Finite Group Databasebeta

← #13155690

Subgroups — #13155690

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic, complemented
2C2×#88792C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic, complemented
5#11722020C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8#10591405—31nomaximal
9#10590743—31nomaximal
10#10590478—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88813—21no—
21#88792C414yescomplemented
22C3×(C2×D3)⋊Dic3—21no—
23C2×#3754—31no—
24C3×D4×Dic3—31no—
25C3×#893—31no—
26#63898D311yescharacteristic
27#63844—31no—
28#63843—31no—
29Dic3×Dic6—31no—
30C2×#3715—31no—
31C2×#3715—31no—
32C2×#3715—61no—
33#63642—31no—
34#63634—31no—
35C3×#832—31no—
36C3×#821—31no—
37C3×#816—31no—
38C3×#810—31no—
39#63576—31no—
40#63576—31no—
41#63576—61no—
42#63567D311yescharacteristic, complemented
43#63563D311yescharacteristic
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yes—
58C3×D3×Dic3—42no—
59C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
60C32×C3⋊C8—21no—
61#5018—91no—
62#4817—91no—
63#4757—91no—
64C2×C32×D4—31no—
65C2×#634C2×D311yescharacteristic
66C22×C3×Dic3—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×D3×Dic3—21no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77#3754—34no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—121no—
87C3×#217—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104(C2×D3)⋊Dic3—21no—
105Dic32—61no—
106Dic32—61no—
107Dic32—61no—
108#3715—34no—
109#3715—34no—
110#3715—64no—
111C22×C33C2×D411yescharacteristic
112C2×C32×D3—22no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4C22×C411yescharacteristic
116C33×C4#4111yescharacteristic
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—21no—
119C3×C3⋊Dic3—21no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—41no—
122C3×C4×D4—91no—
123Q8×Dic3—91no—
124C2×#215—91no—
125D4×Dic3—31no—
126C2×#214—91no—
127C2×#214—91no—
128C2×#214—181no—
129#893—31no—
130#887—91no—
131C4×Dic6—91no—
132C3×#103—91no—
133C3×#102—91no—
134#832—31no—
135#821—31no—
136#816—31no—
137#815—91no—
138#815—91no—
139#815—181no—
140#810—31no—
141#807—91no—
142#807—91no—
143#807—91no—
144#807—181no—
145#803—91no—
146C23×C32—61no—
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—41no—
150C32×D4—32no—
151C32×D4—61no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—41no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3#6711yescharacteristic
163C2×C3⋊Dic3#6711yescharacteristic
164#634—21no—
165#634C4×D312yescomplemented
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—121no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—122no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—22no—
185C3×D12—41no—
186C3×D12C4×D312yes—
187C3×D12C4×D312yes—
188C3×C4×D3—122no—
189C3×Dic6—32no—
190C3×Dic6—32no—
191C3×Dic6—62no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—121no—
195D3×Dic3—42no—
196C32×C8—61no—
197C3⋊(C3⋊C8)—32no—
198C3⋊(C3⋊C8)—32no—
199C3⋊(C3⋊C8)—32no—
200C3⋊(C3⋊C8)—62no—
201C3⋊(C3⋊C8)—62no—
202C3⋊(C3⋊C8)—62no—
203C3⋊(C3⋊C8)—122no—
204C3×C3⋊C8—22no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—21no—
207#440—271noSylow
208C2×C33C4×D411yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C2×C33#13711yescharacteristic
211C32×D3—42no—
212C2×C3×Q8—91no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C22×C3×C4—181no—
217C22×Dic3—61no—
218C2×D12D3211yescharacteristic
219C2×C4×D3—61no—
220C2×Dic6—31no—
221C2×Dic6—31no—
222C2×Dic6—61no—
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×C22⋊C4—181no—
231C3×C42—91no—
232#217—61no—
233#215—94no—
234#214—94no—
235#214—94no—
236#214—184no—
237(C2×D3)⋊C4—61no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—181no—
242C4⋊Dic3D3211yescharacteristic
243Dic3⋊C4—181no—
244Dic3⋊C4—181no—
245C4×Dic3—31no—
246C4×Dic3—31no—
247C4×Dic3—61no—
248C4×Dic3—181no—
249C4×Dic3—181no—
250C4×Dic3—181no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—91no—
253C2×C3⋊C8—91no—
254C2×C3⋊C8—181no—
255C2×C3⋊C8—181no—
256C2×C3⋊C8—181no—
257C2×C3⋊C8—361no—
258C2×C3⋊C8D3211yescharacteristic
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32—62no—
264C22×C32—121no—
265C22×C32—121no—
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—42no—
272C2×C3×D3—41no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—81no—
277C2×C3×D3—81no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—121no—
287C32×C4C2×C4×D311yescharacteristic
288C32×C4C2×C4×D311yescharacteristic
289C32×C4C2×C4×D311yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4#23711yescharacteristic
293C3⋊Dic3—22no—
294C3⋊Dic3—21no—
295C3⋊Dic3—21no—
296C3⋊Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—62no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—61no—
