Finite Group Databasebeta

← C3×#71870 #11796293

Subgroups — C3×#71870 #11796293

Order 1152.

603 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#536C211yesmaximal, characteristic
2C3×#5240C211yesmaximal, characteristic
3C3×#5237C211yesmaximal, characteristic
4C3×#4539C214yesmaximal, complemented
5C3×#2123—31nomaximal
6#71870C311yesmaximal, characteristic, direct product factor
7C2×C32×#41C2211yescharacteristic
8C22×C32×Q8C411yescharacteristic, complemented
9C32×C4⋊Q8—21no—
10C32×#124—21no—
11C32×#123C412yescomplemented
12C32×#123C2212yes—
13C32×C22⋊Q8—21no—
14C32×C22⋊Q8—21no—
15C32×#117C411yescharacteristic, complemented
16C3×#926—21no—
17C2×C3×#208—21no—
18C3×#836—21no—
19C3×#834—21no—
20C3×#834C2212yes—
21C3×#833C2212yes—
22C3×#536—21no—
23C3×#536D311yescharacteristic
24C3×#536C2×C311yescharacteristic
25C3×#413—31no—
26C3×#410—31no—
27#5240C2×C311yescharacteristic
28#5237C2×C311yescharacteristic
29C3×#352—34no—
30#4539C2×C314yescomplemented
31C32×#41—41no—
32C32×#41—41no—
33C2×C32×Q8—21no—
34C2×C32×Q8—21no—
35C2×C32×Q8—41no—
36C2×C32×Q8—41no—
37C2×C32×Q8D411yescharacteristic
38C2×C32×D4C2×C411yescharacteristic
39C2×C32×D4D411yescharacteristic
40C2×C32×D4C2311yescharacteristic
41C22×C32×C4C2×C411yescharacteristic
42C22×C32×C4C2×C412yes—
43C22×C32×C4D411yescharacteristic
44C22×C32×C4D411yescharacteristic
45C32×C4⋊C4—21no—
46C32×C4⋊C4—22no—
47C32×C4⋊C4—22no—
48C32×C22⋊C4—22no—
49C32×C22⋊C4—22no—
50C32×C22⋊C4C2×C412yes—
51C32×C42—21no—
52C3×#217—22no—
53C3×C4⋊Dic3—41no—
54C3×C4×Dic3—41no—
55C3×#208—22no—
56C3×#208—41no—
57C2×C3×C3⋊C8—41no—
58#2123—31noSylow
59C2×C3×#41—21no—
60C2×C3×#41C2×D311yescharacteristic
61C2×C3×#41C22×C311yescharacteristic
62C22×C3×Q8—21no—
63C22×C3×Q8Dic311yescharacteristic, complemented
64C22×C3×Q8C3×C411yescharacteristic, complemented
65C2×C3×M16—61no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×#124—22no—
70C3×#124—21no—
71C3×#124—21no—
72C3×#123—22no—
73C3×#123—22no—
74C3×#123Dic312yescomplemented
75C3×#123C3×C412yescomplemented
76C3×#123C2×D312yes—
77C3×#123C22×C312yes—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—21no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×#117—21no—
85C3×#117—61no—
86C3×#117Dic311yescharacteristic, complemented
87C3×#117C3×C411yescharacteristic, complemented
88#926—21no—
89C2×#208—21no—
90C3×#101—61no—
91C3×#100—32no—
92C3×#99—32no—
93C3×#99—61no—
94#836—21no—
95#834—21no—
96#834C22×C312yes—
97#833C22×C312yes—
98C23×C32C22⋊C411yescharacteristic
99C23×C32C22×C411yescharacteristic
100C23×C32C2×D411yescharacteristic
101C32×Q8—41no—
102C32×Q8—42no—
103C32×Q8—41no—
104C32×Q8—41no—
105C32×D4—41no—
106C32×D4—41no—
107C32×D4—41no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—42no—
117C2×C32×C4C22⋊C411yescharacteristic
118C2×C32×C4C22⋊C411yescharacteristic
119C2×C32×C4C22⋊C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C3×Dic3—42no—
122C3×C3⋊C8—42no—
123#536C3×D311yescharacteristic
124#413—31no—
125#410—31no—
126#352—34no—
127C3×#41—41no—
128C3×#41—41no—
129C3×#41—41no—
130C3×#41—41no—
131C3×#41—42no—
132C3×#41—42no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—41no—
141C2×C3×Q8—41no—
142C2×C3×Q8—41no—
143C2×C3×Q8—41no—
144C2×C3×Q8—42no—
145C2×C3×Q8—42no—
146C2×C3×Q8#6711yescharacteristic
147C2×C3×Q8C3×D411yescharacteristic
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4C2×Dic311yescharacteristic
152C2×C3×D4#6711yescharacteristic
153C2×C3×D4C2×C3×C411yescharacteristic
154C2×C3×D4C3×D411yescharacteristic
155C2×C3×D4C22×D311yescharacteristic
156C2×C3×D4C23×C311yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×Dic311yescharacteristic
162C22×C3×C4C2×Dic312yes—
163C22×C3×C4#6711yescharacteristic
164C22×C3×C4#6711yescharacteristic
165C22×C3×C4C2×C3×C411yescharacteristic
166C22×C3×C4C2×C3×C412yes—
167C22×C3×C4C3×D411yescharacteristic
168C22×C3×C4C3×D411yescharacteristic
169C3×M16—62no—
170C3×M16—121no—
171C2×C3×C8—121no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—121no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C22⋊C4—22no—
189C3×C22⋊C4—62no—
190C3×C22⋊C4C2×Dic312yes—
191C3×C22⋊C4C2×C3×C412yes—
192C3×C42—22no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—121no—
196#217—22no—
197C4⋊Dic3—41no—
198C4×Dic3—41no—
199#208—22no—
200#208—41no—
201C2×C3⋊C8—41no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32C2×C22⋊C411yescharacteristic
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—42no—
213C32×C4—42no—
214C32×C4#9911yescharacteristic
215C32×C4#9911yescharacteristic
