Finite Group Databasebeta

← C3×#10592520 #13162859

Subgroups — C3×#10592520 #13162859

Order 1728.

1100 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#64081C211yesmaximal, characteristic, complemented
4C33×#104C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8#10592520C311yesmaximal, characteristic, direct product factor
9C3×#4821—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832C411yescharacteristic, complemented
12C3×Q8×C3⋊D3C411yescharacteristic, complemented
13C3×#3830—21no—
14C3×#3830C411yescharacteristic, complemented
15C3×D4×C3⋊D3—21no—
16C3×D4×C3⋊D3C411yescharacteristic, complemented
17C3×#3828—21no—
18C3×#3828C2211yescharacteristic
19C2×C3×C4×C3⋊D3—21no—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C33×M16C2211yescharacteristic, complemented
22C33×C42C2211yescharacteristic, complemented
23C3×#3752—21no—
24C3×#3750—21no—
25C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
26C3×#3748C2211yescharacteristic, complemented
27C3×#3743—21no—
28C3×C8×C3⋊D3—21no—
29#41×C3⋊D3C2×C311yescharacteristic
30C3×D3×#41—34no—
31M16×C3⋊D3C2×C311yescharacteristic, complemented
32#64081C2×C311yescharacteristic, complemented
33C3×D3×M16—34no—
34C3×#872—34no—
35C32×#104—24no—
36C32×#104D314yescomplemented
37C32×#104C2×C311yescharacteristic, complemented
38#63664C2×C311yescharacteristic, complemented
39#63652C2×C311yescharacteristic, complemented
40#63632C2×C311yescharacteristic, complemented
41C3×#837—34no—
42C3×#825—34no—
43C3×#805—34no—
44C22×C3×C3⋊D3—21no—
45C22×C3×C3⋊D3D411yescharacteristic
46C33×Q8C2×C411yescharacteristic
47C33×D4—21no—
48C33×D4C2×C411yescharacteristic
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×#638—21no—
53C3×#638—21no—
54C3×#638—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3D411yescharacteristic
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
60C3×C4×C3⋊D3—21no—
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C3×C4×C3⋊D3C2×C411yescharacteristic
64C3×C4×C3⋊D3C2×C411yescharacteristic
65C3×C4×C3⋊D3D411yescharacteristic
66C3×C4×C3⋊D3D411yescharacteristic
67C3×#634—21no—
68C3×#634C2×C411yescharacteristic
69C33×C8—21no—
70C3×C3⋊(C3⋊C8)—21no—
71C3×#421—91no—
72#4821—34no—
73C32×#41—24no—
74C32×#41C2×D314yes—
75C32×#41C22×C311yescharacteristic
76#3832C3×C411yescharacteristic, complemented
77Q8×C3⋊D3C3×C411yescharacteristic, complemented
78#3830—21no—
79#3830C3×C411yescharacteristic, complemented
80D4×C3⋊D3—21no—
81D4×C3⋊D3C3×C411yescharacteristic, complemented
82#3828—21no—
83#3828C22×C311yescharacteristic
84C2×C4×C3⋊D3—21no—
85C2×C4×C3⋊D3C22×C311yescharacteristic
86C3×#239—34no—
87C3×D3×Q8—34no—
88C3×#237—34no—
89C3×#237—64no—
90C3×D3×D4—34no—
91C3×D3×D4—64no—
92C3×#235—34no—
93C3×#235—64no—
94C2×C3×C4×D3—34no—
95C2×C3×C4×D3—64no—
96C32×M16—24no—
97C32×M16C2×D314yescomplemented
98C32×M16C22×C311yescharacteristic, complemented
99C32×C42—24no—
100C32×C42C2×D314yescomplemented
101C32×C42C22×C311yescharacteristic, complemented
102#3752—21no—
103#3750—21no—
104C4×C3⋊Dic3C22×C311yescharacteristic, complemented
