Finite Group Databasebeta

← #13155716

Subgroups — #13155716

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
1#11723691C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic
4#11722052C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63555C211yesmaximal, characteristic, complemented
8#10591436—31nomaximal
9#10590771—31nomaximal
10#10590746—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6294—21no—
14C2×D3×C3⋊Dic3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88814—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3714C414yescomplemented
23D4×C3⋊Dic3—31no—
24C2×C3×#213—31no—
25C2×C3×D24—31no—
26#63898—31no—
27#63894—31no—
28Dic3×D12—31no—
29Dic3×D12—31no—
30Dic3×D12—31no—
31Dic3×D12—61no—
32C2×#3714D311yescharacteristic
33#63648—31no—
34#63644—31no—
35C3×#832—31no—
36C3×#821—31no—
37C3×#815—31no—
38C3×#815—31no—
39#63573—31no—
40#63573—31no—
41#63573—61no—
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63563D311yescharacteristic, complemented
46#63555D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)C2×C412yescomplemented
56C32×D12—21no—
57C32×D12C2×C412yescomplemented
58#6294—42no—
59D3×C3⋊Dic3—42no—
60C32×C3⋊C8—21no—
61C32×C3⋊C8C2×C412yescomplemented
62#5152—91no—
63#4785—91no—
64#4711—91no—
65C2×C32×D4—31no—
66C22×C3⋊Dic3—61no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C2×D311yescharacteristic
74C2×#624—61no—
75C2×D3×Dic3—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C2×D3×Dic3—62no—
79C2×D3×Dic3—61no—
80C2×C32×C8—31no—
81C2×C32×C8—31no—
82#3757—61no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—61no—
87C4⋊(C3⋊Dic3)—61no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—121no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91C4×C3⋊Dic3C2×D311yescharacteristic
92C3×#213—34no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C3×D24—34no—
106#3722—61no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—61no—
112#3714Dic314yescomplemented
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C2×C32×D3—22no—
118C2×C32×D3—41no—
119C2×C32×D3—41no—
120C33×C4C22×C411yescharacteristic
121C33×C4#4111yescharacteristic
122C3⋊(C3⋊Dic3)—41no—
123C3×C3⋊Dic3—21no—
124C3×C3⋊Dic3—21no—
125C3×C3⋊Dic3—41no—
126C2×C3×D8—91no—
127D4×Dic3—91no—
128D4×Dic3—91no—
129D4×Dic3—91no—
130D4×Dic3—181no—
131C2×#213—31no—
132C2×D24—31no—
133#893—91no—
134C4×D12—91no—
135C4×D12—91no—
136C3×#102—91no—
137C3×#97—91no—
138#832—31no—
139#832—91no—
140#821—31no—
141#821—91no—
142#821—91no—
143#821—181no—
144#817—91no—
145#817—91no—
146#817—181no—
147#815—31no—
148#815—31no—
149#808—91no—
150C23×C32—61no—
151C22×C3⋊D3—21no—
152C22×C3×D3—22no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C32×D4—32no—
159C32×D4—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—21no—
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—
175C2×C3⋊Dic3—122no—
176C2×C3⋊Dic3D1211yescharacteristic
177C2×C3⋊Dic3D1211yescharacteristic
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×Dic312yescomplemented
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C3×D12—22no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—22no—
193C3×D12—32no—
194C3×D12—32no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—121no—
199C3×D12C4×D312yescomplemented
200C3×D12C2×Dic312yescomplemented
201#624—122no—
202D3×Dic3—122no—
203D3×Dic3—124no—
204D3×Dic3—122no—
205D3×Dic3—122no—
206D3×Dic3—122no—
207C32×C8—32no—
208C32×C8—61no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8C4×D312yescomplemented
