Finite Group Databasebeta

← #13156649

Subgroups — #13156649

Order 1728.

558 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
1#11723534C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic
4#11722054C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8#10590889—31nomaximal
9#10590412—31nomaximal
10#10590303—31nomaximal
11C2×#6293—21no—
12C2×C3×#627—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C3×Dic3⋊Dic3—21no—
22#64078—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63918D311yescharacteristic
29#63644—31no—
30#63635—31no—
31C3×#817—31no—
32C3×C8×Dic3—31no—
33C3×#808—31no—
34C3×C4×C3⋊C8—31no—
35#63574—31no—
36#63573—31no—
37#63573—31no—
38#63573—31no—
39#63573—31no—
40#63573—31no—
41#63573—31no—
42#63573—61no—
43#63573—61no—
44#63573D311yescharacteristic
45Dic3×C3⋊C8D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53#6293—42no—
54C3×#627—42no—
55C3×C3⋊(C3⋊C8)Q812yescomplemented
56C32×C3⋊C8Q812yescomplemented
57#5080—91no—
58#4855—91no—
59#4680—91no—
60C2×#634—61no—
61C2×C3×Dic6—61no—
62C2×C3×Dic6—61no—
63C2×#627—21no—
64C2×#627—62no—
65C2×#627—61no—
66C2×#627—61no—
67C2×C32×C8—31no—
68C32×C4⋊C4—31no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—121no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
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×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105C22×C33C2×D411yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3SD1612yes—
113C32×Dic3SD1612yes—
114C3×C4⋊Q8—91no—
115#888—31no—
116#888—91no—
117#869—31no—
118#869—91no—
119#869—91no—
120#869—181no—
121C3×#106—91no—
122C3×C4×C8—91no—
123#817—31no—
124#817—91no—
125#817—91no—
126#817—91no—
127#817—91no—
128#817—181no—
129C8×Dic3—31no—
130#808—31no—
131#808—91no—
132#808—91no—
133#808—91no—
134#808—91no—
135#808—91no—
136#808—181no—
137C4×C3⋊C8—31no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—31no—
142C2×C32×C4—31no—
143C2×C32×C4—41no—
144C2×C32×C4—61no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—121no—
155C2×C3⋊Dic3—121no—
156C2×C3⋊Dic3—122no—
157#634—122no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3D1211yescharacteristic
170C2×C3×Dic3D1211yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C3×Dic6—122no—
174C3×Dic6—122no—
175#627—42no—
176#627—124no—
177#627—122no—
178#627—122no—
179C32×C8—32no—
180C3⋊(C3⋊C8)Dic612yescomplemented
181C3×C3⋊C8—22no—
182C3×C3⋊C8—32no—
183C3×C3⋊C8—32no—
184C3×C3⋊C8—62no—
185C3×C3⋊C8Dic612yescomplemented
186C3×C3⋊C8Dic612yescomplemented
187#498—271noSylow
188C2×C33C4⋊Q811yescharacteristic
189C2×C33C2×SD1611yescharacteristic
190C2×C33C2×SD1611yescharacteristic
191C2×C3×Q8—181no—
192C2×Dic6—61no—
193C2×Dic6—61no—
194C2×Dic6—181no—
195C2×Dic6—181no—
196C2×Dic6—182no—
197C2×Dic6—181no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—61no—
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×C42—31no—
209C3×C42—31no—
210C3×C42—61no—
211C4⋊Dic3—31no—
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3—61no—
218C4⋊Dic3—91no—
219C4⋊Dic3—91no—
220C4⋊Dic3—91no—
221C4⋊Dic3—181no—
222C4⋊Dic3—181no—
223C4⋊Dic3—181no—
224C4⋊Dic3—181no—
225C4⋊Dic3—181no—
226C4⋊Dic3—182no—
227C4⋊Dic3—361no—
228C4⋊Dic3D3211yescharacteristic
229Dic3⋊C4—61no—
230Dic3⋊C4—181no—
231C4×Dic3D3211yescharacteristic
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—61no—
235C2×C3⋊C8D3211yescharacteristic
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C2×D1211yescharacteristic
241C22×C32D3×D411yescharacteristic
242C22×C32C2×#6711yescharacteristic
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—32no—
250C32×C4—32no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—121no—
254C32×C4C2×Dic611yescharacteristic
255C32×C4C2×Dic611yescharacteristic
256C32×C4C2×Dic611yescharacteristic
257C32×C4D3×Q811yescharacteristic
258C32×C4D3×Q811yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C3⋊Dic3—41no—
261C3⋊Dic3—41no—
262C3⋊Dic3—121no—
263C3⋊Dic3—122no—
264C3⋊Dic3—121no—
265C3⋊Dic3—121no—
266C3⋊Dic3—241no—
267C3⋊Dic3—241no—
268C3⋊Dic3—242no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—41no—
272C3×Dic3—41no—
273C3×Dic3—81no—
274C3×Dic3—122no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C3×Dic3—121no—
278C3×Dic3—121no—
279C3×Dic3—241no—
280C3×Dic3#20412yes—
