Finite Group Databasebeta

← C3×C3⋊D3×#104 #13162858

Subgroups — C3×C3⋊D3×#104 #13162858

Order 1728.

1196 conjugacy classes of subgroups.

The subgroup lattice has 504 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×C42×C3⋊D3C211yesmaximal, characteristic, complemented
4C33×#104C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8C3⋊D3×#104C311yesmaximal, characteristic, direct product factor
9C3×D3×#104—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⋊D3—21no—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C33×M16C2211yescharacteristic, complemented
23C33×C42C2211yescharacteristic, complemented
24C3×C4×C3⋊Dic3—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
32C42×C3⋊D3C2×C311yescharacteristic, complemented
33C3×D3×M16—34no—
34C3×C42×D3—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, complemented
47C33×D4—21no—
48C33×D4C2×C411yescharacteristic, complemented
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, complemented
60C3×C4×C3⋊D3—24no—
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C3×C4×C3⋊D3—21no—
64C3×C4×C3⋊D3C2×C411yescharacteristic
65C3×C4×C3⋊D3C2×C411yescharacteristic
66C3×C4×C3⋊D3D411yescharacteristic
67C3×C4×C3⋊D3D411yescharacteristic
68C3×#634—21no—
69C3×#634C2×C411yescharacteristic, complemented
70C33×C8—21no—
71C3×C3⋊(C3⋊C8)—21no—
72C2×C3×#104—91no—
73D3×#104—34no—
74C32×#41—24no—
75C32×#41C2×D314yes—
76C32×#41C22×C311yescharacteristic
77#3832C3×C411yescharacteristic, complemented
78Q8×C3⋊D3C3×C411yescharacteristic, complemented
79#3830—21no—
80#3830C3×C411yescharacteristic, complemented
81D4×C3⋊D3—21no—
82D4×C3⋊D3C3×C411yescharacteristic, complemented
83#3828—21no—
84#3828C22×C311yescharacteristic
85C2×C4×C3⋊D3—21no—
86C2×C4×C3⋊D3—21no—
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C3×#239—34no—
89C3×D3×Q8—34no—
90C3×#237—34no—
91C3×#237—64no—
92C3×D3×D4—34no—
93C3×D3×D4—64no—
94C3×#235—34no—
95C3×#235—64no—
96C2×C3×C4×D3—34no—
97C2×C3×C4×D3—64no—
98C2×C3×C4×D3—64no—
99C32×M16—24no—
100C32×M16C2×D314yescomplemented
101C32×M16C22×C311yescharacteristic, complemented
102C32×C42—24no—
103C32×C42C2×D314yescomplemented
104C32×C42C22×C311yescharacteristic, complemented
105C4×C3⋊Dic3—21no—
106C4×C3⋊Dic3C22×C311yescharacteristic, complemented
107#3748C22×C311yescharacteristic, complemented
108#3743—21no—
109C8×C3⋊D3—21no—
110C3×C4×Dic3—34no—
111C3×C4×Dic3—64no—
112C3×#208—34no—
113C3×ΓL(1,𝔽25)—64no—
114C3×D3×C8—64no—
115C22×C33—21no—
116C22×C33C2×D411yescharacteristic
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—21no—
119C2×C3×C3⋊D3—42no—
120C2×C3×C3⋊D3C22⋊C411yescharacteristic
121C2×C3×C3⋊D3C22⋊C411yescharacteristic
122C33×C4—22no—
123C33×C4—21no—
124C33×C4C22×C411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3C22⋊C411yescharacteristic
129C3×C3⋊Dic3C22⋊C411yescharacteristic
130C2×C3×#41—91no—
131D3×#41—34no—
132C2×C3×M16—91no—
133C2×C3×C42—91no—
134D3×M16—34no—
135C42×D3—34no—
136C3×#104—24no—
137C3×#104—91no—
138C3×#104—91no—
139C3×#104—91no—
140C3×#104C3×D314yescomplemented
141C3×#104C3⋊D311yescharacteristic, direct product factor
