Finite Group Databasebeta

← C3×#10590421 #13157521

Subgroups — C3×#10590421 #13157521

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic
3C3×#63634C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63574C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8C3×#5081—31nomaximal
9C3×#4856—31nomaximal
10#10590421C311yesmaximal, characteristic, direct product factor
11C2×C32×Dic6—21no—
12C33×C4⋊C4C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×C4⋊Q8—31no—
23C3×#890—31no—
24C3×#888—21no—
25C3×#888D311yescharacteristic
26C3×#888C2×C311yescharacteristic
27C3×#870—31no—
28#63869C2×C311yescharacteristic
29C32×#106—31no—
30#63634C2×C311yescharacteristic
31C3×#818—31no—
32C3×#815—31no—
33C3×#808—21no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#808D311yescharacteristic
37C3×#808C2×C311yescharacteristic
38C3×#807—31no—
39C3×#807—31no—
40C3×#807—31no—
41C3×#807—61no—
42C3×#803—31no—
43#63575C2×C311yescharacteristic
44#63574C2×C311yescharacteristic
45#63554C2×C311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—42no—
54C3×C3⋊(C3⋊C8)Q812yescomplemented
55C32×C3⋊C8Q812yescomplemented
56C3×#501—91no—
57#5081—31no—
58#4856—31no—
59C2×C32×Q8—61no—
60C2×C3×Dic6—22no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C32×C8—31no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4—31no—
68C32×C4⋊C4—31no—
69C32×C4⋊C4—41no—
70C32×C4⋊C4—61no—
71C32×C4⋊C4C2×D311yescharacteristic
72C32×C4⋊C4C2×D311yescharacteristic
73C32×C4⋊C4C22×C311yescharacteristic
74C32×C42—31no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C22×C311yescharacteristic
86C3×C4⋊Dic3C22×C311yescharacteristic
87C3×Dic3⋊C4—22no—
88C3×Dic3⋊C4—21no—
89C3×Dic3⋊C4—21no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C22×C311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C22×C311yescharacteristic
102#3724—21no—
103Dic3⋊Dic3—21no—
104C22×C33C2×D411yescharacteristic
105C33×C4—41no—
106C33×C4C2×Q811yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C3×C3⋊Dic3—41no—
109C32×Dic3—21no—
110C32×Dic3—21no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C3×C4⋊Q8—31no—
114C3×C4⋊Q8—31no—
115C3×C4⋊Q8—61no—
116C3×#125—91no—
117#890—31no—
118#888C3×D311yescharacteristic
119#870—31no—
120C3×#107—91no—
121C3×#107—91no—
122C3×#106—31no—
123C3×#106—31no—
124C3×#106—61no—
125C3×#106—91no—
126C3×#97—91no—
127#818—31no—
128#815—31no—
129#808—31no—
130#808—31no—
131#808C3×D311yescharacteristic
132#807—31no—
133#807—31no—
134#807—31no—
135#807—61no—
136#803—31no—
137C32×Q8—122no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—31no—
146C2×C32×C4—31no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4—42no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3#6711yescharacteristic
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3C3×D411yescharacteristic
171C2×C3×Dic3C3×D411yescharacteristic
172C3×Dic6—44no—
173C3×Dic6—42no—
174C3×Dic6—42no—
175C32×C8—32no—
176C3⋊(C3⋊C8)C3×Q812yescomplemented
177C3×C3⋊C8—22no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—62no—
181C3×C3⋊C8Dic612yescomplemented
182C3×C3⋊C8C3×Q812yescomplemented
183#501—91noSylow
184C2×C33C4⋊Q811yescharacteristic
185C2×C33AGL(1,ℤ/8)11yescharacteristic
186C2×C33#13711yescharacteristic
187C2×C3×Q8—61no—
188C2×C3×Q8—61no—
189C2×C3×Q8—62no—
190C2×Dic6—21no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—61no—
194C2×C3×C8—91no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—62no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—181no—
210C3×C4⋊C4—181no—
211C3×C4⋊C4D3211yescharacteristic
212C3×C4⋊C4C2×C3×D311yescharacteristic
213C3×C4⋊C4C2×C3×D311yescharacteristic
214C3×C42—31no—
215C3×C42—31no—
216C3×C42—61no—
217C4⋊Dic3—31no—
218C4⋊Dic3—31no—
219C4⋊Dic3—61no—
220C4⋊Dic3—61no—
221C4⋊Dic3C2×C3×D311yescharacteristic
222C4⋊Dic3C2×C3×D311yescharacteristic
223Dic3⋊C4—21no—
224Dic3⋊C4—61no—
225Dic3⋊C4—61no—
226Dic3⋊C4—61no—
227C4×Dic3C2×C3×D311yescharacteristic
228C2×C3⋊C8—31no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8C2×C3×D311yescharacteristic
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—41no—
236C22×C32D3×D411yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C22×C32C2×C3×D411yescharacteristic
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—42no—
249C32×C4—41no—
250C32×C4—41no—
251C32×C4—61no—
252C32×C4—61no—
253C32×C4—81no—
254C32×C4—81no—
255C32×C4—82no—
256C32×C4—121no—
