Finite Group Databasebeta

← #13155722

Subgroups — #13155722

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88810C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic, complemented
4#11722054C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590746—31nomaximal
9#10590455—31nomaximal
10#10590346—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×C3×#624—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88810C414yescomplemented
21#88803—21no—
22C3×#3722—21no—
23C4×C4⋊(C3⋊D3)—31no—
24C3×#893—31no—
25C3×C4×D12—31no—
26#63894D311yescharacteristic, complemented
27Dic3×D12—31no—
28Dic3×D12—31no—
29Dic3×D12—61no—
30C2×#3714—31no—
31C2×#3714—31no—
32C2×#3713—31no—
33#63648—31no—
34#63635—31no—
35C3×#821—31no—
36C3×#815—31no—
37C3×#809—31no—
38C3×#803—31no—
39#63573—31no—
40#63573—31no—
41#63573—61no—
42#63564D311yescharacteristic, complemented
43#63563—31no—
44#63563—31no—
45#63563—61no—
46#63554D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yescomplemented
54C3×C4⋊(C3⋊D3)C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58Dic3×C3⋊D3—42no—
59C3×#624—42no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61C32×C3⋊C8—21no—
62#5014—91no—
63#4785—91no—
64#4711—91no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—61no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×D12—61no—
74C2×C3×C4×D3—61no—
75C2×C3×C4×D3—61no—
76C2×#624—21no—
77C2×D3×Dic3—62no—
78C2×D3×Dic3—61no—
79C2×D3×Dic3—61no—
80C2×C32×C8—31no—
81C32×C4⋊C4—31no—
82C32×C42—31no—
83#3752—61no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—61no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—121no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106#3722—21no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110#3714—34no—
111#3714—34no—
112#3713—34no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3—41no—
118C2×C3×C3⋊D3—41no—
119C2×C3×C3⋊D3—41no—
120C33×C4C22×C411yescharacteristic
121C33×C4#4111yescharacteristic
122C3⋊(C3⋊Dic3)—41no—
123C32×Dic3—21no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C3×C4×D4—91no—
127D4×Dic3—91no—
128C2×#213—91no—
129C2×#213—91no—
130C2×D24—91no—
131#893—31no—
132C4×D12—31no—
133C4×D12—91no—
134C4×D12—91no—
135C4×D12—181no—
136C3×#102—91no—
137C3×#97—91no—
138#832—91no—
139#821—31no—
140#821—91no—
141#821—91no—
142#821—181no—
143#817—91no—
144#815—31no—
145#809—31no—
146#808—91no—
147#808—91no—
148#808—181no—
149#803—31no—
150C22×C3⋊D3—21no—
151C22×C3⋊D3—21no—
152C22×C3×D3—62no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—62no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—121no—
174C2×C3⋊Dic3—122no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)C4×D312yescomplemented
178C4⋊(C3⋊D3)C4×D312yescomplemented
179C4×C3⋊D3—122no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3D1211yescharacteristic
186C2×C3×Dic3D1211yescharacteristic
187C2×C3×Dic3#6711yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C3×D12—32no—
190C3×D12—32no—
191C3×D12—32no—
192C3×D12—32no—
193C3×D12—62no—
194C3×D12—62no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×C4×D3—122no—
202C3×C4×D3—122no—
203#624—42no—
204D3×Dic3—124no—
205D3×Dic3—122no—
206D3×Dic3—122no—
207C32×C8—61no—
208C3⋊(C3⋊C8)C4×D312yescomplemented
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—32no—
212C3×C3⋊C8—32no—
213C3×C3⋊C8—41no—
214C3×C3⋊C8—62no—
