Finite Group Databasebeta

← #13155672

Subgroups — #13155672

Order 1728.

678 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic
4#11722075C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591566—31nomaximal
9#10590766—31nomaximal
10#10590362—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3717C414yescomplemented
23C4×C4⋊(C3⋊D3)—31no—
24C3×Q8×Dic3—31no—
25C2×C3×#215—31no—
26C2×C3×#204—31no—
27C3×C4×Dic6—31no—
28Dic3×D12—31no—
29Dic3×D12—31no—
30Dic3×D12—61no—
31Dic3×Dic6D311yescharacteristic, complemented
32C2×#3717D311yescharacteristic
33#63662—31no—
34#63645—31no—
35C3×#815—31no—
36C3×#803—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63566—31no—
41#63566—31no—
42#63566—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—
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yes—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58Dic3×C3⋊D3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5164—91no—
62#5026—91no—
63#4710—91no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—61no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C2×D311yescharacteristic
73C2×D3×Dic3—62no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—61no—
76C2×C32×C8—31no—
77C32×C42—31no—
78#3752—61no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—121no—
86C4⋊(C3⋊Dic3)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#215—34no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C3×#204—34no—
106Dic3⋊Dic3—21no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110Dic32—21no—
111#3717Dic314yescomplemented
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—41no—
120C32×Dic3—22no—
121C32×Dic3—21no—
122C32×Dic3—21no—
123C2×C3×SD16—91no—
124C3×C4×Q8—91no—
125Q8×Dic3—31no—
126C2×#215—31no—
127D4×Dic3—91no—
128C2×#204—31no—
129C4×D12—91no—
130C4×D12—91no—
131C4×D12—181no—
132C4×Dic6—31no—
133C3×#97—91no—
134#835—91no—
135#835—91no—
136#835—181no—
137#832—91no—
138#821—91no—
139#818—91no—
140#818—91no—
141#818—181no—
142#816—91no—
143#815—31no—
144#809—91no—
145#807—91no—
146#803—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—62no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C32×Q8—32no—
152C32×Q8—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—31no—
157C2×C32×C4—31no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—122no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)C2×Dic312yes—
173C4×C3⋊D3—122no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×D12—32no—
187C3×D12—32no—
188C3×D12—62no—
189C3×D12—62no—
190C3×D12—61no—
191C3×D12—61no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—22no—
195C3×Dic6—41no—
196C3×Dic6C4×D312yescomplemented
197C3×Dic6C2×Dic312yescomplemented
198D3×Dic3—124no—
199D3×Dic3—122no—
200D3×Dic3—122no—
201C32×C8—32no—
202C3⋊(C3⋊C8)—21no—
203C3×C3⋊C8—22no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—121no—
207C3×C3⋊C8C4×D312yescomplemented
208C3×C3⋊C8C2×Dic312yescomplemented
209#440—271noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33AGL(1,ℤ/8)11yescharacteristic
212C2×C33#13711yescharacteristic
213C3×C3⋊D3—42no—
214C2×C3×Q8—31no—
215C2×C3×Q8—31no—
216C2×C3×Q8—61no—
217C2×C3×D4—91no—
218C22×Dic3—181no—
219C2×D12—31no—
220C2×D12—31no—
221C2×D12—61no—
222C2×C4×D3—181no—
223C2×C4×D3—181no—
224C2×C4×D3—182no—
225C2×Dic6D3211yescharacteristic
226C3×SD16—94no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—61no—
230C2×C3×C8—91no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—181no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—61no—
236C3×C42—181no—
237#217—181no—
238#215—34no—
239(C2×D3)⋊C4—181no—
240(C2×D3)⋊C4—181no—
241(C2×D3)⋊C4—182no—
242C4⋊Dic3—31no—
243C4⋊Dic3—31no—
244C4⋊Dic3—61no—
245C4⋊Dic3—61no—
246C4⋊Dic3—91no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3—181no—
250C4⋊Dic3—181no—
251C4⋊Dic3—181no—
252C4⋊Dic3—361no—
253Dic3⋊C4—61no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3D3211yescharacteristic
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8D3211yescharacteristic
261#204—34no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32C2×D1211yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3×D3—64no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—122no—
276C2×C3×D3—122no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—61no—
290C32×C4—62no—
291C32×C4—61no—
292C32×C4C2×C4×D311yescharacteristic
293C32×C4C2×C4×D311yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4#23511yescharacteristic
296C32×C4#23711yescharacteristic
297C32×C4C22×Dic311yescharacteristic
298C3⋊Dic3—41no—
299C3⋊Dic3—121no—
300C3⋊Dic3—122no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—241no—
304C3⋊Dic3—241no—
305C3⋊Dic3—242no—
306C3×Dic3—22no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—21no—
