Finite Group Databasebeta

← C3×#70826 #11773889

Subgroups — C3×#70826 #11773889

Order 1152.

559 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic
2C32×#432C211yesmaximal, characteristic, complemented
3C3×#4926C211yesmaximal, characteristic, complemented
4C3×#4469C212yesmaximal, complemented
5C3×#4468C212yesmaximal
6C3×#1464—31nomaximal
7#70826C311yesmaximal, characteristic, direct product factor
8C22×C3×C4×D3C411yescharacteristic
9C2×C32×C42C2211yescharacteristic
10C2×C3×(C2×D3)⋊C4C411yescharacteristic
11C2×C3×(C2×D3)⋊C4C2211yescharacteristic
12C2×C3×Dic3⋊C4C411yescharacteristic
13C2×C3×Dic3⋊C4C2211yescharacteristic
14C2×C3×C4×Dic3C411yescharacteristic
15C3×#872—21no—
16C3×#872—21no—
17C3×#872—21no—
18C3×#872—21no—
19C32×C4⋊C8—21no—
20C32×C22⋊C8C2212yes—
21C32×#97—21no—
22C3×#830C2212yes—
23C3×C4⋊(C3⋊C8)—21no—
24C3×#803—21no—
25C2×C3×#117—31no—
26C2×#872C2×C311yescharacteristic
27C3×#432—21no—
28C3×#432—31no—
29C3×#432D311yescharacteristic, complemented
30C3×#432C2×C311yescharacteristic, complemented
31#4926C2×C311yescharacteristic, complemented
32C3×#324—32no—
33C3×#323—32no—
34#4469C2×C312yescomplemented
35#4468C2×C312yes—
36C23×C3×D3C2×C411yescharacteristic
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C22×C3×Dic3C2×C411yescharacteristic
41C22×C3×Dic3C2×C411yescharacteristic
42C22×C3×Dic3C2×C411yescharacteristic
43C2×C3×C4×D3—22no—
44C2×C3×C4×D3—21no—
45C2×C3×C4×D3—21no—
46C2×C3×C4×D3—22no—
47C2×C32×C8—22no—
48C32×C42D411yescharacteristic
49C32×C42D411yescharacteristic
50C32×C42D411yescharacteristic
51C32×C42D411yescharacteristic
52C3×(C2×D3)⋊C4—21no—
53C3×(C2×D3)⋊C4—21no—
54C3×(C2×D3)⋊C4—21no—
55C3×(C2×D3)⋊C4—21no—
56C3×Dic3⋊C4—21no—
57C3×Dic3⋊C4—21no—
58C3×Dic3⋊C4—21no—
59C3×Dic3⋊C4—21no—
60C3×C4×Dic3—21no—
61C3×C4×Dic3—21no—
62C2×C3×C3⋊C8—22no—
63#1464—31noSylow
64C23×C3×C4—31no—
65C22×C4×D3C3×C411yescharacteristic
66C3×#117—61no—
67C3×#117—61no—
68C3×#117—61no—
69C3×#117—61no—
70C2×C3×C4⋊C4—31no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C22⋊C4—31no—
73C2×C3×C22⋊C4—31no—
74C2×C3×C42—21no—
75C2×C3×C42—31no—
76C2×C3×C42C2×D311yescharacteristic
77C2×C3×C42C22×C311yescharacteristic
78C2×(C2×D3)⋊C4C3×C411yescharacteristic
79C2×(C2×D3)⋊C4C22×C311yescharacteristic
80C2×Dic3⋊C4C3×C411yescharacteristic
81C2×Dic3⋊C4C22×C311yescharacteristic
82C2×C4×Dic3C3×C411yescharacteristic
83#872—21no—
84#872—21no—
85#872—21no—
86#872—21no—
87C3×C4⋊C8—22no—
88C3×C4⋊C8—21no—
89C3×C4⋊C8—21no—
90C3×C4⋊C8—61no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—32no—
93C3×C22⋊C8C2×D312yes—
94C3×C22⋊C8C22×C312yes—
95C3×#97—22no—
96C3×#97—21no—
97C3×#97—21no—
98C3×#97—61no—
99#830C22×C312yes—
100C4⋊(C3⋊C8)—21no—
101#803—21no—
102C23×C32C22×C411yescharacteristic
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—41no—
106C22×C3×D3—41no—
107C22×C3×D3M1612yes—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4C22⋊C411yescharacteristic
113C2×C32×C4C22⋊C411yescharacteristic
114C2×C32×C4C22⋊C411yescharacteristic
115C2×C32×C4C22⋊C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3M1612yes—
125C3×C4×D3—24no—
126C3×C4×D3—24no—
127C32×C8—42no—
128C3×C3⋊C8—42no—
129C2×#117—31no—
130#432—31no—
131#432C3×D311yescharacteristic, complemented
