Finite Group Databasebeta

← C3×#10596195 #13179476

Subgroups — C3×#10596195 #13179476

Order 1728.

1892 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C33×C4⋊Q8C211yesmaximal, characteristic, complemented
4C3×#64156C211yesmaximal, characteristic, complemented
5C3×#64153C211yesmaximal, characteristic, complemented
6C3×#64105C214yesmaximal, complemented
7C3×#64103C214yesmaximal, complemented
8C3×#64084C211yesmaximal, characteristic, complemented
9C3×#64081C211yesmaximal, characteristic, complemented
10#10596195C311yesmaximal, characteristic, direct product factor
11C3×#5724—34nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C33×Q8C2211yescharacteristic
14C3×#3832—24no—
15C3×Q8×C3⋊D3—24no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)C2212yes—
18C2×C3×C4×C3⋊D3C2212yes—
19C2×C3×C4×C3⋊D3C2212yes—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C3×#634C2211yescharacteristic
22C2×C3×#634C2212yes—
23C33×C4⋊C4C2214yes—
24C33×C42C2211yescharacteristic
25C3×#3752C2212yes—
26C3×#3752C2214yes—
27C3×#3752C2214yes—
28C3×#3750C2212yes—
29C3×#3750C2214yes—
30C3×C4×C3⋊Dic3C2211yescharacteristic
31C2×#3832C2×C311yescharacteristic
32C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—34no—
34C2×C3×D3×Q8—34no—
35C32×C4⋊Q8—24no—
36C32×C4⋊Q8D314yescomplemented
37C32×C4⋊Q8C2×C311yescharacteristic, complemented
38#64156C2×C311yescharacteristic, complemented
39#64153C2×C311yescharacteristic, complemented
40#64105C2×C314yescomplemented
41#64103C2×C314yescomplemented
42#64084C2×C311yescharacteristic, complemented
43#64081C2×C311yescharacteristic, complemented
44C3×#947—34no—
45C3×#944—34no—
46C3×#896—316no—
47C3×#894—316no—
48C3×#875—34no—
49C3×#872—34no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C33×Q8—22no—
53C33×Q8—22no—
54C2×C33×C4C2312yes—
55C2×C33×C4C2312yes—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C3×C4⋊(C3⋊D3)—24no—
62C3×C4⋊(C3⋊D3)—22no—
63C3×C4×C3⋊D3—24no—
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D414yes—
66C3×#634—22no—
67C3×#634—24no—
68C3×#536—91no—
69#5724—34no—
70C2×C32×Q8—24no—
71C2×C32×Q8—24no—
72C2×C32×Q8C2×D314yes—
73C2×C32×Q8C2×D314yes—
74C2×C32×Q8C22×C311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76#3832—24no—
77Q8×C3⋊D3—24no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C2×C4⋊(C3⋊D3)C22×C312yes—
80C2×C4×C3⋊D3C22×C312yes—
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C311yescharacteristic
83C2×#634C22×C311yescharacteristic
84C2×#634C22×C312yes—
85C3×#239—616no—
86C3×D3×Q8—616no—
87C2×C3×D12—38no—
88C2×C3×D12—34no—
89C2×C3×C4×D3—34no—
90C2×C3×C4×D3—38no—
91C2×C3×C4×D3—38no—
92C2×C3×Dic6—38no—
93C2×C3×Dic6—34no—
94C32×C4⋊C4—216no—
95C32×C4⋊C4C2×D3116yes—
96C32×C4⋊C4C22×C314yes—
97C32×C42—24no—
98C32×C42C2×D314yes—
99C32×C42C22×C311yescharacteristic
100#3752C22×C312yes—
101#3752C22×C314yes—
102#3752C22×C314yes—
103#3750C22×C312yes—
104#3750C22×C314yes—
105C4×C3⋊Dic3C22×C311yescharacteristic
106C3×(C2×D3)⋊C4—38no—
107C3×(C2×D3)⋊C4—316no—
108C3×(C2×D3)⋊C4—316no—
109C3×Dic3⋊C4—316no—
110C3×Dic3⋊C4—38no—
111C3×C4×Dic3—34no—
112C22×C33C2411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—24no—
116C2×C3×C3⋊D3C2×D412yes—
117C33×C4—22no—
118C33×C4—22no—
119C33×C4—22no—
120C33×C4C2×D412yes—
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3C2×D412yes—
124C2×C3×#41—91no—
125C22×C3×Q8—91no—
126C2×#239—34no—
127C2×D3×Q8—34no—
128C3×C4⋊Q8—24no—
