Finite Group Databasebeta

← #13155688

Subgroups — #13155688

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723406C211yesmaximal, characteristic, complemented
2C2×#88794C211yesmaximal, characteristic
3C3×#63898C211yesmaximal, characteristic
4#11722071C211yesmaximal, characteristic, complemented
5#11722041C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8#10591405—31nomaximal
9#10590590—31nomaximal
10#10590461—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#88794C414yescomplemented
22C3×(C2×D3)⋊Dic3—21no—
23C2×#3754—31no—
24#64096—31no—
25C3×D4×Dic3—31no—
26C3×#893—31no—
27#63898D311yescharacteristic
28#63844—31no—
29#63844—31no—
30#63844—61no—
31C2×#3716—31no—
32C2×#3716—31no—
33C2×#3716—61no—
34#63637—31no—
35C3×#832—31no—
36C3×#809—31no—
37C3×#807—31no—
38C3×#807—31no—
39#63576D311yescharacteristic, complemented
40#63567—31no—
41#63567—31no—
42#63567—61no—
43#63562D311yescharacteristic
44#63555—31no—
45#63554—31no—
46#63554—31no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308C2×C412yescomplemented
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yes—
58C3×D3×Dic3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61#5018—91no—
62#4889—91no—
63#4757—91no—
64C2×C32×D4—31no—
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—31no—
68C2×#634—61no—
69C2×#634—61no—
70C2×#634—61no—
71C2×#634—121no—
72C22×C3×Dic3—61no—
73C2×C3×D12—21no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×C4×D3—61no—
77C2×D3×Dic3—21no—
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80#3754—34no—
81#3750—61no—
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#217—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8—61no—
103#3724—61no—
104#3724—61no—
105#3724—62no—
106(C2×D3)⋊Dic3—21no—
107Dic32—61no—
108Dic32—61no—
109Dic32—62no—
110#3716—34no—
111#3716—34no—
112#3716—64no—
113C22×C33C2×D411yescharacteristic
114C2×C32×D3—22no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—21no—
122C32×Dic3—41no—
123C32×Dic3—41no—
124C3×C4×D4—91no—
125D4×Dic3—31no—
126C2×#214—91no—
127C2×#214—91no—
128C2×#214—181no—
129C2×#204—91no—
130#893—31no—
131#887—91no—
132#887—91no—
133#887—181no—
134C4×Dic6—91no—
135C3×#107—91no—
136C3×#102—91no—
137#832—31no—
138#816—91no—
139#815—91no—
140#815—91no—
141#810—91no—
142#810—91no—
143#810—181no—
144#809—31no—
145#807—31no—
146#807—31no—
147#803—91no—
148C23×C32—61no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—41no—
152C32×D4—32no—
153C32×D4—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—122no—
170C2×C3⋊Dic3#6711yescharacteristic
171C2×C3⋊Dic3#6711yescharacteristic
172#634—32no—
173#634—32no—
174#634—32no—
175#634—62no—
176#634—62no—
177#634—62no—
178#634—61no—
179#634—62no—
180#634—61no—
181#634—61no—
182#634—122no—
183#634—121no—
184#634—121no—
185#634—122no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—41no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—122no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—22no—
202C3×D12—41no—
203C3×D12C4×D312yes—
204C3×D12C4×D312yes—
205C3×C4×D3—122no—
206D3×Dic3—42no—
207C3⋊(C3⋊C8)—21no—
208C3⋊(C3⋊C8)C4×D312yescomplemented
209C3×C3⋊C8—32no—
210C3×C3⋊C8—32no—
211C3×C3⋊C8—62no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—121no—
215#440—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33#13711yescharacteristic
219C32×D3—42no—
220C2×C3×D4—31no—
221C2×C3×D4—31no—
222C2×C3×D4—61no—
223C22×C3×C4—181no—
224C22×Dic3—61no—
225C2×D12D3211yescharacteristic
226C2×C4×D3—61no—
227C2×Dic6—91no—
228C2×Dic6—91no—
229C2×Dic6—91no—
230C2×Dic6—181no—
231C2×Dic6—181no—
232C2×Dic6—181no—
233C2×Dic6—361no—
234C2×C3×C8—91no—
235C2×C3×C8—91no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—61no—
242C3×C22⋊C4—181no—
243C3×C42—91no—
244#217—61no—
245#214—94no—
246#214—94no—
247#214—184no—
248(C2×D3)⋊C4—61no—
249C4⋊Dic3D3211yescharacteristic
250C4⋊Dic3D3211yescharacteristic
251Dic3⋊C4—181no—
252Dic3⋊C4—181no—
253Dic3⋊C4—181no—
254Dic3⋊C4—182no—
255C4×Dic3—31no—
256C4×Dic3—31no—
257C4×Dic3—61no—
258C4×Dic3—181no—
259C4×Dic3—181no—
260C4×Dic3—181no—
261C4×Dic3—182no—
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8—61no—
268#204—94no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32D3×D411yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32C2×#6711yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—42no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—121no—
297C32×C4—121no—
298C32×C4C2×C4×D311yescharacteristic
299C32×C4C2×C4×D311yescharacteristic
300C32×C4C2×C4×D311yescharacteristic
301C32×C4#23511yescharacteristic
302C32×C4#23711yescharacteristic
303C32×C4#23911yescharacteristic
304C3⋊Dic3—21no—
305C3⋊Dic3—21no—
306C3⋊Dic3—62no—
307C3⋊Dic3—64no—
308C3⋊Dic3—62no—
