Finite Group Databasebeta

← C3×#10590280 #13157293

Subgroups — C3×#10590280 #13157293

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
1C32×#875C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C3×#63630C211yesmaximal, characteristic, complemented
4C3×#63565C212yesmaximal, complemented
5C3×#63562C212yesmaximal
6C3×#5089—31nomaximal
7C3×#5055—31nomaximal
8#10590280C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C2211yescharacteristic
10C2×C32×Dic6—21no—
11C2×C32×Dic6C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×(C2×D3)⋊C4—21no—
15C32×(C2×D3)⋊C4—21no—
16C32×C4⋊Dic3—21no—
17C32×C4⋊Dic3C2212yes—
18C32×C4⋊Q8—31no—
19C32×#124—31no—
20C3×#875—21no—
21C3×#875D311yescharacteristic
22C3×#875C2×C311yescharacteristic
23C3×#869—21no—
24C3×#869D311yescharacteristic, complemented
25C3×#869C2×C311yescharacteristic, complemented
26#63630C2×C311yescharacteristic, complemented
27C3×#835—32no—
28C3×#832—32no—
29C3×#810—32no—
30C3×#809—32no—
31C3×#803—31no—
32C3×#803—31no—
33C3×#803—61no—
34#63565C2×C312yescomplemented
35#63562C2×C312yes—
36C22×C32×D3—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C32×Dic3—22no—
41C2×C32×Dic3—21no—
42C32×D12—22no—
43C32×Dic6—22no—
44C32×Dic6—42no—
45C3×C3⋊(C3⋊C8)—22no—
46C3×#489—91no—
47#5089—31no—
48#5055—31no—
49C2×C32×Q8—31no—
50C2×C32×Q8—61no—
51C2×C32×D4—31no—
52C2×C3×D12—21no—
53C2×C3×D12C2×D311yescharacteristic
54C2×C3×D12C22×C311yescharacteristic
55C2×C3×Dic6—22no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6C2×D311yescharacteristic
60C2×C3×Dic6C22×C311yescharacteristic
61C32×C4⋊C4—32no—
62C32×C4⋊C4—61no—
63C32×C22⋊C4—61no—
64C32×C22⋊C4—61no—
65C32×C42—21no—
66C32×C42—21no—
67C32×C42—21no—
68C32×C42—41no—
69C32×C42C2×D311yescharacteristic
70C32×C42C2×D311yescharacteristic
71C32×C42C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C312yes—
73C3×(C2×D3)⋊C4—22no—
74C3×(C2×D3)⋊C4—22no—
75C3×(C2×D3)⋊C4—21no—
76C3×(C2×D3)⋊C4—21no—
77C3×(C2×D3)⋊C4—21no—
78C3×(C2×D3)⋊C4—21no—
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—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—62no—
88C22×C33C2×D411yescharacteristic
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C2×C32×D3—41no—
92C33×C4—21no—
93C33×C4—21no—
94C33×C4#4112yes—
95C32×Dic3—41no—
96C32×Dic3—42no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—31no—
99C3×C4⋊Q8—61no—
100C3×#124—31no—
101C3×#124—31no—
102C3×#124—61no—
103#875C3×D311yescharacteristic
104#869C3×D311yescharacteristic, complemented
105C3×#103—92no—
106C3×#102—92no—
107C3×#97—91no—
108#835—32no—
109#832—32no—
110#810—32no—
111#809—32no—
112#803—31no—
113#803—31no—
114#803—61no—
115C23×C32—61no—
116C22×C3×D3—22no—
117C22×C3×D3—21no—
118C22×C3×D3—21no—
119C32×Q8—62no—
120C32×Q8—122no—
121C32×D4—62no—
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—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—61no—
135C2×C32×C4—62no—
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4#6711yescharacteristic
140C2×C32×C4C3×D411yescharacteristic
141C2×C32×C4C3×D411yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3×Dic3—24no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—21no—
151C3×D12—22no—
152C3×D12—22no—
153C3×D12—42no—
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—42no—
157C3×Dic6—44no—
158C3×Dic6—42no—
159C3×Dic6—42no—
160C3⋊(C3⋊C8)—22no—
161C3×C3⋊C8—64no—
162C3×C3⋊C8—62no—
163C3×C3⋊C8—62no—
164#489—91noSylow
165C2×C33#12411yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C2×C33#13711yescharacteristic
168C32×D3—81no—
169C2×C3×Q8—31no—
170C2×C3×Q8—31no—
171C2×C3×Q8—61no—
172C2×C3×Q8—62no—
173C2×C3×Q8—61no—
174C2×C3×Q8—61no—
175C2×C3×D4—31no—
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×D12C2×C3×D311yescharacteristic
179C2×Dic6—21no—
180C2×Dic6C2×C3×D311yescharacteristic
181C2×C3×C8—92no—
182C3×C4⋊C4—32no—
183C3×C4⋊C4—32no—
184C3×C4⋊C4—62no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4—61no—
187C3×C4⋊C4—61no—
188C3×C22⋊C4—62no—
189C3×C22⋊C4—62no—
190C3×C22⋊C4—61no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—61no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—41no—
198C3×C42D3211yescharacteristic
199C3×C42C2×C3×D311yescharacteristic
200C3×C42C2×C3×D311yescharacteristic
201(C2×D3)⋊C4—21no—
202(C2×D3)⋊C4—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×C3×D312yes—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—32no—
207C2×C3⋊C8—62no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—41no—
212C22×C32—121no—
213C22×C32—121no—
