242
C3×#639572
C3×#637432
C3×#637422
#105935073
C3×#4289