Finite Group Databasebeta

← #13155674

Subgroups — #13155674

Order 1728.

634 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic
2C3×#63843C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic
4#11722037C211yesmaximal, characteristic, complemented
5#11722033C211yesmaximal, characteristic
6#11722020C211yesmaximal, characteristic, complemented
7C3×#63575C211yesmaximal, characteristic, complemented
8#10591566—31nomaximal
9#10590779—31nomaximal
10#10590457—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6294—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3717C414yescomplemented
23C3×Q8×Dic3—31no—
24C2×C3×#215—31no—
25C2×C3×#204—31no—
26C3×#887—31no—
27#63898—31no—
28#63894—31no—
29Dic3×D12—31no—
30#63843D311yescharacteristic, complemented
31C2×#3717D311yescharacteristic
32#63662—31no—
33#63642—31no—
34#63636—31no—
35C3×#818—31no—
36C3×#807—31no—
37#63575D311yescharacteristic, complemented
38#63565—31no—
39#63565—31no—
40#63565—61no—
41#63562—31no—
42#63562—31no—
43#63562—61no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58#6294—42no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5164—91no—
62#5026—91no—
63#4889—91no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C3×Dic6C2×D311yescharacteristic
72C2×#624—61no—
73C2×D3×Dic3—61no—
74C2×D3×Dic3—61no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4×C3⋊Dic3C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—121no—
86C3×#215—34no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×#204—34no—
104Dic3⋊Dic3—21no—
105#3722—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108Dic32—21no—
109#3717Dic314yescomplemented
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—21no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—22no—
120C32×Dic3—41no—
121C2×C3×SD16—91no—
122C3×C4×Q8—91no—
123Q8×Dic3—31no—
124C2×#215—31no—
125D4×Dic3—91no—
126C2×#204—31no—
127#893—91no—
128#887—31no—
129C4×D12—91no—
130C3×#107—91no—
131#835—91no—
132#835—91no—
133#835—181no—
134#832—91no—
135#818—31no—
136#815—91no—
137#815—91no—
138#815—181no—
139#810—91no—
140#809—91no—
141#809—91no—
142#809—181no—
143#807—31no—
144#803—91no—
145C22×C3⋊D3—21no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—121no—
149C32×Q8—32no—
150C32×Q8—61no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3#6711yescharacteristic
162C2×C3⋊Dic3#6711yescharacteristic
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)C2×Dic312yes—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—121no—
180C3×D12—32no—
181C3×D12—32no—
182C3×D12—62no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—121no—
186C3×Dic6—21no—
187C3×Dic6—21no—
188C3×Dic6—22no—
189C3×Dic6—41no—
190C3×Dic6C4×D312yescomplemented
191C3×Dic6C2×Dic312yescomplemented
192#624—122no—
193D3×Dic3—122no—
194D3×Dic3—122no—
195C32×C8—32no—
196C3⋊(C3⋊C8)—61no—
197C3⋊(C3⋊C8)—62no—
198C3⋊(C3⋊C8)—61no—
199C3⋊(C3⋊C8)—61no—
200C3⋊(C3⋊C8)—121no—
201C3⋊(C3⋊C8)—121no—
202C3⋊(C3⋊C8)—122no—
203C3×C3⋊C8—22no—
204C3×C3⋊C8C4×D312yescomplemented
205C3×C3⋊C8C2×Dic312yescomplemented
206#440—271noSylow
207C2×C33C4×D411yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C2×C33#13711yescharacteristic
210C3×C3⋊D3—42no—
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—61no—
214C2×C3×D4—91no—
215C22×Dic3—181no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—61no—
219C2×C4×D3—181no—
220C2×C4×D3—181no—
221C2×Dic6D3211yescharacteristic
222C3×SD16—94no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—61no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—181no—
231C3×C42—91no—
232C3×C42—181no—
233#217—181no—
234#215—34no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3D3211yescharacteristic
242Dic3⋊C4—61no—
243C4×Dic3—31no—
244C4×Dic3—31no—
245C4×Dic3—61no—
246C4×Dic3—61no—
247C4×Dic3—61no—
248C2×C3⋊C8—91no—
249C2×C3⋊C8—91no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—181no—
252C2×C3⋊C8—181no—
253C2×C3⋊C8—181no—
254C2×C3⋊C8—361no—
255C2×C3⋊C8D3211yescharacteristic
256#204—34no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32D3×D411yescharacteristic
262C22×C32D3×D411yescharacteristic
263C22×C32C2×#6711yescharacteristic
264C2×C3⋊D3—22no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—122no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—241no—
275C2×C3×D3—241no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—62no—
285C32×C4—121no—
286C32×C4C2×C4×D311yescharacteristic
287C32×C4C2×C4×D311yescharacteristic
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C32×C4C22×Dic311yescharacteristic
292C3⋊Dic3—21no—
293C3⋊Dic3—21no—
294C3⋊Dic3—41no—
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—42no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—61no—
