Finite Group Databasebeta

← C3×#69531 #11775289

Subgroups — C3×#69531 #11775289

Order 1152.

761 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#930C211yesmaximal, characteristic
2C32×#381C211yesmaximal, characteristic, complemented
3C3×#5207C211yesmaximal, characteristic, complemented
4C3×#4451C214yesmaximal, complemented
5C3×#1510—31nomaximal
6#69531C311yesmaximal, characteristic, direct product factor
7C23×C32×C4C2211yescharacteristic
8C22×C3×#67—21no—
9C22×C3×D12C411yescharacteristic, complemented
10C2×C32×C22⋊C4—21no—
11C3×#930—21no—
12C3×#930—21no—
13C3×#930C412yescomplemented
14C3×#930C2212yes—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×C4⋊Dic3C411yescharacteristic, complemented
17C2×C3×#208—21no—
18C32×#95—21no—
19C32×#95C2212yescomplemented
20C3×#830—21no—
21C3×#830C2212yescomplemented
22C2×C3×#121—31no—
23C2×#930C2×C311yescharacteristic
24C3×#407—31no—
25C3×#381—21no—
26C3×#381D311yescharacteristic, complemented
27C3×#381C2×C311yescharacteristic, complemented
28#5207C2×C311yescharacteristic, complemented
29C3×#327—34no—
30#4451C2×C314yescomplemented
31C24×C32D411yescharacteristic
32C23×C3×D3—21no—
33C22×C32×C4—22no—
34C22×C32×C4—42no—
35C22×C32×C4C2×C411yescharacteristic
36C22×C32×C4C2×C411yescharacteristic
37C22×C32×C4D411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D411yescharacteristic
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—22no—
42C2×C3×#67—22no—
43C2×C3×#67—44no—
44C22×C3×Dic3—21no—
45C2×C3×D12—21no—
46C2×C3×D12—42no—
47C2×C3×D12C2×C412yes—
48C32×C22⋊C4—42no—
49C3×(C2×D3)⋊C4—22no—
50C3×(C2×D3)⋊C4—22no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3C2×C412yes—
53C3×#208—42no—
54C2×C3×C3⋊C8—22no—
55#1510—31noSylow
56C22×C3×D4—31no—
57C22×C3×D4—61no—
58C23×C3×C4—21no—
59C23×C3×C4C2×D311yescharacteristic
60C23×C3×C4C22×C311yescharacteristic
61C22×#67—21no—
62C22×D12C3×C411yescharacteristic, complemented
63C2×C3×M16—61no—
64C3×#121—32no—
65C3×#121—32no—
66C3×#121—61no—
67C3×#121—61no—
68C2×C3×C4⋊C4—31no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—61no—
73#930—21no—
74#930—21no—
75#930C3×C412yescomplemented
76#930C22×C312yes—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3C3×C411yescharacteristic, complemented
79C2×#208—21no—
80C3×C22⋊C8—32no—
81C3×C22⋊C8—61no—
82C3×#95—22no—
83C3×#95—21no—
84C3×#95—21no—
85C3×#95—22no—
86C3×#95C2×D312yescomplemented
87C3×#95C22×C312yescomplemented
88#830—21no—
89#830C22×C312yescomplemented
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—41no—
93C23×C32C22⋊C411yescharacteristic
94C23×C32C22⋊C411yescharacteristic
95C23×C32C2×D411yescharacteristic
96C22×C3×D3—22no—
97C22×C3×D3—42no—
98C22×C3×D3—42no—
99C22×C3×D3—42no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—41no—
105C2×C32×C4—41no—
106C2×C32×C4—42no—
107C2×C32×C4—42no—
108C2×C32×C4—42no—
109C2×C32×C4C22⋊C411yescharacteristic
110C2×C32×C4C22⋊C411yescharacteristic
111C2×C32×C4C22×C411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C3×#67—44no—
114C3×#67—44no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—42no—
117C3×D12—42no—
118C3×D12—42no—
119C3×C3⋊C8—42no—
120C2×#121—31no—
121#407—31no—
122#381C3×D311yescharacteristic, complemented
123#327—34no—
124C24×C3—21no—
125C24×C3—61no—
126C24×C3#6711yescharacteristic
127C24×C3C3×D411yescharacteristic
