Finite Group Databasebeta

← C3×#10592426 #13162766

Subgroups — C3×#10592426 #13162766

Order 1728.

1148 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
4C33×C22⋊C8C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63647C211yesmaximal, characteristic, complemented
8#10592426C311yesmaximal, characteristic, direct product factor
9C3×#4727—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638C411yescharacteristic
12C2×C3×#638C411yescharacteristic
13C3×#3828—22no—
14C3×#3828—22no—
15C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C411yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C2×C33×C8C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3743—22no—
24C3×C8×C3⋊D3—22no—
25C2×#3828C2×C311yescharacteristic
26C2×C3×#235—34no—
27C2×#3743C2×C311yescharacteristic, complemented
28C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
29C2×C3×ΓL(1,𝔽25)—34no—
30C2×C3×D3×C8—34no—
31C32×C22⋊C8—24no—
32C32×C22⋊C8D314yescomplemented
33C32×C22⋊C8C2×C311yescharacteristic, complemented
34#63657C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37C3×#830—34no—
38C3×(C2×D3)⋊C8—34no—
39C3×(C2×D3)⋊C8—34no—
40C23×C33C2×C411yescharacteristic
41C22×C3×C3⋊D3C2×C411yescharacteristic
42C22×C3×C3⋊D3C2×C411yescharacteristic
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2×C411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C3×#638—22no—
48C3×#638—22no—
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C3×C4⋊(C3⋊D3)—22no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3D412yes—
55C3×C4×C3⋊D3D412yes—
56C3×#634—22no—
57C33×C8—21no—
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×C3⋊(C3⋊C8)—21no—
61C3×#408—91no—
62#4727—34no—
63C22×C32×C4—24no—
64C22×C32×C4C2×D314yes—
65C22×C32×C4C22×C311yescharacteristic
66C2×#638C3×C411yescharacteristic
67C2×#638C3×C411yescharacteristic
68#3828—22no—
69#3828—22no—
70C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
71C2×C4×C3⋊D3C22×C311yescharacteristic
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×#634C3×C411yescharacteristic
74C2×C3×#67—34no—
75C2×C3×#67—34no—
76C3×#235—68no—
77C3×#235—68no—
78C2×C3×D12—34no—
79C2×C3×C4×D3—34no—
80C2×C3×C4×D3—34no—
81C2×C3×Dic6—34no—
82C2×C32×C8—24no—
83C2×C32×C8—24no—
84C2×C32×C8C2×D314yes—
85C2×C32×C8C2×D314yes—
86C2×C32×C8C22×C311yescharacteristic
87C2×C32×C8C22×C311yescharacteristic
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90#3743—22no—
91C8×C3⋊D3—22no—
92C2×C3×C3⋊C8—34no—
93C2×C3×C3⋊C8—34no—
94C3×ΓL(1,𝔽25)—68no—
95C3×D3×C8—68no—
96C22×C33—21no—
97C22×C33—21no—
98C22×C33—21no—
99C22×C33C22×C411yescharacteristic
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—21no—
104C2×C3×C3⋊D3—21no—
105C2×C3×C3⋊D3C22⋊C412yes—
106C33×C4—21no—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—21no—
110C3×C3⋊Dic3C22⋊C412yes—
111C2×C3×#41—91no—
112C2×#235—34no—
113C2×C3×M16—91no—
114C22×C3×C8—91no—
115C2×ΓL(1,𝔽25)—34no—
116C2×D3×C8—34no—
117C3×C22⋊C8—24no—
118C3×C22⋊C8—91no—
119C3×C22⋊C8—91no—
120C3×C22⋊C8—91no—
121C3×C22⋊C8C3×D314yescomplemented
122C3×C22⋊C8C3⋊D311yescharacteristic, complemented
123#830—34no—
124(C2×D3)⋊C8—34no—
125(C2×D3)⋊C8—34no—
126C23×C32—24no—
127C23×C32C4×D314yes—
128C23×C32C2×C3×C411yescharacteristic
129C22×C3⋊D3C2×C3×C411yescharacteristic
130C22×C3⋊D3C2×C3×C411yescharacteristic
131C22×C3×D3—34no—
132C22×C3×D3—34no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—44no—
