Finite Group Databasebeta

← C3×#10590294 #13157308

Subgroups — C3×#10590294 #13157308

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64084C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C32×#803C211yesmaximal, characteristic, complemented
4C3×#63567C212yesmaximal, complemented
5C3×#63564C212yesmaximal
6C3×#5089—31nomaximal
7C3×#4714—31nomaximal
8#10590294C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6—21no—
12C33×C42C2211yescharacteristic
13C3×#3752—21no—
14C3×#3752—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2212yes—
17C2×C32×C3⋊C8C2212yes—
18C32×C4⋊Q8—31no—
19#64084C2×C311yescharacteristic
20C3×#875—31no—
21C3×#875—31no—
22C3×#875—61no—
23C3×#869—21no—
24C3×#869D311yescharacteristic, complemented
25C3×#869C2×C311yescharacteristic, complemented
26C32×#97—31no—
27C3×#821—32no—
28C3×#816—32no—
29C3×#810—32no—
30C3×#809—32no—
31C3×#803—21no—
32C3×#803D311yescharacteristic, complemented
33C3×#803C2×C311yescharacteristic, complemented
34#63567C2×C312yescomplemented
35#63564C2×C312yes—
36C22×C3×C3⋊D3—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—22no—
42C3×#634—22no—
43C2×C32×Dic3—22no—
44C32×Dic6—42no—
45C32×C3⋊C8—22no—
46C3×#489—91no—
47#5089—31no—
48#4714—31no—
49C2×C32×Q8—61no—
50C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
51C2×#634C22×C311yescharacteristic
52C2×C3×D12—31no—
53C2×C3×D12—31no—
54C2×C3×D12—61no—
55C2×C3×Dic6—22no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6—31no—
60C2×C3×Dic6—61no—
61C2×C32×C8—32no—
62C32×C4⋊C4—32no—
63C32×C4⋊C4—61no—
64C32×C42—21no—
65C32×C42—21no—
66C32×C42—21no—
67C32×C42—41no—
68C32×C42C2×D311yescharacteristic
69C32×C42C2×D311yescharacteristic
70C32×C42C22×C311yescharacteristic
71#3752—21no—
72#3752—21no—
73C3×(C2×D3)⋊C4—62no—
74C3×(C2×D3)⋊C4—62no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×(C2×D3)⋊C4—61no—
78C3×(C2×D3)⋊C4—61no—
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3—22no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3C2×D312yes—
84C3×C4⋊Dic3C22×C312yes—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2×D312yes—
87C2×C3×C3⋊C8C22×C312yes—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C33×C4—21no—
93C33×C4—21no—
94C33×C4#4112yes—
95C3×C3⋊Dic3—41no—
96C32×Dic3—42no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—31no—
99C3×C4⋊Q8—61no—
100C3×#124—91no—
101#875—31no—
102#875—31no—
103#875—61no—
104#869C3×D311yescharacteristic, complemented
105C3×#103—92no—
106C3×#102—92no—
107C3×#97—31no—
108C3×#97—31no—
109C3×#97—61no—
110#821—32no—
111#816—32no—
112#810—32no—
113#809—32no—
114#803C3×D311yescharacteristic, complemented
115C22×C3⋊D3—21no—
116C22×C3×D3—62no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C32×Q8—122no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—62no—
133C2×C32×C4D1211yescharacteristic
134C2×C32×C4D1211yescharacteristic
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C3×D411yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C23×C311yescharacteristic
142C2×C3⋊Dic3—21no—
143C4⋊(C3⋊D3)—22no—
144#634—22no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C3×D12—62no—
152C3×D12—62no—
153C3×D12—122no—
154C3×Dic6—44no—
155C3×Dic6—42no—
156C3×Dic6—42no—
157C3×Dic6—62no—
158C3×Dic6—62no—
159C3×Dic6—122no—
160C32×C8—62no—
161C3×C3⋊C8—24no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8—22no—
164#489—91noSylow
165C2×C33#12411yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C33#13711yescharacteristic
168C3×C3⋊D3—81no—
169C2×C3×Q8—62no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—91no—
173C2×C3×D4—91no—
174C2×D12—31no—
175C2×D12—31no—
176C2×D12—61no—
177C2×Dic6—21no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180C2×Dic6—61no—
181C2×C3×C8—32no—
182C2×C3×C8—32no—
183C2×C3×C8—62no—
184C3×C4⋊C4—32no—
185C3×C4⋊C4—32no—
186C3×C4⋊C4—62no—
187C3×C4⋊C4—62no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4—61no—
190C3×C22⋊C4—181no—
191C3×C22⋊C4—181no—
192C3×C42—21no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—41no—
196C3×C42D3211yescharacteristic
197C3×C42C2×C3×D311yescharacteristic
198C3×C42C2×C3×D311yescharacteristic
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—62no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—61no—
204(C2×D3)⋊C4—61no—
205C4⋊Dic3—21no—
206C4⋊Dic3C2×C3×D312yes—
207C2×C3⋊C8C2×C3×D312yes—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—41no—
212C22×C32C2×D1211yescharacteristic
213C22×C32C2×#6711yescharacteristic