214C3×C3⋊C8C2×Dic312yescomplemented
215#442—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33AGL(1,ℤ/8)11yescharacteristic
219C3×C3⋊D3—42no—
220C32×D3—42no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—61no—
224C2×C3×D4—91no—
225C22×Dic3—181no—
226C22×Dic3—181no—
227C22×Dic3—181no—
228C22×Dic3—182no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×D12D3211yescharacteristic
233C2×C4×D3—181no—
234C2×C4×D3—181no—
235C2×C4×D3—181no—
236C3×D8—94no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—61no—
243C3×C4⋊C4—91no—
244C3×C42—91no—
245#217—181no—
246#217—181no—
247#217—181no—
248#217—182no—
249#213—34no—
250(C2×D3)⋊C4—181no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—181no—
253C4⋊Dic3—31no—
254C4⋊Dic3—31no—
255C4⋊Dic3—61no—
256C4⋊Dic3—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—91no—
259C4⋊Dic3—181no—
260C4⋊Dic3—181no—
261C4⋊Dic3—181no—
262C4⋊Dic3—361no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—61no—
266C2×C3⋊C8D3211yescharacteristic
267C2×C3⋊C8D3211yescharacteristic
268D24—34no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32C2×D1211yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32D3×D411yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3×D3—24no—
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C2×C3×D3—41no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—122no—
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×C4C2×C4×D311yescharacteristic
309C32×C4C2×C4×D311yescharacteristic
310C32×C4#23511yescharacteristic
311C32×C4#23711yescharacteristic
312C32×C4#23711yescharacteristic
313C32×C4C22×Dic311yescharacteristic
314C3⋊Dic3—21no—
315C3⋊Dic3—21no—
316C3⋊Dic3—41no—
317C3⋊Dic3—121no—
318C3⋊Dic3—122no—
319C3⋊Dic3—121no—
320C3⋊Dic3—121no—
321C3⋊Dic3—241no—
322C3⋊Dic3—241no—
323C3⋊Dic3—242no—
324C3×Dic3—62no—
325C3×Dic3—61no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—121no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C3×Dic3—241no—
333C2×D8—91no—
334C4×D4—271no—
335C4×D4—271no—
336#106—271no—
337#102—91no—
338#102—271no—
339#97—91no—
340C33#44211yescharacteristic, complemented, Sylow
341C23×C3—62no—
342C23×C3—61no—
343C23×C3—61no—
344C23×C3—181no—
345C22×D3—21no—
346C22×D3—61no—
347C22×D3—61no—
348C22×D3—121no—
349C3×D4—32no—
350C3×D4—32no—
351C3×D4—62no—
352C3×D4—62no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—92no—
356C3×D4—181no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—181no—
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×Dic3—31no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—121no—
376C2×Dic3—182no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—361no—
381C2×Dic3—361no—
382C2×Dic3—362no—
383C2×Dic3—362no—
384C2×Dic3—364no—
385C2×Dic3—362no—
386C2×Dic3—362no—
387D12—21no—
388D12—32no—
389D12—32no—
390D12—62no—
391D12—61no—
392D12—61no—
393D12—121no—
394D12D3×Dic312yescomplemented
395C4×D3—362no—
396C4×D3—362no—
397C4×D3—362no—
398C3×C8—32no—
399C3×C8—32no—
400C3×C8—62no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—121no—
404C3⋊C8—21no—
405C3⋊C8D3×Dic312yescomplemented
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×C32—122no—
419C2×C32C4×D1211yescharacteristic
420C2×C32#67⋊C411yescharacteristic
421C2×C32#90811yescharacteristic
422C2×C32#90811yescharacteristic
423C2×C32#91111yescharacteristic
424C2×C32#91111yescharacteristic
425C2×C32#91111yescharacteristic
426C2×C32#91411yescharacteristic
427C2×C32D4×Dic311yescharacteristic
