Finite Group Databasebeta

← C3×#10591620 #13158655

Subgroups — C3×#10591620 #13158655

Order 1728.

810 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 140 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×#64153C211yesmaximal, characteristic, complemented
2C2×C32×#215C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic
4C2×C3×#3716C212yesmaximal, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#5175—32nomaximal
7#10591620C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C3×#625—22no—
12C3×#3750—22no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×#215—24no—
15C2×C32×C3⋊C8C2212yes—
16C3×#3716—24no—
17C2×C32×SD16—32no—
18#64153C2×C311yescharacteristic, complemented
19C3×#944—32no—
20C3×#944—61no—
21C2×C3×#215—22no—
22C2×C3×#215D312yescomplemented
23C2×C3×#215C2×C312yescomplemented
24C3×#940—32no—
25C2×C3×#214—32no—
26C2×C3×#204—32no—
27#63914C2×C311yescharacteristic
28C2×#3716C2×C312yescomplemented
29C3×C8×Dic3—32no—
30#63553C2×C311yescharacteristic, complemented
31C22×C32×D3—22no—
32C33×Q8—22no—
33C2×C33×C4—21no—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3D412yes—
37C3×#634—22no—
38C32×D12—22no—
39C32×D12—22no—
40C3×#625—44no—
41C32×C3⋊C8D414yescomplemented
42C3×#492—91no—
43#5175—32no—
44C2×C32×Q8—21no—
45C2×C32×Q8—22no—
46C2×C32×Q8—41no—
47C2×C32×Q8C2×D312yes—
48C2×C32×Q8C22×C311yescharacteristic
49C2×C32×D4—32no—
50C2×#634C22×C311yescharacteristic
51C2×C3×#67—64no—
52C2×C3×D12—22no—
53C2×C3×D12C2×D312yes—
54C2×C3×D12C22×C312yes—
55C2×C3×Dic6—32no—
56C2×C3×Dic6—61no—
57C2×#625—22no—
58C32×SD16—64no—
59C2×C32×C8—32no—
60#3750—22no—
61C4×C3⋊Dic3C22×C311yescharacteristic
62C3×#215—28no—
63C3×#215—24no—
64C3×#215—24no—
65C3×#214—64no—
66C3×Dic3⋊C4—64no—
67C3×Dic3⋊C4—64no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8C2×D312yes—
72C2×C3×C3⋊C8C22×C312yes—
73C3×#204—64no—
74#3716—24no—
75C22×C33C2×D411yescharacteristic
76C2×C32×D3—42no—
77C2×C32×D3—44no—
78C33×C4—41no—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3SD1614yescomplemented
82C2×C3×SD16—32no—
83C2×C3×SD16—32no—
84C2×C3×SD16—62no—
85C2×C3×SD16—92no—
86C3×C4⋊Q8—91no—
87C3×#127—91no—
88#944—32no—
89#944—61no—
90C2×#215C3×D312yescomplemented
91#940—32no—
92C2×#214—32no—
93C2×#204—32no—
94C3×C4×C8—91no—
95C8×Dic3—32no—
96C23×C32—62no—
97C22×C3×D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C32×Q8—22no—
101C32×Q8—24no—
102C32×Q8—22no—
103C32×Q8—22no—
104C32×Q8—42no—
105C32×Q8—42no—
106C32×Q8—44no—
107C32×D4—62no—
108C32×D4—62no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—41no—
115C2×C32×C4—42no—
116C2×C32×C4—42no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3C3×D412yes—
121#634—22no—
122C3×#67—128no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—62no—
127C3×D12—24no—
128C3×D12—22no—
129C3×D12—22no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—42no—
133C3×Dic6—62no—
134C3×Dic6—62no—
135C3×Dic6—122no—
136#625—44no—
137C32×C8—34no—
138C3×C3⋊C8—24no—
139C3×C3⋊C8#6714yescomplemented
140C3×C3⋊C8C3×D414yescomplemented
141#492—91noSylow
142C2×C33#12711yescharacteristic
143C2×C33C2×SD1612yes—
144C32×D3—82no—
145C2×C3×Q8—21no—
146C2×C3×Q8—22no—
147C2×C3×Q8—41no—
148C2×C3×Q8—91no—
149C2×C3×Q8D3211yescharacteristic
150C2×C3×Q8C2×C3×D312yes—
151C2×C3×D4—32no—
152C2×C3×D4—32no—
153C2×C3×D4—62no—
154C2×C3×D4—182no—
155C2×#67—64no—
156C2×D12C2×C3×D312yes—
157C2×Dic6—32no—
158C2×Dic6—61no—
159C3×SD16—64no—
160C3×SD16—64no—
161C3×SD16—68no—
162C3×SD16—184no—
163C2×C3×C8—32no—
164C2×C3×C8—32no—
165C2×C3×C8—62no—
166C3×C4⋊C4—182no—
167C3×C42—91no—
168#215—24no—
169#214—64no—
170Dic3⋊C4—64no—
171Dic3⋊C4—64no—
172C4×Dic3—32no—
173C4×Dic3—61no—
174C2×C3⋊C8C2×C3×D312yes—
175#204—64no—
176C22×C32—21no—
177C22×C32—22no—
178C22×C32—41no—
179C22×C32—122no—
180C22×C32—124no—
181C22×C32D3×D412yes—
182C22×C32C2×C3×D411yescharacteristic
