Finite Group Databasebeta

← C3×#10590418 #13157543

Subgroups — C3×#10590418 #13157543

Order 1728.

514 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#890C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3C3×#63634C211yesmaximal, characteristic
4C32×#807C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8C3×#5064—31nomaximal
9C3×#4876—31nomaximal
10#10590418C311yesmaximal, characteristic, direct product factor
11C2×C3×#627—21no—
12C33×C4⋊C4C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×#125—31no—
23C3×#890—21no—
24C3×#890D311yescharacteristic
25C3×#890C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic
29C32×#107—31no—
30#63634C2×C311yescharacteristic
31C3×#817—31no—
32C3×#815—31no—
33C3×#808—31no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#807—21no—
37C3×#807—31no—
38C3×#807—31no—
39C3×#807—61no—
40C3×#807D311yescharacteristic
41C3×#807C2×C311yescharacteristic
42C3×#803—31no—
43#63574C2×C311yescharacteristic
44#63573C2×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
53C3×#627—42no—
54C3×C3⋊(C3⋊C8)Q812yescomplemented
55C32×C3⋊C8Q812yescomplemented
56C3×#501—91no—
57#5064—31no—
58#4876—31no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—61no—
61C2×#627—21no—
62C2×C32×C8—31no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—31no—
67C32×C4⋊C4—31no—
68C32×C4⋊C4—41no—
69C32×C4⋊C4—61no—
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—22no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C22×C311yescharacteristic
89C3×C4⋊Dic3C22×C311yescharacteristic
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4—21no—
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C22×C311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
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—91no—
114C3×#125—31no—
115C3×#125—31no—
116C3×#125—61no—
117#890C3×D311yescharacteristic
118#888—31no—
119#869—31no—
120C3×#107—31no—
121C3×#107—31no—
122C3×#107—61no—
123C3×#107—91no—
124C3×#106—91no—
125C3×#106—91no—
126C3×#97—91no—
127#817—31no—
128#815—31no—
129#808—31no—
130#808—31no—
131#808—31no—
132#807—31no—
133#807—31no—
134#807—61no—
135#807C3×D311yescharacteristic
136#803—31no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—31no—
145C2×C32×C4—31no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—42no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3#6711yescharacteristic
168C2×C3×Dic3#6711yescharacteristic
169C2×C3×Dic3C3×D411yescharacteristic
170C2×C3×Dic3C3×D411yescharacteristic
171C3×Dic6—122no—
172C3×Dic6—122no—
173#627—42no—
174C32×C8—32no—
175C3⋊(C3⋊C8)C3×Q812yescomplemented
176C3×C3⋊C8—22no—
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—62no—
180C3×C3⋊C8Dic612yescomplemented
181C3×C3⋊C8C3×Q812yescomplemented
182#501—91noSylow
183C2×C33C4⋊Q811yescharacteristic
184C2×C33AGL(1,ℤ/8)11yescharacteristic
185C2×C33#13711yescharacteristic
186C2×C3×Q8—181no—
187C2×Dic6—61no—
188C2×Dic6—61no—
189C2×C3×C8—31no—
190C2×C3×C8—31no—
191C2×C3×C8—61no—
192C2×C3×C8—91no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—62no—
208C3×C4⋊C4—62no—
209C3×C4⋊C4—91no—
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—21no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220C4⋊Dic3—61no—
221C4⋊Dic3—61no—
222C4⋊Dic3C2×C3×D311yescharacteristic
223C4⋊Dic3C2×C3×D311yescharacteristic
224Dic3⋊C4—21no—
225Dic3⋊C4—61no—
226C4×Dic3C2×C3×D311yescharacteristic
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8C2×C3×D311yescharacteristic
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32D3×D411yescharacteristic
236C22×C32C2×#6711yescharacteristic
237C22×C32C2×C3×D411yescharacteristic
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—42no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—61no—
251C32×C4—61no—
252C32×C4—81no—
253C32×C4—81no—
254C32×C4—82no—
255C32×C4—121no—
256C32×C4—121no—
257C32×C4C2×Dic611yescharacteristic
258C32×C4D3×Q811yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4D3×Q811yescharacteristic
261C32×C4C2×C3×Q811yescharacteristic
262C32×C4C2×C3×Q811yescharacteristic