303C3×Dic3—61no—
304C3×Dic3—62no—
305C3×Dic3—81no—
306C3×Dic3—122no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C2×SD16—271no—
313C4×Q8—271no—
314C4×D4—91no—
315#107—271no—
316#103—91no—
317#102—91no—
318#97—271no—
319C33#44011yescharacteristic, complemented, Sylow
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—121no—
323C22×D3—21no—
324C3×Q8—92no—
325C3×Q8—181no—
326C3×D4—32no—
327C3×D4—32no—
328C3×D4—62no—
329C3×D4—61no—
330C3×D4—61no—
331C3×D4—121no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—41no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—121no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—362no—
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
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—61no—
357C2×Dic3—61no—
358C2×Dic3—121no—
359C2×Dic3—121no—
360C2×Dic3—122no—
361D12—21no—
362D12#62412yes—
363C4×D3—122no—
364Dic6—32no—
365Dic6—32no—
366Dic6—62no—
367Dic6—61no—
368Dic6—61no—
369Dic6—121no—
370C3×C8—62no—
371C3×C8—61no—
372C3×C8—61no—
373C3⋊C8—21no—
374C3⋊C8—92no—
375C3⋊C8—92no—
376C3⋊C8—92no—
377C3⋊C8—182no—
378C3⋊C8—182no—
379C3⋊C8—182no—
380C3⋊C8—362no—
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—122no—
394C2×C32#67⋊C411yescharacteristic
395C2×C32#67⋊C411yescharacteristic
396C2×C32#91111yescharacteristic
397C2×C32#91411yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91811yescharacteristic
400C2×C32C4×#6711yescharacteristic
401C2×C32#93211yescharacteristic
402C2×C32#95111yescharacteristic
403C3×D3—42no—
404C3×D3—42no—
405C3×D3—82no—
406C2×Q8—91no—
407C2×D4—31no—
408C22×C4—181no—
409SD16—274no—
410C2×C8—31no—
411C2×C8—271no—
412C4⋊C4—31no—
413C4⋊C4—91no—
414C4⋊C4—541no—
415C22⋊C4—181no—
416C42—91no—
417C42—541no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—41no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—122no—
425C22×C3—121no—
426C22×C3—121no—
427C22×C3—121no—
428C22×C3—121no—
429C22×C3—241no—
430C22×C3—241no—
431C22×C3#380211yescharacteristic
432C22×C3D3×#6711yescharacteristic
433C22×C3D3×#6711yescharacteristic
434C2×D3—22no—
435C2×D3—41no—
436C2×D3—41no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—181no—
448C3×C4—181no—
449C3×C4—182no—
450C3×C4—241no—
451C3×C4—361no—
452C3×C4#379511yescharacteristic
453C3×C4C4×D3211yescharacteristic
454C3×C4C4×D3211yescharacteristic
455C3×C4#380411yescharacteristic
456C3×C4#380511yescharacteristic
457C3×C4C2×#62411yescharacteristic
458Dic3—41no—
459Dic3—62no—
460Dic3—62no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—61no—
465Dic3—122no—
466Dic3—121no—
467Dic3—121no—
468Dic3—121no—
469Dic3—121no—
470Dic3—241no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#475711yescharacteristic, complemented
476C32#481711yescharacteristic, complemented
477C32#501811yescharacteristic, complemented
478C23—61no—
479Q8—92no—
480Q8—181no—
481D4—32no—
482D4—61no—
483C2×C4—61no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—362no—
489C2×C4D3311yescharacteristic
490C8—61no—
491C8—272no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—122no—
505C2×C3—122no—
506C2×C3—242no—
507C2×C3#6379811yescharacteristic
508C2×C3#6381111yescharacteristic
509C2×C3#6387911yescharacteristic
510C2×C3#6390211yescharacteristic
511C2×C3#6392911yescharacteristic
512C2×C3#6393211yescharacteristic
513C2×C3#6393411yescharacteristic
514C2×C3#6394311yescharacteristic
515C2×C3#6395211yescharacteristic
516D3—42no—
517C22—62no—
518C22—121no—
519C22—121no—
520C22#8895311yescharacteristic
521C4—121no—
522C4—181no—
523C4—181no—
524C4—182no—
525C4—361no—
526C4D3×#62411yescharacteristic
527C4#8896111yescharacteristic
528C3—21no—
529C3—21no—
530C3—21no—
531C3—41no—
532C3#1059047811yescharacteristic, complemented
533C3#1059074311yescharacteristic, complemented
534C3#1059140511yescharacteristic, complemented
535C2—122no—
536C2#1172287511yescharacteristic
537C2#1172289111yescharacteristic
538C2#1172293611yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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
1. #11723668C2semidirect
4. #11722056C2semidirect
5. #11722020C2semidirect
6. C3×#63567C2semidirect
21. #88792C4semidirect
42. #63567D3semidirect
54. C3×#634C2×C4semidirect
59. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
165. #634C4×D3semidirect
319. C33#440semidirect
475. C32#4757semidirect
476. C32#4817semidirect
477. C32#5018semidirect
532. C3#10590478semidirect
533. C3#10590743semidirect
534. C3#10591405semidirect

Maximal subgroups

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

subgroupindex
#117236682
C2×#887922
C3×#638982
#117220562
#117220202
C3×#635672
C3×#635632
#105914053
#105907433
#105904783