Finite Group Databasebeta

← #13155705

Subgroups — #13155705

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723399C211yesmaximal, characteristic
2C2×#88800C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8#10590810—31nomaximal
9#10590480—31nomaximal
10#10590383—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806—21no—
19Dic3×C3⋊Dic3—21no—
20#88800C414yescomplemented
21C3×#3724—21no—
22C3×Dic32—21no—
23C2×#3746—31no—
24C4×#634—31no—
25C3×#887—31no—
26C3×C4×Dic6—31no—
27#63844—31no—
28#63844—31no—
29#63844—61no—
30#63844D311yescharacteristic
31C2×#3719—31no—
32C2×#3719—31no—
33C2×#3719—61no—
34#63637—31no—
35C3×#817—31no—
36C3×#815—31no—
37C3×#808—31no—
38C3×#803—31no—
39#63573D311yescharacteristic
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—61no—
44#63566—31no—
45#63565—31no—
46#63554D311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49#6308—21no—
50#6308C2×C412yes—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8C2×C412yes—
60#5030—91no—
61#4853—91no—
62#4716—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×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C42—31no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750—61no—
79C4×C3⋊Dic3—61no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81#3746—34no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3—61no—
92C3×C4×Dic3—61no—
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
101#3724—21no—
102#3724—62no—
103#3724—61no—
104#3724—61no—
105Dic32—21no—
106Dic32—62no—
107Dic32—61no—
108Dic32—61no—
109#3719—34no—
110#3719—34no—
111#3719—64no—
112C22×C33C2×D411yescharacteristic
113C33×C4C22×C411yescharacteristic
114C33×C4#4111yescharacteristic
115C3⋊(C3⋊Dic3)—22no—
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—21no—
119C32×Dic3—21no—
120C32×Dic3—41no—
121C3×C4×Q8—91no—
122C2×#216—91no—
123C2×Dic12—91no—
124C2×Dic12—91no—
125C2×Dic12—181no—
126#887—31no—
127#887—91no—
128C4×Dic6—31no—
129C4×Dic6—91no—
130C4×Dic6—91no—
131C4×Dic6—181no—
132C3×#106—91no—
133C3×#97—91no—
134#835—91no—
135#817—31no—
136#816—91no—
137#816—91no—
138#815—31no—
139#810—91no—
140#810—91no—
141#810—91no—
142#810—181no—
143#808—31no—
144#803—31no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—122no—
164#634—21no—
165#634—32no—
166#634—32no—
167#634—32no—
168#634—62no—
169#634—62no—
170#634—62no—
171#634—61no—
172#634—62no—
173#634—61no—
174#634—61no—
175#634—122no—
176#634—121no—
177#634—121no—
178#634—122no—
179#634C4×D312yes—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3D1211yescharacteristic
192C2×C3×Dic3D1211yescharacteristic
193C2×C3×Dic3#6711yescharacteristic
194C2×C3×Dic3#6711yescharacteristic
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—121no—
201C32×C8—32no—
202C3⋊(C3⋊C8)—21no—
203C3×C3⋊C8—22no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—121no—
207C3×C3⋊C8C4×D312yes—
208C3×C3⋊C8C4×D312yes—
209#441—271noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33#13711yescharacteristic
212C2×C33#13711yescharacteristic
213C2×C3×Q8—91no—
214C2×Dic6—31no—
215C2×Dic6—31no—
216C2×Dic6—61no—
217C2×Dic6—91no—
218C2×Dic6—91no—
219C2×Dic6—91no—
220C2×Dic6—181no—
221C2×Dic6—181no—
222C2×Dic6—181no—
223C2×Dic6—361no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—181no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—61no—
236C3×C42—181no—
237#216—94no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3D3211yescharacteristic
242Dic3⋊C4—61no—
243Dic3⋊C4—61no—
244Dic3⋊C4—181no—
245Dic3⋊C4—181no—
246Dic3⋊C4—182no—
247Dic3⋊C4—181no—
248C4×Dic3—61no—
249C4×Dic3—61no—
250C4×Dic3—181no—
251C4×Dic3—181no—
252C4×Dic3—182no—
253C4×Dic3—181no—
254C4×Dic3D3211yescharacteristic
255C2×C3⋊C8—31no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—61no—
258C2×C3⋊C8D3211yescharacteristic
259Dic12—94no—
260Dic12—94no—
261Dic12—184no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32C2×D1211yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—61no—
278C32×C4—61no—
279C32×C4—121no—
280C32×C4C2×C4×D311yescharacteristic
281C32×C4C2×C4×D311yescharacteristic
282C32×C4C2×C4×D311yescharacteristic
283C32×C4#23511yescharacteristic
284C32×C4#23511yescharacteristic
285C32×C4#23911yescharacteristic
286C3⋊Dic3—22no—
287C3⋊Dic3—41no—
288C3⋊Dic3—62no—
289C3⋊Dic3—64no—
290C3⋊Dic3—62no—