129C3×C4⋊Q8—91no—
130C3×C4⋊Q8C3×D314yescomplemented
131C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
132C3×#124—91no—
133C3×#124—91no—
134C3×#123—94no—
135C3×C22⋊Q8—94no—
136C3×#117—91no—
137#947—34no—
138#944—34no—
139#896—316no—
140#894—316no—
141#875—34no—
142#872—34no—
143C22×C3⋊D3C23×C312yes—
144C22×C3⋊D3C23×C311yescharacteristic
145C22×C3×D3—34no—
146C22×C3×D3—38no—
147C32×Q8—22no—
148C32×Q8—22no—
149C32×Q8—28no—
150C32×Q8—28no—
151C32×Q8—48no—
152C32×Q8—48no—
153C2×C32×C4—28no—
154C2×C32×C4—24no—
155C2×C32×C4—28no—
156C2×C32×C4—28no—
157C2×C32×C4C22×D318yes—
158C2×C32×C4C22×D318yes—
159C2×C32×C4C22×D318yes—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C23×C312yes—
162C2×C32×C4C23×C312yes—
163C2×C32×C4C23×C312yes—
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3C23×C312yes—
166C2×C3⋊Dic3C23×C312yes—
167C2×C3⋊Dic3C23×C311yescharacteristic
168C4⋊(C3⋊D3)—24no—
169C4⋊(C3⋊D3)—22no—
170C4×C3⋊D3—24no—
171C4×C3⋊D3—24no—
172C4×C3⋊D3C3×D414yes—
173#634—22no—
174#634—24no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—38no—
177C2×C3×Dic3—38no—
178C3×D12—616no—
179C3×D12—68no—
180C3×C4×D3—316no—
181C3×C4×D3—616no—
182C3×C4×D3—616no—
183C3×Dic6—616no—
184C3×Dic6—68no—
185#536—91noSylow
186C2×C33C22×D411yescharacteristic
187C2×C332−1+412yes—
188C3×C3⋊D3—22no—
189C3×C3⋊D3—42no—
190C3×#41—184no—
191C2×C3×Q8—24no—
192C2×C3×Q8—24no—
193C2×C3×Q8—92no—
194C2×C3×Q8—91no—
195C2×C3×Q8—184no—
196C2×C3×Q8C2×C3×D314yes—
197C2×C3×Q8C2×C3×D314yes—
198C2×C3×Q8C2×C3⋊D311yescharacteristic
199C2×C3×Q8C2×C3⋊D311yescharacteristic
200C2×C3×D4—92no—
201C2×C3×D4—91no—
202C22×C3×C4—91no—
203C22×C3×C4—92no—
204C22×C3×C4—92no—
205#239—616no—
206D3×Q8—616no—
207C2×D12—34no—
208C2×D12—38no—
209C2×C4×D3—34no—
210C2×C4×D3—38no—
211C2×C4×D3—38no—
212C2×Dic6—34no—
213C2×Dic6—38no—
214C3×C4⋊C4—216no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—94no—
217C3×C4⋊C4C2×C3×D3116yes—
218C3×C4⋊C4C2×C3⋊D314yes—
219C3×C22⋊C4—92no—
220C3×C22⋊C4—94no—
221C3×C22⋊C4—94no—
222C3×C42—24no—
223C3×C42—91no—
224C3×C42C2×C3×D314yes—
225C3×C42C2×C3⋊D311yescharacteristic
226(C2×D3)⋊C4—38no—
227(C2×D3)⋊C4—316no—
228(C2×D3)⋊C4—316no—
229Dic3⋊C4—38no—
230Dic3⋊C4—316no—
231C4×Dic3—34no—
232C22×C32—24no—
233C22×C32C23×D314yes—
234C22×C32C24×C311yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3—22no—
237C2×C3⋊D3—24no—
238C2×C3⋊D3C2×C3×D412yes—
239C2×C3×D3—38no—
240C2×C3×D3—68no—
241C2×C3×D3—68no—
242C2×C3×D3—616no—
243C32×C4—216no—
244C32×C4—28no—
245C32×C4—22no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—28no—
249C32×C4—28no—
250C32×C4—28no—
251C32×C4—48no—
252C32×C4—48no—
253C32×C4D3×D418yes—
254C32×C4C2×C3×D412yes—
255C3⋊Dic3—22no—
256C3⋊Dic3—22no—
257C3⋊Dic3C2×C3×D412yes—
258C3×Dic3—38no—
259C3×Dic3—68no—
260C3×Dic3—68no—
261C2×#41—91no—
262C22×Q8—91no—
263C4⋊Q8—91no—
264C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
265#124—91no—
266#124—91no—
267#123—94no—
268C22⋊Q8—94no—
269#117—91no—
270C33#53611yescharacteristic, complemented, Sylow
271C23×C3—91no—
272C23×C3—92no—
273C22×D3—34no—
274C22×D3—38no—
275C3×Q8—28no—
276C3×Q8—28no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—48no—
280C3×Q8—48no—
281C3×Q8—184no—
282C3×Q8—182no—
283C3×D4—184no—
284C3×D4—182no—
285C2×C3×C4—28no—
286C2×C3×C4—24no—
287C2×C3×C4—28no—
288C2×C3×C4—28no—
289C2×C3×C4—91no—
290C2×C3×C4—94no—
291C2×C3×C4—92no—
292C2×C3×C4—92no—
293C2×C3×C4—184no—
