Finite Group Databasebeta

← #13155701

Subgroups — #13155701

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723410C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4#11722076C211yesmaximal, characteristic
5#11722021C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591582—31nomaximal
9#10590819—31nomaximal
10#10590485—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21Dic3×C3⋊Dic3—21no—
22C3×#3719C414yescomplemented
23Q8×C3⋊Dic3—31no—
24#64096—31no—
25C2×C3×#216—31no—
26C2×C3×Dic12—31no—
27#63843—31no—
28#63843—31no—
29#63843—31no—
30#63843—31no—
31#63843—61no—
32#63843—61no—
33C2×#3719D311yescharacteristic
34#63642—31no—
35C3×#835—31no—
36C3×#817—31no—
37C3×#810—31no—
38C3×#808—31no—
39#63573D311yescharacteristic
40#63567—31no—
41#63566—31no—
42#63565—31no—
43#63565D311yescharacteristic
44#63555—31no—
45#63555—31no—
46#63555—61no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yes—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8C2×C412yes—
60#5184—91no—
61#4853—91no—
62#4789—91no—
63C2×C32×Q8—31no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3750—61no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3—61no—
84C4×C3⋊Dic3—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#216—34no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C3×Dic12—34no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—62no—
105Dic3⋊Dic3—62no—
106Dic32—61no—
107Dic32—61no—
108Dic32—61no—
109Dic32—61no—
110Dic32—62no—
111Dic32—62no—
112#3719Dic314yescomplemented
113C22×C33C2×D411yescharacteristic
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3⋊(C3⋊Dic3)—21no—
117C3⋊(C3⋊Dic3)—21no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—41no—
120C32×Dic3—22no—
121C32×Dic3—41no—
122C2×C3×Q16—91no—
123Q8×Dic3—91no—
124Q8×Dic3—91no—
125Q8×Dic3—91no—
126Q8×Dic3—181no—
127C2×#216—31no—
128C2×Dic12—31no—
129#887—91no—
130#887—91no—
131#887—91no—
132#887—181no—
133C3×#106—91no—
134C3×#103—91no—
135#835—31no—
136#835—91no—
137#817—31no—
138#816—91no—
139#815—91no—
140#815—91no—
141#815—91no—
142#815—181no—
143#810—31no—
144#810—91no—
145#808—31no—
146C32×Q8—32no—
147C32×Q8—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—31no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173#634—21no—
174#634C2×Dic312yes—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C3×Dic6—22no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—22no—
191C3×Dic6—32no—
192C3×Dic6—32no—
193C3×Dic6—62no—
194C3×Dic6—62no—
195C3×Dic6—61no—
196C3×Dic6—61no—
197C3×Dic6C4×D312yes—
198C3×Dic6C2×Dic312yes—
199C32×C8—32no—
200C3⋊(C3⋊C8)—21no—
201C3×C3⋊C8—22no—
202C3×C3⋊C8—62no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8C4×D312yes—
206C3×C3⋊C8C2×Dic312yes—
207#441—271noSylow
208C2×C33C4×D411yescharacteristic
209C2×C33#13711yescharacteristic
210C2×C33#13711yescharacteristic
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—61no—
214C2×C3×Q8—91no—
215C2×Dic6—31no—
216C2×Dic6—31no—
217C2×Dic6—61no—
218C2×Dic6D3211yescharacteristic
219C3×Q16—94no—
220C2×C3×C8—31no—
221C2×C3×C8—31no—
222C2×C3×C8—61no—
223C2×C3×C8—91no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—91no—
228#216—34no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—182no—
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—182no—
241C4×Dic3—91no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—181no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—181no—
248C4×Dic3—181no—
249C4×Dic3—181no—
250C4×Dic3—181no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C4×Dic3—182no—
254C4×Dic3—182no—
255C4×Dic3—361no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260Dic12—34no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32D3×D411yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—62no—
277C32×C4—121no—
278C32×C4C2×C4×D311yescharacteristic
279C32×C4C2×C4×D311yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4#23711yescharacteristic
282C32×C4#23911yescharacteristic
283C32×C4C22×Dic311yescharacteristic
284C3⋊Dic3—22no—
285C3⋊Dic3—41no—
286C3⋊Dic3—61no—
287C3⋊Dic3—61no—
288C3⋊Dic3—62no—
289C3⋊Dic3—62no—
290C3⋊Dic3—61no—
291C3⋊Dic3—61no—
292C3⋊Dic3—61no—
293C3⋊Dic3—61no—
294C3⋊Dic3—121no—
295C3⋊Dic3—121no—