263C3⋊Dic3—41no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—21no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—42no—
271C3×Dic3—41no—
272C3×Dic3—41no—
273C3×Dic3—41no—
274C3×Dic3—41no—
275C3×Dic3—81no—
276C3×Dic3—121no—
277C3×Dic3—121no—
278C3×Dic3—241no—
279C4⋊Q8—91no—
280#125—31no—
281#107—31no—
282#107—91no—
283#106—91no—
284#106—91no—
285#97—91no—
286C33#50111yescharacteristic, complemented, Sylow
287C3×Q8—362no—
288C2×C3×C4—22no—
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—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—62no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—121no—
311C2×C3×C4—181no—
312C2×C3×C4C2×D3211yescharacteristic
313C2×C3×C4C22×C3×D311yescharacteristic
314C2×C3×C4C22×C3×D311yescharacteristic
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—121no—
320C2×Dic3C3×#6711yescharacteristic
321C2×Dic3C3×#6711yescharacteristic
322Dic6—122no—
323Dic6—122no—
324C3×C8—32no—
325C3×C8—32no—
326C3×C8—62no—
327C3×C8—92no—
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—62no—
331C3⋊C8C3×Dic612yescomplemented
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32#88811yescharacteristic
345C2×C32#91111yescharacteristic
346C2×C32#91811yescharacteristic
347C2×C32#94411yescharacteristic
348C2×C32#94911yescharacteristic
349C2×C32#95111yescharacteristic
350C2×C32C3×C4⋊Q811yescharacteristic
351C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
352C2×C32C3×#13711yescharacteristic
353C2×Q8—181no—
354C2×C8—31no—
355C2×C8—91no—
356C4⋊C4—31no—
357C4⋊C4—31no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—91no—
361C4⋊C4—181no—
362C4⋊C4C3×D3211yescharacteristic
363C42—31no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3D3×#6711yescharacteristic
369C22×C3C3×D3×D411yescharacteristic
370C22×C3C2×C3×#6711yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—42no—
378C3×C4—41no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4—82no—
390C3×C4—121no—
391C3×C4—121no—
392C3×C4—122no—
393C3×C4—121no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—241no—
398C3×C4—361no—
399C3×C4D3×Dic611yescharacteristic
400C3×C4#379711yescharacteristic
401C3×C4C2×C3×Dic611yescharacteristic
402C3×C4C3×D3×Q811yescharacteristic
403C3×C4C3×D3×Q811yescharacteristic
404C3×C4C3×D3×Q811yescharacteristic
405Dic3—21no—
406Dic3—21no—
407Dic3—41no—
408Dic3—41no—
409Dic3—121no—
410Dic3—121no—
411Dic3—241no—
412C32—21no—
413C32—21no—
414C32—21no—
415C32—41no—
416C32#487611yescharacteristic, complemented
417C32#506411yescharacteristic, complemented
418C32C3×#50111yescharacteristic, complemented
419Q8—362no—
420C2×C4—21no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—181no—
426C2×C4C2×C3×D3211yescharacteristic
427C8—32no—
428C8—92no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3#6383911yescharacteristic
442C2×C3#6392511yescharacteristic
443C2×C3#6394311yescharacteristic
444C2×C3C3×#88811yescharacteristic
445C2×C3C3×#91111yescharacteristic
446C2×C3C3×#91811yescharacteristic
447C2×C3C3×#94411yescharacteristic
448C2×C3C3×#94911yescharacteristic
449C2×C3C3×#95111yescharacteristic
450C22C3×D3×#6711yescharacteristic
451C4—41no—
452C4—61no—
453C4—61no—
454C4—121no—
455C4—121no—
456C4—361no—
457C4C3×D3×Dic611yescharacteristic
458C4C3×#379711yescharacteristic
459C3—21no—
460C3—21no—
461C3—21no—
462C3—41no—
463C3#1059041811yescharacteristic, direct product factor
464C3C3×#487611yescharacteristic, complemented
465C3C3×#506411yescharacteristic, complemented
466C2C3×#6383911yescharacteristic
467C2C3×#6392511yescharacteristic
468C2C3×#6394311yescharacteristic
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. #10590418C3direct (N × H)
54. C3×C3⋊(C3⋊C8)Q8semidirect
55. C32×C3⋊C8Q8semidirect
175. C3⋊(C3⋊C8)C3×Q8semidirect
180. C3×C3⋊C8Dic6semidirect
181. C3×C3⋊C8C3×Q8semidirect
286. C33#501semidirect
331. C3⋊C8C3×Dic6semidirect
416. C32#4876semidirect
417. C32#5064semidirect
418. C32C3×#501semidirect
463. C3#10590418direct (N × H)
464. C3C3×#4876semidirect
465. C3C3×#5064semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
C3×#639182
C3×#636342
C32×#8072
C3×#635742
C3×#635732
C3×#635542
C3×#50643
C3×#48763
#105904183