Finite Group Databasebeta

← C3×#10590566 #13157559

Subgroups — C3×#10590566 #13157559

Order 1728.

612 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64096C211yesmaximal, characteristic
2C3×#63873C211yesmaximal, characteristic
3C2×C3×#3719C212yesmaximal
4C32×#810C212yesmaximal
5C3×#63572C211yesmaximal, characteristic
6C3×#4795—32nomaximal
7#10590566C311yesmaximal, characteristic, direct product factor
8C2×C3×#634C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C33×C4⋊C4C2211yescharacteristic
11C3×#3750—21no—
12C3×C4×C3⋊Dic3—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15C3×#3724—21no—
16C3×#3724—21no—
17C3×#3719—24no—
18#64096C2×C311yescharacteristic
19C3×#944—32no—
20C2×C3×#216—32no—
21C2×C3×Dic12—32no—
22C3×#887—32no—
23C3×#887—61no—
24#63873C2×C311yescharacteristic
25C2×#3719C2×C312yes—
26C32×#103—32no—
27C3×Dic3⋊C8—32no—
28C3×#810—22no—
29C3×#810D312yes—
30C3×#810C2×C312yes—
31#63572C2×C311yescharacteristic
32C2×C33×C4—21no—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×#634—21no—
38C3×#634D412yes—
39C2×C32×Dic3—22no—
40C32×Dic6—22no—
41C32×Dic6—22no—
42C32×C3⋊C8—22no—
43C3×#462—91no—
44#4795—32no—
45C2×C32×Q8—32no—
46C2×#634C22×C311yescharacteristic
47C2×C3×Dic6—22no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—61no—
50C2×C3×Dic6C2×D312yes—
51C2×C3×Dic6C22×C312yes—
52C2×C32×C8—32no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—41no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C22×C311yescharacteristic
58#3750—21no—
59C4×C3⋊Dic3—21no—
60C4×C3⋊Dic3C22×C311yescharacteristic
61C3×#216—64no—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—32no—
67C3×C4×Dic3—62no—
68C3×C4×Dic3—61no—
69C3×C4×Dic3—62no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C3×Dic12—64no—
74#3724—21no—
75#3724—21no—
76#3719—24no—
77C22×C33C2×D411yescharacteristic
78C33×C4—41no—
79C33×C4C2×D411yescharacteristic
80C33×C4#4111yescharacteristic
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3—21no—
83C3×C3⋊Dic3Q1612yescomplemented
84C32×Dic3—42no—
85C2×C3×Q16—92no—
86C3×C4⋊Q8—91no—
87C3×C4×Q8—91no—
88#944—32no—
89C2×#216—32no—
90C2×Dic12—32no—
91#887—32no—
92#887—61no—
93C3×C4⋊C8—91no—
94C3×#103—32no—
95C3×#103—32no—
96C3×#103—62no—
97Dic3⋊C8—32no—
98#810C3×D312yes—
99C32×Q8—62no—
100C32×Q8—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—22no—
106C2×C32×C4—41no—
107C2×C32×C4—42no—
108C2×C32×C4—42no—
109C2×C32×C4—62no—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C23×C311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3C3×D411yescharacteristic
114C2×C3⋊Dic3C3×D411yescharacteristic
115#634—21no—
116#634C3×D412yes—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C3×Dic6—24no—
127C3×Dic6—22no—
128C3×Dic6—22no—
129C3×Dic6—22no—
130C3×Dic6—22no—
131C3×Dic6—34no—
132C3×Dic6—42no—
133C3×Dic6—62no—
134C3×Dic6—62no—
135C3×Dic6—62no—
136C32×C8—62no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—42no—
140#462—91noSylow
141C2×C33#12111yescharacteristic
142C2×C33C2×Q1611yescharacteristic
143C2×C33#13711yescharacteristic
144C2×C3×Q8—32no—
145C2×C3×Q8—32no—
146C2×C3×Q8—62no—
147C2×C3×Q8—91no—
148C2×Dic6—32no—
149C2×Dic6—61no—
150C2×Dic6C2×C3×D312yes—
151C3×Q16—184no—
152C2×C3×C8—32no—
153C2×C3×C8—32no—
154C2×C3×C8—62no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—181no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4D3211yescharacteristic
162C3×C4⋊C4C2×C3×D312yes—
163C3×C42—91no—
164C3×C42—181no—
165#216—64no—
166Dic3⋊C4—62no—
167Dic3⋊C4—62no—
168Dic3⋊C4—62no—
169Dic3⋊C4—62no—
170C4×Dic3—32no—
171C4×Dic3—61no—
172C4×Dic3—62no—
173C4×Dic3—62no—
174C2×C3⋊C8C2×C3×D312yes—