142#837—34no—
143#825—34no—
144#805—34no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3C3×D411yescharacteristic
147C22×C3×D3—34no—
148C22×C3×D3—64no—
149C32×Q8—24no—
150C32×Q8C4×D314yescomplemented
151C32×Q8C2×C3×C411yescharacteristic, complemented
152C32×D4—21no—
153C32×D4—24no—
154C32×D4—24no—
155C32×D4—44no—
156C32×D4C4×D314yescomplemented
157C32×D4C2×C3×C411yescharacteristic, complemented
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—44no—
164C2×C32×C4—44no—
165C2×C32×C4C22×D314yes—
166C2×C32×C4C23×C311yescharacteristic
167#638—21no—
168#638—21no—
169#638—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3C3×D411yescharacteristic
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)C2×C3×C411yescharacteristic, complemented
175C4×C3⋊D3—24no—
176C4×C3⋊D3—21no—
177C4×C3⋊D3—21no—
178C4×C3⋊D3—21no—
179C4×C3⋊D3C2×C3×C411yescharacteristic
180C4×C3⋊D3C2×C3×C411yescharacteristic
181C4×C3⋊D3C3×D411yescharacteristic
182C4×C3⋊D3C3×D411yescharacteristic
183#634—21no—
184#634C2×C3×C411yescharacteristic, complemented
185C3×#67—64no—
186C3×#67—64no—
187C3×#67—64no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191C3×D12—34no—
192C3×D12—64no—
193C3×C4×D3—34no—
194C3×C4×D3—34no—
195C3×C4×D3—34no—
196C3×C4×D3—34no—
197C3×C4×D3—64no—
198C3×C4×D3—64no—
199C3×C4×D3—616no—
200C3×C4×D3—64no—
201C3×Dic6—34no—
202C3×Dic6—64no—
203C32×C8—21no—
204C32×C8—24no—
205C32×C8—44no—
206C3⋊(C3⋊C8)—21no—
207C3×C3⋊C8—64no—
208C2×#104—91noSylow
209C2×C33—21no—
210C2×C33—41no—
211C2×C33C2×C22⋊C411yescharacteristic
212C3×C3⋊D3—21no—
213C3×C3⋊D3—41no—
214C3×C3⋊D3#10412yesdirect product factor
215C3×#41—24no—
216C3×#41—91no—
217C3×#41—91no—
218C3×#41—91no—
219C3×#41—181no—
220C3×#41—181no—
221C3×#41C2×C3×D314yes—
222C3×#41C2×C3⋊D311yescharacteristic
223C2×C3×Q8—91no—
224C2×C3×D4—91no—
225C2×C3×D4—181no—
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C22×C3×C4—181no—
229#239—34no—
230D3×Q8—34no—
231#237—34no—
232#237—64no—
233D3×D4—34no—
234D3×D4—64no—
235#235—34no—
236#235—64no—
237C2×C4×D3—34no—
238C2×C4×D3—64no—
239C2×C4×D3—64no—
240C3×M16—24no—
241C3×M16—91no—
242C3×M16—181no—
243C3×M16C2×C3×D314yescomplemented
244C3×M16C2×C3⋊D311yescharacteristic, complemented
245C2×C3×C8—181no—
246C3×C42—24no—
247C3×C42—91no—
248C3×C42—181no—
249C3×C42C2×C3×D314yescomplemented
250C3×C42C2×C3⋊D311yescharacteristic, complemented
251C4×Dic3—34no—
252C4×Dic3—64no—
253#208—34no—
254ΓL(1,𝔽25)—64no—
255D3×C8—64no—
256C22×C32—21no—
257C22×C32—24no—
258C22×C32—24no—
259C22×C32—44no—
260C22×C32D3×D414yes—
261C22×C32C2×C3×D411yescharacteristic
262C2×C3⋊D3—22no—
263C2×C3⋊D3—21no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3C3×C22⋊C411yescharacteristic
266C2×C3⋊D3C3×C22⋊C411yescharacteristic
267C2×C3×D3—34no—
268C2×C3×D3—34no—
269C2×C3×D3—68no—
270C2×C3×D3—64no—
271C2×C3×D3—128no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—24no—
275C32×C4—24no—
276C32×C4—28no—
277C32×C4—24no—
278C32×C4—44no—
279C32×C4—48no—
280C32×C4C2×C4×D314yes—
281C32×C4D3×D414yes—
282C32×C4C22×C3×C411yescharacteristic