214C22×C32C2×C3×D411yescharacteristic
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4—42no—
244C32×C4—42no—
245C32×C4—122no—
246C32×C4#23512yes—
247C32×C4#23912yes—
248C32×C4C3×#4112yes—
249C3⋊Dic3—41no—
250C3×Dic3—44no—
251C3×Dic3—42no—
252C3×Dic3—42no—
253C3×Dic3—122no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C4⋊Q8—31no—
257#124—91no—
258#103—92no—
259#102—92no—
260#97—31no—
261C33#48911yescharacteristic, complemented, Sylow
262C23×C3—181no—
263C22×D3—62no—
264C22×D3—61no—
265C22×D3—61no—
266C3×Q8—124no—
267C3×Q8—122no—
268C3×Q8—122no—
269C3×Q8—182no—
270C3×D4—182no—
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—21no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—64no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—181no—
287C2×C3×C4#62611yescharacteristic
288C2×C3×C4#62611yescharacteristic
289C2×C3×C4C3×D1211yescharacteristic
290C2×C3×C4C3×D1211yescharacteristic
291C2×C3×C4C3×#6711yescharacteristic
292C2×C3×C4C3×#6711yescharacteristic
293C2×C3×C4C2×D3211yescharacteristic
294C2×C3×C4C22×C3×D311yescharacteristic
295C2×C3×C4C22×C3×D311yescharacteristic
296C2×Dic3—22no—
297C2×Dic3—62no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300D12—62no—
301D12—62no—
302D12—122no—
303Dic6—42no—
304Dic6—62no—
305Dic6—62no—
306Dic6—122no—
307C3×C8—64no—
308C3×C8—62no—
309C3×C8—62no—
310C3⋊C8—22no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32#87511yescharacteristic
324C2×C32#90811yescharacteristic
325C2×C32#90911yescharacteristic
326C2×C32#93211yescharacteristic
327C2×C32#94211yescharacteristic
328C2×C32#94711yescharacteristic
329C2×C32C3×#12411yescharacteristic
330C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
331C2×C32C3×#13711yescharacteristic
332C3⋊D3—81no—
333C3×D3—242no—
334C3×D3—241no—
335C3×D3—241no—
336C2×Q8—61no—
337C2×Q8—91no—
338C2×D4—91no—
339C2×C8—32no—
340C4⋊C4—32no—
341C4⋊C4—61no—
342C22⋊C4—181no—
343C22⋊C4—181no—
344C42C3×D3211yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—41no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3C2×#62611yescharacteristic
353C22×C3C2×C3×D1211yescharacteristic
354C22×C3C2×C3×#6711yescharacteristic
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—121no—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—124no—
383C3×C4—122no—
384C3×C4—122no—
385C3×C4—361no—
386C3×C4#379912yes—
387C3×C4C3×#23512yes—
388C3×C4C3×#23912yes—
389Dic3—42no—
390Dic3—122no—
391Dic3—121no—
392Dic3—121no—
393C32—21no—
394C32—21no—
395C32—21no—
396C32—41no—
397C32#471411yescharacteristic, complemented
398C32#508911yescharacteristic, complemented
399C32C3×#48911yescharacteristic, complemented
400C23—181no—
401Q8—122no—
402Q8—182no—
403D4—182no—
404C2×C4—62no—
405C2×C4—181no—
406C2×C4C3×#62611yescharacteristic
407C2×C4C3×#62611yescharacteristic
408C2×C4C2×C3×D3211yescharacteristic
409C8—62no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—721no—
423C2×C3#6383011yescharacteristic
424C2×C3#6383311yescharacteristic
425C2×C3#6386411yescharacteristic
426C2×C3C3×#87511yescharacteristic
427C2×C3C3×#90811yescharacteristic
428C2×C3C3×#90911yescharacteristic
429C2×C3C3×#93211yescharacteristic
430C2×C3C3×#94211yescharacteristic
431C2×C3C3×#94711yescharacteristic
432D3—242no—
433D3—241no—
434D3—241no—
435C22—361no—
436C22—361no—
437C22—361no—
438C22C2×C3×#62611yescharacteristic
439C4—21no—
440C4—21no—
441C4—122no—
442C4—361no—
443C4C3×#379912yes—
444C3—21no—
445C3—21no—
446C3—21no—
447C3—41no—
448C3#1059029411yescharacteristic, direct product factor
449C3C3×#471411yescharacteristic, complemented
450C3C3×#508911yescharacteristic, complemented
451C2—721no—
452C2C3×#6383011yescharacteristic
453C2C3×#6383311yescharacteristic
454C2C3×#6386411yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C42D32
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#869C2semidirect
3. C32×#803C2semidirect
4. C3×#63567C2semidirect
8. #10590294C3direct (N × H)
24. C3×#869D3semidirect
25. C3×#869C2×C3semidirect
32. C3×#803D3semidirect
33. C3×#803C2×C3semidirect
34. #63567C2×C3semidirect
104. #869C3×D3semidirect
114. #803C3×D3semidirect
261. C33#489semidirect
397. C32#4714semidirect
398. C32#5089semidirect
399. C32C3×#489semidirect
448. C3#10590294direct (N × H)
449. C3C3×#4714semidirect
450. C3C3×#5089semidirect

Maximal subgroups

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

subgroupindex
C3×#640842
C32×#8692
C32×#8032
C3×#635672
C3×#635642
C3×#50893
C3×#47143
#105902943