Finite Group Databasebeta

← #13155604

Subgroups — #13155604

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88809C211yesmaximal, characteristic, complemented
2C4×#6290C211yesmaximal, characteristic
3C2×C3×#3711C211yesmaximal, characteristic, complemented
4#11722050C211yesmaximal, characteristic, complemented
5#11722018C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591051—31nomaximal
9#10590735—31nomaximal
10#10590734—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6290C411yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88809—22no—
20#88806C411yescharacteristic
21#88803C411yescharacteristic
22(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
23C3×#3711—22no—
24C4×#638—31no—
25C2×C3×#208—31no—
26C2×C3×ΓL(1,𝔽25)—31no—
27C4×#625—31no—
28C4×#625—31no—
29C4×#625—61no—
30C2×#3712—31no—
31C2×#3711—31no—
32C2×#3711—31no—
33C2×#3711D311yescharacteristic, complemented
34#63641—31no—
35C8×C3⋊Dic3—31no—
36C3×#830—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×(C2×D3)⋊C8—31no—
40#63572—31no—
41#63572—31no—
42#63572—61no—
43#63557D311yescharacteristic, complemented
44(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
45#63553—31no—
46#63553—31no—
47#63553—61no—
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
52C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
53C2×C3×C3⋊Dic3C2×C411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×C4×C3⋊D3—21no—
56C2×C32×Dic3C2×C411yescharacteristic
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59#6290—22no—
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)D412yescomplemented
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64#5128—91no—
65#4729—91no—
66#4729—91no—
67C22×C32×C4—31no—
68C2×#638—31no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3C2×D311yescharacteristic
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×#625—31no—
77C2×#625—31no—
78C2×#625—61no—
79C2×C32×C8—31no—
80C2×C32×C8—31no—
81#3757—31no—
82#3752—31no—
83#3750—31no—
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
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×#208—62no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×ΓL(1,𝔽25)—62no—
107#3724—31no—
108#3724—31no—
109#3724—61no—
110(C2×D3)⋊Dic3—31no—
111(C2×D3)⋊Dic3—31no—
112(C2×D3)⋊Dic3—31no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—61no—
116#3712—62no—
117#3711—22no—
118#3711—62no—
119#3711—62no—
120C22×C33C22×C411yescharacteristic
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3—21no—
124C2×C32×D3—21no—
125C2×C32×D3—21no—
126C2×C32×D3—21no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3⋊(C3⋊Dic3)—21no—
130C3⋊(C3⋊Dic3)M1612yescomplemented
131C3×C3⋊Dic3—21no—
132C32×Dic3—21no—
133C2×C3×M16—91no—
134C4×#67—91no—
135C4×#67—91no—
136C4×#67—91no—
137C4×#67—181no—
138C2×#208—31no—
139C2×#208—91no—
140C2×ΓL(1,𝔽25)—31no—
141C2×ΓL(1,𝔽25)—91no—
142C2×ΓL(1,𝔽25)—91no—
143C3×C22⋊C8—91no—
144C3×C22⋊C8—91no—
145#830—31no—
146(C2×D3)⋊C8—31no—
147(C2×D3)⋊C8—31no—
148(C2×D3)⋊C8—31no—
149Dic3⋊C8—91no—
150Dic3⋊C8—91no—
151Dic3⋊C8—91no—
152Dic3⋊C8—181no—
153C8×Dic3—91no—
154C8×Dic3—91no—
155C8×Dic3—91no—
156C8×Dic3—181no—
157C23×C32—31no—
158C22×C3⋊D3C4×D311yescharacteristic
159C22×C3×D3—21no—
160C22×C3×D3—31no—
161C22×C3×D3—31no—
162C22×C3×D3—61no—
163C22×C3×D3C4×D311yescharacteristic
164C22×C3×D3C4×D311yescharacteristic
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175#638—62no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—61no—
183C2×C3⋊Dic3—61no—
184C2×C3⋊Dic3—61no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—61no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—121no—
189C2×C3⋊Dic3—121no—
190C2×C3⋊Dic3C4×D311yescharacteristic
191C4×C3⋊D3—21no—
192C4×C3⋊D3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3C4×D311yescharacteristic
198C2×C3×Dic3C4×D311yescharacteristic
199C3×C4×D3—21no—
200C3×C4×D3—21no—
201C3×C4×D3—21no—
202C3×C4×D3—21no—
203C3×C4×D3—41no—
204C3×C4×D3—41no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—121no—
210C3×C4×D3—121no—
211#625—62no—
212#625—62no—
213#625—64no—
214C32×C8—61no—
215C32×C8—61no—
216C3⋊(C3⋊C8)—21no—
217C3⋊(C3⋊C8)#6712yescomplemented
218C3×C3⋊C8—22no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—41no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—121no—
230#436—271noSylow
231C2×C33C4×D411yescharacteristic
232C2×C33C2×M1611yescharacteristic
233C2×C33#13111yescharacteristic
234C3×C3⋊D3—41no—
235C32×D3—41no—
236C22×C3×C4—31no—
237C22×C3×C4—31no—
238C22×C3×C4—61no—
239C22×C3×C4—91no—
240C2×#67—91no—
241C2×#67—91no—
242C2×#67—91no—
243C2×#67—181no—
244C2×C4×D3—31no—
245C2×C4×D3—31no—
246C2×C4×D3—61no—
247C2×C4×D3D3211yescharacteristic
