Finite Group Databasebeta

← C3×#71034 #11779152

Subgroups — C3×#71034 #11779152

Order 1152.

484 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5567C211yesmaximal, characteristic, complemented
2C32×#460C211yesmaximal, characteristic
3C3×#5075C211yesmaximal, characteristic
4C3×#5070C211yesmaximal, characteristic
5C3×#4535C211yesmaximal, characteristic, complemented
6C3×#4484C211yesmaximal, characteristic, complemented
7C3×#4482C211yesmaximal, characteristic, complemented
8C3×#1626—31nomaximal
9#71034C311yesmaximal, characteristic, direct product factor
10C2×C32×SD16—21no—
11C32×#127C2211yescharacteristic
12C32×C4×Q8C2211yescharacteristic
13C3×#944—21no—
14C2×C3×#214—21no—
15C3×#889—22no—
16C3×#887—21no—
17C3×#870—21no—
18C3×#869C2211yescharacteristic
19C3×C4×Dic6—21no—
20C3×C4×Dic6C2211yescharacteristic
21C32×C4⋊C8C2211yescharacteristic
22C32×#102—21no—
23C3×#832—22no—
24C3×#832—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27C3×#557—31no—
28#5567C2×C311yescharacteristic, complemented
29C3×#486—31no—
30C3×#460—21no—
31C3×#460—31no—
32C3×#460D311yescharacteristic
33C3×#460C2×C311yescharacteristic
34#5075C2×C311yescharacteristic
35#5070C2×C311yescharacteristic
36C3×#332—31no—
37C3×Q8⋊C8—31no—
38C3×Q8⋊C8—31no—
39#4535C2×C311yescharacteristic, complemented
40#4484C2×C311yescharacteristic, complemented
41#4482C2×C311yescharacteristic, complemented
42C2×C32×Q8D411yescharacteristic
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C3×Dic6—21no—
47C2×C3×Dic6D411yescharacteristic
48C32×SD16—42no—
49C2×C32×C8—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4D411yescharacteristic
52C32×C42—21no—
53C32×C42C2311yescharacteristic
54C3×#214—42no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3D412yes—
57C3×C4⋊Dic3D411yescharacteristic
58C3×Dic3⋊C4—22no—
59C3×Dic3⋊C4—22no—
60C3×Dic3⋊C4—22no—
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3—22no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#1626—31noSylow
67C2×C3×SD16—22no—
68C2×C3×SD16—21no—
69C2×C3×SD16—21no—
70C2×C3×SD16—61no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8—61no—
73C3×#127—21no—
74C3×#127C2×D311yescharacteristic
75C3×#127C22×C311yescharacteristic
76C3×#125—61no—
77C3×#125—62no—
78C3×C4×Q8—21no—
79C3×C4×Q8—31no—
80C3×C4×Q8—61no—
81C3×C4×Q8—61no—
82C3×C4×Q8C2×D311yescharacteristic
83C3×C4×Q8C22×C311yescharacteristic
84#944—21no—
85C2×#214—21no—
86#889—22no—
87#887—21no—
88#870—21no—
89#869C22×C311yescharacteristic
90C4×Dic6—21no—
91C4×Dic6C22×C311yescharacteristic
92C3×C4⋊C8—21no—
93C3×C4⋊C8—31no—
94C3×C4⋊C8C2×D311yescharacteristic
95C3×C4⋊C8C22×C311yescharacteristic
96C3×#102—22no—
97C3×#102—21no—
98C3×#102—21no—
99C3×#102—61no—
100C3×#102—62no—
101C3×C4×C8—31no—
102#832—22no—
103#832—21no—
104C4⋊(C3⋊C8)C22×C311yescharacteristic
105C4×C3⋊C8C22×C311yescharacteristic
106C23×C32—41no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×D4—41no—
110C32×D4—41no—
111C32×D4—42no—
112C32×D4—41no—
113C32×D4—41no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—22no—
122C3×Dic6—21no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6SD1612yescomplemented
126C32×C8—41no—
127C3×C3⋊C8—41no—
128C3×C3⋊C8—41no—
129#557—31no—
130#486—31no—
131#460—31no—
132#460C3×D311yescharacteristic
133#332—31no—
134Q8⋊C8—31no—
135Q8⋊C8—31no—
136C2×C3×Q8—21no—
137C2×C3×Q8—31no—
138C2×C3×Q8—61no—
139C2×C3×Q8#6711yescharacteristic
140C2×C3×Q8C3×D411yescharacteristic
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—41no—
149C2×C3×D4—41no—
150C2×Dic6—21no—
151C2×Dic6C3×D411yescharacteristic
152C3×SD16—44no—
153C3×SD16—42no—
154C3×SD16—42no—
155C3×SD16—122no—
156C2×C3×C8—22no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—61no—
160C2×C3×C8—61no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—41no—
167C3×C4⋊C4—62no—
168C3×C4⋊C4—62no—
169C3×C4⋊C4—62no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—61no—
172C3×C4⋊C4#6711yescharacteristic
173C3×C4⋊C4C3×D411yescharacteristic
174C3×C42—21no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42—41no—
178C3×C42—62no—
179C3×C42—61no—
180C3×C42C22×D311yescharacteristic
181C3×C42C23×C311yescharacteristic
182#214—42no—
183C4⋊Dic3—21no—
184C4⋊Dic3C3×D411yescharacteristic
185C4⋊Dic3C3×D412yes—
186Dic3⋊C4—22no—
187Dic3⋊C4—22no—
188Dic3⋊C4—22no—
189Dic3⋊C4—21no—
190C4×Dic3—22no—
191C4×Dic3—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C22×C32—81no—
195C22×C32—81no—
196C22×C32—81no—
197C22×C32#12011yescharacteristic