309C3⋊Dic3—62no—
310C3⋊Dic3—122no—
311C3⋊Dic3—122no—
312C3⋊Dic3—124no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—62no—
319C3×Dic3—61no—
320C3×Dic3—61no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C3×Dic3—81no—
324C3×Dic3—121no—
325C2×SD16—271no—
326C4×Q8—271no—
327C4×D4—91no—
328#107—91no—
329#103—271no—
330#102—91no—
331#97—271no—
332C33#44011yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C22×D3—21no—
337C3×D4—32no—
338C3×D4—32no—
339C3×D4—62no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—121no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—41no—
347C2×C3×C4—62no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—121no—
355C2×C3×C4—362no—
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—122no—
368C2×Dic3—182no—
369C2×Dic3—181no—
370C2×Dic3—181no—
371C2×Dic3—181no—
372C2×Dic3—361no—
373C2×Dic3—361no—
374C2×Dic3—362no—
375D12—21no—
376D12#62412yes—
377C4×D3—122no—
378Dic6—92no—
379Dic6—92no—
380Dic6—92no—
381Dic6—182no—
382Dic6—182no—
383Dic6—182no—
384Dic6—182no—
385Dic6—181no—
386Dic6—181no—
387Dic6—181no—
388Dic6—362no—
389Dic6—361no—
390Dic6—361no—
391Dic6—362no—
392C3×C8—92no—
393C3×C8—181no—
394C3⋊C8—32no—
395C3⋊C8—32no—
396C3⋊C8—62no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—121no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—122no—
413C2×C32#67⋊C411yescharacteristic
414C2×C32#89311yescharacteristic
415C2×C32#91111yescharacteristic
416C2×C32#91111yescharacteristic
417C2×C32#91511yescharacteristic
418C2×C32#91811yescharacteristic
419C2×C32C4×#6711yescharacteristic
420C2×C32#93211yescharacteristic
421C2×C32#95111yescharacteristic
422C3×D3—42no—
423C3×D3—42no—
424C3×D3—82no—
425C2×Q8—271no—
426C2×D4—31no—
427C22×C4—181no—
428SD16—274no—
429C2×C8—91no—
430C2×C8—91no—
431C4⋊C4—31no—
432C4⋊C4—31no—
433C4⋊C4—541no—
434C22⋊C4—181no—
435C42—91no—
436C42—541no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—122no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3—241no—
450C22×C3D3×#6711yescharacteristic
451C22×C3D3×#6711yescharacteristic
452C22×C3#381111yescharacteristic
453C2×D3—22no—
454C2×D3—41no—
455C2×D3—41no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—122no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—121no—
469C3×C4—181no—
470C3×C4—181no—
471C3×C4—241no—
472C3×C4#379611yescharacteristic
473C3×C4#379911yescharacteristic
474C3×C4C4×D3211yescharacteristic
475C3×C4C4×D3211yescharacteristic
476C3×C4#380411yescharacteristic
477C3×C4C2×#62411yescharacteristic
478Dic3—41no—
479Dic3—41no—
480Dic3—62no—
481Dic3—61no—
482Dic3—61no—
483Dic3—61no—
484Dic3—61no—
485Dic3—121no—
486Dic3—184no—
487Dic3—182no—
488Dic3—182no—
489Dic3—182no—
490Dic3—362no—
491Dic3—362no—
492Dic3—364no—
493C32—21no—
494C32—21no—
495C32—21no—
496C32—41no—
497C32#475711yescharacteristic, complemented
498C32#488911yescharacteristic, complemented
499C32#501811yescharacteristic, complemented
500C23—61no—
501Q8—272no—
502Q8—541no—
503D4—32no—
504D4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—362no—
510C2×C4—541no—
511C2×C4D3311yescharacteristic
512C8—92no—
513C8—181no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—122no—
527C2×C3—122no—
528C2×C3—242no—
529C2×C3#6388111yescharacteristic
530C2×C3#6390411yescharacteristic
531C2×C3#6392711yescharacteristic
532C2×C3#6392911yescharacteristic
533C2×C3#6393211yescharacteristic
534C2×C3#6393211yescharacteristic
535C2×C3#6394311yescharacteristic
536C2×C3#6395111yescharacteristic
537C2×C3#6395211yescharacteristic
538D3—42no—
539C22—62no—
540C22—121no—
541C22—121no—
542C22#8895211yescharacteristic
543C4—121no—
544C4—121no—
545C4—181no—
546C4—181no—
547C4—542no—
548C4D3×#62411yescharacteristic
549C4#8896211yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059046111yescharacteristic, complemented
555C3#1059059011yescharacteristic, complemented
556C3#1059140511yescharacteristic, complemented
557C2—122no—
558C2#1172287211yescharacteristic
559C2#1172288811yescharacteristic
560C2#1172293511yescharacteristic
subgroupG/H
centerC22#88952
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. #11723406C2semidirect
4. #11722071C2semidirect
5. #11722041C2semidirect
6. C3×#63576C2semidirect
21. #88794C4semidirect
39. #63576D3semidirect
51. #6308C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
208. C3⋊(C3⋊C8)C4×D3semidirect
332. C33#440semidirect
497. C32#4757semidirect
498. C32#4889semidirect
499. C32#5018semidirect
554. C3#10590461semidirect
555. C3#10590590semidirect
556. C3#10591405semidirect

Maximal subgroups

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

subgroupindex
#117234062
C2×#887942
C3×#638982
#117220712
#117220412
C3×#635762
C3×#635622
#105914053
#105905903
#105904613