Finite Group Databasebeta

← C3×#10592901 #13163242

Subgroups — C3×#10592901 #13163242

Order 1728.

1152 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic
2C33×#137C211yesmaximal, characteristic, complemented
3C3×#64156C211yesmaximal, characteristic, complemented
4C3×#64151C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63650C211yesmaximal, characteristic, complemented
8#10592901C311yesmaximal, characteristic, direct product factor
9C3×#5202—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C33×Q8C2211yescharacteristic, complemented
12C3×#3832—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—21no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×C4⋊(C3⋊D3)—21no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C33×Q16—21no—
20C33×SD16—21no—
21C33×M16C2211yescharacteristic, complemented
22C3×#3756—21no—
23C3×#3755—21no—
24C3×#3752—22no—
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C3×#3748C2211yescharacteristic, complemented
27#64366C2×C311yescharacteristic
28C3×#1009—34no—
29C32×#137—24no—
30C32×#137D314yescomplemented
31C32×#137C2×C311yescharacteristic, complemented
32#64156C2×C311yescharacteristic, complemented
33#64151C2×C311yescharacteristic, complemented
34C3×#947—34no—
35C3×#942—34no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39C3×#837—34no—
40C3×#825—34no—
41C3×#823—34no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D412yes—
44C33×Q8—21no—
45C33×Q8—21no—
46C33×Q8D411yescharacteristic
47C33×D4—21no—
48C33×D4D411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—22no—
53C3×#638—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—22no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)D411yescharacteristic
59C3×C4×C3⋊D3—22no—
60C3×C4×C3⋊D3—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#455—91no—
65#5202—34no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C22×C311yescharacteristic
69C2×C32×Q8—24no—
70C2×C32×Q8C2×D314yescomplemented
71C2×C32×Q8C22×C311yescharacteristic, complemented
72#3832—21no—
73D4×C3⋊D3—22no—
74D4×C3⋊D3—21no—
75#3828—21no—
76#3828C22×C311yescharacteristic
77C2×C4⋊(C3⋊D3)—21no—
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
79C3×#239—64no—
80C3×D3×D4—68no—
81C3×D3×D4—64no—
82C3×#235—34no—
83C3×#235—64no—
84C2×C3×D12—34no—
85C2×C3×D12—64no—
86C32×Q16—28no—
87C32×Q16—24no—
88C32×Q16—21no—
89C32×SD16—28no—
90C32×SD16—24no—
91C32×SD16—21no—
92C32×M16—24no—
93C32×M16C2×D314yescomplemented
94C32×M16C22×C311yescharacteristic, complemented
95#3756—21no—
96#3755—21no—
97#3752—22no—
98C4×C3⋊Dic3C22×C311yescharacteristic, complemented
99#3748C22×C311yescharacteristic, complemented
100C3×#216—64no—
101C3×#215—64no—
102C3×(C2×D3)⋊C4—68no—
103C3×C4×Dic3—34no—
104C3×#208—34no—
105C22×C33—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—42no—
110C2×C3×C3⋊D3—82no—
111C33×C4—21no—
112C33×C4—41no—
113C33×C4C2×D411yescharacteristic
114C33×C4C2×D411yescharacteristic
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C3×H(5,𝔽2)—91no—
118#1009—34no—
119C3×#137—24no—
120C3×#137—91no—
121C3×#137C3×D314yescomplemented
122C3×#137C3⋊D311yescharacteristic, complemented
123C3×#124—91no—
124#947—34no—
125#942—34no—
126C3×#104—91no—
127C3×#104—91no—
128C3×#100—91no—
129#837—34no—
130#825—34no—
131#823—34no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3C3×D412yes—
134C22×C3×D3—38no—
135C22×C3×D3—68no—
136C32×Q8—28no—
137C32×Q8—24no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—24no—
141C32×Q8—24no—