257C32×C4—121no—
258C32×C4C2×Dic611yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4D3×Q811yescharacteristic
261C32×C4D3×Q811yescharacteristic
262C32×C4C2×C3×Q811yescharacteristic
263C32×C4C2×C3×Q811yescharacteristic
264C3⋊Dic3—41no—
265C3×Dic3—22no—
266C3×Dic3—22no—
267C3×Dic3—21no—
268C3×Dic3—21no—
269C3×Dic3—21no—
270C3×Dic3—21no—
271C3×Dic3—42no—
272C3×Dic3—41no—
273C3×Dic3—41no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—81no—
277C3×Dic3—121no—
278C3×Dic3—121no—
279C3×Dic3—241no—
280C4⋊Q8—31no—
281#125—91no—
282#107—91no—
283#107—91no—
284#106—31no—
285#106—91no—
286#97—91no—
287C33#50111yescharacteristic, complemented, Sylow
288C3×Q8—124no—
289C3×Q8—122no—
290C3×Q8—122no—
291C2×C3×C4—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—41no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—62no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—121no—
314C2×C3×C4—181no—
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C22×C3×D311yescharacteristic
317C2×C3×C4C22×C3×D311yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—121no—
323C2×Dic3C3×#6711yescharacteristic
324C2×Dic3C3×#6711yescharacteristic
325Dic6—42no—
326C3×C8—32no—
327C3×C8—32no—
328C3×C8—62no—
329C3×C8—92no—
330C3⋊C8—32no—
331C3⋊C8—32no—
332C3⋊C8—62no—
333C3⋊C8C3×Dic612yescomplemented
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32#88811yescharacteristic
347C2×C32#91411yescharacteristic
348C2×C32#91511yescharacteristic
349C2×C32#93211yescharacteristic
350C2×C32#94211yescharacteristic
351C2×C32#94411yescharacteristic
352C2×C32C3×C4⋊Q811yescharacteristic
353C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
354C2×C32C3×#13711yescharacteristic
355C2×Q8—61no—
356C2×C8—31no—
357C2×C8—91no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—61no—
361C4⋊C4—91no—
362C4⋊C4—181no—
363C4⋊C4—181no—
364C4⋊C4C3×D3211yescharacteristic
365C42—31no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3D3×#6711yescharacteristic
371C22×C3C3×D3×D411yescharacteristic
372C22×C3C2×C3×#6711yescharacteristic
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—61no—
391C3×C4—81no—
392C3×C4—81no—
393C3×C4—82no—
394C3×C4—122no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4—121no—
399C3×C4—241no—
400C3×C4—361no—
401C3×C4D3×Dic611yescharacteristic
402C3×C4#379711yescharacteristic
403C3×C4C2×C3×Dic611yescharacteristic
404C3×C4C3×D3×Q811yescharacteristic
405C3×C4C3×D3×Q811yescharacteristic
406C3×C4C3×D3×Q811yescharacteristic
407Dic3—21no—
408Dic3—21no—
409Dic3—41no—
410Dic3—41no—
411Dic3—121no—
412Dic3—121no—
413Dic3—241no—
414C32—21no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32#485611yescharacteristic, complemented
419C32#508111yescharacteristic, complemented
420C32C3×#50111yescharacteristic, complemented
421Q8—122no—
422C2×C4—21no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—181no—
428C2×C4C2×C3×D3211yescharacteristic
429C8—32no—
430C8—92no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3#6383911yescharacteristic
444C2×C3#6393411yescharacteristic
445C2×C3#6394611yescharacteristic
446C2×C3C3×#88811yescharacteristic
447C2×C3C3×#91411yescharacteristic
448C2×C3C3×#91511yescharacteristic
449C2×C3C3×#93211yescharacteristic
450C2×C3C3×#94211yescharacteristic
451C2×C3C3×#94411yescharacteristic
452C22C3×D3×#6711yescharacteristic
453C4—41no—
454C4—61no—
455C4—61no—
456C4—121no—
457C4—121no—
458C4—361no—
459C4C3×D3×Dic611yescharacteristic
460C4C3×#379711yescharacteristic
461C3—21no—
462C3—21no—
463C3—21no—
464C3—41no—
465C3#1059042111yescharacteristic, direct product factor
466C3C3×#485611yescharacteristic, complemented
467C3C3×#508111yescharacteristic, complemented
468C2C3×#6383911yescharacteristic
469C2C3×#6393411yescharacteristic
470C2C3×#6394611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#501
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #10590421C3direct (N × H)
54. C3×C3⋊(C3⋊C8)Q8semidirect
55. C32×C3⋊C8Q8semidirect
176. C3⋊(C3⋊C8)C3×Q8semidirect
181. C3×C3⋊C8Dic6semidirect
182. C3×C3⋊C8C3×Q8semidirect
287. C33#501semidirect
333. C3⋊C8C3×Dic6semidirect
418. C32#4856semidirect
419. C32#5081semidirect
420. C32C3×#501semidirect
465. C3#10590421direct (N × H)
466. C3C3×#4856semidirect
467. C3C3×#5081semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#638692
C3×#636342
C32×#8082
C3×#635752
C3×#635742
C3×#635542
C3×#50813
C3×#48563
#105904213