105#3748C22×C311yescharacteristic, complemented
106#3743—21no—
107C8×C3⋊D3—21no—
108C3×(C2×D3)⋊C4—64no—
109C3×Dic3⋊C4—64no—
110C3×C4×Dic3—34no—
111C3×#208—34no—
112C3×ΓL(1,𝔽25)—64no—
113C3×D3×C8—64no—
114C22×C33—21no—
115C22×C33C2×D411yescharacteristic
116C2×C3×C3⋊D3—21no—
117C2×C3×C3⋊D3—41no—
118C2×C3×C3⋊D3—42no—
119C2×C3×C3⋊D3C22⋊C411yescharacteristic
120C2×C3×C3⋊D3C22⋊C411yescharacteristic
121C33×C4—21no—
122C33×C4—41no—
123C33×C4C22×C411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3—41no—
127C3×C3⋊Dic3C22⋊C411yescharacteristic
128C3×C3⋊Dic3C22⋊C411yescharacteristic
129C2×C3×#41—91no—
130D3×#41—34no—
131C2×C3×M16—91no—
132C3×#117—91no—
133D3×M16—34no—
134#872—34no—
135C3×#104—24no—
136C3×#104—91no—
137C3×#104—91no—
138C3×#104—91no—
139C3×#104C3×D314yescomplemented
140C3×#104C3⋊D311yescharacteristic, complemented
141#837—34no—
142#825—34no—
143#805—34no—
144C22×C3⋊D3—21no—
145C22×C3⋊D3C3×D411yescharacteristic
146C22×C3×D3—34no—
147C22×C3×D3—64no—
148C32×Q8—24no—
149C32×Q8C4×D314yes—
150C32×Q8C2×C3×C411yescharacteristic
151C32×D4—21no—
152C32×D4—24no—
153C32×D4—24no—
154C32×D4—44no—
155C32×D4C4×D314yes—
156C32×D4C2×C3×C411yescharacteristic
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—24no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—44no—
163C2×C32×C4—44no—
164C2×C32×C4C22×D314yes—
165C2×C32×C4C23×C311yescharacteristic
166#638—21no—
167#638—21no—
168#638—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3C3×D411yescharacteristic
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)C2×C3×C411yescharacteristic
174C4×C3⋊D3—21no—
175C4×C3⋊D3—21no—
176C4×C3⋊D3—21no—
177C4×C3⋊D3C2×C3×C411yescharacteristic
178C4×C3⋊D3C2×C3×C411yescharacteristic
179C4×C3⋊D3C3×D411yescharacteristic
180C4×C3⋊D3C3×D411yescharacteristic
181#634—21no—
182#634C2×C3×C411yescharacteristic
183C3×#67—64no—
184C3×#67—64no—
185C3×#67—64no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C3×D12—34no—
190C3×D12—64no—
191C3×C4×D3—34no—
192C3×C4×D3—34no—
193C3×C4×D3—34no—
194C3×C4×D3—34no—
195C3×C4×D3—64no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×Dic6—34no—
199C3×Dic6—64no—
200C32×C8—21no—
201C32×C8—24no—
202C32×C8—44no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—64no—
205#421—91noSylow
206C2×C33—21no—
207C2×C33—41no—
208C2×C33C2×C22⋊C411yescharacteristic
209C3×C3⋊D3—21no—
210C3×C3⋊D3—21no—
211C3×C3⋊D3—41no—
212C3×#41—24no—
213C3×#41—91no—
214C3×#41—91no—
215C3×#41—91no—
216C3×#41—181no—
217C3×#41—181no—
218C3×#41C2×C3×D314yes—
219C3×#41C2×C3⋊D311yescharacteristic
220C2×C3×Q8—91no—
221C2×C3×D4—91no—
222C2×C3×D4—181no—
223C22×C3×C4—91no—
224C22×C3×C4—181no—
225#239—34no—
226D3×Q8—34no—
227#237—34no—
228#237—64no—
229D3×D4—34no—
230D3×D4—64no—
231#235—34no—
232#235—64no—
233C2×C4×D3—34no—
234C2×C4×D3—64no—
235C3×M16—24no—
236C3×M16—91no—
237C3×M16—181no—
238C3×M16C2×C3×D314yescomplemented
239C3×M16C2×C3⋊D311yescharacteristic, complemented
240C2×C3×C8—181no—
241C3×C4⋊C4—181no—
242C3×C22⋊C4—181no—
243C3×C42—24no—
244C3×C42—91no—
245C3×C42C2×C3×D314yescomplemented