294C2×C3×C4—184no—
295C2×C3×C4C22×C3×D318yes—
296C2×C3×C4C22×C3×D318yes—
297C2×C3×C4C22×C3×D318yes—
298C2×C3×C4C22×C3×D314yes—
299C2×C3×C4C22×C3⋊D312yes—
300C2×C3×C4C22×C3⋊D312yes—
301C2×C3×C4C22×C3⋊D312yes—
302C2×C3×C4C22×C3⋊D311yescharacteristic
303C2×Dic3—34no—
304C2×Dic3—38no—
305C2×Dic3—38no—
306D12—616no—
307D12—68no—
308C4×D3—316no—
309C4×D3—616no—
310C4×D3—616no—
311Dic6—616no—
312Dic6—68no—
313C2×C32—24no—
314C2×C32—28no—
315C2×C32C2×D3×D414yes—
316C2×C32#100718yes—
317C2×C32C22×C3×D411yescharacteristic
318C2×C32C3×2−1+412yes—
319C3⋊D3—22no—
320C3⋊D3—42no—
321C3×D3—68no—
322C3×D3—128no—
323#41—184no—
324C2×Q8—92no—
325C2×Q8—91no—
326C2×Q8—184no—
327C2×Q8C2×C3×C3⋊D311yescharacteristic
328C2×Q8C2×C3×C3⋊D311yescharacteristic
329C2×D4—92no—
330C2×D4—91no—
331C22×C4—91no—
332C22×C4—92no—
333C22×C4—92no—
334C4⋊C4—92no—
335C4⋊C4—94no—
336C4⋊C4C2×C3×C3⋊D314yes—
337C22⋊C4—92no—
338C22⋊C4—94no—
339C22⋊C4—94no—
340C42—91no—
341C42C2×C3×C3⋊D311yescharacteristic
342C22×C3—24no—
343C22×C3—92no—
344C22×C3—182no—
345C22×C3—182no—
346C22×C3—184no—
347C22×C3C23×C3×D314yes—
348C22×C3C23×C3⋊D311yescharacteristic
349C2×D3—38no—
350C2×D3—68no—
351C2×D3—68no—
352C2×D3—616no—
353C3×C4—216no—
354C3×C4—28no—
355C3×C4—28no—
356C3×C4—28no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—28no—
361C3×C4—48no—
362C3×C4—48no—
363C3×C4—92no—
364C3×C4—182no—
365C3×C4—182no—
366C3×C4C3×D3×D418yes—
367C3×C4D4×C3⋊D312yes—
368Dic3—38no—
369Dic3—68no—
370Dic3—68no—
371C32—24no—
372C32#572414yescomplemented
373C32C3×#53611yescharacteristic, complemented
374C23—91no—
375C23—92no—
376Q8—22no—
377Q8—22no—
378Q8—184no—
379Q8—182no—
380D4—184no—
381D4—182no—
382C2×C4—91no—
383C2×C4—94no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—184no—
387C2×C4—184no—
388C2×C4C22×C3×C3⋊D312yes—
389C2×C4C22×C3×C3⋊D312yes—
390C2×C4C22×C3×C3⋊D312yes—
391C2×C4C22×C3×C3⋊D311yescharacteristic
392C2×C3—24no—
393C2×C3—28no—
394C2×C3—182no—
395C2×C3—362no—
396C2×C3C2×C3×D3×D414yes—
397C2×C3C3×#100718yes—
398C2×C3C2×D4×C3⋊D311yescharacteristic
399C2×C3#6436412yes—
400D3—68no—
401D3—128no—
402C22—92no—
403C22—182no—
404C22—182no—
405C22—184no—
406C22C23×C3×C3⋊D311yescharacteristic
407C4—22no—
408C4—22no—
409C4—22no—
410C4—92no—
411C4—182no—
412C4—182no—
413C4C3×D4×C3⋊D312yes—
414C3—24no—
415C3C3×#572414yescomplemented
416C3#1059619511yescharacteristic, direct product factor
417C2—182no—
418C2—362no—
419C2C2×C3×D4×C3⋊D311yescharacteristic
420C2C3×#6436412yes—
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC2×C3×#3832C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C33×C4⋊Q8C2semidirect
4. C3×#64156C2semidirect
5. C3×#64153C2semidirect
6. C3×#64105C2semidirect
7. C3×#64103C2semidirect
8. C3×#64084C2semidirect
9. C3×#64081C2semidirect
10. #10596195C3direct (N × H)
32. C2×Q8×C3⋊D3C2×C3semidirect
36. C32×C4⋊Q8D3semidirect
37. C32×C4⋊Q8C2×C3semidirect
38. #64156C2×C3semidirect
39. #64153C2×C3semidirect
40. #64105C2×C3semidirect
41. #64103C2×C3semidirect
42. #64084C2×C3semidirect
43. #64081C2×C3semidirect
130. C3×C4⋊Q8C3×D3semidirect
131. C3×C4⋊Q8C3⋊D3semidirect
264. C4⋊Q8C3×C3⋊D3semidirect
270. C33#536semidirect
372. C32#5724semidirect
373. C32C3×#536semidirect
415. C3C3×#5724semidirect
416. C3#10596195direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×Q8×C3⋊D32
C33×C4⋊Q82
C3×#641562
C3×#641532
C3×#641052
C3×#641032
C3×#640842
C3×#640812
#105961953
C3×#57243