Finite Group Databasebeta

← C3×#71369 #11780082

Subgroups — C3×#71369 #11780082

Order 1152.

529 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5726C211yesmaximal, characteristic
2C32×#500C211yesmaximal, characteristic, complemented
3C3×#5087C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5C3×#4547C211yesmaximal, characteristic, complemented
6C3×#4459C211yesmaximal, characteristic, complemented
7C3×#4459C211yesmaximal, characteristic, complemented
8C3×#1647—31nomaximal
9#71369C311yesmaximal, characteristic, direct product factor
10C2×C3×#239—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C32×C4⋊Q8C2211yescharacteristic
13C3×#946—21no—
14C2×C3×#215—21no—
15C3×#895—21no—
16C3×#894—22no—
17C3×D3×C4⋊C4—21no—
18C3×C4×D12C2211yescharacteristic
19C3×C4×D12C2211yescharacteristic
20C32×#107—22no—
21C32×C4×C8C2211yescharacteristic
22C3×#835—21no—
23C3×#809—21no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×#547—31no—
28#5726C2×C311yescharacteristic
29C3×#500—21no—
30C3×#500D311yescharacteristic, complemented
31C3×#500C2×C311yescharacteristic, complemented
32C3×#474—31no—
33C3×#461—31no—
34#5087C2×C311yescharacteristic, complemented
35#5083C2×C311yescharacteristic, complemented
36C3×#332—31no—
37C3×D4⋊C8—31no—
38C3×D4⋊C8—31no—
39#4547C2×C311yescharacteristic, complemented
40#4459C2×C311yescharacteristic, complemented
41#4459C2×C311yescharacteristic, complemented
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C3×#239—42no—
45C3×#239—42no—
46C2×C3×D12—22no—
47C2×C3×D12D411yescharacteristic
48C2×C3×D12D411yescharacteristic
49C2×C3×C4×D3—22no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—21no—
52C2×C32×C8—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4D412yes—
57C32×C42C2311yescharacteristic
58C3×#215—42no—
59C3×(C2×D3)⋊C4—22no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×C4⋊Dic3D411yescharacteristic
63C3×C4⋊Dic3D411yescharacteristic
64C3×Dic3⋊C4—22no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67#1647—31noSylow
68C2×C3×#41—61no—
69C2×#239—21no—
70C2×C3×SD16—61no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4⋊Q8C2×D311yescharacteristic
75C3×C4⋊Q8C22×C311yescharacteristic
76C3×C4⋊Q8C22×C311yescharacteristic
77C3×#123—62no—
78C3×C22⋊Q8—61no—
79C3×#121—61no—
80C3×C4×D4—31no—
81C3×C4×D4—31no—
82C2×C3×C4⋊C4—61no—
83#946—21no—
84C2×#215—21no—
85#895—21no—
86#894—22no—
87D3×C4⋊C4—21no—
88C4×D12C22×C311yescharacteristic
89C4×D12C22×C311yescharacteristic
90C3×#107—24no—
91C3×#107—22no—
92C3×#107—22no—
93C3×#107—61no—
94C3×C4⋊C8—31no—
95C3×C4⋊C8—31no—
96C3×#103—61no—
97C3×#102—61no—
98C3×C4×C8—21no—
99C3×C4×C8C2×D311yescharacteristic
100C3×C4×C8C22×C311yescharacteristic
101#835—21no—
102#809—21no—
103#807—21no—
104C4⋊(C3⋊C8)C22×C311yescharacteristic
105C4⋊(C3⋊C8)C22×C311yescharacteristic
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C32×Q8—41no—
109C32×Q8—41no—
110C32×Q8—41no—
111C32×Q8—41no—
112C2×C32×C4—22no—
113C2×C32×C4—41no—
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C3×D12—21no—
120C3×D12—21no—
121C3×D12—21no—
122C3×D12—44no—