128C23×D3—21no—
129C2×C3×D4—32no—
130C2×C3×D4—62no—
131C2×C3×D4—62no—
132C2×C3×D4—61no—
133C2×C3×D4—122no—
134C2×C3×D4—124no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—44no—
147C22×C3×C4—61no—
148C22×C3×C4C4×D311yescharacteristic
149C22×C3×C4C4×D311yescharacteristic
150C22×C3×C4D1211yescharacteristic
151C22×C3×C4D1211yescharacteristic
152C22×C3×C4#6711yescharacteristic
153C22×C3×C4C2×C3×C411yescharacteristic
154C22×C3×C4C2×C3×C411yescharacteristic
155C22×C3×C4C3×D411yescharacteristic
156C22×C3×C4C3×D411yescharacteristic
157C22×C3×C4C3×D411yescharacteristic
158C22×C3×C4C22×D311yescharacteristic
159C22×C3×C4C23×C311yescharacteristic
160C2×#67—22no—
161C2×#67—22no—
162C2×#67—44no—
163C22×Dic3—21no—
164C2×D12—21no—
165C2×D12—42no—
166C2×D12C2×C3×C412yes—
167C3×M16—122no—
168C2×C3×C8—62no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—61no—
171C3×C22⋊C4—44no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—62no—
175C3×C22⋊C4—62no—
176(C2×D3)⋊C4—22no—
177(C2×D3)⋊C4—22no—
178C4⋊Dic3—21no—
179C4⋊Dic3C2×C3×C412yes—
180#208—42no—
181C2×C3⋊C8—22no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—44no—
190C22×C32#9912yes—
191C22×C32C2×C22⋊C411yescharacteristic
192C22×C32AGL(1,ℤ/8)12yes—
193C2×C3×D3—42no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—84no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—82no—
202C3×Dic3—42no—
203C22×D4—31no—
204C22×D4—61no—
205C23×C4C2×C3×D311yescharacteristic
206C2×M16—61no—
207#121—32no—
208#121—32no—
209#121—61no—
210#121—61no—
211C2×C4⋊C4—31no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—61no—
214C22⋊C8—32no—
215C22⋊C8—61no—
216#95—21no—
217#95C2×C3×D312yescomplemented
218C23×C3—24no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—22no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—42no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—62no—
231C23×C3—122no—
232C23×C3—122no—
233C23×C3—122no—
234C23×C3(C2×D3)⋊C411yescharacteristic
235C23×C3(C2×D3)⋊C411yescharacteristic
236C23×C3C3×C22⋊C411yescharacteristic
237C23×C3C3×C22⋊C411yescharacteristic
238C23×C3C2×#6711yescharacteristic
239C23×C3C2×C3×D411yescharacteristic
240C22×D3—22no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C3×D4—122no—
245C3×D4—122no—
246C3×D4—124no—
247C3×D4—124no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—44no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—62no—
280C2×C3×C4—122no—
281C2×C3×C4(C2×D3)⋊C411yescharacteristic
282C2×C3×C4(C2×D3)⋊C411yescharacteristic
283C2×C3×C4C3×C22⋊C411yescharacteristic
284C2×C3×C4C3×C22⋊C411yescharacteristic
285C2×C3×C4C2×C4×D311yescharacteristic
286C2×C3×C4C2×D1211yescharacteristic
287C2×C3×C4C22×C3×C411yescharacteristic
288C2×C3×C4C2×C3×D411yescharacteristic
289#67—44no—
290#67—44no—
291C2×Dic3—22no—
292C2×Dic3—42no—
293D12—42no—
294D12—42no—
295C3×C8—122no—
296C3⋊C8—42no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—41no—
300C2×C32C2×#9911yescharacteristic
301C2×C32#41811yescharacteristic
302C2×C32#42111yescharacteristic
303C3×D3—82no—
304C24—61no—
305C24C3×#6711yescharacteristic
306C2×D4—32no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—61no—
310C2×D4—122no—
311C2×D4—124no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—61no—
315C22×C4C3×C4×D311yescharacteristic
316C22×C4C3×C4×D311yescharacteristic