142C32×Q8—44no—
143C32×Q8#6714yes—
144C32×Q8C3×D411yescharacteristic
145C32×D4—21no—
146C32×D4—24no—
147C32×D4—24no—
148C32×D4—44no—
149C32×D4#6714yes—
150C32×D4C3×D411yescharacteristic
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—44no—
157C2×C32×C4—44no—
158C2×C32×C4C22×D314yes—
159C2×C32×C4C23×C311yescharacteristic
160#638—22no—
161#638—21no—
162C2×C3⋊Dic3—21no—
163C4⋊(C3⋊D3)—22no—
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)C3×D411yescharacteristic
167C4×C3⋊D3—22no—
168C4×C3⋊D3—21no—
169#634C3×D411yescharacteristic
170C3×#67—68no—
171C3×#67—64no—
172C2×C3×Dic3—64no—
173C3×D12—34no—
174C3×D12—68no—
175C3×D12—64no—
176C3×D12—64no—
177C3×C4×D3—68no—
178C3×C4×D3—64no—
179C3×Dic6—34no—
180C32×C8—28no—
181C32×C8—24no—
182C32×C8—21no—
183C3⋊(C3⋊C8)—21no—
184C3×C3⋊C8—64no—
185#455—91noSylow
186C2×C33—21no—
187C2×C33—41no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—42no—
190C3×C3⋊D3—41no—
191C3×#41—24no—
192C3×#41—91no—
193C3×#41—181no—
194C3×#41—181no—
195C3×#41C2×C3×D314yes—
196C3×#41C2×C3⋊D311yescharacteristic
197C2×C3×Q8—24no—
198C2×C3×Q8C2×C3×D314yescomplemented
199C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
200C2×C3×D4—91no—
201C2×C3×D4—181no—
202C2×C3×D4—182no—
203C2×C3×D4—181no—
204#239—64no—
205D3×D4—68no—
206D3×D4—64no—
207#235—34no—
208#235—64no—
209C2×D12—34no—
210C2×D12—64no—
211C3×Q16—28no—
212C3×Q16—24no—
213C3×Q16—21no—
214C3×Q16—181no—
215C3×SD16—28no—
216C3×SD16—24no—
217C3×SD16—21no—
218C3×SD16—181no—
219C3×M16—24no—
220C3×M16—91no—
221C3×M16C2×C3×D314yescomplemented
222C3×M16C2×C3⋊D311yescharacteristic, complemented
223C3×C22⋊C4—182no—
224C3×C42—91no—
225#216—64no—
226#215—64no—
227(C2×D3)⋊C4—68no—
228C4×Dic3—34no—
229#208—34no—
230C22×C32—21no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—44no—
234C22×C32D3×D414yes—
235C22×C32C2×C3×D411yescharacteristic
236C2×C3⋊D3—22no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—82no—
240C2×C3×D3—64no—
241C2×C3×D3—68no—
242C2×C3×D3—128no—
243C2×C3×D3—248no—
244C32×C4—21no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—24no—
248C32×C4—44no—
249C32×C4—48no—
250C32×C4—44no—
251C32×C4—41no—
252C32×C4D3×D414yes—
253C32×C4C2×#6714yes—
254C32×C4C2×C3×D411yescharacteristic
255C32×C4C2×C3×D411yescharacteristic
256C3⋊Dic3—21no—
257C3⋊Dic3—41no—
258C3×Dic3—64no—
259C3×Dic3—124no—
260H(5,𝔽2)—91no—
261#137—91no—
262#137C3×C3⋊D311yescharacteristic, complemented
263#124—91no—
264#104—91no—
265#104—91no—
266#100—91no—
267C33#45511yescharacteristic, complemented, Sylow
268C23×C3—92no—
269C23×C3—182no—
270C22×D3—38no—
271C22×D3—68no—
272C3×Q8—28no—
273C3×Q8—24no—
274C3×Q8—24no—
275C3×Q8—21no—
276C3×Q8—21no—
277C3×Q8—24no—
278C3×Q8—44no—
279C3×Q8—91no—
280C3×Q8C3×#6714yes—
281C3×Q8#63811yescharacteristic
282C3×D4—24no—
283C3×D4—21no—
284C3×D4—24no—
285C3×D4—44no—
286C3×D4—91no—
287C3×D4—182no—
288C3×D4—182no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—181no—
292C3×D4C3×#6714yes—
293C3×D4#63811yescharacteristic
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4—182no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4C22×C3×D314yes—
305C2×C3×C4C22×C3⋊D311yescharacteristic
306#67—68no—
307#67—64no—
308C2×Dic3—64no—
309D12—34no—
310D12—68no—
311D12—64no—
312D12—64no—
313C4×D3—68no—
314C4×D3—64no—
315Dic6—34no—
316C3×C8—28no—
317C3×C8—24no—
318C3×C8—21no—
319C3×C8—181no—
320C3⋊C8—64no—
321C2×C32—21no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—44no—
325C2×C32—41no—
326C2×C32—44no—