198C32×C4—21no—
199C32×C4—41no—
200C32×C4—41no—
201C32×C4C2×SD1611yescharacteristic
202C32×C4#13511yescharacteristic
203C32×C4AGL(1,ℤ/8)11yescharacteristic
204C32×C4#13711yescharacteristic
205C3×Dic3—22no—
206C3×Dic3—42no—
207C3×Dic3—41no—
208C2×SD16—21no—
209C2×SD16—61no—
210C4⋊Q8—31no—
211C4⋊Q8—61no—
212#127C2×C3×D311yescharacteristic
213#125—61no—
214#125—62no—
215C4×Q8—31no—
216C4×Q8—61no—
217C4×Q8—61no—
218C4×Q8C2×C3×D311yescharacteristic
219C4⋊C8—31no—
220C4⋊C8C2×C3×D311yescharacteristic
221#102—21no—
222#102—61no—
223#102—62no—
224C4×C8—31no—
225C23×C3—42no—
226C23×C3—41no—
227C23×C3—41no—
228C3×Q8—22no—
229C3×Q8—21no—
230C3×Q8—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—32no—
234C3×Q8—41no—
235C3×Q8—61no—
236C3×Q8—121no—
237C3×Q8—121no—
238C3×D4—42no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—82no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—62no—
265C2×C3×C4D3×D411yescharacteristic
266C2×C3×C4D3×D411yescharacteristic
267C2×C3×C4C2×#6711yescharacteristic
268C2×C3×C4C2×C3×D411yescharacteristic
269C2×C3×C4C2×C3×D411yescharacteristic
270C2×C3×C4C2×C3×D411yescharacteristic
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—22no—
274Dic6—21no—
275Dic6—41no—
276Dic6—41no—
277Dic6C3×SD1612yescomplemented
278C3×C8—42no—
279C3×C8—41no—
280C3×C8—41no—
281C3×C8—121no—
282C3×C8—121no—
283C3⋊C8—41no—
284C3⋊C8—41no—
285C2×C32—161no—
286C2×C32#44811yescharacteristic
287C2×C32#44911yescharacteristic
288C2×C32#45411yescharacteristic
289C2×Q8—31no—
290C2×Q8—61no—
291C2×Q8C3×#6711yescharacteristic
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295SD16—42no—
296SD16—122no—
297C2×C8—21no—
298C2×C8—61no—
299C2×C8—61no—
300C4⋊C4—21no—
301C4⋊C4—31no—
302C4⋊C4—32no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4C3×#6711yescharacteristic
309C42—21no—
310C42—62no—
311C42—61no—
312C42C22×C3×D311yescharacteristic
313C22×C3—21no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3#93711yescharacteristic
324C22×C3C3×#12011yescharacteristic
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—42no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—62no—
338C3×C4—81no—
339C3×C4—121no—
340C3×C4—122no—
341C3×C4D3×SD1611yescharacteristic
342C3×C4#91511yescharacteristic
343C3×C4#93211yescharacteristic
344C3×C4#95011yescharacteristic
345C3×C4C2×C3×SD1611yescharacteristic
346C3×C4C3×#13511yescharacteristic
347C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
348C3×C4C3×#13711yescharacteristic
349Dic3—22no—
350Dic3—42no—
351Dic3—41no—
352C32#162611yescharacteristic, complemented, Sylow
353C23—41no—
354Q8—21no—
355Q8—21no—
356Q8—32no—
357Q8—61no—
358Q8—121no—
359Q8—121no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—62no—
370C2×C4C3×D3×D411yescharacteristic
371C2×C4C3×D3×D411yescharacteristic
372C2×C4C2×C3×#6711yescharacteristic
373C8—41no—
374C8—121no—
375C8—121no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—162no—
380C2×C3—161no—
381C2×C3—161no—
382C2×C3#503711yescharacteristic
383C2×C3#517011yescharacteristic
384C2×C3#520111yescharacteristic
385C2×C3C3×#44811yescharacteristic
386C2×C3C3×#44911yescharacteristic
387C2×C3C3×#45411yescharacteristic
388C22—81no—
389C22—81no—
390C22—81no—
391C22C3×#93711yescharacteristic
392C4—21no—
393C4—41no—
394C4—41no—
395C4—62no—
396C4—121no—
397C4—122no—
398C4C3×D3×SD1611yescharacteristic
399C4C3×#91511yescharacteristic
400C4C3×#93211yescharacteristic
401C4C3×#95011yescharacteristic
402C3—21no—
403C3#7103411yescharacteristic, direct product factor
404C3C3×#162611yescharacteristic, complemented
405C2—161no—
406C2C3×#503711yescharacteristic
407C2C3×#517011yescharacteristic
408C2C3×#520111yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#460C2
socleC22×C32#120
prime radicalC32×#127C22
hypercenterC3×#460D3
hypocenterC3C3×#1626
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#460

Complements

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

NH ≅ G/Ntype
1. C3×#5567C2semidirect
5. C3×#4535C2semidirect
6. C3×#4484C2semidirect
7. C3×#4482C2semidirect
9. #71034C3direct (N × H)
28. #5567C2×C3semidirect
39. #4535C2×C3semidirect
40. #4484C2×C3semidirect
41. #4482C2×C3semidirect
125. C3×Dic6SD16semidirect
277. Dic6C3×SD16semidirect
352. C32#1626semidirect
403. C3#71034direct (N × H)
404. C3C3×#1626semidirect

Maximal subgroups

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

subgroupindex
C3×#55672
C32×#4602
C3×#50752
C3×#50702
C3×#45352
C3×#44842
C3×#44822
C3×#16263
#710343