175Dic12—64no—
176C22×C32—21no—
177C22×C32—22no—
178C22×C32—41no—
179C22×C32D3×D412yes—
180C22×C32C2×C3×D411yescharacteristic
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4—42no—
189C32×C4—42no—
190C32×C4—82no—
191C32×C4—82no—
192C32×C4—122no—
193C32×C4#23512yes—
194C32×C4D3×D412yes—
195C32×C4C2×C3×D411yescharacteristic
196C32×C4C3×#4111yescharacteristic
197C3⋊Dic3—22no—
198C3⋊Dic3—21no—
199C3⋊Dic3C3×Q1612yescomplemented
200C3×Dic3—34no—
201C3×Dic3—44no—
202C3×Dic3—42no—
203C3×Dic3—42no—
204C3×Dic3—62no—
205C3×Dic3—64no—
206C3×Dic3—64no—
207C3×Dic3—62no—
208C3×Dic3—121no—
209C2×Q16—92no—
210C4⋊Q8—91no—
211C4×Q8—91no—
212C4⋊C8—91no—
213#103—32no—
214C33#46211yescharacteristic, complemented, Sylow
215C3×Q8—64no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—62no—
220C3×Q8—92no—
221C3×Q8—122no—
222C3×Q8—181no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—21no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—122no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3211yescharacteristic
238C2×C3×C4C22×C3×D312yes—
239C2×Dic3—22no—
240C2×Dic3—32no—
241C2×Dic3—32no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246Dic6—22no—
247Dic6—22no—
248Dic6—34no—
249Dic6—62no—
250Dic6—62no—
251Dic6—62no—
252C3×C8—62no—
253C3×C8—62no—
254C3×C8—122no—
255C3⋊C8—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32#88412yes—
266C2×C32#91512yes—
267C2×C32D3×Q1612yes—
268C2×C32C3×#12111yescharacteristic
269C2×C32C2×C3×Q1611yescharacteristic
270C2×C32C3×#13711yescharacteristic
271C2×Q8—32no—
272C2×Q8—91no—
273Q16—184no—
274C2×C8—32no—
275C4⋊C4—181no—
276C4⋊C4—181no—
277C4⋊C4—181no—
278C4⋊C4C3×D3211yescharacteristic
279C42—91no—
280C42—181no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3#381111yescharacteristic
285C22×C3C3×D3×D412yes—
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—42no—
291C3×C4—42no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—82no—
296C3×C4—82no—
297C3×C4—92no—
298C3×C4—124no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4—182no—
302C3×C4—181no—
303C3×C4#379811yescharacteristic
304C3×C4#380211yescharacteristic
305C3×C4C3×#23512yes—
306C3×C4C3×D3×D412yes—
307Dic3—34no—
308Dic3—42no—
309Dic3—62no—
310Dic3—64no—
311Dic3—64no—
312Dic3—62no—
313Dic3—121no—
314C32—21no—
315C32—22no—
316C32—41no—
317C32#479512yescomplemented
318C32C3×#46211yescharacteristic, complemented
319Q8—62no—
320Q8—62no—
321Q8—92no—
322Q8—181no—
323C2×C4—21no—
324C2×C4—62no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—181no—
328C2×C4C2×C3×D3211yescharacteristic
329C8—62no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—22no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3#6391211yescharacteristic
340C2×C3#6393111yescharacteristic
341C2×C3#6394411yescharacteristic
342C2×C3C3×#88412yes—
343C2×C3C3×#91512yes—
344C2×C3C3×D3×Q1612yes—
345C22C3×#381111yescharacteristic
346C4—41no—
347C4—92no—
348C4—122no—
349C4—182no—
350C4—181no—
351C4C3×#379811yescharacteristic
352C4C3×#380211yescharacteristic
353C3—22no—
354C3—21no—
355C3—41no—
356C3#1059056611yescharacteristic, direct product factor
357C3C3×#479512yescomplemented
358C2C3×#6391211yescharacteristic
359C2C3×#6393111yescharacteristic
360C2C3×#6394411yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. #10590566C3direct (N × H)
83. C3×C3⋊Dic3Q16semidirect
199. C3⋊Dic3C3×Q16semidirect
214. C33#462semidirect
317. C32#4795semidirect
318. C32C3×#462semidirect
356. C3#10590566direct (N × H)
357. C3C3×#4795semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
C3×#638732
C2×C3×#37192
C32×#8102
C3×#635722
C3×#47953
#105905663