Finite Group Databasebeta

← #13155682

Subgroups — #13155682

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590795—31nomaximal
9#10590478—31nomaximal
10#10590360—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#624—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811C414yescomplemented
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22C3×#3722—21no—
23C4×#634—31no—
24C3×#893—31no—
25C3×C4×D12—31no—
26#63894D311yescharacteristic
27Dic3×Dic6—31no—
28Dic3×Dic6—31no—
29Dic3×Dic6—61no—
30C2×#3717—31no—
31C2×#3716—31no—
32C2×#3715—31no—
33#63648—31no—
34#63634—31no—
35C3×#816—31no—
36C3×#815—31no—
37C3×#810—31no—
38C3×#803—31no—
39#63575—31no—
40#63575—31no—
41#63575—61no—
42#63567D311yescharacteristic, complemented
43#63563—31no—
44#63563—31no—
45#63563—61no—
46#63554D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yes—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×#624—42no—
59C3×C3⋊(C3⋊C8)C2×C412yescomplemented
60C32×C3⋊C8—21no—
61#5018—91no—
62#4817—91no—
63#4710—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634C2×D311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×#624—21no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77C32×C42—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—121no—
85#3750—61no—
86C4×C3⋊Dic3—61no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×(C2×D3)⋊C4—61no—
89C3×(C2×D3)⋊C4—61no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4×Dic3—21no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105#3722—21no—
106Dic32—62no—
107Dic32—61no—
108Dic32—61no—
109#3717—34no—
110#3716—34no—
111#3715—34no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4C22×C411yescharacteristic
117C33×C4#4111yescharacteristic
118C3⋊(C3⋊Dic3)—41no—
119C3×C3⋊Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—21no—
122C32×Dic3—41no—
123C3×C4×D4—91no—
124Q8×Dic3—91no—
125C2×#215—91no—
126C2×#214—91no—
127C2×#204—91no—
128#893—31no—
129C4×D12—31no—
130C4×Dic6—91no—
131C4×Dic6—91no—
132C4×Dic6—181no—
133C3×#103—91no—
134C3×#97—91no—
135#832—91no—
136#821—91no—
137#821—91no—
138#821—181no—
139#818—91no—
140#816—31no—
141#815—31no—
142#810—31no—
143#807—91no—
144#807—91no—
145#807—181no—
146#803—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—121no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—62no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—121no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—122no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)C4×D312yes—
171#634—21no—
172#634C4×D312yescomplemented
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3D1211yescharacteristic
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3#6711yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C3×D12—32no—
186C3×D12—32no—
187C3×D12—62no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—121no—
191C3×C4×D3—122no—
192C3×C4×D3—122no—
193C3×Dic6—32no—
194C3×Dic6—32no—
195C3×Dic6—62no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199#624—42no—
200C32×C8—61no—
201C3⋊(C3⋊C8)C4×D312yescomplemented
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—32no—
205C3×C3⋊C8—32no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—62no—
208#440—271noSylow
209C2×C33C4×D411yescharacteristic
210C2×C33AGL(1,ℤ/8)11yescharacteristic
211C2×C33#13711yescharacteristic
212C3×C3⋊D3—42no—
213C2×C3×Q8—91no—
214C2×C3×D4—91no—
215C22×C3×C4—181no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—61no—
219C2×C4×D3—61no—
220C2×C4×D3—61no—
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C22⋊C4—181no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42—61no—
235#215—94no—
236#214—94no—
237(C2×D3)⋊C4—61no—
238(C2×D3)⋊C4—61no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—181no—
243C4⋊Dic3—181no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—361no—
247C4⋊Dic3D3211yescharacteristic
248Dic3⋊C4—181no—
249Dic3⋊C4—181no—
250Dic3⋊C4—182no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C4×Dic3—182no—
254C4×Dic3—181no—
255C4×Dic3D3211yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8D3211yescharacteristic
260#204—94no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32C2×D1211yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32C2×#6711yescharacteristic
268C2×C3⋊D3—22no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—241no—
279C2×C3×D3—241no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—61no—
289C32×C4—61no—
290C32×C4—121no—
291C32×C4C2×C4×D311yescharacteristic
292C32×C4C2×C4×D311yescharacteristic
293C32×C4C2×C4×D311yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4#23511yescharacteristic
296C32×C4#23711yescharacteristic
297C3⋊Dic3—22no—
298C3⋊Dic3—121no—
299C3⋊Dic3—122no—
300C3⋊Dic3—121no—
301C3⋊Dic3—121no—