327C2×C32—84no—
328C2×C32#93714yes—
329C2×C32C3×#12011yescharacteristic
330C3⋊D3—42no—
331C3⋊D3—41no—
332C3×D3—128no—
333C3×D3—124no—
334#41—91no—
335#41—181no—
336#41—181no—
337#41C2×C3×C3⋊D311yescharacteristic
338C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
339C2×D4—91no—
340C2×D4—181no—
341C2×D4—182no—
342C2×D4—181no—
343Q16—21no—
344Q16—181no—
345SD16—21no—
346SD16—181no—
347M16—91no—
348M16C2×C3×C3⋊D311yescharacteristic, complemented
349C22⋊C4—182no—
350C42—91no—
351C22×C3—24no—
352C22×C3—21no—
353C22×C3—24no—
354C22×C3—44no—
355C22×C3—181no—
356C22×C3—182no—
357C22×C3—362no—
358C22×C3—722no—
359C22×C3C3×D3×D414yes—
360C22×C3D4×C3⋊D311yescharacteristic
361C2×D3—64no—
362C2×D3—68no—
363C2×D3—128no—
364C2×D3—248no—
365C3×C4—24no—
366C3×C4—21no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—44no—
370C3×C4—48no—
371C3×C4—44no—
372C3×C4—41no—
373C3×C4—181no—
374C3×C4—361no—
375C3×C4C3×D3×D414yes—
376C3×C4C2×C3×#6714yes—
377C3×C4D4×C3⋊D311yescharacteristic
378C3×C4C2×#63811yescharacteristic
379Dic3—64no—
380Dic3—124no—
381C32—24no—
382C32#520214yescomplemented
383C32C3×#45511yescharacteristic, complemented
384C23—92no—
385C23—182no—
386Q8—21no—
387Q8—21no—
388Q8—91no—
389Q8C3×#63811yescharacteristic
390D4—21no—
391D4—91no—
392D4—182no—
393D4—182no—
394D4—181no—
395D4—181no—
396D4—181no—
397D4C3×#63811yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—182no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4C22×C3×C3⋊D311yescharacteristic
404C8—21no—
405C8—181no—
406C2×C3—24no—
407C2×C3—21no—
408C2×C3—24no—
409C2×C3—44no—
410C2×C3—44no—
411C2×C3—41no—
412C2×C3—84no—
413C2×C3—362no—
414C2×C3—361no—
415C2×C3C3×#93714yes—
416C2×C3#6414611yescharacteristic
417D3—128no—
418D3—124no—
419C22—21no—
420C22—181no—
421C22—182no—
422C22—362no—
423C22—722no—
424C22C3×D4×C3⋊D311yescharacteristic
425C4—21no—
426C4—41no—
427C4—181no—
428C4—361no—
429C4C3×D4×C3⋊D311yescharacteristic
430C4C2×C3×#63811yescharacteristic
431C3—24no—
432C3C3×#520214yescomplemented
433C3#1059290111yescharacteristic, direct product factor
434C2—21no—
435C2—41no—
436C2—362no—
437C2—361no—
438C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#137C2
socleC2×C33#120
prime radicalC3×#64366C2
hypercenterC3×#137C3⋊D3
hypocenterC32C3×#455
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
2. C33×#137C2semidirect
3. C3×#64156C2semidirect
4. C3×#64151C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63650C2semidirect
8. #10592901C3direct (N × H)
11. C2×C33×Q8C22semidirect
21. C33×M16C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C3×#3748C22semidirect
30. C32×#137D3semidirect
31. C32×#137C2×C3semidirect
32. #64156C2×C3semidirect
33. #64151C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63650C2×C3semidirect
70. C2×C32×Q8C2×D3semidirect
71. C2×C32×Q8C22×C3semidirect
93. C32×M16C2×D3semidirect
94. C32×M16C22×C3semidirect
98. C4×C3⋊Dic3C22×C3semidirect
99. #3748C22×C3semidirect
121. C3×#137C3×D3semidirect
122. C3×#137C3⋊D3semidirect
198. C2×C3×Q8C2×C3×D3semidirect
199. C2×C3×Q8C2×C3⋊D3semidirect
221. C3×M16C2×C3×D3semidirect
222. C3×M16C2×C3⋊D3semidirect
262. #137C3×C3⋊D3semidirect
267. C33#455semidirect
338. C2×Q8C2×C3×C3⋊D3semidirect
348. M16C2×C3×C3⋊D3semidirect
382. C32#5202semidirect
383. C32C3×#455semidirect
432. C3C3×#5202semidirect
433. C3#10592901direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643662
C33×#1372
C3×#641562
C3×#641512
C3×#636642
C3×#636522
C3×#636502
#105929013
C3×#52023