246C3×C42C2×C3⋊D311yescharacteristic, complemented
247(C2×D3)⋊C4—64no—
248Dic3⋊C4—64no—
249C4×Dic3—34no—
250#208—34no—
251ΓL(1,𝔽25)—64no—
252D3×C8—64no—
253C22×C32—21no—
254C22×C32—24no—
255C22×C32—24no—
256C22×C32—44no—
257C22×C32D3×D414yes—
258C22×C32C2×C3×D411yescharacteristic
259C2×C3⋊D3—21no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—42no—
262C2×C3⋊D3C3×C22⋊C411yescharacteristic
263C2×C3⋊D3C3×C22⋊C411yescharacteristic
264C2×C3×D3—34no—
265C2×C3×D3—34no—
266C2×C3×D3—64no—
267C2×C3×D3—124no—
268C2×C3×D3—128no—
269C32×C4—21no—
270C32×C4—24no—
271C32×C4—24no—
272C32×C4—24no—
273C32×C4—44no—
274C32×C4—41no—
275C32×C4—44no—
276C32×C4—48no—
277C32×C4C2×C4×D314yes—
278C32×C4D3×D414yes—
279C32×C4C22×C3×C411yescharacteristic
280C32×C4C2×C3×D411yescharacteristic
281C3⋊Dic3—21no—
282C3⋊Dic3—41no—
283C3⋊Dic3C3×C22⋊C411yescharacteristic
284C3⋊Dic3C3×C22⋊C411yescharacteristic
285C3×Dic3—34no—
286C3×Dic3—34no—
287C3×Dic3—64no—
288C3×Dic3—124no—
289C2×#41—91no—
290C2×M16—91no—
291#117—91no—
292#104—91no—
293#104—91no—
294#104—91no—
295#104C3×C3⋊D311yescharacteristic, complemented
296C33#42111yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C23×C3—181no—
299C22×D3—34no—
300C22×D3—64no—
301C3×Q8—24no—
302C3×Q8—91no—
303C3×Q8—181no—
304C3×Q8C3×C4×D314yes—
305C3×Q8C4×C3⋊D311yescharacteristic
306C3×D4—24no—
307C3×D4—21no—
308C3×D4—24no—
309C3×D4—44no—
310C3×D4—91no—
311C3×D4—181no—
312C3×D4—181no—
313C3×D4—181no—
314C3×D4—181no—
315C3×D4C3×C4×D314yes—
316C3×D4C4×C3⋊D311yescharacteristic
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4—91no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C22×C3×D314yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336#67—64no—
337#67—64no—
338#67—64no—
339C2×Dic3—34no—
340C2×Dic3—64no—
341C2×Dic3—64no—
342D12—34no—
343D12—64no—
344C4×D3—34no—
345C4×D3—34no—
346C4×D3—34no—
347C4×D3—34no—
348C4×D3—64no—
349C4×D3—64no—
350C4×D3—64no—
351Dic6—34no—
352Dic6—64no—
353C3×C8—24no—
354C3×C8—21no—
355C3×C8—44no—
356C3×C8—181no—
357C3⋊C8—64no—
358C2×C32—21no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—44no—
362C2×C32—41no—
363C2×C32—44no—
364C2×C32—84no—
365C2×C32D3×C22⋊C414yes—
366C2×C32C2×C3×C22⋊C411yescharacteristic
367C3⋊D3—21no—
368C3⋊D3—21no—
369C3⋊D3—41no—
370C3×D3—64no—
371C3×D3—64no—
372C3×D3—124no—
373#41—91no—
374#41—91no—
375#41—91no—
376#41—181no—
377#41—181no—
378#41C2×C3×C3⋊D311yescharacteristic
379C2×Q8—91no—
380C2×D4—91no—
381C2×D4—181no—
382C22×C4—91no—
383C22×C4—181no—
384M16—91no—
385M16—181no—
386M16C2×C3×C3⋊D311yescharacteristic, complemented
387C2×C8—181no—
388C4⋊C4—181no—
389C22⋊C4—181no—
390C42—91no—
391C42C2×C3×C3⋊D311yescharacteristic, complemented
392C22×C3—24no—
393C22×C3—21no—
394C22×C3—24no—
395C22×C3—44no—
396C22×C3—91no—
397C22×C3—91no—
398C22×C3—181no—
399C22×C3—361no—
400C22×C3—362no—
401C22×C3C3×D3×D414yes—
402C22×C3D4×C3⋊D311yescharacteristic
403C2×D3—34no—
404C2×D3—34no—
405C2×D3—64no—
406C2×D3—124no—
407C2×D3—128no—
408C3×C4—24no—
409C3×C4—21no—