215#442—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33AGL(1,ℤ/8)11yescharacteristic
219C3×C3⋊D3—42no—
220C3×C3⋊D3—42no—
221C2×C3×D4—91no—
222C2×C3×D4—91no—
223C22×C3×C4—181no—
224C22×Dic3—181no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—31no—
228C2×D12—31no—
229C2×D12—61no—
230C2×D12—61no—
231C2×C4×D3—61no—
232C2×C4×D3—61no—
233C2×C4×D3—181no—
234C2×C4×D3—181no—
235C2×C4×D3—182no—
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—61no—
239C2×C3×C8—91no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C22⋊C4—181no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247#217—181no—
248#213—94no—
249#213—94no—
250(C2×D3)⋊C4—61no—
251(C2×D3)⋊C4—61no—
252(C2×D3)⋊C4—181no—
253(C2×D3)⋊C4—181no—
254(C2×D3)⋊C4—182no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—181no—
259C4⋊Dic3—181no—
260C4⋊Dic3—181no—
261C4⋊Dic3—361no—
262C4⋊Dic3D3211yescharacteristic
263C4×Dic3D3211yescharacteristic
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8D3211yescharacteristic
268D24—94no—
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—22no—
278C2×C3⋊D3—41no—
279C2×C3⋊D3—41no—
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3×D3—64no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—241no—
299C2×C3×D3—241no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—61no—
309C32×C4—61no—
310C32×C4—121no—
311C32×C4C2×C4×D311yescharacteristic
312C32×C4C2×C4×D311yescharacteristic
313C32×C4C2×C4×D311yescharacteristic
314C32×C4#23511yescharacteristic
315C32×C4#23511yescharacteristic
316C32×C4#23711yescharacteristic
317C3⋊Dic3—121no—
318C3⋊Dic3—122no—
319C3⋊Dic3—121no—
320C3⋊Dic3—121no—
321C3⋊Dic3—241no—
322C3⋊Dic3—241no—
323C3⋊Dic3—242no—
324C3×Dic3—22no—
325C3×Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C3×Dic3—41no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—81no—
333C2×D8—271no—
334C4×D4—91no—
335C4×D4—271no—
336#106—271no—
337#102—91no—
338#102—271no—
339#97—91no—
340C33#44211yescharacteristic, complemented, Sylow
341C23×C3—181no—
342C23×C3—181no—
343C22×D3—62no—
344C22×D3—61no—
345C22×D3—61no—
346C22×D3—61no—
347C22×D3—61no—
348C22×D3—121no—
349C3×D4—92no—
350C3×D4—92no—
351C3×D4—181no—
352C3×D4—181no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—121no—
366C2×C3×C4—362no—
367C2×C3×C4C2×D3211yescharacteristic
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—182no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3—361no—
376C2×Dic3—361no—
377C2×Dic3—362no—
378C2×Dic3—362no—
379C2×Dic3#62611yescharacteristic
380C2×Dic3#62611yescharacteristic
381D12—32no—
382D12—32no—
383D12—32no—
384D12—32no—
385D12—62no—
386D12—62no—
387D12—62no—
388D12—61no—
389D12—61no—
390D12—61no—
391D12—61no—
392D12—121no—
393C4×D3—122no—
394C4×D3—122no—
395C4×D3—364no—
396C4×D3—362no—
397C4×D3—362no—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—92no—
401C3×C8—121no—
402C3⋊C8—21no—
403C3⋊C8—32no—
404C3⋊C8—32no—
405C3⋊C8—62no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32C4×D1211yescharacteristic
419C2×C32#67⋊C411yescharacteristic
420C2×C32#90811yescharacteristic
421C2×C32#90811yescharacteristic
422C2×C32#91111yescharacteristic
423C2×C32#91411yescharacteristic
424C2×C32C4×#6711yescharacteristic
425C2×C32#93211yescharacteristic
426C2×C32#94911yescharacteristic
427C3⋊D3—42no—
428C3⋊D3—42no—
429C3×D3—124no—
430C3×D3—122no—
431C3×D3—122no—
432C3×D3—122no—
433C3×D3—122no—
434C3×D3—242no—