302C3⋊Dic3—241no—
303C3⋊Dic3—241no—
304C3⋊Dic3—242no—
305C3×Dic3—22no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—64no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—81no—
317C2×SD16—271no—
318C4×Q8—271no—
319C4×D4—91no—
320#107—271no—
321#103—91no—
322#102—271no—
323#97—91no—
324C33#44011yescharacteristic, complemented, Sylow
325C23×C3—181no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—121no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×D4—92no—
332C3×D4—181no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—121no—
346C2×C3×C4—181no—
347C2×C3×C4—362no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×Dic3—21no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—182no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—361no—
360C2×Dic3—361no—
361C2×Dic3—362no—
362C2×Dic3#62611yescharacteristic
363C2×Dic3#62611yescharacteristic
364D12—32no—
365D12—32no—
366D12—62no—
367D12—61no—
368D12—61no—
369D12—121no—
370C4×D3—122no—
371C4×D3—122no—
372Dic6—32no—
373Dic6—32no—
374Dic6—62no—
375Dic6—62no—
376Dic6—61no—
377Dic6—61no—
378C3×C8—61no—
379C3×C8—61no—
380C3×C8—92no—
381C3×C8—121no—
382C3⋊C8—21no—
383C3⋊C8—32no—
384C3⋊C8—32no—
385C3⋊C8—62no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32C4×D1211yescharacteristic
399C2×C32#67⋊C411yescharacteristic
400C2×C32#90811yescharacteristic
401C2×C32#90911yescharacteristic
402C2×C32#91411yescharacteristic
403C2×C32#91811yescharacteristic
404C2×C32C4×#6711yescharacteristic
405C2×C32#93211yescharacteristic
406C2×C32#95111yescharacteristic
407C3⋊D3—42no—
408C3×D3—122no—
409C3×D3—122no—
410C3×D3—242no—
411C2×Q8—91no—
412C2×D4—91no—
413C22×C4—181no—
414SD16—274no—
415C2×C8—31no—
416C2×C8—91no—
417C4⋊C4—31no—
418C4⋊C4—271no—
419C4⋊C4—541no—
420C22⋊C4—181no—
421C42—31no—
422C42—541no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—182no—
428C22×C3—361no—
429C22×C3—361no—
430C22×C3D3×D1211yescharacteristic
431C22×C3C2×#62611yescharacteristic
432C22×C3D3×#6711yescharacteristic
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—122no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—121no—
439C2×D3—121no—
440C2×D3—241no—
441C2×D3—241no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—62no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—121no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—182no—
459C3×C4—241no—
460C3×C4#379511yescharacteristic
461C3×C4#379811yescharacteristic
462C3×C4C4×D3211yescharacteristic
463C3×C4C4×D3211yescharacteristic
464C3×C4C4×D3211yescharacteristic
465C3×C4#380411yescharacteristic
466Dic3—21no—
467Dic3—21no—
468Dic3—41no—
469Dic3—64no—
470Dic3—62no—
471Dic3—62no—
472Dic3—362no—
473Dic3—361no—
474Dic3—361no—
475Dic3—361no—
476Dic3—721no—
477Dic3—721no—
478Dic3—722no—
479C32—21no—
480C32—21no—
481C32—21no—
482C32—41no—
483C32#471011yescharacteristic, complemented
484C32#481711yescharacteristic, complemented
485C32#501811yescharacteristic, complemented
486C23—181no—
487Q8—92no—
488Q8—181no—
489D4—92no—
490D4—181no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—61no—
494C2×C4—181no—
495C2×C4—362no—
496C2×C4—541no—
497C2×C4D3311yescharacteristic
498C8—61no—
499C8—92no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—362no—
513C2×C3#6379411yescharacteristic
514C2×C3#6380111yescharacteristic
515C2×C3#6383011yescharacteristic
516C2×C3#6383111yescharacteristic
517C2×C3#6387911yescharacteristic
518C2×C3#6388011yescharacteristic
519C2×C3C4×#62611yescharacteristic
520C2×C3#6393411yescharacteristic
521C2×C3#6394311yescharacteristic
522D3—122no—
523D3—122no—
524D3—242no—
525C22—182no—
526C22—361no—
527C22—361no—
528C22D3×#62611yescharacteristic
529C4—61no—
530C4—61no—
531C4—121no—
532C4—182no—
533C4—1081no—
534C4#8894911yescharacteristic
535C4#8896011yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059036011yescharacteristic, complemented
541C3#1059047811yescharacteristic, complemented
542C3#1059079511yescharacteristic, complemented
543C2—362no—
544C2#1172286811yescharacteristic
545C2#1172288711yescharacteristic
546C2#1172293411yescharacteristic
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
2. Dic3×#634C2semidirect
4. #11722057C2semidirect
6. C3×#63567C2semidirect
7. C3×#63554C2semidirect
19. #88811C4semidirect
42. #63567D3semidirect
46. #63554D3semidirect
54. C3×#634C2×C4semidirect
59. C3×C3⋊(C3⋊C8)C2×C4semidirect
172. #634C4×D3semidirect
201. C3⋊(C3⋊C8)C4×D3semidirect
324. C33#440semidirect
483. C32#4710semidirect
484. C32#4817semidirect
485. C32#5018semidirect
540. C3#10590360semidirect
541. C3#10590478semidirect
542. C3#10590795semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
Dic3×#6342
C3×#638942
#117220572
#117220342
C3×#635672
C3×#635542
#105907953
#105904783
#105903603