Finite Group Databasebeta

← #13156517

Subgroups — #13156517

Order 1728.

582 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723443C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8#10590890—31nomaximal
9#10590560—31nomaximal
10#10590288—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21C3×Dic3⋊Dic3—21no—
22#64078—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic
26#63866—31no—
27#63866—31no—
28#63866—61no—
29#63643—31no—
30#63637—31no—
31C3×#816—31no—
32C3×C8×Dic3—31no—
33C3×#810—31no—
34C3×C4×C3⋊C8—31no—
35#63567—31no—
36#63567—31no—
37#63567—31no—
38#63567—31no—
39#63567—31no—
40#63567—61no—
41#63567—61no—
42#63567D311yescharacteristic
43#63566—31no—
44#63565—31no—
45Dic3×C3⋊C8D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5093—91no—
61#4792—91no—
62#4681—91no—
63C2×#634—31no—
64C2×#634—31no—
65C2×#634—31no—
66C2×#634—61no—
67C2×#634—61no—
68C2×#634—61no—
69C2×#634—121no—
70C2×#634C2×D311yescharacteristic
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
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—62no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3⋊(C3⋊Dic3)—41no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3SD1612yes—
116C32×Dic3Q1612yes—
117C3×C4⋊Q8—91no—
118#944—91no—
119#888—31no—
120#869—31no—
121#869—91no—
122#869—91no—
123#869—181no—
124C3×#103—91no—
125C3×C4×C8—91no—
126#835—91no—
127#816—31no—
128#816—91no—
129#816—91no—
130#816—91no—
131#816—91no—
132#816—181no—
133C8×Dic3—31no—
134#810—31no—
135#810—91no—
136#810—91no—
137#810—91no—
138#810—91no—
139#810—181no—
140C4×C3⋊C8—31no—
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—61no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—61no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—121no—
158C2×C3⋊Dic3—121no—
159C2×C3⋊Dic3—122no—
160#634—21no—
161#634—21no—
162#634—61no—
163#634—61no—
164#634—62no—
165#634—62no—
166#634—61no—
167#634—61no—
168#634—61no—
169#634—61no—
170#634—121no—
171#634—121no—
172#634—121no—
173#634—121no—
174#634—122no—
175#634—122no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3D1211yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—121no—
196C3×Dic6—121no—
197C3×Dic6—122no—
198C3×Dic6—122no—
199#627—42no—
200C32×C8—61no—
201C3⋊(C3⋊C8)—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—121no—
208#487—271noSylow
209C2×C33#12411yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33C2×Q1611yescharacteristic
212C2×C3×Q8—91no—
213C2×C3×Q8—181no—
214C2×Dic6—31no—
215C2×Dic6—31no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×Dic6—61no—
219C2×Dic6—91no—
220C2×Dic6—91no—
221C2×Dic6—91no—
222C2×Dic6—181no—
223C2×Dic6—181no—
224C2×Dic6—181no—
225C2×Dic6—361no—
226C2×C3×C8—31no—
227C2×C3×C8—31no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—181no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42—61no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—181no—
243C4⋊Dic3—181no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—182no—
247C4⋊Dic3—182no—
248C4⋊Dic3D3211yescharacteristic
249Dic3⋊C4—61no—
250Dic3⋊C4—181no—
251Dic3⋊C4—181no—
252C4×Dic3D3211yescharacteristic
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8D3211yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32C2×D1211yescharacteristic
262C22×C32D3×D411yescharacteristic
263C22×C32C2×#6711yescharacteristic
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—32no—
271C32×C4—32no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4—121no—
275C32×C4#23511yescharacteristic
276C32×C4#23511yescharacteristic
277C32×C4#23511yescharacteristic
278C32×C4#23711yescharacteristic
279C32×C4#23911yescharacteristic
280C32×C4#23911yescharacteristic
281C3⋊Dic3—41no—
282C3⋊Dic3—41no—
283C3⋊Dic3—122no—
284C3⋊Dic3—122no—
285C3⋊Dic3—122no—
286C3⋊Dic3—124no—
287C3⋊Dic3—121no—
288C3⋊Dic3—121no—
289C3⋊Dic3—121no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—81no—
295C3×Dic3—122no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301C3×Dic3#20412yes—
302C3×Dic3Dic1212yes—
303C3×Dic3#21412yes—
304C3×Dic3#21612yes—
305C4⋊Q8—91no—