435C2×D4—91no—
436C2×D4—91no—
437C22×C4—181no—
438C22×C4—541no—
439D8—274no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4—271no—
444C22⋊C4—181no—
445C22⋊C4—541no—
446C42—31no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—361no—
454C22×C3—361no—
455C22×C3—361no—
456C22×C3—361no—
457C22×C3D3×D1211yescharacteristic
458C22×C3C2×#62611yescharacteristic
459C22×C3D3×#6711yescharacteristic
460C2×D3—64no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—121no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—121no—
476C2×D3—241no—
477C2×D3—241no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—62no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—61no—
490C3×C4—61no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—241no—
495C3×C4#379511yescharacteristic
496C3×C4#379811yescharacteristic
497C3×C4C4×D3211yescharacteristic
498C3×C4C4×D3211yescharacteristic
499C3×C4C4×D3211yescharacteristic
500C3×C4#380411yescharacteristic
501Dic3—21no—
502Dic3—21no—
503Dic3—41no—
504Dic3—362no—
505Dic3—361no—
506Dic3—361no—
507Dic3—361no—
508Dic3—721no—
509Dic3—721no—
510Dic3—722no—
511C32—21no—
512C32—21no—
513C32—21no—
514C32—41no—
515C32#471111yescharacteristic, complemented
516C32#478511yescharacteristic, complemented
517C32#501411yescharacteristic, complemented
518C23—181no—
519C23—181no—
520D4—92no—
521D4—92no—
522D4—181no—
523D4—181no—
524C2×C4—31no—
525C2×C4—31no—
526C2×C4—61no—
527C2×C4—362no—
528C2×C4—541no—
529C2×C4—1082no—
530C2×C4D3311yescharacteristic
531C8—61no—
532C8—92no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—362no—
546C2×C3—362no—
547C2×C3#6379411yescharacteristic
548C2×C3#6379511yescharacteristic
549C2×C3#6382511yescharacteristic
550C2×C3#6383011yescharacteristic
551C2×C3#6387911yescharacteristic
552C2×C3#6388011yescharacteristic
553C2×C3C4×#62611yescharacteristic
554C2×C3#6393211yescharacteristic
555C2×C3#6393911yescharacteristic
556D3—124no—
557D3—122no—
558D3—122no—
559D3—122no—
560D3—122no—
561D3—242no—
562C22—182no—
563C22—182no—
564C22—361no—
565C22—361no—
566C22—361no—
567C22—361no—
568C22D3×#62611yescharacteristic
569C4—61no—
570C4—61no—
571C4—121no—
572C4—1081no—
573C4#8894911yescharacteristic
574C4#8896011yescharacteristic
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059034611yescharacteristic, complemented
580C3#1059045511yescharacteristic, complemented
581C3#1059074611yescharacteristic, complemented
582C2—362no—
583C2—362no—
584C2#1172286911yescharacteristic
585C2#1172287111yescharacteristic
586C2#1172293411yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypocenterC33#442
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×C4⋊(C3⋊D3)C2semidirect
3. C3×#63894C2semidirect
4. #11722054C2semidirect
5. #11722034C2semidirect
6. C3×#63564C2semidirect
7. C3×#63554C2semidirect
20. #88810C4semidirect
26. #63894D3semidirect
42. #63564D3semidirect
46. #63554D3semidirect
53. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
177. C4⋊(C3⋊D3)C4×D3semidirect
178. C4⋊(C3⋊D3)C4×D3semidirect
208. C3⋊(C3⋊C8)C4×D3semidirect
340. C33#442semidirect
515. C32#4711semidirect
516. C32#4785semidirect
517. C32#5014semidirect
579. C3#10590346semidirect
580. C3#10590455semidirect
581. C3#10590746semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
Dic3×C4⋊(C3⋊D3)2
C3×#638942
#117220542
#117220342
C3×#635642
C3×#635542
#105907463
#105904553
#105903463