281C3×Dic3#20412yes—
282C3×Dic3#21412yes—
283C3×Dic3#21512yes—
284C4⋊Q8—91no—
285C4⋊Q8—271no—
286#106—91no—
287#106—271no—
288#106—271no—
289#106—271no—
290C4×C8—91no—
291C33#49811yescharacteristic, complemented, Sylow
292C3×Q8—362no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—41no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—121no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4C2×D3211yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C2×D3211yescharacteristic
311C2×Dic3—21no—
312C2×Dic3—62no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—121no—
318C2×Dic3—182no—
319C2×Dic3—181no—
320C2×Dic3—181no—
321C2×Dic3—181no—
322C2×Dic3—361no—
323C2×Dic3—361no—
324C2×Dic3—362no—
325C2×Dic3#62611yescharacteristic
326C2×Dic3#62611yescharacteristic
327Dic6—122no—
328Dic6—122no—
329Dic6—362no—
330Dic6—362no—
331Dic6—364no—
332Dic6—362no—
333C3×C8—32no—
334C3×C8—32no—
335C3×C8—62no—
336C3×C8—92no—
337C3⋊C8—32no—
338C3⋊C8—32no—
339C3⋊C8—62no—
340C3⋊C8#62712yescomplemented
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32#86911yescharacteristic
354C2×C32#88811yescharacteristic
355C2×C32C2×#20411yescharacteristic
356C2×C32C2×#20411yescharacteristic
357C2×C32D3×SD1611yescharacteristic
358C2×C32D3×SD1611yescharacteristic
359C2×C32C2×#21411yescharacteristic
360C2×C32C2×#21511yescharacteristic
361C2×C32#94411yescharacteristic
362C2×Q8—181no—
363C2×Q8—541no—
364C2×C8—31no—
365C2×C8—91no—
366C4⋊C4—31no—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C4⋊C4—181no—
370C4⋊C4—271no—
371C4⋊C4—541no—
372C42—31no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3D3×D1211yescharacteristic
378C22×C3C2×#62611yescharacteristic
379C22×C3D3×#6711yescharacteristic
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—32no—
387C3×C4—32no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—62no—
393C3×C4—62no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—241no—
397C3×C4—361no—
398C3×C4—361no—
399C3×C4D3×Dic611yescharacteristic
400C3×C4D3×Dic611yescharacteristic
401C3×C4D3×Dic611yescharacteristic
402C3×C4D3×Dic611yescharacteristic
403C3×C4#379711yescharacteristic
404C3×C4C2×#62711yescharacteristic
405Dic3—41no—
406Dic3—122no—
407Dic3—121no—
408Dic3—121no—
409Dic3—121no—
410Dic3—121no—
411Dic3—241no—
412Dic3—362no—
413Dic3—361no—
414Dic3—361no—
415Dic3—361no—
416Dic3—721no—
417Dic3—721no—
418Dic3—722no—
419Dic3#371612yes—
420Dic3#371712yes—
421C32—21no—
422C32—21no—
423C32—21no—
424C32—41no—
425C32#468011yescharacteristic, complemented
426C32#485511yescharacteristic, complemented
427C32#508011yescharacteristic, complemented
428Q8—362no—
429Q8—1082no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—61no—
433C2×C4—181no—
434C2×C4—181no—
435C2×C4—541no—
436C2×C4D3311yescharacteristic
437C8—32no—
438C8—92no—
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×C3D3×#20411yescharacteristic
452C2×C3D3×#20411yescharacteristic
453C2×C3C2×#371611yescharacteristic
454C2×C3C2×#371711yescharacteristic
455C2×C3#6383911yescharacteristic
456C2×C3#6386611yescharacteristic
457C2×C3#6391811yescharacteristic
458C2×C3D3×#21411yescharacteristic
459C2×C3D3×#21511yescharacteristic
460C22D3×#62611yescharacteristic
461C4—32no—
462C4—32no—
463C4—121no—
464C4—361no—
465C4—361no—
466C4—1081no—
467C4D3×#62711yescharacteristic
468C4#8895711yescharacteristic
469C3—21no—
470C3—21no—
471C3—21no—
472C3—41no—
473C3#1059030311yescharacteristic, complemented
474C3#1059041211yescharacteristic, complemented
475C3#1059088911yescharacteristic, complemented
476C2D3×#371611yescharacteristic
477C2D3×#371711yescharacteristic
478C2#1172303311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#498
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
55. C3×C3⋊(C3⋊C8)Q8semidirect
56. C32×C3⋊C8Q8semidirect
180. C3⋊(C3⋊C8)Dic6semidirect
185. C3×C3⋊C8Dic6semidirect
186. C3×C3⋊C8Dic6semidirect
291. C33#498semidirect
340. C3⋊C8#627semidirect
425. C32#4680semidirect
426. C32#4855semidirect
427. C32#5080semidirect
473. C3#10590303semidirect
474. C3#10590412semidirect
475. C3#10590889semidirect

Maximal subgroups

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

subgroupindex
#117235342
C3×#639182
#117220792
#117220542
#117220522
C3×#635732
C3×Dic3×C3⋊C82
#105908893
#105904123
#105903033