216C3×Dic3—82no—
217C2×#41C2×C3×D311yescharacteristic
218C22×Q8C3×Dic311yescharacteristic, complemented
219C2×M16—61no—
220C4⋊Q8—21no—
221#124—21no—
222#123C3×Dic312yescomplemented
223#123C2×C3×D312yes—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226#117—61no—
227#117C3×Dic311yescharacteristic, complemented
228#101—61no—
229#100—32no—
230#99—32no—
231#99—61no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3#21711yescharacteristic
236C23×C3C3×C22⋊C411yescharacteristic
237C23×C3C22×Dic311yescharacteristic
238C23×C3C2×#6711yescharacteristic
239C23×C3C22×C3×C411yescharacteristic
240C23×C3C2×C3×D411yescharacteristic
241C3×Q8—41no—
242C3×Q8—41no—
243C3×Q8—44no—
244C3×Q8—42no—
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×Q8—41no—
252C3×Q8—41no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—42no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
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—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—122no—
294C2×C3×C4#21711yescharacteristic
295C2×C3×C4#21711yescharacteristic
296C2×C3×C4#21711yescharacteristic
297C2×C3×C4C3×C22⋊C411yescharacteristic
298C2×C3×C4C3×C22⋊C411yescharacteristic
299C2×C3×C4C3×C22⋊C411yescharacteristic
300C2×C3×C4C2×#6711yescharacteristic
301C2×C3×C4C2×C3×D411yescharacteristic
302C2×Dic3—42no—
303C3×C8—122no—
304C3⋊C8—42no—
305C2×C32—21no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32C2×#9911yescharacteristic
310#41—41no—
311#41—41no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8C3×#6711yescharacteristic
317C2×D4C2×C3×Dic311yescharacteristic
318C2×D4C3×#6711yescharacteristic
319C2×D4C22×C3×D311yescharacteristic
320C22×C4C2×C3×Dic311yescharacteristic
321C22×C4C2×C3×Dic312yes—
322C22×C4C3×#6711yescharacteristic
323C22×C4C3×#6711yescharacteristic
324M16—62no—
325M16—121no—
326C2×C8—121no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—121no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—62no—
334C22⋊C4C2×C3×Dic312yes—
335C42—21no—
336C42—121no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3C2×#21711yescharacteristic
357C22×C3C2×C3×C22⋊C411yescharacteristic
358C3×C4—22no—
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—41no—
368C3×C4—41no—
369C3×C4—44no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—42no—
375C3×C4—242no—
376C3×C4#83411yescharacteristic
377C3×C4#83411yescharacteristic
378C3×C4C3×#9911yescharacteristic
379C3×C4C3×#9911yescharacteristic
380Dic3—82no—
381C32#212311yescharacteristic, complemented, Sylow
382C23C3×#21711yescharacteristic
383C23C22×C3×Dic311yescharacteristic
384C23C2×C3×#6711yescharacteristic
385Q8—41no—
386Q8—42no—
387Q8—41no—
388Q8—41no—
389D4—41no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—122no—
402C2×C4C3×#21711yescharacteristic
403C2×C4C3×#21711yescharacteristic
404C2×C4C3×#21711yescharacteristic
405C2×C4C2×C3×#6711yescharacteristic
406C8—122no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—42no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—81no—
419C2×C3—81no—
420C2×C3C2×#83411yescharacteristic
421C2×C3C2×C3×#9911yescharacteristic
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22C2×C3×#21711yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—42no—
433C4—42no—
434C4—242no—
435C4C3×#83411yescharacteristic
436C4C3×#83411yescharacteristic
437C3—21no—
438C3#7187011yescharacteristic, direct product factor
439C3C3×#212311yescharacteristic, complemented
440C2—21no—
441C2—41no—
442C2—41no—
443C2—41no—
444C2C2×C3×#83411yescharacteristic
subgroupG/H
centerC2×C3C2×#834
commutatorC23×C3C22×C3×C4
FrattiniC2×D4C22×C3×D3
FittingC32×#536C2
socleC2×C32C2×#99
prime radicalC2×C32×#41C22
cosocleC2×C3×D4C23×C3
hypercenterC3×#536D3
hypocenterC3C3×#2123
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ C3×#536

Complements

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

NH ≅ G/Ntype
4. C3×#4539C2semidirect
6. #71870C3direct (N × H)
8. C22×C32×Q8C4semidirect
11. C32×#123C4semidirect
15. C32×#117C4semidirect
30. #4539C2×C3semidirect
63. C22×C3×Q8Dic3semidirect
64. C22×C3×Q8C3×C4semidirect
74. C3×#123Dic3semidirect
75. C3×#123C3×C4semidirect
86. C3×#117Dic3semidirect
87. C3×#117C3×C4semidirect
218. C22×Q8C3×Dic3semidirect
222. #123C3×Dic3semidirect
227. #117C3×Dic3semidirect
381. C32#2123semidirect
438. C3#71870direct (N × H)
439. C3C3×#2123semidirect

Maximal subgroups

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

subgroupindex
C32×#5362
C3×#52402
C3×#52372
C3×#45392
C3×#21233
#718703