Finite Group Databasebeta

← #13155684

Subgroups — #13155684

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#11723399C211yesmaximal, characteristic, complemented
2C2×#88796C211yesmaximal, characteristic
3C3×#63894C211yesmaximal, characteristic
4#11722075C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590773—31nomaximal
9#10590590—31nomaximal
10#10590360—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#624—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#88796C414yescomplemented
22C3×#3722—21no—
23C2×#3744—31no—
24C4×#634—31no—
25C3×#893—31no—
26C3×C4×D12—31no—
27#63894D311yescharacteristic
28#63844—31no—
29#63844—31no—
30#63844—61no—
31C2×#3716—31no—
32C2×#3716—31no—
33C2×#3716—61no—
34#63637—31no—
35C3×#818—31no—
36C3×#815—31no—
37C3×#807—31no—
38C3×#803—31no—
39#63575D311yescharacteristic, complemented
40#63567—31no—
41#63567—31no—
42#63567—61no—
43#63564—31no—
44#63563—31no—
45#63562—31no—
46#63554D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308C2×C412yescomplemented
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)C2×C412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×#624—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5018—91no—
62#4889—91no—
63#4710—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—31no—
68C2×#634—61no—
69C2×#634—61no—
70C2×#634—61no—
71C2×#634—121no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×#624—21no—
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80C32×C42—31no—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82#3750—61no—
83C4×C3⋊Dic3—61no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85#3744—34no—
86C3×(C2×D3)⋊C4—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103#3724—62no—
104#3724—61no—
105#3724—61no—
106#3722—21no—
107Dic32—62no—
108Dic32—61no—
109Dic32—61no—
110#3716—34no—
111#3716—34no—
112#3716—64no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—21no—
122C32×Dic3—21no—
123C32×Dic3—41no—
124C3×C4×D4—91no—
125C2×#214—91no—
126C2×#204—91no—
127C2×#204—91no—
128C2×#204—181no—
129#893—31no—
130#887—91no—
131C4×D12—31no—
132C4×Dic6—91no—
133C4×Dic6—91no—
134C4×Dic6—181no—
135C3×#107—91no—
136C3×#97—91no—
137#832—91no—
138#821—91no—
139#818—31no—
140#816—91no—
141#815—31no—
142#810—91no—
143#810—91no—
144#810—181no—
145#809—91no—
146#807—31no—
147#803—31no—
148C22×C3⋊D3—21no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
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—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)C4×D312yes—
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—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—41no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3D1211yescharacteristic
195C2×C3×Dic3D1211yescharacteristic
196C2×C3×Dic3#6711yescharacteristic
197C2×C3×Dic3#6711yescharacteristic
198C3×D12—32no—
199C3×D12—32no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204C3×C4×D3—122no—
205C3×C4×D3—122no—
206#624—42no—
207C32×C8—32no—
208C3⋊(C3⋊C8)—21no—
209C3×C3⋊C8—22no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8—121no—
213C3×C3⋊C8C4×D312yescomplemented
214C3×C3⋊C8C4×D312yescomplemented
215#440—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33#13711yescharacteristic
219C3×C3⋊D3—42no—
220C2×C3×D4—91no—
221C22×C3×C4—181no—
222C2×D12—31no—
223C2×D12—31no—
224C2×D12—61no—
225C2×C4×D3—61no—
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—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—91no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—91no—
242C3×C22⋊C4—181no—
243C3×C42—31no—
244C3×C42—31no—
245C3×C42—61no—
246#214—94no—
247(C2×D3)⋊C4—61no—
248(C2×D3)⋊C4—61no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3D3211yescharacteristic
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—182no—
256Dic3⋊C4—181no—
257C4×Dic3—181no—
258C4×Dic3—181no—
259C4×Dic3—182no—
260C4×Dic3—181no—
261C4×Dic3D3211yescharacteristic
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8D3211yescharacteristic
266#204—94no—
267#204—94no—
268#204—184no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—122no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—241no—
287C2×C3×D3—241no—
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—61no—
297C32×C4—61no—
298C32×C4—121no—
299C32×C4C2×C4×D311yescharacteristic
300C32×C4C2×C4×D311yescharacteristic
301C32×C4C2×C4×D311yescharacteristic
302C32×C4#23511yescharacteristic
303C32×C4#23511yescharacteristic
304C32×C4#23911yescharacteristic
305C3⋊Dic3—41no—
306C3⋊Dic3—62no—
307C3⋊Dic3—64no—
308C3⋊Dic3—62no—
309C3⋊Dic3—62no—
310C3⋊Dic3—122no—
311C3⋊Dic3—122no—
312C3⋊Dic3—124no—