283C32×C4C2×C3×D411yescharacteristic
284C3⋊Dic3—22no—
285C3⋊Dic3—21no—
286C3⋊Dic3C3×C22⋊C411yescharacteristic
287C3⋊Dic3C3×C22⋊C411yescharacteristic
288C3×Dic3—34no—
289C3×Dic3—34no—
290C3×Dic3—68no—
291C3×Dic3—64no—
292C2×#41—91no—
293C2×M16—91no—
294C2×C42—91no—
295#104—91no—
296#104—91no—
297#104—91no—
298#104C3×C3⋊D311yescharacteristic, direct product factor
299C33C2×#10411yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C23×C3—181no—
302C22×D3—34no—
303C22×D3—64no—
304C3×Q8—24no—
305C3×Q8—91no—
306C3×Q8—181no—
307C3×Q8C3×C4×D314yescomplemented
308C3×Q8C4×C3⋊D311yescharacteristic, complemented
309C3×D4—24no—
310C3×D4—21no—
311C3×D4—24no—
312C3×D4—44no—
313C3×D4—91no—
314C3×D4—181no—
315C3×D4—181no—
316C3×D4—181no—
317C3×D4—181no—
318C3×D4C3×C4×D314yescomplemented
319C3×D4C4×C3⋊D311yescharacteristic, complemented
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—21no—
324C2×C3×C4—24no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—184no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4C22×C3×D314yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340#67—64no—
341#67—64no—
342#67—64no—
343C2×Dic3—34no—
344C2×Dic3—64no—
345C2×Dic3—64no—
346D12—34no—
347D12—64no—
348C4×D3—34no—
349C4×D3—34no—
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—64no—
353C4×D3—64no—
354C4×D3—616no—
355C4×D3—64no—
356Dic6—34no—
357Dic6—64no—
358C3×C8—24no—
359C3×C8—21no—
360C3×C8—44no—
361C3×C8—181no—
362C3⋊C8—64no—
363C2×C32—21no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—44no—
367C2×C32—41no—
368C2×C32—44no—
369C2×C32—84no—
370C2×C32D3×C22⋊C414yes—
371C2×C32C2×C3×C22⋊C411yescharacteristic
372C3⋊D3—21no—
373C3⋊D3—41no—
374C3⋊D3C3×#10412yesdirect product factor
375C3×D3—38no—
376C3×D3—64no—
377C3×D3—124no—
378#41—91no—
379#41—91no—
380#41—91no—
381#41—181no—
382#41—181no—
383#41C2×C3×C3⋊D311yescharacteristic
384C2×Q8—91no—
385C2×D4—91no—
386C2×D4—181no—
387C22×C4—91no—
388C22×C4—181no—
389C22×C4—181no—
390M16—91no—
391M16—181no—
392M16C2×C3×C3⋊D311yescharacteristic, complemented
393C2×C8—181no—
394C42—91no—
395C42—181no—
396C42C2×C3×C3⋊D311yescharacteristic, complemented
397C22×C3—24no—
398C22×C3—21no—
399C22×C3—24no—
400C22×C3—44no—
401C22×C3—91no—
402C22×C3—91no—
403C22×C3—182no—
404C22×C3—181no—
405C22×C3—362no—
406C22×C3C3×D3×D414yes—
407C22×C3D4×C3⋊D311yescharacteristic
408C2×D3—34no—
409C2×D3—34no—
410C2×D3—68no—
411C2×D3—64no—
412C2×D3—128no—
413C3×C4—28no—
414C3×C4—22no—
415C3×C4—24no—
416C3×C4—21no—
417C3×C4—24no—
418C3×C4—24no—
419C3×C4—44no—
420C3×C4—48no—
421C3×C4—91no—
422C3×C4—91no—
423C3×C4—182no—
424C3×C4—181no—
425C3×C4C2×C3×C4×D314yes—
426C3×C4C3×D3×D414yes—
427C3×C4C2×C4×C3⋊D311yescharacteristic
428C3×C4D4×C3⋊D311yescharacteristic
429Dic3—34no—
430Dic3—34no—
431Dic3—68no—
432Dic3—64no—
433C32—24no—
434C32D3×#10414yescomplemented
435C32C2×C3×#10411yescharacteristic, complemented
436C23—91no—
437C23—181no—
438Q8—91no—
439Q8—181no—
440Q8C3×C4×C3⋊D311yescharacteristic, complemented
441D4—21no—
442D4—91no—
443D4—181no—
444D4—181no—
445D4—181no—
446D4—181no—