317C22×C4C3×D1211yescharacteristic
318C22×C4C3×D1211yescharacteristic
319C22×C4C3×#6711yescharacteristic
320C22×C4C22×C3×D311yescharacteristic
321M16—122no—
322C2×C8—62no—
323C4⋊C4—32no—
324C4⋊C4—61no—
325C22⋊C4—42no—
326C22⋊C4—62no—
327C22⋊C4—62no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—48no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—244no—
359C22×C3#80612yes—
360C22×C3C3×#9912yes—
361C22×C3C2×(C2×D3)⋊C411yescharacteristic
362C22×C3#94912yes—
363C22×C3C2×C3×C22⋊C411yescharacteristic
364C22×C3C3×AGL(1,ℤ/8)12yes—
365C2×D3—42no—
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—84no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—84no—
382C3×C4—82no—
383C3×C4—82no—
384C3×C4—122no—
385Dic3—42no—
386C32#151011yescharacteristic, complemented, Sylow
387C23—22no—
388C23—22no—
389C23—41no—
390C23—62no—
391C23—122no—
392C23—122no—
393C23—122no—
394C23C3×(C2×D3)⋊C411yescharacteristic
395C23C3×(C2×D3)⋊C411yescharacteristic
396C23C2×C3×#6711yescharacteristic
397D4—122no—
398D4—122no—
399D4—124no—
400D4—124no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—62no—
411C2×C4—122no—
412C2×C4C3×(C2×D3)⋊C411yescharacteristic
413C2×C4C3×(C2×D3)⋊C411yescharacteristic
414C2×C4C2×C3×C4×D311yescharacteristic
415C2×C4C2×C3×D1211yescharacteristic
416C8—122no—
417C2×C3—24no—
418C2×C3—22no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—42no—
426C2×C3—42no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—242no—
430C2×C3C2×#80611yescharacteristic
431C2×C3#500111yescharacteristic
432C2×C3#500511yescharacteristic
433C2×C3C2×C3×#9911yescharacteristic
434C2×C3C3×#41811yescharacteristic
435C2×C3C3×#42111yescharacteristic
436D3—82no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—44no—
445C22—122no—
446C22—122no—
447C22—122no—
448C22—244no—
449C22C3×#80612yes—
450C22C2×C3×(C2×D3)⋊C411yescharacteristic
451C22C3×#94912yes—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—82no—
457C4—122no—
458C3—21no—
459C3#6953111yescharacteristic, direct product factor
460C3C3×#151011yescharacteristic, complemented
461C2—22no—
462C2—22no—
463C2—41no—
464C2—242no—
465C2C2×C3×#80611yescharacteristic
466C2C3×#500111yescharacteristic
467C2C3×#500511yescharacteristic
subgroupG/H
centerC22×C3C2×(C2×D3)⋊C4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#381C2
socleC22×C32C2×C22⋊C4
prime radicalC2×C3×#930C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#381D3
hypocenterC3C3×#1510
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#381

Complements

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

NH ≅ G/Ntype
2. C32×#381C2semidirect
3. C3×#5207C2semidirect
4. C3×#4451C2semidirect
6. #69531C3direct (N × H)
9. C22×C3×D12C4semidirect
13. C3×#930C4semidirect
16. C2×C3×C4⋊Dic3C4semidirect
19. C32×#95C22semidirect
21. C3×#830C22semidirect
26. C3×#381D3semidirect
27. C3×#381C2×C3semidirect
28. #5207C2×C3semidirect
30. #4451C2×C3semidirect
62. C22×D12C3×C4semidirect
75. #930C3×C4semidirect
78. C2×C4⋊Dic3C3×C4semidirect
86. C3×#95C2×D3semidirect
87. C3×#95C22×C3semidirect
89. #830C22×C3semidirect
122. #381C3×D3semidirect
217. #95C2×C3×D3semidirect
386. C32#1510semidirect
459. C3#69531direct (N × H)
460. C3C3×#1510semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9302
C32×#3812
C3×#52072
C3×#44512
C3×#15103
#695313