Finite Group Databasebeta

← C3×#10590269 #13157281

Subgroups — C3×#10590269 #13157281

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4C3×#63576C212yesmaximal, complemented
5C3×#63562C212yesmaximal
6C3×#5087—31nomaximal
7C3×#5007—31nomaximal
8#10590269C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C32×Dic6—21no—
12C33×C42C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×(C2×D3)⋊C4—21no—
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2212yes—
18C32×C4⋊Q8—31no—
19C32×C4×D4—31no—
20C3×C4×D12—21no—
21C3×C4×D12D311yescharacteristic
22C3×C4×D12C2×C311yescharacteristic
23C3×#869—21no—
24C3×#869D311yescharacteristic, complemented
25C3×#869C2×C311yescharacteristic, complemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#832—32no—
28C3×#809—32no—
29C3×#807—32no—
30C3×#807—32no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63576C2×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—21no—
43C32×D12Q812yescomplemented
44C32×C4×D3—42no—
45C32×Dic6—42no—
46C3×C3⋊(C3⋊C8)—22no—
47C3×#474—91no—
48#5087—31no—
49#5007—31no—
50C2×C32×Q8—61no—
51C2×C32×D4—31no—
52C22×C32×C4—61no—
53C2×C3×D12—21no—
54C2×C3×D12C2×D311yescharacteristic
55C2×C3×D12C22×C311yescharacteristic
56C2×C3×C4×D3—22no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×Dic6—22no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4—32no—
64C32×C4⋊C4—61no—
65C32×C22⋊C4—61no—
66C32×C42—21no—
67C32×C42—21no—
68C32×C42—21no—
69C32×C42—41no—
70C32×C42C2×D311yescharacteristic
71C32×C42C2×D311yescharacteristic
72C32×C42C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C312yes—
74C3×(C2×D3)⋊C4—22no—
75C3×(C2×D3)⋊C4—21no—
76C3×(C2×D3)⋊C4—21no—
77C3×C4⋊Dic3—22no—
78C3×C4⋊Dic3—22no—
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D312yes—
84C3×C4⋊Dic3C22×C312yes—
85C3×C4⋊Dic3C22×C311yescharacteristic
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—62no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—22no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4—21no—
94C33×C4D812yes—
95C33×C4C2×Q811yescharacteristic
96C33×C4#4111yescharacteristic
97C32×Dic3—42no—
98C32×Dic3—41no—
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8—31no—
101C3×C4⋊Q8—61no—
102C3×C4×D4—31no—
103C3×C4×D4—31no—
104C3×C4×D4—61no—
105C4×D12C3×D311yescharacteristic
106#869C3×D311yescharacteristic, complemented
107C3×#107—92no—
108C3×C4⋊C8—91no—
109C3×#102—92no—
110#832—32no—
111#809—32no—
112#807—32no—
113#807—32no—
114C4⋊(C3⋊C8)—31no—
115C4⋊(C3⋊C8)—31no—
116C4⋊(C3⋊C8)—61no—
117C23×C32—61no—
118C22×C3×D3—22no—
119C22×C3×D3—21no—
120C22×C3×D3—21no—
121C32×Q8—122no—
122C32×D4—32no—
123C32×D4—61no—
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—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4—61no—
137C2×C32×C4—62no—
138C2×C32×C4—122no—
139C2×C32×C4#6711yescharacteristic
140C2×C32×C4#6711yescharacteristic
141C2×C32×C4#6711yescharacteristic
142C2×C32×C4#6711yescharacteristic
143C2×C32×C4C3×D411yescharacteristic
144C2×C32×C4C3×D411yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C23×C311yescharacteristic
148C2×C3×Dic3—24no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—21no—
154C3×D12—22no—
155C3×D12—21no—
156C3×D12—21no—
157C3×D12—22no—
158C3×D12Dic612yescomplemented
159C3×D12C3×Q812yescomplemented
160C3×C4×D3—44no—
