Finite Group Databasebeta

← C3×#10590869 #13157832

Subgroups — C3×#10590869 #13157832

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×Dic12C211yesmaximal, characteristic, complemented
2C2×C32×D24C211yesmaximal, characteristic
3C3×#63863C212yesmaximal, complemented
4C2×C3×#3715C212yesmaximal, complemented
5C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#5191—31nomaximal
7C3×#5155—31nomaximal
8#10590869C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C2212yes—
10C2×C32×Dic6C2212yes—
11C2×C33×C8C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C32×Dic12—22no—
15C32×D24—22no—
16C3×(C2×D3)⋊Dic3—24no—
17C3×#3715—24no—
18C2×C32×Q16—31no—
19C2×C32×D8—31no—
20C3×#947—32no—
21C2×C3×#215—32no—
22C3×#936—32no—
23C2×C3×#214—32no—
24C2×C3×Dic12—21no—
25C2×C3×Dic12D311yescharacteristic, complemented
26C2×C3×Dic12C2×C311yescharacteristic, complemented
27C2×C3×D24—21no—
28C2×C3×D24D311yescharacteristic
29C2×C3×D24C2×C311yescharacteristic
30#63863C2×C312yescomplemented
31C2×#3715C2×C312yescomplemented
32C8×C3⋊Dic3C2×C311yescharacteristic, complemented
33C3×C8×Dic3—31no—
34C3×C8×Dic3—31no—
35C3×C8×Dic3—61no—
36C22×C32×D3—22no—
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3D412yes—
39C2×C32×Dic3—22no—
40C32×D12—22no—
41C32×D12—22no—
42C32×Dic6—22no—
43C32×Dic6—22no—
44C33×C8D412yes—
45C3×C3⋊(C3⋊C8)D412yes—
46C3×#495—91no—
47#5191—31no—
48#5155—31no—
49C2×C32×Q8—32no—
50C2×C32×D4—32no—
51C2×C3×D12—22no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C22×C312yes—
54C2×C3×Dic6—22no—
55C2×C3×Dic6C2×D312yes—
56C2×C3×Dic6C22×C312yes—
57C32×Q16—62no—
58C32×D8—62no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8—41no—
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C4×C3⋊Dic3C22×C311yescharacteristic
67C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
68C3×#217—64no—
69C3×#215—64no—
70C3×#214—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—31no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78C3×Dic12—24no—
79C3×Dic12—22no—
80C3×Dic12—22no—
81C3×D24—24no—
82C3×D24—22no—
83C3×D24—22no—
84(C2×D3)⋊Dic3—24no—
85#3715—24no—
86C22×C33C2×D411yescharacteristic
87C2×C32×D3—42no—
88C2×C32×D3—44no—
89C33×C4C2×D411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3×C3⋊Dic3—22no—
92C32×Dic3—42no—
93C2×C3×Q16—31no—
94C2×C3×Q16—31no—
95C2×C3×Q16—61no—
96C2×C3×SD16—92no—
97C2×C3×D8—31no—
98C2×C3×D8—31no—
99C2×C3×D8—61no—
100C3×#124—92no—
101#947—32no—
102C2×#215—32no—
103#936—32no—
104C2×#214—32no—
105C2×Dic12C3×D311yescharacteristic, complemented
106C2×D24C3×D311yescharacteristic
107C3×C4×C8—91no—
108C8×Dic3—31no—
109C8×Dic3—31no—
110C8×Dic3—61no—
111C23×C32—62no—
112C22×C3×D3—22no—
113C22×C3×D3—22no—
114C22×C3×D3—42no—
115C32×Q8—62no—
116C32×Q8—62no—
117C32×D4—62no—
118C32×D4—62no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—62no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3C3×D412yes—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—62no—
134C3×D12—24no—
135C3×D12—22no—
136C3×D12—22no—
137C3×D12—22no—
138C3×D12—22no—
139C3×D12—42no—
140C3×Dic6—24no—
141C3×Dic6—22no—
142C3×Dic6—22no—
143C3×Dic6—22no—
144C3×Dic6—22no—
145C3×Dic6—42no—
146C32×C8—22no—
147C32×C8—22no—
148C32×C8—22no—
149C32×C8—42no—
150C32×C8#6712yes—
151C32×C8#6712yes—
152C32×C8C3×D412yes—
153C3⋊(C3⋊C8)C3×D412yes—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—32no—
156C3×C3⋊C8—62no—
157#495—91noSylow
158C2×C33#12711yescharacteristic
159C2×C33#13512yes—
160C32×D3—82no—
161C2×C3×Q8—32no—
162C2×C3×Q8—32no—
163C2×C3×Q8—62no—
164C2×C3×D4—32no—
165C2×C3×D4—32no—
166C2×C3×D4—62no—
167C2×D12C2×C3×D312yes—
168C2×Dic6C2×C3×D312yes—
169C3×Q16—64no—
170C3×Q16—62no—
171C3×Q16—62no—
172C3×SD16—184no—
173C3×D8—64no—
174C3×D8—62no—
175C3×D8—62no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—41no—
180C2×C3×C8—91no—
181C2×C3×C8D3211yescharacteristic
182C2×C3×C8C2×C3×D311yescharacteristic
183C2×C3×C8C2×C3×D311yescharacteristic
184C3×C22⋊C4—184no—
185C3×C42—91no—
186#217—64no—
187#215—64no—
188#214—64no—
189(C2×D3)⋊C4—64no—
190C4×Dic3—31no—
191C4×Dic3—31no—
192C4×Dic3—61no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—61no—
196Dic12—22no—
197D24—22no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—41no—
202C22×C32—122no—
203C22×C32—124no—
204C22×C32D3×D411yescharacteristic
205C22×C32D3×D411yescharacteristic
