Finite Group Databasebeta

← C3×#10591546 #13158626

Subgroups — C3×#10591546 #13158626

Order 1728.

576 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×Q8×Dic3C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8C3×#5021—31nomaximal
9C3×#4793—31nomaximal
10#10591546C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#627—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×C4×Q8—31no—
23C3×Q8×Dic3—21no—
24C3×Q8×Dic3D311yescharacteristic
25C3×Q8×Dic3C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic
29C32×#103—31no—
30#63662C2×C311yescharacteristic
31C3×#835—21no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35C3×#835D311yescharacteristic
36C3×#835C2×C311yescharacteristic
37C3×#818—31no—
38C3×Dic3⋊C8—31no—
39C3×#807—31no—
40C3×#807—31no—
41C3×#807—31no—
42C3×C4⋊(C3⋊C8)—31no—
43#63576C2×C311yescharacteristic
44#63575C2×C311yescharacteristic
45Dic3⋊(C3⋊C8)C2×C311yescharacteristic
46C33×Q8—21no—
47C33×Q8Q812yescomplemented
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C3×#627—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C3×#477—91no—
59#5021—31no—
60#4793—31no—
61C2×C32×Q8—21no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8—41no—
65C2×C32×Q8C2×D311yescharacteristic
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×Q8C22×C311yescharacteristic
68C2×C3×Dic6—61no—
69C2×C3×Dic6—61no—
70C2×#627—21no—
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—61no—
75C32×C42—31no—
76C32×C42—61no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×C4⋊Dic3—22no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C22×C311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—22no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C22×C311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4—22no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—41no—
111C32×Dic3—21no—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3Q1612yes—
115C3×C4⋊Q8—91no—
116C3×C4×Q8—31no—
117C3×C4×Q8—31no—
118C3×C4×Q8—61no—
119Q8×Dic3C3×D311yescharacteristic
120#888—31no—
121#869—31no—
122C3×#107—91no—
123C3×#107—91no—
124C3×C4⋊C8—91no—
125C3×#103—31no—
126C3×#103—31no—
127C3×#103—61no—
128C3×#103—91no—
129#835—31no—
130#835—31no—
131#835—61no—
132#835C3×D311yescharacteristic
133#818—31no—
134Dic3⋊C8—31no—
135#807—31no—
136#807—31no—
137#807—31no—
138C4⋊(C3⋊C8)—31no—
139C32×Q8—21no—
140C32×Q8—22no—
141C32×Q8—21no—
142C32×Q8—21no—
143C32×Q8—22no—
144C32×Q8—22no—
145C32×Q8—22no—
146C32×Q8—42no—
147C32×Q8—41no—
148C32×Q8—41no—
149C32×Q8—42no—
150C32×Q8Dic612yescomplemented
151C32×Q8Dic612yescomplemented
152C32×Q8C3×Q812yescomplemented
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C311yescharacteristic
171C2×C3⋊Dic3—21no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3#6711yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3C3×D411yescharacteristic
186C2×C3×Dic3C3×D411yescharacteristic
187C3×Dic6—122no—
188C3×Dic6—122no—
189#627—42no—
190C32×C8—61no—
191C3⋊(C3⋊C8)—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—41no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—121no—
198#477—91noSylow
199C2×C33C22⋊Q811yescharacteristic
200C2×C33C2×Q1611yescharacteristic
201C2×C33AGL(1,ℤ/8)11yescharacteristic
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—41no—
206C2×C3×Q8—181no—
207C2×C3×Q8D3211yescharacteristic
208C2×C3×Q8C2×C3×D311yescharacteristic
209C2×C3×Q8C2×C3×D311yescharacteristic
210C2×Dic6—61no—
211C2×Dic6—61no—
212C2×C3×C8—31no—
213C2×C3×C8—31no—
214C2×C3×C8—61no—
215C2×C3×C8—91no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—62no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—181no—
227C3×C42—31no—
228C3×C42—31no—
229C3×C42—61no—
230C3×C42—61no—
231C3×C42—61no—
232C3×C42—62no—
233C4⋊Dic3—21no—
234C4⋊Dic3—31no—
235C4⋊Dic3—31no—
236C4⋊Dic3—61no—
237C4⋊Dic3—61no—
238C4⋊Dic3C2×C3×D311yescharacteristic
239C4⋊Dic3C2×C3×D311yescharacteristic
240Dic3⋊C4—61no—
241C4×Dic3—21no—
242C4×Dic3C2×C3×D311yescharacteristic
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8C2×C3×D311yescharacteristic
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32D3×D411yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C32×C4—22no—
255C32×C4—24no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—32no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—42no—
268C32×C4—42no—
269C32×C4—44no—
270C32×C4—61no—
271C32×C4—121no—
272C32×C4—121no—
273C32×C4C2×Dic611yescharacteristic
274C32×C4C2×Dic611yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4C2×C3×Q811yescharacteristic