140C2×C32×C4—44no—
141C2×C32×C4C4×D314yes—
142C2×C32×C4C2×C3×C411yescharacteristic
143C2×C32×C4C22×D314yes—
144C2×C32×C4C23×C311yescharacteristic
145#638—22no—
146#638—22no—
147C2×C3⋊Dic3C2×C3×C411yescharacteristic
148C2×C3⋊Dic3C2×C3×C411yescharacteristic
149C4⋊(C3⋊D3)—22no—
150C4×C3⋊D3—21no—
151C4×C3⋊D3—21no—
152C4×C3⋊D3C3×D412yes—
153C4×C3⋊D3C3×D412yes—
154#634—22no—
155C3×#67—68no—
156C3×#67—68no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—34no—
159C3×D12—68no—
160C3×C4×D3—38no—
161C3×C4×D3—38no—
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164C3×Dic6—68no—
165C32×C8—21no—
166C32×C8—21no—
167C32×C8—24no—
168C32×C8—24no—
169C32×C8—44no—
170C32×C8—44no—
171C3⋊(C3⋊C8)—21no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—64no—
174C3×C3⋊C8—64no—
175#408—91noSylow
176C2×C33—41no—
177C2×C33C2×C22⋊C411yescharacteristic
178C2×C33#13111yescharacteristic
179C2×C33#13111yescharacteristic
180C3×C3⋊D3—22no—
181C3×C3⋊D3—41no—
182C3×#41—182no—
183C3×#41—182no—
184C2×C3×Q8—91no—
185C2×C3×D4—91no—
186C2×C3×D4—91no—
187C2×C3×D4—91no—
188C22×C3×C4—24no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4C2×C3×D314yes—
192C22×C3×C4C2×C3⋊D311yescharacteristic
193C2×#67—34no—
194C2×#67—34no—
195#235—68no—
196#235—68no—
197C2×D12—34no—
198C2×C4×D3—34no—
199C2×C4×D3—34no—
200C2×Dic6—34no—
201C3×M16—182no—
202C2×C3×C8—24no—
203C2×C3×C8—24no—
204C2×C3×C8—91no—
205C2×C3×C8—91no—
206C2×C3×C8—182no—
207C2×C3×C8C2×C3×D314yes—
208C2×C3×C8C2×C3×D314yes—
209C2×C3×C8C2×C3⋊D311yescharacteristic
210C2×C3×C8C2×C3⋊D311yescharacteristic
211C2×C3⋊C8—34no—
212C2×C3⋊C8—34no—
213ΓL(1,𝔽25)—68no—
214D3×C8—68no—
215C22×C32—28no—
216C22×C32—24no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—24no—
223C22×C32—44no—
224C22×C32—44no—
225C22×C32C2×C4×D314yes—
226C22×C32C22×C3×C411yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—21no—
230C2×C3⋊D3—21no—
231C2×C3⋊D3—21no—
232C2×C3⋊D3C3×C22⋊C412yes—
233C2×C3×D3—38no—
234C2×C3×D3—64no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C32×C4—21no—
240C32×C4—24no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—44no—
244C32×C4D3×D414yes—
245C32×C4D3×D414yes—
246C32×C4C2×C3×D411yescharacteristic
247C32×C4C2×C3×D411yescharacteristic
248C3⋊Dic3—21no—
249C3⋊Dic3C3×C22⋊C412yes—
250C3×Dic3—38no—
251C3×Dic3—64no—
252C2×#41—91no—
253C2×M16—91no—
254C22×C8—91no—
255C22⋊C8—91no—
256C22⋊C8—91no—
257C22⋊C8—91no—
258C22⋊C8C3×C3⋊D311yescharacteristic, complemented
259C33#40811yescharacteristic, complemented, Sylow
260C23×C3—24no—
261C23×C3—91no—
262C23×C3—91no—
263C23×C3C3×C4×D314yes—
264C23×C3C4×C3⋊D311yescharacteristic
265C22×D3—34no—
266C22×D3—34no—
267C3×Q8—182no—
268C3×D4—182no—
269C3×D4—182no—
270C3×D4—182no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—91no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—91no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C3×C4×D314yes—
286C2×C3×C4C4×C3⋊D311yescharacteristic
287C2×C3×C4C22×C3×D314yes—
288C2×C3×C4C22×C3⋊D311yescharacteristic
289#67—68no—
290#67—68no—
291C2×Dic3—34no—
292C2×Dic3—34no—
293D12—68no—
294C4×D3—38no—
295C4×D3—38no—
296C4×D3—64no—
297C4×D3—64no—
298Dic6—68no—
299C3×C8—24no—
300C3×C8—24no—
301C3×C8—21no—
302C3×C8—21no—
303C3×C8—44no—
304C3×C8—44no—
305C3×C8—181no—
306C3×C8—181no—
307C3⋊C8—64no—
308C3⋊C8—64no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—24no—
312C2×C32—48no—