447D4C3×C4×C3⋊D311yescharacteristic, complemented
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—91no—
451C2×C4—91no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—91no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—184no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4C22×C3×C3⋊D311yescharacteristic
462C8—21no—
463C8—181no—
464C2×C3—21no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—44no—
468C2×C3—41no—
469C2×C3—44no—
470C2×C3—84no—
471C2×C3—92no—
472C2×C3—181no—
473C2×C3—361no—
474C2×C3C3×D3×C22⋊C414yes—
475C2×C3C22⋊C4×C3⋊D311yescharacteristic
476D3—38no—
477D3—64no—
478D3—124no—
479C22—21no—
480C22—91no—
481C22—91no—
482C22—182no—
483C22—181no—
484C22—362no—
485C22C3×D4×C3⋊D311yescharacteristic
486C4—22no—
487C4—21no—
488C4—91no—
489C4—91no—
490C4—182no—
491C4—181no—
492C4C2×C3×C4×C3⋊D311yescharacteristic
493C4C3×D4×C3⋊D311yescharacteristic
494C3—24no—
495C3C3×D3×#10414yescomplemented
496C3C3⋊D3×#10411yescharacteristic, direct product factor
497C2—21no—
498C2—41no—
499C2—92no—
500C2—181no—
501C2—361no—
502C2C3×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
hypocenterC32C2×C3×#104
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×C42×C3⋊D3C2semidirect
4. C33×#104C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63632C2semidirect
8. C3⋊D3×#104C3direct (N × H)
11. C3×#3832C4semidirect
12. C3×Q8×C3⋊D3C4semidirect
14. C3×#3830C4semidirect
16. C3×D4×C3⋊D3C4semidirect
22. C33×M16C22semidirect
23. C33×C42C22semidirect
25. C3×C4×C3⋊Dic3C22semidirect
26. C3×#3748C22semidirect
31. M16×C3⋊D3C2×C3semidirect
32. C42×C3⋊D3C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63664C2×C3semidirect
39. #63652C2×C3semidirect
40. #63632C2×C3semidirect
46. C33×Q8C2×C4semidirect
48. C33×D4C2×C4semidirect
59. C3×C4⋊(C3⋊D3)C2×C4semidirect
69. C3×#634C2×C4semidirect
77. #3832C3×C4semidirect
78. Q8×C3⋊D3C3×C4semidirect
80. #3830C3×C4semidirect
82. D4×C3⋊D3C3×C4semidirect
100. C32×M16C2×D3semidirect
101. C32×M16C22×C3semidirect
103. C32×C42C2×D3semidirect
104. C32×C42C22×C3semidirect
106. C4×C3⋊Dic3C22×C3semidirect
107. #3748C22×C3semidirect
140. C3×#104C3×D3semidirect
141. C3×#104C3⋊D3direct (N × H)
150. C32×Q8C4×D3semidirect
151. C32×Q8C2×C3×C4semidirect
156. C32×D4C4×D3semidirect
157. C32×D4C2×C3×C4semidirect
174. C4⋊(C3⋊D3)C2×C3×C4semidirect
184. #634C2×C3×C4semidirect
214. C3×C3⋊D3#104direct (N × H)
243. C3×M16C2×C3×D3semidirect
244. C3×M16C2×C3⋊D3semidirect
249. C3×C42C2×C3×D3semidirect
250. C3×C42C2×C3⋊D3semidirect
298. #104C3×C3⋊D3direct (N × H)
299. C33C2×#104semidirect
307. C3×Q8C3×C4×D3semidirect
308. C3×Q8C4×C3⋊D3semidirect
318. C3×D4C3×C4×D3semidirect
319. C3×D4C4×C3⋊D3semidirect
374. C3⋊D3C3×#104direct (N × H)
392. M16C2×C3×C3⋊D3semidirect
396. C42C2×C3×C3⋊D3semidirect
434. C32D3×#104semidirect
435. C32C2×C3×#104semidirect
440. Q8C3×C4×C3⋊D3semidirect
447. D4C3×C4×C3⋊D3semidirect
495. C3C3×D3×#104semidirect
496. C3C3⋊D3×#104direct (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×C42×C3⋊D32
C33×#1042
C3×#636642
C3×#636522
C3×#636322
C3⋊D3×#1043
C3×D3×#1043