123C3×D12D812yes—
124C3×C4×D3—44no—
125C3×C4×D3—41no—
126C3×C4×D3—41no—
127C3×C4×D3—41no—
128C3×C4×D3—41no—
129C32×C8—22no—
130C3×C3⋊C8—41no—
131C3×C3⋊C8—41no—
132#547—31no—
133#500C3×D311yescharacteristic, complemented
134#474—31no—
135#461—31no—
136#332—31no—
137D4⋊C8—31no—
138D4⋊C8—31no—
139C3×#41—122no—
140C3×#41—122no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—21no—
144C2×C3×Q8—21no—
145C2×C3×Q8—41no—
146C2×C3×Q8—41no—
147C2×C3×D4—31no—
148C2×C3×D4—31no—
149C2×C3×D4—62no—
150C22×C3×C4—62no—
151C22×C3×C4—61no—
152C22×C3×C4—61no—
153#239—42no—
154#239—42no—
155C2×D12—22no—
156C2×D12C3×D411yescharacteristic
157C2×D12C3×D411yescharacteristic
158C2×C4×D3—22no—
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C3×SD16—122no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—61no—
166C2×C3×C8—61no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—41no—
179C3×C4⋊C4—62no—
180C3×C4⋊C4#6712yes—
181C3×C4⋊C4C3×D412yes—
182C3×C22⋊C4—62no—
183C3×C22⋊C4—61no—
184C3×C22⋊C4—61no—
185C3×C42—21no—
186C3×C42C22×D311yescharacteristic
187C3×C42C23×C311yescharacteristic
188#215—42no—
189(C2×D3)⋊C4—22no—
190(C2×D3)⋊C4—21no—
191(C2×D3)⋊C4—21no—
192C4⋊Dic3C3×D411yescharacteristic
193C4⋊Dic3C3×D411yescharacteristic
194Dic3⋊C4—22no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C22×C32#12011yescharacteristic
198C2×C3×D3—22no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—41no—
204C32×C4—21no—
205C32×C4—42no—
206C32×C4—81no—
207C32×C4C2×D811yescharacteristic
208C32×C4#13511yescharacteristic
209C32×C4AGL(1,ℤ/8)11yescharacteristic
210C32×C4#13711yescharacteristic
211C3×Dic3—41no—
212C3×Dic3—41no—
213C2×#41—61no—
214C2×SD16—61no—
215C4⋊Q8C2×C3×D311yescharacteristic
216C4⋊Q8C2×C3×D311yescharacteristic
217#123—62no—
218C22⋊Q8—61no—
219#121—61no—
220C4×D4—31no—
221C4×D4—31no—
222C2×C4⋊C4—61no—
223#107—22no—
224#107—61no—
225C4⋊C8—31no—
226C4⋊C8—31no—
227#103—61no—
228#102—61no—
229C4×C8C2×C3×D311yescharacteristic
230C23×C3—61no—
231C23×C3—61no—
232C22×D3—21no—
233C22×D3—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8—81no—
243C3×Q8—81no—
244C3×Q8—81no—
245C3×Q8—81no—
246C3×D4—32no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—124no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
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—61no—
261C2×C3×C4—61no—
262C2×C3×C4—124no—
263C2×C3×C4—121no—
264C2×C3×C4—121no—
265C2×C3×C4—121no—
266C2×C3×C4—121no—
267C2×C3×C4D3×D411yescharacteristic
268C2×C3×C4D3×D411yescharacteristic
269C2×C3×C4C2×#6711yescharacteristic
270C2×C3×C4C2×C3×D411yescharacteristic
271C2×C3×C4C2×C3×D411yescharacteristic
272C2×C3×C4C2×C3×D411yescharacteristic
273C2×Dic3—21no—
274C2×Dic3—21no—
275D12—21no—
276D12—21no—
277D12—21no—
278D12—44no—
279D12C3×D812yes—
280C4×D3—44no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—41no—
285C3×C8—24no—
286C3×C8—22no—
287C3×C8—22no—
288C3×C8—121no—
289C3×C8—121no—
290C3⋊C8—41no—
291C3⋊C8—41no—
292C2×C32#44711yescharacteristic
293C2×C32#45211yescharacteristic
294C2×C32#45611yescharacteristic
295C3×D3—42no—
296C3×D3—81no—