428C3⋊D3—42no—
429C3×D3—44no—
430C3×D3—42no—
431C3×D3—42no—
432C3×D3—122no—
433C3×D3—122no—
434C3×D3—242no—
435C2×D4—31no—
436C2×D4—91no—
437C22×C4—541no—
438C22×C4—541no—
439D8—94no—
440C2×C8—31no—
441C2×C8—31no—
442C4⋊C4—91no—
443C4⋊C4—271no—
444C22⋊C4—541no—
445C22⋊C4—541no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—64no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—122no—
455C22×C3—122no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3—182no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3D3×D1211yescharacteristic
464C22×C3D3×D1211yescharacteristic
465C22×C3#381111yescharacteristic
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—122no—
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—181no—
487C3×C4—181no—
488C3×C4—361no—
489C3×C4#379511yescharacteristic
490C3×C4C4×D3211yescharacteristic
491C3×C4C2×D3×Dic311yescharacteristic
492C3×C4C2×D3×Dic311yescharacteristic
493C3×C4#380411yescharacteristic
494C3×C4#380511yescharacteristic
495Dic3—62no—
496Dic3—61no—
497Dic3—61no—
498Dic3—61no—
499Dic3—61no—
500Dic3—121no—
501Dic3—121no—
502Dic3—121no—
503Dic3—241no—
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#515211yescharacteristic, complemented
518C23—61no—
519C23—181no—
520D4—32no—
521D4—61no—
522D4—92no—
523D4—181no—
524C2×C4—91no—
525C2×C4—91no—
526C2×C4—181no—
527C2×C4—541no—
528C2×C4—1082no—
529C2×C4—1082no—
530C2×C4D3311yescharacteristic
531C8—32no—
532C8—61no—
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—124no—
546C2×C3—122no—
547C2×C3—122no—
548C2×C3—362no—
549C2×C3#6379411yescharacteristic
550C2×C3#6379511yescharacteristic
551C2×C3#6379511yescharacteristic
552C2×C3#6379511yescharacteristic
553C2×C3#6388011yescharacteristic
554C2×C3Dic3×D1211yescharacteristic
555C2×C3#6392711yescharacteristic
556C2×C3#6393711yescharacteristic
557C2×C3#67⋊Dic311yescharacteristic
558D3—42no—
559D3—122no—
560D3—122no—
561D3—242no—
562C22—62no—
563C22—121no—
564C22—121no—
565C22—182no—
566C22—361no—
567C22—361no—
568C22#8895511yescharacteristic
569C4—181no—
570C4—181no—
571C4—361no—
572C4—1081no—
573C4D32×Dic311yescharacteristic
574C4#8896111yescharacteristic
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059074611yescharacteristic, complemented
580C3#1059077111yescharacteristic, complemented
581C3#1059143611yescharacteristic, complemented
582C2—122no—
583C2—362no—
584C2#1172287911yescharacteristic
585C2#1172288011yescharacteristic
586C2#1172292811yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
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
1. #11723691C2semidirect
2. D12×C3⋊Dic3C2semidirect
4. #11722052C2semidirect
5. #11722034C2semidirect
6. C3×#63563C2semidirect
7. C3×#63555C2semidirect
22. C3×#3714C4semidirect
45. #63563D3semidirect
46. #63555D3semidirect
55. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
112. #3714Dic3semidirect
179. C4⋊(C3⋊D3)C2×Dic3semidirect
199. C3×D12C4×D3semidirect
200. C3×D12C2×Dic3semidirect
213. C3×C3⋊C8C4×D3semidirect
214. C3×C3⋊C8C2×Dic3semidirect
340. C33#442semidirect
394. D12D3×Dic3semidirect
405. C3⋊C8D3×Dic3semidirect
515. C32#4711semidirect
516. C32#4785semidirect
517. C32#5152semidirect
579. C3#10590746semidirect
580. C3#10590771semidirect
581. C3#10591436semidirect

Maximal subgroups

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

subgroupindex
#117236912
D12×C3⋊Dic32
C2×C3×#37142
#117220522
#117220342
C3×#635632
C3×#635552
#105914363
#105907713
#105907463