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—44no—
413C3×C4—41no—
414C3×C4—44no—
415C3×C4—48no—
416C3×C4—91no—
417C3×C4—91no—
418C3×C4—181no—
419C3×C4—361no—
420C3×C4C2×C3×C4×D314yes—
421C3×C4C3×D3×D414yes—
422C3×C4C2×C4×C3⋊D311yescharacteristic
423C3×C4D4×C3⋊D311yescharacteristic
424Dic3—34no—
425Dic3—34no—
426Dic3—64no—
427Dic3—124no—
428C32—24no—
429C32#482114yescomplemented
430C32C3×#42111yescharacteristic, complemented
431C23—91no—
432C23—181no—
433Q8—91no—
434Q8—181no—
435Q8C3×C4×C3⋊D311yescharacteristic
436D4—21no—
437D4—91no—
438D4—181no—
439D4—181no—
440D4—181no—
441D4—181no—
442D4C3×C4×C3⋊D311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4C22×C3×C3⋊D311yescharacteristic
456C8—21no—
457C8—181no—
458C2×C3—21no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3—44no—
462C2×C3—41no—
463C2×C3—44no—
464C2×C3—84no—
465C2×C3—181no—
466C2×C3—181no—
467C2×C3—361no—
468C2×C3C3×D3×C22⋊C414yes—
469C2×C3C22⋊C4×C3⋊D311yescharacteristic
470D3—64no—
471D3—64no—
472D3—124no—
473C22—21no—
474C22—91no—
475C22—91no—
476C22—181no—
477C22—361no—
478C22—362no—
479C22C3×D4×C3⋊D311yescharacteristic
480C4—21no—
481C4—41no—
482C4—91no—
483C4—91no—
484C4—181no—
485C4—361no—
486C4C2×C3×C4×C3⋊D311yescharacteristic
487C4C3×D4×C3⋊D311yescharacteristic
488C3—24no—
489C3C3×#482114yescomplemented
490C3#1059252011yescharacteristic, direct product factor
491C2—21no—
492C2—41no—
493C2—181no—
494C2—181no—
495C2—361no—
496C2C3×C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC3×C4D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#104C2
socleC2×C33C2×C22⋊C4
prime radicalC3×#41×C3⋊D3C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#104C3⋊D3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C3×#64081C2semidirect
4. C33×#104C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63632C2semidirect
8. #10592520C3direct (N × H)
11. C3×#3832C4semidirect
12. C3×Q8×C3⋊D3C4semidirect
14. C3×#3830C4semidirect
16. C3×D4×C3⋊D3C4semidirect
21. C33×M16C22semidirect
22. C33×C42C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C3×#3748C22semidirect
31. M16×C3⋊D3C2×C3semidirect
32. #64081C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63664C2×C3semidirect
39. #63652C2×C3semidirect
40. #63632C2×C3semidirect
76. #3832C3×C4semidirect
77. Q8×C3⋊D3C3×C4semidirect
79. #3830C3×C4semidirect
81. D4×C3⋊D3C3×C4semidirect
97. C32×M16C2×D3semidirect
98. C32×M16C22×C3semidirect
100. C32×C42C2×D3semidirect
101. C32×C42C22×C3semidirect
104. C4×C3⋊Dic3C22×C3semidirect
105. #3748C22×C3semidirect
139. C3×#104C3×D3semidirect
140. C3×#104C3⋊D3semidirect
238. C3×M16C2×C3×D3semidirect
239. C3×M16C2×C3⋊D3semidirect
245. C3×C42C2×C3×D3semidirect
246. C3×C42C2×C3⋊D3semidirect
295. #104C3×C3⋊D3semidirect
296. C33#421semidirect
386. M16C2×C3×C3⋊D3semidirect
391. C42C2×C3×C3⋊D3semidirect
429. C32#4821semidirect
430. C32C3×#421semidirect
489. C3C3×#4821semidirect
490. C3#10592520direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
C3×M16×C3⋊D32
C3×#640812
C33×#1042
C3×#636642
C3×#636522
C3×#636322
#105925203
C3×#48213