183C2×C3×D3—44no—
184C2×C3×D3—44no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—44no—
188C2×C3×D3—84no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—42no—
194C32×C4—44no—
195C32×C4—44no—
196C32×C4—42no—
197C32×C4—41no—
198C32×C4D3×D412yes—
199C32×C4C2×#6712yes—
200C32×C4C2×C3×D412yes—
201C3⋊Dic3—41no—
202C3⋊Dic3C3×SD1614yescomplemented
203C3×Dic3—38no—
204C3×Dic3—64no—
205C3×Dic3—122no—
206C3×Dic3—122no—
207C2×SD16—32no—
208C2×SD16—92no—
209C4⋊Q8—91no—
210#127—91no—
211C4×C8—91no—
212C33#49211yescharacteristic, complemented, Sylow
213C23×C3—62no—
214C23×C3—62no—
215C23×C3—122no—
216C22×D3—22no—
217C3×Q8—24no—
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—22no—
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—44no—
224C3×Q8—182no—
225C3×D4—64no—
226C3×D4—62no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—62no—
230C3×D4—122no—
231C3×D4—364no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—92no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3211yescharacteristic
243C2×C3×C4C22×C3×D312yes—
244#67—128no—
245C2×Dic3—34no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249D12—22no—
250D12—22no—
251Dic6—62no—
252Dic6—62no—
253Dic6—122no—
254C3×C8—34no—
255C3×C8—34no—
256C3×C8—64no—
257C3⋊C8C3×#6714yescomplemented
258C2×C32—24no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—242no—
265C2×C32D3×SD1614yes—
266C2×C32#94012yes—
267C2×C32C3×#12711yescharacteristic
268C2×C32C2×C3×SD1612yes—
269C3×D3—84no—
270C3×D3—82no—
271C3×D3—82no—
272C2×Q8—91no—
273C2×Q8C3×D3211yescharacteristic
274C2×D4—32no—
275C2×D4—182no—
276SD16—64no—
277SD16—184no—
278C2×C8—32no—
279C4⋊C4—182no—
280C42—91no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—124no—
285C22×C3—124no—
286C22×C3—122no—
287C22×C3—124no—
288C22×C3—122no—
289C22×C3—244no—
290C22×C3#381111yescharacteristic
291C22×C3C3×D3×D412yes—
292C2×D3—44no—
293C2×D3—42no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—42no—
298C3×C4—44no—
299C3×C4—44no—
300C3×C4—42no—
301C3×C4—41no—
302C3×C4—42no—
303C3×C4—94no—
304C3×C4—361no—
305C3×C4D3×#6712yes—
306C3×C4C3×D3×D412yes—
307C3×C4C2×C3×#6712yes—
308Dic3—38no—
309Dic3—64no—
310Dic3—122no—
311Dic3—122no—
312C32—22no—
313C32—21no—
314C32—41no—
315C32#517512yescomplemented
316C32C3×#49211yescharacteristic, complemented
317C23—62no—
318Q8—22no—
319Q8—182no—
320D4—62no—
321D4—62no—
322D4—364no—
323C2×C4—21no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4C2×C3×D3211yescharacteristic
327C8—34no—
328C2×C3—22no—
329C2×C3—24no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—42no—
333C2×C3—41no—
334C2×C3—244no—
335C2×C3—242no—
336C2×C3—242no—
337C2×C3#6394012yes—
338C2×C3#6397911yescharacteristic
339C2×C3C3×D3×SD1614yes—
340C2×C3C3×#94012yes—
341D3—82no—
342C22—124no—
343C22—122no—
344C22C3×#381111yescharacteristic
345C4—41no—
346C4—94no—
347C4—361no—
348C4C3×D3×#6712yes—
349C3—22no—
350C3—21no—
351C3—41no—
352C3#1059162011yescharacteristic, direct product factor
353C3C3×#517512yescomplemented
354C2—242no—
355C2C3×#6394012yes—
356C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#492
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64153C2semidirect
2. C2×C32×#215C2semidirect
4. C2×C3×#3716C2semidirect
5. C3×#63553C2semidirect
7. #10591620C3direct (N × H)
18. #64153C2×C3semidirect
22. C2×C3×#215D3semidirect
23. C2×C3×#215C2×C3semidirect
28. C2×#3716C2×C3semidirect
30. #63553C2×C3semidirect
41. C32×C3⋊C8D4semidirect
81. C3×C3⋊Dic3SD16semidirect
90. C2×#215C3×D3semidirect
139. C3×C3⋊C8#67semidirect
140. C3×C3⋊C8C3×D4semidirect
202. C3⋊Dic3C3×SD16semidirect
212. C33#492semidirect
257. C3⋊C8C3×#67semidirect
315. C32#5175semidirect
316. C32C3×#492semidirect
352. C3#10591620direct (N × H)
353. C3C3×#5175semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C2×C32×#2152
C3×#639142
C2×C3×#37162
C3×#635532
C3×#51753
#105916203