Finite Group Databasebeta

← C3×#71746 #11775706

Subgroups — C3×#71746 #11775706

Order 1152.

585 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#529C211yesmaximal, characteristic
2C3×C4×#208C211yesmaximal, characteristic, complemented
3C2×C3×#803C211yesmaximal, characteristic, complemented
4C3×#4543C212yesmaximal, complemented
5C3×#4540C212yesmaximal, complemented
6C3×#1524—31nomaximal
7#71746C311yesmaximal, characteristic, direct product factor
8C32×#126—21no—
9C32×#125C2211yescharacteristic
10C32×#124C2211yescharacteristic
11C32×#123—21no—
12C32×C22⋊Q8C411yescharacteristic
13C32×#121C411yescharacteristic
14C32×C4×Q8C411yescharacteristic
15C32×C4×D4—21no—
16C32×C4×D4C411yescharacteristic
17C32×#117—21no—
18C2×C32×C42C2211yescharacteristic
19C2×C3×#208—21no—
20C22×C3×C3⋊C8—21no—
21C3×#803—21no—
22C3×#803C2212yes—
23C3×#803C2212yes—
24C3×C4×C3⋊C8—21no—
25C3×#529—21no—
26C3×#529D311yescharacteristic
27C3×#529C2×C311yescharacteristic
28C3×C4×M16—31no—
29C2×C3×#97—31no—
30C4×#208C2×C311yescharacteristic, complemented
31C2×#803C2×C311yescharacteristic, complemented
32C3×#330—32no—
33C3×#329—32no—
34#4543C2×C312yescomplemented
35#4540C2×C312yescomplemented
36C2×C32×Q8C2×C411yescharacteristic
37C2×C32×D4—21no—
38C2×C32×D4C2×C411yescharacteristic
39C22×C32×C4—21no—
40C22×C32×C4C2×C411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4C2×C411yescharacteristic
48C32×C4⋊C4C2×C411yescharacteristic
49C32×C22⋊C4—21no—
50C32×C22⋊C4—21no—
51C32×C22⋊C4—21no—
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C42—21no—
55C32×C42C2×C411yescharacteristic
56C32×C42D411yescharacteristic
57C32×C42D411yescharacteristic
58C32×C42C2311yescharacteristic
59C3×#208—24no—
60C2×C3×C3⋊C8—24no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8—22no—
63#1524—31noSylow
64C2×C3×M16—61no—
65C22×C3×C8—61no—
66C3×#126—22no—
67C3×#126—21no—
68C3×#126—21no—
69C3×#125—21no—
70C3×#125C2×D311yescharacteristic
71C3×#125C22×C311yescharacteristic
72C3×#124—21no—
73C3×#124C2×D311yescharacteristic
74C3×#124C22×C311yescharacteristic
75C3×#123—22no—
76C3×#123—21no—
77C3×#123—21no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8Dic311yescharacteristic
80C3×C22⋊Q8C3×C411yescharacteristic
81C3×#121—21no—
82C3×#121Dic311yescharacteristic
83C3×#121C3×C411yescharacteristic
84C3×C4×Q8—21no—
85C3×C4×Q8Dic311yescharacteristic
86C3×C4×Q8C3×C411yescharacteristic
87C3×C4×D4—22no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4Dic311yescharacteristic
92C3×C4×D4C3×C411yescharacteristic
93C3×#117—22no—
94C3×#117—21no—
95C3×#117—21no—
96C2×C3×C42—21no—
97C2×C3×C42C2×D311yescharacteristic
98C2×C3×C42C22×C311yescharacteristic
99C2×#208—21no—
100C22×C3⋊C8—21no—
101C3×#97—32no—
102C3×#97—32no—
103C3×#97—61no—
104C3×C4×C8—61no—
105#803—21no—
106#803C22×C312yes—
107#803C22×C312yes—
108C4×C3⋊C8—21no—
109C23×C32—21no—
110C23×C32C22⋊C411yescharacteristic
111C32×Q8—41no—
112C32×D4—41no—
113C32×D4—42no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—41no—
120C2×C32×C4—42no—
121C2×C32×C4C22⋊C411yescharacteristic
122C2×C32×C4C22⋊C411yescharacteristic
123C2×C32×C4C22⋊C411yescharacteristic
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C3×C3⋊C8—24no—
128C3×C3⋊C8—24no—
129#529C3×D311yescharacteristic
130C4×M16—31no—
131C2×#97—31no—
132#330—32no—
133#329—32no—
134C2×C3×Q8—21no—
135C2×C3×Q8C2×Dic311yescharacteristic
136C2×C3×Q8C2×C3×C411yescharacteristic
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4C2×Dic311yescharacteristic
142C2×C3×D4C2×C3×C411yescharacteristic
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4C2×Dic311yescharacteristic
150C22×C3×C4#6711yescharacteristic
151C22×C3×C4#6711yescharacteristic
152C22×C3×C4C2×C3×C411yescharacteristic
153C22×C3×C4C3×D411yescharacteristic
154C22×C3×C4C3×D411yescharacteristic
155C3×M16—64no—
156C2×C3×C8—64no—
157C2×C3×C8—62no—
158C2×C3×C8—62no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4C2×Dic311yescharacteristic
174C3×C4⋊C4C2×Dic311yescharacteristic
175C3×C4⋊C4C2×C3×C411yescharacteristic
176C3×C4⋊C4C2×C3×C411yescharacteristic
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—21no—
192C3×C42—22no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42C2×Dic311yescharacteristic
200C3×C42#6711yescharacteristic
201C3×C42#6711yescharacteristic
202C3×C42C2×C3×C411yescharacteristic
203C3×C42C3×D411yescharacteristic
204C3×C42C3×D411yescharacteristic
205C3×C42C22×D311yescharacteristic
206C3×C42C23×C311yescharacteristic