214C22×C32—121no—
215C22×C32C2×#6711yescharacteristic
216C22×C32C2×#6711yescharacteristic
217C22×C32C2×C3×D411yescharacteristic
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—42no—
221C2×C3×D3—41no—
222C2×C3×D3—41no—
223C2×C3×D3—41no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
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—121no—
246C32×C4—122no—
247C32×C4#23712yes—
248C32×C4#23912yes—
249C32×C4C3×#4112yes—
250C3×Dic3—44no—
251C3×Dic3—42no—
252C3×Dic3—42no—
253C3×Dic3—41no—
254C3×Dic3—41no—
255C3×Dic3—42no—
256C4⋊Q8—31no—
257#124—31no—
258#103—92no—
259#102—92no—
260#97—91no—
261C33#48911yescharacteristic, complemented, Sylow
262C23×C3—62no—
263C23×C3—61no—
264C23×C3—61no—
265C22×D3—21no—
266C3×Q8—62no—
267C3×Q8—62no—
268C3×Q8—122no—
269C3×Q8—124no—
270C3×Q8—122no—
271C3×Q8—122no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—122no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—62no—
288C2×C3×C4—64no—
289C2×C3×C4—61no—
290C2×C3×C4—62no—
291C2×C3×C4—61no—
292C2×C3×C4—62no—
293C2×C3×C4#62511yescharacteristic
294C2×C3×C4#62511yescharacteristic
295C2×C3×C4C3×#6711yescharacteristic
296C2×C3×C4C3×#6711yescharacteristic
297C2×C3×C4C3×#6711yescharacteristic
298C2×C3×C4C3×#6711yescharacteristic
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C22×C3×D311yescharacteristic
301C2×C3×C4C22×C3×D311yescharacteristic
302C2×Dic3—22no—
303C2×Dic3—21no—
304D12—22no—
305Dic6—22no—
306Dic6—42no—
307C3×C8—182no—
308C3⋊C8—64no—
309C3⋊C8—62no—
310C3⋊C8—62no—
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—241no—
324C2×C32#93211yescharacteristic
325C2×C32#93611yescharacteristic
326C2×C32#94211yescharacteristic
327C2×C32#94711yescharacteristic
328C2×C32#94911yescharacteristic
329C2×C32#95111yescharacteristic
330C2×C32C3×#12411yescharacteristic
331C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
332C2×C32C3×#13711yescharacteristic
333C3×D3—82no—
334C3×D3—81no—
335C3×D3—81no—
336C2×Q8—31no—
337C2×Q8—61no—
338C2×D4—31no—
339C2×C8—92no—
340C4⋊C4—32no—
341C4⋊C4—61no—
342C22⋊C4—61no—
343C22⋊C4—61no—
344C42C3×D3211yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—41no—
349C22×C3—122no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3C2×#62511yescharacteristic
359C22×C3C2×C3×#6711yescharacteristic
360C22×C3C2×C3×#6711yescharacteristic
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
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—121no—
385C3×C4—122no—
386C3×C4—122no—
387C3×C4—121no—
388C3×C4#379612yes—
389C3×C4C3×#23712yes—
390C3×C4C3×#23912yes—
391Dic3—42no—
392Dic3—41no—
393C32—21no—
394C32—21no—
395C32—21no—
396C32—41no—
397C32#505511yescharacteristic, complemented
398C32#508911yescharacteristic, complemented
399C32C3×#48911yescharacteristic, complemented
400C23—61no—
401Q8—62no—
402Q8—122no—
403D4—62no—
404C2×C4—61no—
405C2×C4—62no—
406C2×C4C3×#62511yescharacteristic
407C2×C4C3×#62511yescharacteristic
408C2×C4C2×C3×D3211yescharacteristic
409C8—182no—
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—242no—
423C2×C3—241no—
424C2×C3—241no—
425C2×C3#6382311yescharacteristic
426C2×C3#6382711yescharacteristic
427C2×C3#6386311yescharacteristic
428C2×C3C3×#93211yescharacteristic
429C2×C3C3×#93611yescharacteristic
430C2×C3C3×#94211yescharacteristic
431C2×C3C3×#94711yescharacteristic
432C2×C3C3×#94911yescharacteristic
433C2×C3C3×#95111yescharacteristic
434D3—81no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22C2×C3×#62511yescharacteristic
439C4—21no—
440C4—21no—
441C4—122no—
442C4—121no—
443C4C3×#379612yes—
444C3—21no—
445C3—21no—
446C3—21no—
447C3—41no—
448C3#1059028011yescharacteristic, direct product factor
449C3C3×#505511yescharacteristic, complemented
450C3C3×#508911yescharacteristic, complemented
451C2—241no—
452C2C3×#6382311yescharacteristic
453C2C3×#6382711yescharacteristic
454C2C3×#6386311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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. C3×#63630C2semidirect
4. C3×#63565C2semidirect
8. #10590280C3direct (N × H)
24. C3×#869D3semidirect
25. C3×#869C2×C3semidirect
26. #63630C2×C3semidirect
34. #63565C2×C3semidirect
104. #869C3×D3semidirect
261. C33#489semidirect
397. C32#5055semidirect
398. C32#5089semidirect
399. C32C3×#489semidirect
448. C3#10590280direct (N × H)
449. C3C3×#5055semidirect
450. C3C3×#5089semidirect

Maximal subgroups

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

subgroupindex
C32×#8752
C32×#8692
C3×#636302
C3×#635652
C3×#635622
C3×#50893
C3×#50553
#105902803