303C3×Dic3—61no—
304C3×Dic3—81no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—241no—
310C2×SD16—91no—
311C4×Q8—91no—
312C4×D4—271no—
313#107—91no—
314#103—271no—
315#102—271no—
316#97—271no—
317C33#44011yescharacteristic, complemented, Sylow
318C23×C3—181no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—121no—
322C3×Q8—32no—
323C3×Q8—32no—
324C3×Q8—62no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—121no—
328C3×D4—92no—
329C3×D4—181no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—121no—
341C2×C3×C4—121no—
342C2×C3×C4—181no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—121no—
357C2×Dic3—362no—
358D12—32no—
359D12—32no—
360D12—62no—
361D12—61no—
362D12—61no—
363D12—121no—
364C4×D3—362no—
365C4×D3—362no—
366Dic6—21no—
367Dic6D3×Dic312yescomplemented
368C3×C8—32no—
369C3×C8—32no—
370C3×C8—62no—
371C3⋊C8—182no—
372C3⋊C8—181no—
373C3⋊C8—181no—
374C3⋊C8—181no—
375C3⋊C8—361no—
376C3⋊C8—361no—
377C3⋊C8—362no—
378C3⋊C8D3×Dic312yescomplemented
379C2×C32—21no—
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—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32#89311yescharacteristic
392C2×C32#91111yescharacteristic
393C2×C32#91411yescharacteristic
394C2×C32#91511yescharacteristic
395C2×C32#91811yescharacteristic
396C2×C32C4×#6711yescharacteristic
397C2×C32D4×Dic311yescharacteristic
398C2×C32#94211yescharacteristic
399C2×C32#94911yescharacteristic
400C3⋊D3—42no—
401C3×D3—122no—
402C3×D3—122no—
403C3×D3—242no—
404C2×Q8—31no—
405C2×D4—91no—
406C22×C4—541no—
407SD16—94no—
408C2×C8—31no—
409C2×C8—271no—
410C4⋊C4—31no—
411C4⋊C4—91no—
412C4⋊C4—181no—
413C22⋊C4—541no—
414C42—91no—
415C42—181no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—182no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3#380211yescharacteristic
424C22×C3D3×#6711yescharacteristic
425C22×C3D3×#6711yescharacteristic
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—122no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—241no—
434C2×D3—241no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—122no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—181no—
449C3×C4—181no—
450C3×C4—241no—
451C3×C4—361no—
452C3×C4#379611yescharacteristic
453C3×C4#379911yescharacteristic
454C3×C4C4×D3211yescharacteristic
455C3×C4C2×D3×Dic311yescharacteristic
456C3×C4C2×D3×Dic311yescharacteristic
457C3×C4#380411yescharacteristic
458Dic3—22no—
459Dic3—41no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—121no—
465Dic3—121no—
466Dic3—121no—
467Dic3—121no—
468Dic3—241no—
469C32—21no—
470C32—21no—
471C32—21no—
472C32—41no—
473C32#488911yescharacteristic, complemented
474C32#502611yescharacteristic, complemented
475C32#516411yescharacteristic, complemented
476C23—181no—
477Q8—32no—
478Q8—61no—
479D4—92no—
480D4—181no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—181no—
486C2×C4—1082no—
487C2×C4D3311yescharacteristic
488C8—32no—
489C8—541no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—362no—
503C2×C3#6379811yescharacteristic
504C2×C3#6381111yescharacteristic
505C2×C3#6389811yescharacteristic
506C2×C3#6390411yescharacteristic
507C2×C3#6392511yescharacteristic
508C2×C3#6393911yescharacteristic
509C2×C3#6394611yescharacteristic
510C2×C3#6394811yescharacteristic
511C2×C3Dic3×#6711yescharacteristic
512D3—122no—
513D3—122no—
514D3—242no—
515C22—182no—
516C22—361no—
517C22—361no—
518C22#8895311yescharacteristic
519C4—62no—
520C4—121no—
521C4—181no—
522C4—181no—
523C4—361no—
524C4D32×Dic311yescharacteristic
525C4#8896211yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059045711yescharacteristic, complemented
531C3#1059077911yescharacteristic, complemented
532C3#1059156611yescharacteristic, complemented
533C2—362no—
534C2#1172287611yescharacteristic
535C2#1172289211yescharacteristic
536C2#1172292711yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63843C2semidirect
4. #11722037C2semidirect
6. #11722020C2semidirect
7. C3×#63575C2semidirect
22. C3×#3717C4semidirect
30. #63843D3semidirect
37. #63575D3semidirect
57. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
109. #3717Dic3semidirect
190. C3×Dic6C4×D3semidirect
191. C3×Dic6C2×Dic3semidirect
204. C3×C3⋊C8C4×D3semidirect
205. C3×C3⋊C8C2×Dic3semidirect
317. C33#440semidirect
367. Dic6D3×Dic3semidirect
378. C3⋊C8D3×Dic3semidirect
473. C32#4889semidirect
474. C32#5026semidirect
475. C32#5164semidirect
530. C3#10590457semidirect
531. C3#10590779semidirect
532. C3#10591566semidirect

Maximal subgroups

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

subgroupindex
#117236912
C3×#638432
C2×C3×#37172
#117220372
#117220332
#117220202
C3×#635752
#105915663
#105907793
#105904573