278C32×C4C3×#4111yescharacteristic
279C3⋊Dic3—41no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—22no—
283C3×Dic3—41no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—81no—
290C3×Dic3—121no—
291C3×Dic3—121no—
292C3×Dic3—241no—
293C3×Dic3#21612yes—
294C3×Dic3C3×Q1612yes—
295C4⋊Q8—91no—
296C4×Q8—31no—
297#107—91no—
298#107—91no—
299C4⋊C8—91no—
300#103—31no—
301#103—91no—
302C33#47711yescharacteristic, complemented, Sylow
303C3×Q8—22no—
304C3×Q8—21no—
305C3×Q8—21no—
306C3×Q8—21no—
307C3×Q8—22no—
308C3×Q8—22no—
309C3×Q8—22no—
310C3×Q8—41no—
311C3×Q8—41no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—362no—
315C3×Q8#62712yescomplemented
316C3×Q8C3×Dic612yescomplemented
317C3×Q8C3×Dic612yescomplemented
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—121no—
341C2×C3×C4—181no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C22×C3×D311yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—121no—
350C2×Dic3C3×#6711yescharacteristic
351C2×Dic3C3×#6711yescharacteristic
352Dic6—122no—
353Dic6—122no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—121no—
357C3×C8—181no—
358C3⋊C8—21no—
359C3⋊C8—61no—
360C3⋊C8—61no—
361C3⋊C8—121no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32#87811yescharacteristic
375C2×C32#91411yescharacteristic
376C2×C32D3×Q1611yescharacteristic
377C2×C32#92411yescharacteristic
378C2×C32C2×#21611yescharacteristic
379C2×C32#94911yescharacteristic
380C2×C32C3×C22⋊Q811yescharacteristic
381C2×C32C2×C3×Q1611yescharacteristic
382C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
383C2×Q8—181no—
384C2×Q8C3×D3211yescharacteristic
385C2×C8—31no—
386C2×C8—91no—
387C4⋊C4—31no—
388C4⋊C4—31no—
389C4⋊C4—61no—
390C4⋊C4—91no—
391C4⋊C4—181no—
392C42—31no—
393C42—61no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—41no—
398C22×C3D3×#6711yescharacteristic
399C22×C3C3×D3×D411yescharacteristic
400C22×C3C2×C3×#6711yescharacteristic
401C3×C4—24no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—32no—
412C3×C4—32no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—44no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—62no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—121no—
422C3×C4—122no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—241no—
428C3×C4—361no—
429C3×C4#380411yescharacteristic
430C3×C4C2×#62711yescharacteristic
431C3×C4C2×C3×Dic611yescharacteristic
432C3×C4C2×C3×Dic611yescharacteristic
433C3×C4C3×#23511yescharacteristic
434C3×C4C3×#23711yescharacteristic
435Dic3—21no—
436Dic3—41no—
437Dic3—41no—
438Dic3—121no—
439Dic3—121no—
440Dic3—241no—
441Dic3C3×#21612yes—
442C32—21no—
443C32—21no—
444C32—21no—
445C32—41no—
446C32#479311yescharacteristic, complemented
447C32#502111yescharacteristic, complemented
448C32C3×#47711yescharacteristic, complemented
449Q8—21no—
450Q8—362no—
451Q8C3×#62712yescomplemented
452C2×C4—21no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—181no—
458C2×C4C2×C3×D3211yescharacteristic
459C8—61no—
460C8—181no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3#6393911yescharacteristic
474C2×C3D3×#21611yescharacteristic
475C2×C3#6396411yescharacteristic
476C2×C3C3×#87811yescharacteristic
477C2×C3C3×#91411yescharacteristic
478C2×C3C3×D3×Q1611yescharacteristic
479C2×C3C3×#92411yescharacteristic
480C2×C3C2×C3×#21611yescharacteristic
481C2×C3C3×#94911yescharacteristic
482C22C3×D3×#6711yescharacteristic
483C4—22no—
484C4—32no—
485C4—61no—
486C4—121no—
487C4—121no—
488C4—361no—
489C4C3×#380411yescharacteristic
490C4C2×C3×#62711yescharacteristic
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#1059154611yescharacteristic, direct product factor
496C3C3×#479311yescharacteristic, complemented
497C3C3×#502111yescharacteristic, complemented
498C2C3×#6393911yescharacteristic
499C2C3×D3×#21611yescharacteristic
500C2C3×#6396411yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#477
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
10. #10591546C3direct (N × H)
47. C33×Q8Q8semidirect
150. C32×Q8Dic6semidirect
151. C32×Q8Dic6semidirect
152. C32×Q8C3×Q8semidirect
302. C33#477semidirect
315. C3×Q8#627semidirect
316. C3×Q8C3×Dic6semidirect
317. C3×Q8C3×Dic6semidirect
446. C32#4793semidirect
447. C32#5021semidirect
448. C32C3×#477semidirect
451. Q8C3×#627semidirect
495. C3#10591546direct (N × H)
496. C3C3×#4793semidirect
497. C3C3×#5021semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×Dic32
C3×#639182
C3×#636622
C32×#8352
C3×#635762
C3×#635752
C3×Dic3⋊(C3⋊C8)2
C3×#50213
C3×#47933
#105915463