206C22×C32C2×C3×D411yescharacteristic
207C2×C3×D3—44no—
208C2×C3×D3—44no—
209C2×C3×D3—42no—
210C2×C3×D3—42no—
211C2×C3×D3—44no—
212C2×C3×D3—84no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4—122no—
222C32×C4D3×D411yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C32×C4C2×C3×D411yescharacteristic
227C32×C4C2×C3×D411yescharacteristic
228C3⋊Dic3—22no—
229C3×Dic3—44no—
230C3×Dic3—42no—
231C3×Dic3—42no—
232C3×Dic3—64no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C2×Q16—31no—
236C2×SD16—92no—
237C2×D8—31no—
238#124—92no—
239C4×C8—91no—
240C33#49511yescharacteristic, complemented, Sylow
241C23×C3—62no—
242C23×C3—62no—
243C23×C3—122no—
244C22×D3—22no—
245C3×Q8—64no—
246C3×Q8—62no—
247C3×Q8—62no—
248C3×Q8—62no—
249C3×Q8—62no—
250C3×Q8—122no—
251C3×D4—64no—
252C3×D4—62no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—122no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—92no—
264C2×C3×C4—122no—
265C2×C3×C4C2×D3211yescharacteristic
266C2×C3×C4C22×C3×D311yescharacteristic
267C2×C3×C4C22×C3×D311yescharacteristic
268C2×Dic3—22no—
269C2×Dic3—32no—
270C2×Dic3—32no—
271C2×Dic3—62no—
272D12—22no—
273D12—22no—
274Dic6—22no—
275Dic6—22no—
276C3×C8—22no—
277C3×C8—22no—
278C3×C8—22no—
279C3×C8—42no—
280C3×C8—92no—
281C3×C8#62512yes—
282C3×C8C3×#6712yes—
283C3×C8C3×#6712yes—
284C3⋊C8—32no—
285C3⋊C8—32no—
286C3⋊C8—62no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32—242no—
296C2×C32#91212yes—
297C2×C32#91912yes—
298C2×C32#94011yescharacteristic
299C2×C32#94011yescharacteristic
300C2×C32C3×#12711yescharacteristic
301C2×C32C3×#13512yes—
302C3×D3—84no—
303C3×D3—82no—
304C3×D3—82no—
305C2×Q8—32no—
306C2×D4—32no—
307Q16—62no—
308SD16—184no—
309D8—62no—
310C2×C8—91no—
311C2×C8C3×D3211yescharacteristic
312C22⋊C4—184no—
313C42—91no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—124no—
319C22×C3—124no—
320C22×C3—122no—
321C22×C3—124no—
322C22×C3—122no—
323C22×C3—244no—
324C22×C3#380211yescharacteristic
325C22×C3C3×D3×D411yescharacteristic
326C22×C3C3×D3×D411yescharacteristic
327C2×D3—44no—
328C2×D3—42no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—124no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—182no—
341C3×C4#380211yescharacteristic
342C3×C4C2×#62511yescharacteristic
343C3×C4C3×D3×D411yescharacteristic
344C3×C4C3×D3×D411yescharacteristic
345C3×C4C2×C3×#6711yescharacteristic
346C3×C4C2×C3×#6711yescharacteristic
347Dic3—42no—
348Dic3—64no—
349Dic3—62no—
350Dic3—62no—
351C32—21no—
352C32—21no—
353C32—21no—
354C32—41no—
355C32#515511yescharacteristic, complemented
356C32#519111yescharacteristic, complemented
357C32C3×#49511yescharacteristic, complemented
358C23—62no—
359Q8—62no—
360Q8—62no—
361D4—62no—
362D4—62no—
363C2×C4—62no—
364C2×C4—92no—
365C2×C4C2×C3×D3211yescharacteristic
366C8—92no—
367C8C3×#62512yes—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—42no—
375C2×C3—41no—
376C2×C3—244no—
377C2×C3—242no—
378C2×C3—242no—
379C2×C3#6381012yes—
380C2×C3#6391411yescharacteristic
381C2×C3C3×#91212yes—
382C2×C3C3×#91912yes—
383C2×C3C3×#94011yescharacteristic
384C2×C3C3×#94011yescharacteristic
385D3—82no—
386C22—124no—
387C22—122no—
388C22C3×#380211yescharacteristic
389C4—122no—
390C4—182no—
391C4C3×#380211yescharacteristic
392C4C2×C3×#62511yescharacteristic
393C3—21no—
394C3—21no—
395C3—21no—
396C3—41no—
397C3#1059086911yescharacteristic, direct product factor
398C3C3×#515511yescharacteristic, complemented
399C3C3×#519111yescharacteristic, complemented
400C2—242no—
401C2C3×#6381012yes—
402C2C3×#6391411yescharacteristic
subgroupG/H
centerC22×C3#3802
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D24C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C32×Dic12C2semidirect
3. C3×#63863C2semidirect
4. C2×C3×#3715C2semidirect
5. C3×C8×C3⋊Dic3C2semidirect
8. #10590869C3direct (N × H)
25. C2×C3×Dic12D3semidirect
26. C2×C3×Dic12C2×C3semidirect
30. #63863C2×C3semidirect
31. C2×#3715C2×C3semidirect
32. C8×C3⋊Dic3C2×C3semidirect
105. C2×Dic12C3×D3semidirect
240. C33#495semidirect
355. C32#5155semidirect
356. C32#5191semidirect
357. C32C3×#495semidirect
397. C3#10590869direct (N × H)
398. C3C3×#5155semidirect
399. C3C3×#5191semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic122
C2×C32×D242
C3×#638632
C2×C3×#37152
C3×C8×C3⋊Dic32
C3×#51913
C3×#51553
#105908693