161C3×C4×D3—42no—
162C3×C4×D3—42no—
163C3×Dic6—44no—
164C3×Dic6—42no—
165C3×Dic6—42no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—62no—
168C3×C3⋊C8—62no—
169C3×C3⋊C8—122no—
170#474—91noSylow
171C2×C33C22⋊Q811yescharacteristic
172C2×C33C2×D811yescharacteristic
173C2×C33#13711yescharacteristic
174C32×D3—42no—
175C2×C3×Q8—62no—
176C2×C3×Q8—61no—
177C2×C3×Q8—61no—
178C2×C3×D4—31no—
179C2×C3×D4—31no—
180C2×C3×D4—61no—
181C22×C3×C4—62no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C2×D12C2×C3×D311yescharacteristic
185C2×C4×D3—21no—
186C2×Dic6—21no—
187C2×C3×C8—92no—
188C3×C4⋊C4—32no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—62no—
194C3×C4⋊C4—62no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C22⋊C4—62no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—61no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—41no—
204C3×C42D3211yescharacteristic
205C3×C42C2×C3×D311yescharacteristic
206C3×C42C2×C3×D311yescharacteristic
207(C2×D3)⋊C4—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3C2×C3×D311yescharacteristic
210C4⋊Dic3C2×C3×D312yes—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—32no—
213C2×C3⋊C8—62no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—62no—
219C22×C32—121no—
220C22×C32—121no—
221C22×C32C2×#6711yescharacteristic
222C22×C32C2×#6711yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C2×C3×D3—24no—
225C2×C3×D3—22no—
226C2×C3×D3—22no—
227C2×C3×D3—42no—
228C2×C3×D3—42no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—42no—
247C32×C4—41no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—41no—
251C32×C4—42no—
252C32×C4—121no—
253C32×C4—122no—
254C32×C4#21312yes—
255C32×C4#21312yes—
256C32×C4C3×D812yes—
257C32×C4C2×Dic611yescharacteristic
258C32×C4#23511yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4#23911yescharacteristic
261C32×C4C2×C3×Q811yescharacteristic
262C32×C4C3×#4111yescharacteristic
263C3×Dic3—44no—
264C3×Dic3—42no—
265C3×Dic3—42no—
266C3×Dic3—41no—
267C3×Dic3—42no—
268C3×Dic3—41no—
269C4⋊Q8—31no—
270C4×D4—31no—
271#107—92no—
272C4⋊C8—91no—
273#102—92no—
274C33#47411yescharacteristic, complemented, Sylow
275C23×C3—62no—
276C23×C3—61no—
277C23×C3—61no—
278C22×D3—21no—
279C3×Q8—124no—
280C3×Q8—122no—
281C3×Q8—122no—
282C3×D4—32no—
283C3×D4—32no—
284C3×D4—62no—
285C3×D4—62no—
286C3×D4—61no—
287C3×D4—61no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—62no—
301C2×C3×C4—64no—
302C2×C3×C4—61no—
303C2×C3×C4—62no—
304C2×C3×C4—61no—
305C2×C3×C4—62no—
306C2×C3×C4—124no—
307C2×C3×C4—122no—
308C2×C3×C4—122no—
309C2×C3×C4#62511yescharacteristic
310C2×C3×C4#62511yescharacteristic
311C2×C3×C4C3×#6711yescharacteristic
312C2×C3×C4C3×#6711yescharacteristic
313C2×C3×C4C3×#6711yescharacteristic
314C2×C3×C4C3×#6711yescharacteristic
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D311yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—21no—
320D12—21no—
321D12C3×Dic612yescomplemented
322C4×D3—42no—
323Dic6—42no—
324C3×C8—182no—
325C3⋊C8—62no—
326C3⋊C8—62no—
327C3⋊C8—122no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—122no—
341C2×C32#92411yescharacteristic
342C2×C32C2×#21311yescharacteristic
343C2×C32C2×#21311yescharacteristic
344C2×C32#94211yescharacteristic
345C2×C32#94611yescharacteristic
346C2×C32#95111yescharacteristic