291C3⋊Dic3—62no—
292C3⋊Dic3—122no—
293C3⋊Dic3—122no—
294C3⋊Dic3—124no—
295C3×Dic3—22no—
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—62no—
304C3×Dic3—62no—
305C3×Dic3—81no—
306C3×Dic3—122no—
307C3×Dic3—122no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C2×Q16—271no—
311C4×Q8—91no—
312C4×Q8—271no—
313#106—91no—
314#103—271no—
315#103—271no—
316#97—91no—
317C33#44111yescharacteristic, complemented, Sylow
318C3×Q8—92no—
319C3×Q8—181no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—31no—
327C2×C3×C4—41no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—121no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×Dic3—21no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—62no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3—182no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351C2×Dic3—362no—
352C2×Dic3#62611yescharacteristic
353C2×Dic3#62611yescharacteristic
354Dic6—32no—
355Dic6—32no—
356Dic6—62no—
357Dic6—61no—
358Dic6—61no—
359Dic6—92no—
360Dic6—92no—
361Dic6—92no—
362Dic6—121no—
363Dic6—182no—
364Dic6—182no—
365Dic6—182no—
366Dic6—182no—
367Dic6—181no—
368Dic6—181no—
369Dic6—181no—
370Dic6—362no—
371Dic6—361no—
372Dic6—361no—
373Dic6—362no—
374C3×C8—32no—
375C3×C8—32no—
376C3×C8—62no—
377C3×C8—181no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C3⋊C8#62412yes—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32C4×D1211yescharacteristic
395C2×C32#89311yescharacteristic
396C2×C32#90911yescharacteristic
397C2×C32#90911yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91511yescharacteristic
400C2×C32C4×#6711yescharacteristic
401C2×C32#94211yescharacteristic
402C2×C32#95111yescharacteristic
403C2×Q8—91no—
404C2×Q8—271no—
405Q16—274no—
406C2×C8—31no—
407C2×C8—91no—
408C4⋊C4—31no—
409C4⋊C4—91no—
410C4⋊C4—181no—
411C4⋊C4—541no—
412C42—31no—
413C42—181no—
414C42—541no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3D3×D1211yescharacteristic
420C22×C3C2×#62611yescharacteristic
421C22×C3D3×#6711yescharacteristic
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—62no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—182no—
439C3×C4—241no—
440C3×C4—361no—
441C3×C4#379811yescharacteristic
442C3×C4#379911yescharacteristic
443C3×C4#379911yescharacteristic
444C3×C4C4×D3211yescharacteristic
445C3×C4C4×D3211yescharacteristic
446C3×C4C2×#62411yescharacteristic
447Dic3—21no—
448Dic3—21no—
449Dic3—41no—
450Dic3—62no—
451Dic3—62no—
452Dic3—122no—
453Dic3—122no—
454Dic3—121no—
455Dic3—121no—
456Dic3—184no—
457Dic3—182no—
458Dic3—182no—
459Dic3—182no—
460Dic3—362no—
461Dic3—362no—
462Dic3—364no—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—41no—
467C32#471611yescharacteristic, complemented
468C32#485311yescharacteristic, complemented
469C32#503011yescharacteristic, complemented
470Q8—92no—
471Q8—181no—
472Q8—272no—
473Q8—541no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—61no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—541no—
480C2×C4D3311yescharacteristic
481C8—32no—
482C8—181no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3#6380011yescharacteristic
496C2×C3#6380011yescharacteristic
497C2×C3#6383111yescharacteristic
498C2×C3#6383311yescharacteristic
499C2×C3#6389411yescharacteristic
500C2×C3C4×#62611yescharacteristic
501C2×C3#6390411yescharacteristic
502C2×C3#6392911yescharacteristic
503C2×C3#6394611yescharacteristic
504C22D3×#62611yescharacteristic
505C4—61no—
506C4—61no—
507C4—121no—
508C4—182no—
509C4—361no—
510C4—542no—
511C4D3×#62411yescharacteristic
512C4#8896511yescharacteristic
513C3—21no—
514C3—21no—
515C3—21no—
516C3—41no—
517C3#1059038311yescharacteristic, complemented
518C3#1059048011yescharacteristic, complemented
519C3#1059081011yescharacteristic, complemented
520C2#1172288411yescharacteristic
521C2#1172288611yescharacteristic
522C2#1172293311yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
20. #88800C4semidirect
317. C33#441semidirect
467. C32#4716semidirect
468. C32#4853semidirect
469. C32#5030semidirect
517. C3#10590383semidirect
518. C3#10590480semidirect
519. C3#10590810semidirect

Maximal subgroups

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

subgroupindex
#117233992
C2×#888002
C3×#638442
#117220762
#117220412
C3×#635732
C3×#635542
#105908103
#105904803
#105903833