297#41—122no—
298#41—122no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×D4—31no—
302C2×D4—31no—
303C2×D4—62no—
304C22×C4—62no—
305C22×C4—61no—
306C22×C4—61no—
307SD16—122no—
308C2×C8—21no—
309C2×C8—61no—
310C2×C8—61no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—62no—
317C4⋊C4C3×#6712yes—
318C22⋊C4—62no—
319C22⋊C4—61no—
320C22⋊C4—61no—
321C42C22×C3×D311yescharacteristic
322C22×C3—21no—
323C22×C3—62no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3#93711yescharacteristic
330C22×C3C3×#12011yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—82no—
347C3×C4—82no—
348C3×C4—81no—
349C3×C4—81no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4D3×D811yescharacteristic
353C3×C4#91111yescharacteristic
354C3×C4#91811yescharacteristic
355C3×C4#91911yescharacteristic
356C3×C4C2×C3×D811yescharacteristic
357C3×C4C3×#13511yescharacteristic
358C3×C4C3×AGL(1,ℤ/8)11yescharacteristic
359C3×C4C3×#13711yescharacteristic
360Dic3—41no—
361Dic3—41no—
362C32#164711yescharacteristic, complemented, Sylow
363C23—61no—
364C23—61no—
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369D4—32no—
370D4—61no—
371D4—61no—
372D4—61no—
373D4—124no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—124no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4C3×D3×D411yescharacteristic
384C2×C4C3×D3×D411yescharacteristic
385C2×C4C2×C3×#6711yescharacteristic
386C8—22no—
387C8—121no—
388C8—121no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—122no—
393C2×C3—241no—
394C2×C3#509511yescharacteristic
395C2×C3#515611yescharacteristic
396C2×C3#518711yescharacteristic
397C2×C3C3×#44711yescharacteristic
398C2×C3C3×#45211yescharacteristic
399C2×C3C3×#45611yescharacteristic
400D3—42no—
401D3—81no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22C3×#93711yescharacteristic
409C4—21no—
410C4—42no—
411C4—81no—
412C4—121no—
413C4—121no—
414C4C3×D3×D811yescharacteristic
415C4C3×#91111yescharacteristic
416C4C3×#91811yescharacteristic
417C4C3×#91911yescharacteristic
418C3—21no—
419C3#7136911yescharacteristic, direct product factor
420C3C3×#164711yescharacteristic, complemented
421C2—122no—
422C2—241no—
423C2C3×#509511yescharacteristic
424C2C3×#515611yescharacteristic
425C2C3×#518711yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#500C2
socleC22×C32#120
prime radicalC3×#5726C2
hypercenterC3×#500D3
hypocenterC3C3×#1647
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#500

Complements

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

NH ≅ G/Ntype
2. C32×#500C2semidirect
3. C3×#5087C2semidirect
4. C3×#5083C2semidirect
5. C3×#4547C2semidirect
6. C3×#4459C2semidirect
7. C3×#4459C2semidirect
9. #71369C3direct (N × H)
30. C3×#500D3semidirect
31. C3×#500C2×C3semidirect
34. #5087C2×C3semidirect
35. #5083C2×C3semidirect
39. #4547C2×C3semidirect
40. #4459C2×C3semidirect
41. #4459C2×C3semidirect
133. #500C3×D3semidirect
362. C32#1647semidirect
419. C3#71369direct (N × H)
420. C3C3×#1647semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
C32×#5002
C3×#50872
C3×#50832
C3×#45472
C3×#44592
C3×#16473
#713693