132#324—32no—
133#323—32no—
134C24×C3—31no—
135C23×D3C2×C3×C411yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—31no—
140C22×C3×C4—31no—
141C22×C3×C4—31no—
142C22×C3×C4—62no—
143C22×C3×C4—62no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C22×C3×C4C4×D311yescharacteristic
147C22×C3×C4C4×D311yescharacteristic
148C22×C3×C4C2×C3×C411yescharacteristic
149C22×C3×C4C2×C3×C411yescharacteristic
150C22×C3×C4C22×D311yescharacteristic
151C22×C3×C4C23×C311yescharacteristic
152C22×Dic3C2×C3×C411yescharacteristic
153C22×Dic3C2×C3×C411yescharacteristic
154C22×Dic3C2×C3×C411yescharacteristic
155C2×C4×D3—22no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—22no—
159C2×C3×C8—24no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—62no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—61no—
169C3×C22⋊C4—61no—
170C3×C22⋊C4—61no—
171C3×C42—21no—
172C3×C42—21no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—61no—
176C3×C42—61no—
177C3×C42D1211yescharacteristic
178C3×C42D1211yescharacteristic
179C3×C42#6711yescharacteristic
180C3×C42#6711yescharacteristic
181C3×C42C3×D411yescharacteristic
182C3×C42C3×D411yescharacteristic
183C3×C42C3×D411yescharacteristic
184C3×C42C3×D411yescharacteristic
185(C2×D3)⋊C4—21no—
186(C2×D3)⋊C4—21no—
187(C2×D3)⋊C4—21no—
188(C2×D3)⋊C4—21no—
189Dic3⋊C4—21no—
190Dic3⋊C4—21no—
191Dic3⋊C4—21no—
192Dic3⋊C4—21no—
193C4×Dic3—21no—
194C4×Dic3—21no—
195C2×C3⋊C8—22no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2×C22⋊C411yescharacteristic
199C22×C32C2×M1611yescharacteristic
200C22×C32C2×M1611yescharacteristic
201C2×C3×D3—22no—
202C2×C3×D3—42no—
203C2×C3×D3—41no—
204C2×C3×D3—41no—
205C2×C3×D3—42no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C3×Dic3—22no—
213C3×Dic3—41no—
214C3×Dic3—41no—
215C23×C4—31no—
216#117—61no—
217#117—61no—
218#117—61no—
219#117—61no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—31no—
222C2×C22⋊C4—31no—
223C2×C22⋊C4—31no—
224C2×C42—31no—
225C2×C42C2×C3×D311yescharacteristic
226C4⋊C8—21no—
227C4⋊C8—61no—
228C22⋊C8—32no—
229C22⋊C8C2×C3×D312yes—
230#97—21no—
231#97—61no—
232C23×C3—21no—
233C23×C3—32no—
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C23×C3—121no—
238C23×C3C2×C4×D311yescharacteristic
239C23×C3C22×C3×C411yescharacteristic
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3C3×M1612yes—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—32no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—64no—
265C2×C3×C4—64no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4(C2×D3)⋊C411yescharacteristic
273C2×C3×C4(C2×D3)⋊C411yescharacteristic
274C2×C3×C4(C2×D3)⋊C411yescharacteristic
275C2×C3×C4(C2×D3)⋊C411yescharacteristic
276C2×C3×C4C3×C22⋊C411yescharacteristic
277C2×C3×C4C3×C22⋊C411yescharacteristic
278C2×C3×C4C3×C22⋊C411yescharacteristic
279C2×C3×C4C3×C22⋊C411yescharacteristic
280C2×C3×C4C2×D1211yescharacteristic
281C2×C3×C4C2×#6711yescharacteristic
282C2×C3×C4C2×C3×D411yescharacteristic
283C2×C3×C4C2×C3×D411yescharacteristic
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3C3×M1612yes—
291C4×D3—24no—
292C4×D3—24no—
293C3×C8—44no—
294C3×C8—42no—
295C3×C8—42no—
296C3×C8—122no—
