Finite Group Databasebeta

← #13156315

Subgroups — #13156315

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic
2C3×Dic3×Dic6C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591546—31nomaximal
9#10590489—31nomaximal
10#10590310—31nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×C3×#627—21no—
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×Q8×Dic3—31no—
23C3×#888—31no—
24C3×#869—31no—
25C3×C4×Dic6—31no—
26#63918D311yescharacteristic
27Dic3×Dic6D311yescharacteristic
28#63662—31no—
29#63634—31no—
30C4⋊(C3⋊(C3⋊C8))—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63576—31no—
34#63576—31no—
35#63576—31no—
36#63576—61no—
37#63575—31no—
38#63575—31no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63565—31no—
43#63565—31no—
44#63565—61no—
45#63565D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—21no—
54C32×Dic6Q812yescomplemented
55C3×#627—42no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5091—91no—
59#5021—91no—
60#4793—91no—
61C2×C32×Q8—31no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—61no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6C2×D311yescharacteristic
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#627—21no—
68C32×C4⋊C4—31no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—61no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—121no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—21no—
103Dic32—21no—
104C22×C33C2×D411yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C33×C4#4111yescharacteristic
107C3×C3⋊Dic3—41no—
108C3×C3⋊Dic3—41no—
109C32×Dic3—22no—
110C32×Dic3—21no—
111C32×Dic3—41no—
112C32×Dic3Q1612yes—
113C3×C4⋊Q8—91no—
114C3×C4×Q8—91no—
115Q8×Dic3—31no—
116#888—31no—
117#869—31no—
118C4×Dic6—31no—
119C3×#103—91no—
120#835—31no—
121#835—91no—
122#835—91no—
123#835—181no—
124#818—91no—
125Dic3⋊C8—91no—
126#810—31no—
127#810—91no—
128#807—91no—
129#807—91no—
130#807—91no—
131#807—91no—
132#807—91no—
133#807—181no—
134C4⋊(C3⋊C8)—91no—
135C4⋊(C3⋊C8)—91no—
136C4⋊(C3⋊C8)—181no—
137C32×Q8—32no—
138C32×Q8—61no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—31no—
143C2×C32×C4—31no—
144C2×C32×C4—41no—
145C2×C32×C4—61no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—41no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—121no—
165C2×C3×Dic3—121no—
166C2×C3×Dic3#6711yescharacteristic
167C2×C3×Dic3#6711yescharacteristic
168C2×C3×Dic3#6711yescharacteristic
169C2×C3×Dic3#6711yescharacteristic
170C3×Dic6—21no—
171C3×Dic6—21no—
172C3×Dic6—22no—
173C3×Dic6—41no—
174C3×Dic6—122no—
175C3×Dic6—122no—
176C3×Dic6Dic612yescomplemented
177C3×Dic6Dic612yescomplemented
178#627—42no—
179C3⋊(C3⋊C8)—21no—
180C3⋊(C3⋊C8)—61no—
181C3⋊(C3⋊C8)—62no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—61no—
184C3⋊(C3⋊C8)—121no—
185C3⋊(C3⋊C8)—121no—
186C3⋊(C3⋊C8)—122no—
187C3×C3⋊C8—61no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—121no—
190#477—271noSylow
191C2×C33C22⋊Q811yescharacteristic
192C2×C33C2×Q1611yescharacteristic
193C2×C33AGL(1,ℤ/8)11yescharacteristic
194C2×C3×Q8—31no—
195C2×C3×Q8—31no—
196C2×C3×Q8—61no—
197C2×C3×Q8—181no—
198C2×Dic6—61no—
199C2×Dic6—61no—
200C2×Dic6D3211yescharacteristic
201C2×C3×C8—91no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—91no—
206C3×C4⋊C4—91no—
207C3×C4⋊C4—181no—
208C3×C4⋊C4—181no—
209C3×C42—31no—
210C3×C42—31no—
211C3×C42—61no—
212C3×C42—181no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—31no—
216C4⋊Dic3—31no—
217C4⋊Dic3—61no—
218C4⋊Dic3—61no—
219C4⋊Dic3—61no—
220C4⋊Dic3—61no—
221C4⋊Dic3D3211yescharacteristic
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224C4×Dic3—61no—
225C4×Dic3—61no—
226C4×Dic3D3211yescharacteristic
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—91no—
231C2×C3⋊C8—91no—
232C2×C3⋊C8—91no—
233C2×C3⋊C8—181no—
234C2×C3⋊C8—181no—
235C2×C3⋊C8—181no—
236C2×C3⋊C8—361no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32D3×D411yescharacteristic
242C22×C32C2×#6711yescharacteristic
243C22×C32C2×#6711yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—32no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—61no—
254C32×C4—62no—
255C32×C4—121no—
256C32×C4C2×Dic611yescharacteristic