207#208—24no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32#10412yes—
217C22×C32C2×C22⋊C411yescharacteristic
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4#10412yes—
227C2×M16—61no—
228C22×C8—61no—
229#126—21no—
230#125C2×C3×D311yescharacteristic
231#124C2×C3×D311yescharacteristic
232#123—21no—
233C22⋊Q8C3×Dic311yescharacteristic
234#121C3×Dic311yescharacteristic
235C4×Q8C3×Dic311yescharacteristic
236C4×D4—21no—
237C4×D4C3×Dic311yescharacteristic
238#117—21no—
239C2×C42C2×C3×D311yescharacteristic
240#97—32no—
241#97—32no—
242#97—61no—
243C4×C8—61no—
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3#21711yescharacteristic
249C23×C3C3×C22⋊C411yescharacteristic
250C3×Q8—41no—
251C3×Q8—41no—
252C3×Q8—42no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—42no—
256C3×D4—44no—
257C3×D4—42no—
258C3×D4—42no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—44no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#21711yescharacteristic
287C2×C3×C4#21711yescharacteristic
288C2×C3×C4#21711yescharacteristic
289C2×C3×C4C3×C22⋊C411yescharacteristic
290C2×C3×C4C3×C22⋊C411yescharacteristic
291C2×C3×C4C3×C22⋊C411yescharacteristic
292C2×C3×C4C22×Dic311yescharacteristic
293C2×C3×C4C2×#6711yescharacteristic
294C2×C3×C4C2×#6711yescharacteristic
295C2×C3×C4C22×C3×C411yescharacteristic
296C2×C3×C4C2×C3×D411yescharacteristic
297C2×C3×C4C2×C3×D411yescharacteristic
298C3×C8—64no—
299C3×C8—64no—
300C3⋊C8—24no—
301C3⋊C8—24no—
302C2×C32—22no—
303C2×C32—81no—
304C2×C32#41311yescharacteristic
305C2×C32C2×#10411yescharacteristic
306C2×C32C2×#10411yescharacteristic
307C2×Q8C2×C3×Dic311yescharacteristic
308C2×D4—21no—
309C2×D4C2×C3×Dic311yescharacteristic
310C22×C4—21no—
311C22×C4C2×C3×Dic311yescharacteristic
312C22×C4C3×#6711yescharacteristic
313C22×C4C3×#6711yescharacteristic
314M16—64no—
315C2×C8—64no—
316C2×C8—62no—
317C2×C8—62no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4C2×C3×Dic311yescharacteristic
323C4⋊C4C2×C3×Dic311yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C42—21no—
330C42C2×C3×Dic311yescharacteristic
331C42C3×#6711yescharacteristic
332C42C3×#6711yescharacteristic
333C42C22×C3×D311yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3#83712yes—
352C22×C3C3×#10412yes—
353C22×C3C2×#21711yescharacteristic
354C22×C3C2×C3×C22⋊C411yescharacteristic
355C3×C4—22no—
356C3×C4—21no—
357C3×C4—21no—
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—22no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4#83712yes—
381C3×C4C3×#10412yes—
382C32#152411yescharacteristic, complemented, Sylow
383C23—21no—
384C23C3×#21711yescharacteristic
385Q8—41no—
386D4—41no—
387D4—42no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4C3×#21711yescharacteristic
396C2×C4C3×#21711yescharacteristic
397C2×C4C3×#21711yescharacteristic
398C2×C4C22×C3×Dic311yescharacteristic
399C2×C4C2×C3×#6711yescharacteristic
400C2×C4C2×C3×#6711yescharacteristic
401C8—64no—
402C8—64no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—42no—
409C2×C3—82no—
410C2×C3—81no—
411C2×C3—81no—
412C2×C3C2×#83711yescharacteristic
413C2×C3C2×#83711yescharacteristic
414C2×C3#523711yescharacteristic
415C2×C3C3×#41311yescharacteristic
416C2×C3C2×C3×#10411yescharacteristic
417C2×C3C2×C3×#10411yescharacteristic
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22C3×#83712yes—
424C22C2×C3×#21711yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4C3×#83712yes—
434C3—21no—
435C3#7174611yescharacteristic, direct product factor
436C3C3×#152411yescharacteristic, complemented
437C2—22no—
438C2—81no—
439C2C2×C3×#83711yescharacteristic
440C2C2×C3×#83711yescharacteristic
441C2C3×#523711yescharacteristic
subgroupG/H
centerC2×C3×C4#217
commutatorC2×C3×C4C22×C3×C4
FrattiniC42C22×C3×D3
FittingC32×#529C2
socleC22×C32C2×C22⋊C4
prime radicalC32×#121C4
cosocleC3×C42C23×C3
hypercenterC3×#529D3
hypocenterC3C3×#1524
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C3×#529

Complements

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

NH ≅ G/Ntype
2. C3×C4×#208C2semidirect
3. C2×C3×#803C2semidirect
4. C3×#4543C2semidirect
5. C3×#4540C2semidirect
7. #71746C3direct (N × H)
30. C4×#208C2×C3semidirect
31. C2×#803C2×C3semidirect
34. #4543C2×C3semidirect
35. #4540C2×C3semidirect
382. C32#1524semidirect
435. C3#71746direct (N × H)
436. C3C3×#1524semidirect

Maximal subgroups

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

subgroupindex
C32×#5292
C3×C4×#2082
C2×C3×#8032
C3×#45432
C3×#45402
C3×#15243
#717463