312C3×Dic3—21no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318C2×SD16—91no—
319C4×Q8—91no—
320C4×D4—271no—
321#107—271no—
322#103—271no—
323#102—271no—
324#97—91no—
325C33#44011yescharacteristic, complemented, Sylow
326C23×C3—181no—
327C22×D3—62no—
328C22×D3—61no—
329C22×D3—61no—
330C3×Q8—32no—
331C3×Q8—32no—
332C3×Q8—62no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×Q8—121no—
336C3×D4—92no—
337C3×D4—181no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—41no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—121no—
351C2×C3×C4—181no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—121no—
359C2×Dic3—182no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—361no—
364C2×Dic3—361no—
365C2×Dic3—362no—
366C2×Dic3—362no—
367C2×Dic3#62611yescharacteristic
368C2×Dic3#62611yescharacteristic
369D12—32no—
370D12—32no—
371D12—62no—
372D12—62no—
373D12—61no—
374D12—61no—
375C4×D3—364no—
376C4×D3—362no—
377C4×D3—362no—
378Dic6—21no—
379Dic6D3×Dic312yescomplemented
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—62no—
383C3×C8—181no—
384C3⋊C8—61no—
385C3⋊C8—61no—
386C3⋊C8—121no—
387C3⋊C8D3×Dic312yescomplemented
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32C4×D1211yescharacteristic
401C2×C32#90811yescharacteristic
402C2×C32#90911yescharacteristic
403C2×C32#91411yescharacteristic
404C2×C32#91511yescharacteristic
405C2×C32C4×#6711yescharacteristic
406C2×C32D4×Dic311yescharacteristic
407C2×C32#94211yescharacteristic
408C2×C32#94911yescharacteristic
409C3⋊D3—42no—
410C3×D3—124no—
411C3×D3—122no—
412C3×D3—122no—
413C2×Q8—31no—
414C2×D4—91no—
415C22×C4—541no—
416SD16—94no—
417C2×C8—31no—
418C2×C8—91no—
419C4⋊C4—91no—
420C4⋊C4—181no—
421C4⋊C4—271no—
422C22⋊C4—541no—
423C42—31no—
424C42—181no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—41no—
429C22×C3—182no—
430C22×C3—361no—
431C22×C3—361no—
432C22×C3D3×D1211yescharacteristic
433C22×C3C2×#62611yescharacteristic
434C22×C3D3×#6711yescharacteristic
435C2×D3—64no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—62no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—61no—
458C3×C4—61no—
459C3×C4—121no—
460C3×C4—122no—
461C3×C4—361no—
462C3×C4#379511yescharacteristic
463C3×C4#379811yescharacteristic
464C3×C4C4×D3211yescharacteristic
465C3×C4C2×D3×Dic311yescharacteristic
466C3×C4C2×D3×Dic311yescharacteristic
467C3×C4#380411yescharacteristic
468Dic3—21no—
469Dic3—21no—
470Dic3—22no—
471Dic3—121no—
472Dic3—121no—
473Dic3—241no—
474Dic3—362no—
475Dic3—361no—
476Dic3—361no—
477Dic3—361no—
478Dic3—721no—
479Dic3—721no—
480Dic3—722no—
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#471011yescharacteristic, complemented
486C32#502611yescharacteristic, complemented
487C32#516411yescharacteristic, complemented
488C23—181no—
489Q8—32no—
490Q8—61no—
491D4—92no—
492D4—181no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—61no—
496C2×C4—181no—
497C2×C4—541no—
498C2×C4—1082no—
499C2×C4D3311yescharacteristic
500C8—32no—
501C8—181no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—362no—
515C2×C3#6379411yescharacteristic
516C2×C3#6380011yescharacteristic
517C2×C3#6382511yescharacteristic
518C2×C3#6383311yescharacteristic
519C2×C3Dic3×D1211yescharacteristic
520C2×C3C4×#62611yescharacteristic
521C2×C3#6393911yescharacteristic
522C2×C3#6394611yescharacteristic
523C2×C3Dic3×#6711yescharacteristic
524D3—124no—
525D3—122no—
526D3—122no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22D3×#62611yescharacteristic
531C4—61no—
532C4—61no—
533C4—62no—
534C4—361no—
535C4—1081no—
536C4D32×Dic311yescharacteristic
537C4#8896011yescharacteristic
538C3—21no—
539C3—21no—
540C3—21no—
541C3—41no—
542C3#1059036211yescharacteristic, complemented
543C3#1059076611yescharacteristic, complemented
544C3#1059156611yescharacteristic, complemented
545C2—362no—
546C2#1172287111yescharacteristic
547C2#1172288411yescharacteristic
548C2Dic3×#62611yescharacteristic
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. C3×Dic3×Dic6C2semidirect
5. #11722055C2semidirect
6. #11722038C2semidirect
7. C3×#63554C2semidirect
22. C3×#3717C4semidirect
31. Dic3×Dic6D3semidirect
46. #63554D3semidirect
57. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3717Dic3semidirect
196. C3×Dic6C4×D3semidirect
197. C3×Dic6C2×Dic3semidirect
207. C3×C3⋊C8C4×D3semidirect
208. C3×C3⋊C8C2×Dic3semidirect
325. C33#440semidirect
379. Dic6D3×Dic3semidirect
387. C3⋊C8D3×Dic3semidirect
485. C32#4710semidirect
486. C32#5026semidirect
487. C32#5164semidirect
542. C3#10590362semidirect
543. C3#10590766semidirect
544. C3#10591566semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊(C3⋊D3)2
C3×Dic3×Dic62
C2×C3×#37172
#117220752
#117220552
#117220382
C3×#635542
#105915663
#105907663
#105903623