313C3×Dic3—22no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—21no—
317C3×Dic3—21no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—81no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C2×SD16—271no—
326C4×Q8—271no—
327C4×D4—91no—
328#107—91no—
329#103—271no—
330#102—271no—
331#97—91no—
332C33#44011yescharacteristic, complemented, Sylow
333C23×C3—181no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—121no—
337C3×D4—92no—
338C3×D4—181no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—41no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4—362no—
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—62no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—182no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—361no—
366C2×Dic3—361no—
367C2×Dic3—362no—
368C2×Dic3#62611yescharacteristic
369C2×Dic3#62611yescharacteristic
370D12—32no—
371D12—32no—
372D12—62no—
373D12—61no—
374D12—61no—
375D12—121no—
376C4×D3—122no—
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—32no—
393C3×C8—32no—
394C3×C8—62no—
395C3×C8—181no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—121no—
399C3⋊C8#62412yescomplemented
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×C32C4×D1211yescharacteristic
413C2×C32#89311yescharacteristic
414C2×C32#90811yescharacteristic
415C2×C32#90911yescharacteristic
416C2×C32#91111yescharacteristic
417C2×C32#91811yescharacteristic
418C2×C32C4×#6711yescharacteristic
419C2×C32#93211yescharacteristic
420C2×C32#95111yescharacteristic
421C3⋊D3—42no—
422C3×D3—122no—
423C3×D3—122no—
424C3×D3—242no—
425C2×Q8—271no—
426C2×D4—91no—
427C22×C4—181no—
428SD16—274no—
429C2×C8—31no—
430C2×C8—91no—
431C4⋊C4—31no—
432C4⋊C4—91no—
433C4⋊C4—541no—
434C22⋊C4—181no—
435C42—31no—
436C42—541no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—182no—
442C22×C3—361no—
443C22×C3—361no—
444C22×C3D3×D1211yescharacteristic
445C22×C3C2×#62611yescharacteristic
446C22×C3D3×#6711yescharacteristic
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—122no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—241no—
455C2×D3—241no—
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—62no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—61no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—121no—
472C3×C4—241no—
473C3×C4—361no—
474C3×C4#379811yescharacteristic
475C3×C4#379911yescharacteristic
476C3×C4#379911yescharacteristic
477C3×C4C4×D3211yescharacteristic
478C3×C4C4×D3211yescharacteristic
479C3×C4C2×#62411yescharacteristic
480Dic3—21no—
481Dic3—21no—
482Dic3—41no—
483Dic3—122no—
484Dic3—121no—
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#471011yescharacteristic, complemented
498C32#488911yescharacteristic, complemented
499C32#501811yescharacteristic, complemented
500C23—181no—
501Q8—272no—
502Q8—541no—
503D4—92no—
504D4—181no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—61no—
508C2×C4—181no—
509C2×C4—362no—
510C2×C4—541no—
511C2×C4D3311yescharacteristic
512C8—32no—
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—362no—
527C2×C3#6379511yescharacteristic
528C2×C3#6380111yescharacteristic
529C2×C3#6383011yescharacteristic
530C2×C3#6383111yescharacteristic
531C2×C3#6389411yescharacteristic
532C2×C3C4×#62611yescharacteristic
533C2×C3#6390411yescharacteristic
534C2×C3#6393211yescharacteristic
535C2×C3#6394311yescharacteristic
536D3—122no—
537D3—122no—
538D3—242no—
539C22—182no—
540C22—361no—
541C22—361no—
542C22D3×#62611yescharacteristic
543C4—61no—
544C4—61no—
545C4—121no—
546C4—361no—
547C4—542no—
548C4D3×#62411yescharacteristic
549C4#8896511yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059036011yescharacteristic, complemented
555C3#1059059011yescharacteristic, complemented
556C3#1059077311yescharacteristic, complemented
557C2—362no—
558C2#1172286911yescharacteristic
559C2#1172288711yescharacteristic
560C2#1172293311yescharacteristic
subgroupG/H
centerC22D3×#626
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
1. #11723399C2semidirect
5. #11722041C2semidirect
6. C3×#63575C2semidirect
7. C3×#63554C2semidirect
21. #88796C4semidirect
39. #63575D3semidirect
46. #63554D3semidirect
51. #6308C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
213. C3×C3⋊C8C4×D3semidirect
214. C3×C3⋊C8C4×D3semidirect
332. C33#440semidirect
399. C3⋊C8#624semidirect
497. C32#4710semidirect
498. C32#4889semidirect
499. C32#5018semidirect
554. C3#10590360semidirect
555. C3#10590590semidirect
556. C3#10590773semidirect

Maximal subgroups

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

subgroupindex
#117233992
C2×#887962
C3×#638942
#117220752
#117220412
C3×#635752
C3×#635542
#105907733
#105905903
#105903603