347C2×C32C3×C22⋊Q811yescharacteristic
348C2×C32C2×C3×D811yescharacteristic
349C2×C32C3×#13711yescharacteristic
350C3×D3—44no—
351C3×D3—42no—
352C3×D3—42no—
353C2×Q8—61no—
354C2×D4—31no—
355C22×C4—61no—
356C2×C8—92no—
357C4⋊C4—32no—
358C4⋊C4—31no—
359C4⋊C4—61no—
360C22⋊C4—61no—
361C42C3×D3211yescharacteristic
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—64no—
367C22×C3—62no—
368C22×C3—62no—
369C22×C3—122no—
370C22×C3—122no—
371C22×C3—121no—
372C22×C3—121no—
373C22×C3—121no—
374C22×C3—121no—
375C22×C3C2×#62511yescharacteristic
376C22×C3C2×C3×#6711yescharacteristic
377C22×C3C2×C3×#6711yescharacteristic
378C2×D3—22no—
379C2×D3—41no—
380C2×D3—41no—
381C3×C4—22no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—42no—
400C3×C4—122no—
401C3×C4—124no—
402C3×C4—121no—
403C3×C4—122no—
404C3×C4—121no—
405C3×C4—122no—
406C3×C4#371312yes—
407C3×C4C3×#21312yes—
408C3×C4C3×#21312yes—
409C3×C4D3×Dic611yescharacteristic
410C3×C4#379911yescharacteristic
411C3×C4C2×C3×Dic611yescharacteristic
412C3×C4C3×#23511yescharacteristic
413C3×C4C3×D3×Q811yescharacteristic
414C3×C4C3×#23911yescharacteristic
415Dic3—42no—
416Dic3—41no—
417C32—21no—
418C32—21no—
419C32—21no—
420C32—41no—
421C32#500711yescharacteristic, complemented
422C32#508711yescharacteristic, complemented
423C32C3×#47411yescharacteristic, complemented
424C23—61no—
425Q8—122no—
426D4—32no—
427D4—61no—
428C2×C4—61no—
429C2×C4—62no—
430C2×C4—122no—
431C2×C4C3×#62511yescharacteristic
432C2×C4C3×#62511yescharacteristic
433C2×C4C2×C3×D3211yescharacteristic
434C8—182no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—124no—
448C2×C3—122no—
449C2×C3—122no—
450C2×C3C2×#371311yescharacteristic
451C2×C3#6382711yescharacteristic
452C2×C3#6385611yescharacteristic
453C2×C3C3×#92411yescharacteristic
454C2×C3C2×C3×#21311yescharacteristic
455C2×C3C2×C3×#21311yescharacteristic
456C2×C3C3×#94211yescharacteristic
457C2×C3C3×#94611yescharacteristic
458C2×C3C3×#95111yescharacteristic
459D3—42no—
460C22—62no—
461C22—121no—
462C22—121no—
463C22C2×C3×#62511yescharacteristic
464C4—21no—
465C4—121no—
466C4—122no—
467C4C3×#371312yes—
468C4C3×D3×Dic611yescharacteristic
469C4C3×#379911yescharacteristic
470C3—21no—
471C3—21no—
472C3—21no—
473C3—41no—
474C3#1059026911yescharacteristic, direct product factor
475C3C3×#500711yescharacteristic, complemented
476C3C3×#508711yescharacteristic, complemented
477C2—122no—
478C2C2×C3×#371311yescharacteristic
479C2C3×#6382711yescharacteristic
480C2C3×#6385611yescharacteristic
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×#474
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×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. C3×#63576C2semidirect
8. #10590269C3direct (N × H)
24. C3×#869D3semidirect
25. C3×#869C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
34. #63576C2×C3semidirect
43. C32×D12Q8semidirect
106. #869C3×D3semidirect
158. C3×D12Dic6semidirect
159. C3×D12C3×Q8semidirect
274. C33#474semidirect
321. D12C3×Dic6semidirect
421. C32#5007semidirect
422. C32#5087semidirect
423. C32C3×#474semidirect
474. C3#10590269direct (N × H)
475. C3C3×#5007semidirect
476. C3C3×#5087semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C32×#8692
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#635762
C3×#635622
C3×#50873
C3×#50073
#105902693