306C4⋊Q8—271no—
307#103—91no—
308#103—271no—
309#103—271no—
310#103—271no—
311C4×C8—91no—
312C33#48711yescharacteristic, complemented, Sylow
313C3×Q8—181no—
314C3×Q8—181no—
315C3×Q8—362no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—41no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—121no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—62no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—121no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—361no—
346C2×Dic3—361no—
347C2×Dic3—362no—
348C2×Dic3#62611yescharacteristic
349C2×Dic3#62611yescharacteristic
350Dic6—61no—
351Dic6—61no—
352Dic6—61no—
353Dic6—61no—
354Dic6—121no—
355Dic6—121no—
356Dic6—122no—
357Dic6—122no—
358Dic6—182no—
359Dic6—182no—
360Dic6—181no—
361Dic6—181no—
362Dic6—181no—
363Dic6—181no—
364Dic6—181no—
365Dic6—181no—
366Dic6—361no—
367Dic6—361no—
368Dic6—361no—
369Dic6—361no—
370Dic6—362no—
371Dic6—362no—
372C3×C8—61no—
373C3×C8—61no—
374C3×C8—121no—
375C3×C8—181no—
376C3⋊C8—21no—
377C3⋊C8—61no—
378C3⋊C8—61no—
379C3⋊C8—121no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32#87511yescharacteristic
393C2×C32#88511yescharacteristic
394C2×C32C2×#20411yescharacteristic
395C2×C32C2×Dic1211yescharacteristic
396C2×C32D3×SD1611yescharacteristic
397C2×C32D3×Q1611yescharacteristic
398C2×C32C2×#21411yescharacteristic
399C2×C32C2×#21611yescharacteristic
400C2×C32#94711yescharacteristic
401C2×Q8—91no—
402C2×Q8—181no—
403C2×Q8—271no—
404C2×C8—31no—
405C2×C8—91no—
406C4⋊C4—31no—
407C4⋊C4—91no—
408C4⋊C4—181no—
409C4⋊C4—541no—
410C4⋊C4—541no—
411C42—31no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3D3×D1211yescharacteristic
417C22×C3C2×#62611yescharacteristic
418C22×C3D3×#6711yescharacteristic
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—32no—
426C3×C4—32no—
427C3×C4—32no—
428C3×C4—32no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—241no—
436C3×C4—361no—
437C3×C4—361no—
438C3×C4#379611yescharacteristic
439C3×C4#379811yescharacteristic
440C3×C4#379911yescharacteristic
441C3×C4#379911yescharacteristic
442C3×C4#379911yescharacteristic
443C3×C4#380411yescharacteristic
444Dic3—41no—
445Dic3—122no—
446Dic3—121no—
447Dic3—121no—
448Dic3—121no—
449Dic3—121no—
450Dic3—241no—
451Dic3—362no—
452Dic3—362no—
453Dic3—362no—
454Dic3—364no—
455Dic3—361no—
456Dic3—361no—
457Dic3—361no—
458Dic3#371612yes—
459Dic3#371912yes—
460C32—21no—
461C32—21no—
462C32—21no—
463C32—41no—
464C32#468111yescharacteristic, complemented
465C32#479211yescharacteristic, complemented
466C32#509311yescharacteristic, complemented
467Q8—181no—
468Q8—181no—
469Q8—362no—
470Q8—541no—
471Q8—541no—
472C2×C4—31no—
473C2×C4—31no—
474C2×C4—61no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—541no—
478C2×C4D3311yescharacteristic
479C8—61no—
480C8—181no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3D3×#20411yescharacteristic
494C2×C3D3×Dic1211yescharacteristic
495C2×C3C2×#371611yescharacteristic
496C2×C3C2×#371911yescharacteristic
497C2×C3#6385211yescharacteristic
498C2×C3#6386411yescharacteristic
499C2×C3#6390711yescharacteristic
500C2×C3D3×#21411yescharacteristic
501C2×C3D3×#21611yescharacteristic
502C22D3×#62611yescharacteristic
503C4—32no—
504C4—32no—
505C4—121no—
506C4—361no—
507C4—361no—
508C4—1081no—
509C4#8896211yescharacteristic
510C4#8896511yescharacteristic
511C3—21no—
512C3—21no—
513C3—21no—
514C3—41no—
515C3#1059028811yescharacteristic, complemented
516C3#1059056011yescharacteristic, complemented
517C3#1059089011yescharacteristic, complemented
518C2D3×#371611yescharacteristic
519C2D3×#371911yescharacteristic
520C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#487
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
312. C33#487semidirect
464. C32#4681semidirect
465. C32#4792semidirect
466. C32#5093semidirect
515. C3#10590288semidirect
516. C3#10590560semidirect
517. C3#10590890semidirect

Maximal subgroups

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

subgroupindex
#117234432
C3×#639182
#117220762
#117220422
#117220412
C3×#635672
C3×Dic3×C3⋊C82
#105908903
#105905603
#105902883