257C32×C4C2×Dic611yescharacteristic
258C32×C4#23511yescharacteristic
259C32×C4#23711yescharacteristic
260C32×C4D3×Q811yescharacteristic
261C32×C4#23911yescharacteristic
262C3⋊Dic3—41no—
263C3⋊Dic3—41no—
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—21no—
267C3×Dic3—21no—
268C3×Dic3—22no—
269C3×Dic3—42no—
270C3×Dic3—41no—
271C3×Dic3—41no—
272C3×Dic3—41no—
273C3×Dic3—81no—
274C3×Dic3—121no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C3×Dic3—121no—
278C3×Dic3—241no—
279C3×Dic3—241no—
280C3×Dic3#21612yes—
281C3×Dic3#21612yes—
282C4⋊Q8—91no—
283C4×Q8—91no—
284#107—271no—
285#107—271no—
286C4⋊C8—271no—
287#103—91no—
288#103—271no—
289C33#47711yescharacteristic, complemented, Sylow
290C3×Q8—32no—
291C3×Q8—32no—
292C3×Q8—62no—
293C3×Q8—61no—
294C3×Q8—61no—
295C3×Q8—121no—
296C3×Q8—362no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—41no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3#62511yescharacteristic
327C2×Dic3#62511yescharacteristic
328Dic6—21no—
329Dic6—122no—
330Dic6—122no—
331Dic6#62712yescomplemented
332C3×C8—181no—
333C3⋊C8—61no—
334C3⋊C8—61no—
335C3⋊C8—121no—
336C3⋊C8—182no—
337C3⋊C8—181no—
338C3⋊C8—181no—
339C3⋊C8—181no—
340C3⋊C8—361no—
341C3⋊C8—361no—
342C3⋊C8—362no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#87811yescharacteristic
356C2×C32#91411yescharacteristic
357C2×C32D3×Q1611yescharacteristic
358C2×C32#92411yescharacteristic
359C2×C32#93211yescharacteristic
360C2×C32C2×#21611yescharacteristic
361C2×C32C2×#21611yescharacteristic
362C2×C32#94611yescharacteristic
363C2×C32#94911yescharacteristic
364C2×Q8—31no—
365C2×Q8—181no—
366C2×C8—91no—
367C2×C8—271no—
368C4⋊C4—31no—
369C4⋊C4—91no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C4⋊C4—181no—
373C42—31no—
374C42—181no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3C2×#62511yescharacteristic
380C22×C3D3×#6711yescharacteristic
381C22×C3D3×#6711yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—62no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—121no—
398C3×C4—122no—
399C3×C4—121no—
400C3×C4—121no—
401C3×C4—241no—
402C3×C4—361no—
403C3×C4—361no—
404C3×C4D3×Dic611yescharacteristic
405C3×C4D3×Dic611yescharacteristic
406C3×C4#379611yescharacteristic
407C3×C4#379911yescharacteristic
408C3×C4#380411yescharacteristic
409C3×C4C2×#62711yescharacteristic
410Dic3—21no—
411Dic3—22no—
412Dic3—41no—
413Dic3—121no—
414Dic3—121no—
415Dic3—121no—
416Dic3—121no—
417Dic3—241no—
418Dic3—241no—
419Dic3#371812yes—
420C32—21no—
421C32—21no—
422C32—21no—
423C32—41no—
424C32#479311yescharacteristic, complemented
425C32#502111yescharacteristic, complemented
426C32#509111yescharacteristic, complemented
427Q8—32no—
428Q8—61no—
429Q8—362no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—181no—
435C2×C4—181no—
436C2×C4D3311yescharacteristic
437C8—181no—
438C8—541no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#6382311yescharacteristic
452C2×C3C2×#371811yescharacteristic
453C2×C3#6385611yescharacteristic
454C2×C3#6388711yescharacteristic
455C2×C3#6393411yescharacteristic
456C2×C3#6393911yescharacteristic
457C2×C3D3×#21611yescharacteristic
458C2×C3D3×#21611yescharacteristic
459C2×C3#6396411yescharacteristic
460C22D3×#62511yescharacteristic
461C4—32no—
462C4—61no—
463C4—62no—
464C4—121no—
465C4—361no—
466C4—361no—
467C4D3×#62711yescharacteristic
468C4#8896211yescharacteristic
469C3—21no—
470C3—21no—
471C3—21no—
472C3—41no—
473C3#1059031011yescharacteristic, complemented
474C3#1059048911yescharacteristic, complemented
475C3#1059154611yescharacteristic, complemented
476C2D3×#371811yescharacteristic
477C2#1172286711yescharacteristic
478C2#1172296911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
54. C32×Dic6Q8semidirect
176. C3×Dic6Dic6semidirect
177. C3×Dic6Dic6semidirect
289. C33#477semidirect
331. Dic6#627semidirect
424. C32#4793semidirect
425. C32#5021semidirect
426. C32#5091semidirect
473. C3#10590310semidirect
474. C3#10590489semidirect
475. C3#10591546semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×Dic3×Dic62
#117220802
#117220562
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
C3×#635652
#105915463
#105904893
#105903103