297C3⋊C8—42no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32#40711yescharacteristic
301C2×C32#41011yescharacteristic
302C2×C32#41311yescharacteristic
303C3×D3—42no—
304C24—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4C3×C4×D311yescharacteristic
313C22×C4C3×C4×D311yescharacteristic
314C22×C4C22×C3×D311yescharacteristic
315C2×C8—22no—
316C2×C8—62no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C22⋊C4—61no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C42—61no—
326C42—61no—
327C42C3×D1211yescharacteristic
328C42C3×D1211yescharacteristic
329C42C3×#6711yescharacteristic
330C42C3×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—122no—
342C22×C3—122no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3D3×M1611yescharacteristic
346C22×C3D3×M1611yescharacteristic
347C22×C3C2×(C2×D3)⋊C411yescharacteristic
348C22×C3C2×C3×C22⋊C411yescharacteristic
349C22×C3C2×C3×M1611yescharacteristic
350C22×C3C2×C3×M1611yescharacteristic
351C2×D3—22no—
352C2×D3—42no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—42no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—62no—
375C3×C4—121no—
376C3×C4—121no—
377Dic3—22no—
378Dic3—41no—
379Dic3—41no—
380C32#146411yescharacteristic, complemented, Sylow
381C23—32no—
382C23—61no—
383C23—61no—
384C23—121no—
385C23—121no—
386C23C2×C3×C4×D311yescharacteristic
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—32no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4C3×(C2×D3)⋊C411yescharacteristic
401C2×C4C3×(C2×D3)⋊C411yescharacteristic
402C2×C4C3×(C2×D3)⋊C411yescharacteristic
403C2×C4C3×(C2×D3)⋊C411yescharacteristic
404C2×C4C2×C3×D1211yescharacteristic
405C2×C4C2×C3×#6711yescharacteristic
406C8—42no—
407C8—122no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—122no—
418C2×C3#500611yescharacteristic
419C2×C3#512611yescharacteristic
420C2×C3#513211yescharacteristic
421C2×C3C3×#40711yescharacteristic
422C2×C3C3×#41011yescharacteristic
423C2×C3C3×#41311yescharacteristic
424D3—42no—
425C22—21no—
426C22—21no—
427C22—62no—
428C22—122no—
429C22—122no—
430C22—121no—
431C22—121no—
432C22C3×D3×M1611yescharacteristic
433C22C3×D3×M1611yescharacteristic
434C22C2×C3×(C2×D3)⋊C411yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—62no—
442C4—121no—
443C4—121no—
444C3—21no—
445C3#7082611yescharacteristic, direct product factor
446C3C3×#146411yescharacteristic, complemented
447C2—21no—
448C2—21no—
449C2—122no—
450C2C3×#500611yescharacteristic
451C2C3×#512611yescharacteristic
452C2C3×#513211yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#432C2
socleC22×C32C2×C22⋊C4
prime radicalC23×C3×D3C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC3×#432D3
hypocenterC3C3×#1464
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×#432

Complements

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

NH ≅ G/Ntype
2. C32×#432C2semidirect
3. C3×#4926C2semidirect
4. C3×#4469C2semidirect
7. #70826C3direct (N × H)
29. C3×#432D3semidirect
30. C3×#432C2×C3semidirect
31. #4926C2×C3semidirect
34. #4469C2×C3semidirect
131. #432C3×D3semidirect
380. C32#1464semidirect
445. C3#70826direct (N × H)
446. C3C3×#1464semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C32×#4322
C3×#49262
C3×#44692
C3×#44682
C3×#14643
#708263