313C2×C32—44no—
314C2×C32—41no—
315C2×C32D3×C22⋊C414yes—
316C2×C32#90114yes—
317C2×C32#90714yes—
318C2×C32C2×C3×C22⋊C411yescharacteristic
319C2×C32C3×#13111yescharacteristic
320C2×C32C3×#13111yescharacteristic
321C3⋊D3—22no—
322C3⋊D3—41no—
323C3×D3—68no—
324C3×D3—124no—
325#41—182no—
326#41—182no—
327C2×Q8—91no—
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4C2×C3×C3⋊D311yescharacteristic
334M16—182no—
335C2×C8—91no—
336C2×C8—91no—
337C2×C8—182no—
338C2×C8C2×C3×C3⋊D311yescharacteristic
339C2×C8C2×C3×C3⋊D311yescharacteristic
340C22×C3—28no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—24no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—24no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—92no—
351C22×C3—181no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3—181no—
356C22×C3C2×C3×C4×D314yes—
357C22×C3C2×C4×C3⋊D311yescharacteristic
358C2×D3—38no—
359C2×D3—64no—
360C2×D3—64no—
361C2×D3—64no—
362C2×D3—64no—
363C2×D3—64no—
364C3×C4—24no—
365C3×C4—21no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—44no—
369C3×C4—92no—
370C3×C4—181no—
371C3×C4C3×D3×D414yes—
372C3×C4C3×D3×D414yes—
373C3×C4D4×C3⋊D311yescharacteristic
374C3×C4D4×C3⋊D311yescharacteristic
375Dic3—38no—
376Dic3—64no—
377C32—24no—
378C32#472714yescomplemented
379C32C3×#40811yescharacteristic, complemented
380C23—91no—
381C23—91no—
382C23C3×C4×C3⋊D311yescharacteristic
383Q8—182no—
384D4—182no—
385D4—182no—
386D4—182no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—91no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4C3×C4×C3⋊D311yescharacteristic
396C2×C4C22×C3×C3⋊D311yescharacteristic
397C8—21no—
398C8—21no—
399C8—181no—
400C8—181no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—24no—
404C2×C3—48no—
405C2×C3—44no—
406C2×C3—41no—
407C2×C3—182no—
408C2×C3—361no—
409C2×C3C3×D3×C22⋊C414yes—
410C2×C3C3×#90114yes—
411C2×C3C3×#90714yes—
412C2×C3C22⋊C4×C3⋊D311yescharacteristic
413C2×C3#6411011yescharacteristic
414C2×C3#6411611yescharacteristic
415D3—68no—
416D3—124no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—92no—
421C22—181no—
422C22—181no—
423C22—181no—
424C22—181no—
425C22—181no—
426C22C2×C3×C4×C3⋊D311yescharacteristic
427C4—21no—
428C4—92no—
429C4—181no—
430C4C3×D4×C3⋊D311yescharacteristic
431C4C3×D4×C3⋊D311yescharacteristic
432C3—24no—
433C3C3×#472714yescomplemented
434C3#1059242611yescharacteristic, direct product factor
435C2—41no—
436C2—182no—
437C2—361no—
438C2C3×C22⋊C4×C3⋊D311yescharacteristic
439C2C3×#6411011yescharacteristic
440C2C3×#6411611yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C22×C4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#408
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. C2×C3×C8×C3⋊D3C2semidirect
4. C33×C22⋊C8C2semidirect
5. C3×#63657C2semidirect
6. C3×#63647C2semidirect
7. C3×#63647C2semidirect
8. #10592426C3direct (N × H)
27. C2×#3743C2×C3semidirect
28. C2×C8×C3⋊D3C2×C3semidirect
32. C32×C22⋊C8D3semidirect
33. C32×C22⋊C8C2×C3semidirect
34. #63657C2×C3semidirect
35. #63647C2×C3semidirect
36. #63647C2×C3semidirect
121. C3×C22⋊C8C3×D3semidirect
122. C3×C22⋊C8C3⋊D3semidirect
258. C22⋊C8C3×C3⋊D3semidirect
259. C33#408semidirect
378. C32#4727semidirect
379. C32C3×#408semidirect
433. C3C3×#4727semidirect
434. C3#10592426direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×#37432
C2×C3×C8×C3⋊D32
C33×C22⋊C82
C3×#636572
C3×#636472
#105924263
C3×#47273