296C3⋊Dic3—121no—
297C3⋊Dic3—121no—
298C3⋊Dic3—122no—
299C3⋊Dic3—122no—
300C3×Dic3—24no—
301C3×Dic3—22no—
302C3×Dic3—22no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—64no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C2×Q16—91no—
313C4×Q8—271no—
314C4×Q8—271no—
315#106—91no—
316#103—91no—
317#103—271no—
318#97—271no—
319C33#44111yescharacteristic, complemented, Sylow
320C3×Q8—32no—
321C3×Q8—32no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—92no—
327C3×Q8—181no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—62no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—62no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—62no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—361no—
364C2×Dic3—361no—
365Dic6—21no—
366Dic6—32no—
367Dic6—32no—
368Dic6—62no—
369Dic6—62no—
370Dic6—61no—
371Dic6—61no—
372Dic6D3×Dic312yes—
373C3×C8—32no—
374C3×C8—32no—
375C3×C8—62no—
376C3×C8—181no—
377C3⋊C8—62no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8D3×Dic312yes—
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—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#67⋊C411yescharacteristic
394C2×C32#89311yescharacteristic
395C2×C32#91511yescharacteristic
396C2×C32#91511yescharacteristic
397C2×C32#91511yescharacteristic
398C2×C32#91811yescharacteristic
399C2×C32#91811yescharacteristic
400C2×C32#91811yescharacteristic
401C2×C32D4×Dic311yescharacteristic
402C2×Q8—31no—
403C2×Q8—91no—
404Q16—94no—
405C2×C8—31no—
406C2×C8—91no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—541no—
410C4⋊C4—541no—
411C42—271no—
412C42—541no—
413C42—541no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—41no—
418C22×C3#380211yescharacteristic
419C22×C3#381111yescharacteristic
420C22×C3#381111yescharacteristic
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—64no—
430C3×C4—62no—
431C3×C4—62no—
432C3×C4—122no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—182no—
436C3×C4—361no—
437C3×C4#379611yescharacteristic
438C3×C4#379611yescharacteristic
439C3×C4C4×D3211yescharacteristic
440C3×C4C2×D3×Dic311yescharacteristic
441C3×C4C2×D3×Dic311yescharacteristic
442C3×C4#380511yescharacteristic
443Dic3—22no—
444Dic3—41no—
445Dic3—64no—
446Dic3—62no—
447Dic3—62no—
448Dic3—122no—
449Dic3—121no—
450Dic3—121no—
451Dic3—182no—
452Dic3—182no—
453Dic3—181no—
454Dic3—181no—
455Dic3—181no—
456Dic3—181no—
457Dic3—181no—
458Dic3—181no—
459Dic3—361no—
460Dic3—361no—
461Dic3—361no—
462Dic3—361no—
463Dic3—362no—
464Dic3—362no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32#478911yescharacteristic, complemented
470C32#485311yescharacteristic, complemented
471C32#518411yescharacteristic, complemented
472Q8—32no—
473Q8—61no—
474Q8—92no—
475Q8—181no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—271no—
481C2×C4—271no—
482C2×C4D3311yescharacteristic
483C8—32no—
484C8—181no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3#6381111yescharacteristic
498C2×C3#6381111yescharacteristic
499C2×C3#6388111yescharacteristic
500C2×C3#6389811yescharacteristic
501C2×C3#6393111yescharacteristic
502C2×C3#6394511yescharacteristic
503C2×C3#6395111yescharacteristic
504C2×C3#6395111yescharacteristic
505C2×C3#67⋊Dic311yescharacteristic
506C22#8895411yescharacteristic
507C4—62no—
508C4—121no—
509C4—182no—
510C4—361no—
511C4—541no—
512C4—541no—
513C4D32×Dic311yescharacteristic
514C4#8896411yescharacteristic
515C3—21no—
516C3—21no—
517C3—21no—
518C3—41no—
519C3#1059048511yescharacteristic, complemented
520C3#1059081911yescharacteristic, complemented
521C3#1059158211yescharacteristic, complemented
522C2#1172289311yescharacteristic
523C2#1172289411yescharacteristic
524C2#1172292911yescharacteristic
subgroupG/H
centerC22#88954
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
22. C3×#3719C4semidirect
112. #3719Dic3semidirect
319. C33#441semidirect
469. C32#4789semidirect
470. C32#4853semidirect
471. C32#5184semidirect
519. C3#10590485semidirect
520. C3#10590819semidirect
521. C3#10591582semidirect

Maximal subgroups

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

subgroupindex
#117234102
#117233972
C2×C3×#37192
#117220762
#117220212
C3×#635732
C3×#635652
#105915823
#105908193
#105904853