248C3×M16—182no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C2×C3×C8—91no—
256C2×C3×C8—91no—
257#217—91no—
258#217—91no—
259#217—91no—
260#217—181no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—181no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—181no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—181no—
273C4×Dic3—181no—
274C4×Dic3—181no—
275C4×Dic3—361no—
276#208—62no—
277#208—182no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8D3211yescharacteristic
285C2×C3⋊C8D3211yescharacteristic
286ΓL(1,𝔽25)—62no—
287ΓL(1,𝔽25)—182no—
288ΓL(1,𝔽25)—182no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—61no—
294C22×C32—61no—
295C22×C32—61no—
296C22×C32C2×C4×D311yescharacteristic
297C22×C32C2×C4×D311yescharacteristic
298C22×C32C2×C4×D311yescharacteristic
299C2×C3⋊D3—21no—
300C2×C3⋊D3—21no—
301C2×C3⋊D3—21no—
302C2×C3×D3—22no—
303C2×C3×D3—21no—
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—62no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4—61no—
329C32×C4#23511yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4#23711yescharacteristic
334C32×C4C2×#6711yescharacteristic
335C3⋊Dic3—21no—
336C3⋊Dic3—32no—
337C3⋊Dic3—32no—
338C3⋊Dic3—32no—
339C3⋊Dic3—62no—
340C3⋊Dic3—62no—
341C3⋊Dic3—62no—
342C3⋊Dic3—61no—
343C3⋊Dic3—62no—
344C3⋊Dic3—61no—
345C3⋊Dic3—61no—
346C3⋊Dic3—122no—
347C3⋊Dic3—121no—
348C3⋊Dic3—121no—
349C3⋊Dic3—122no—
350C3×Dic3—21no—
351C3×Dic3—21no—
352C3×Dic3—41no—
353C3×Dic3—61no—
354C3×Dic3—61no—
355C3×Dic3—121no—
356C2×M16—91no—
357C2×M16—271no—
358C4×D4—271no—
359C4⋊C8—271no—
360C22⋊C8—91no—
361C22⋊C8—91no—
362C4×C8—271no—
363C33#43611yescharacteristic, complemented, Sylow
364C23×C3—31no—
365C23×C3—31no—
366C23×C3—61no—
367C23×C3—91no—
368C22×D3—31no—
369C22×D3—31no—
370C22×D3—61no—
371C22×D3#62411yescharacteristic
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—41no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—61no—
383C2×C3×C4—91no—
384C2×C3×C4—121no—
385C2×C3×C4—121no—
386C2×C3×C4—181no—
387C2×C3×C4—181no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—182no—
392#67—182no—
393#67—182no—
394#67—184no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—61no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—91no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—361no—
411C2×Dic3—361no—
412C2×Dic3#62411yescharacteristic
413C4×D3—21no—
414C4×D3—21no—
415C4×D3—61no—
416C4×D3—61no—
417C4×D3—61no—
418C4×D3—61no—
419C4×D3—121no—
420C4×D3—121no—
421C3×C8—62no—
422C3×C8—61no—
423C3×C8—61no—
424C3×C8—61no—
425C3×C8—61no—
426C3×C8—92no—
427C3×C8—121no—
428C3×C8—181no—
429C3⋊C8—21no—
430C3⋊C8—21no—
431C3⋊C8—32no—
432C3⋊C8—32no—
433C3⋊C8—62no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C3⋊C8—121no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—121no—
450C2×C32#67⋊C411yescharacteristic
451C2×C32#67⋊C411yescharacteristic
452C2×C32#90111yescharacteristic
453C2×C32#90111yescharacteristic
454C2×C32D3×M1611yescharacteristic
455C2×C32D3×M1611yescharacteristic
456C2×C32D3×M1611yescharacteristic
457C2×C32#90711yescharacteristic
458C2×C32C4×#6711yescharacteristic
459C3⋊D3—41no—
460C3×D3—42no—
461C3×D3—41no—
462C3×D3—41no—
463C3×D3—122no—
464C3×D3—121no—
465C3×D3—121no—
466C2×D4—271no—
467C22×C4—31no—
468C22×C4—91no—
469M16—182no—
470M16—542no—
471C2×C8—31no—
472C2×C8—31no—
473C2×C8—91no—
474C2×C8—91no—
475C4⋊C4—271no—
476C22⋊C4—271no—
477C22⋊C4—271no—
478C42—271no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—41no—
483C22×C3—62no—
484C22×C3—61no—
485C22×C3—61no—
486C22×C3—61no—
487C22×C3—61no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—121no—
491C22×C3—121no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3—181no—
495C22×C3C4×D3211yescharacteristic
496C22×C3C4×D3211yescharacteristic
497C22×C3C2×#62411yescharacteristic
498C2×D3—21no—
499C2×D3—21no—
500C2×D3—21no—
501C2×D3—62no—
502C2×D3—61no—
503C2×D3—61no—
504C2×D3—61no—
505C2×D3—61no—
506C2×D3—61no—
507C2×D3—61no—
508C2×D3—121no—
509C2×D3—121no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—61no—
519C3×C4—61no—
520C3×C4—121no—
521C3×C4—181no—
522C3×C4#379511yescharacteristic
523C3×C4#380211yescharacteristic
524C3×C4#380411yescharacteristic
525C3×C4#380511yescharacteristic
526C3×C4D3×#6711yescharacteristic
527C3×C4D3×#6711yescharacteristic
528Dic3—21no—
529Dic3—61no—
530Dic3—61no—
531Dic3—92no—
532Dic3—92no—
533Dic3—92no—
534Dic3—121no—
535Dic3—182no—
536Dic3—182no—
537Dic3—182no—
538Dic3—182no—
539Dic3—181no—
540Dic3—181no—
541Dic3—181no—
542Dic3—362no—
543Dic3—361no—
544Dic3—361no—
545Dic3—362no—
546C32—21no—
547C32—21no—
548C32—21no—
549C32—41no—
550C32#472911yescharacteristic, complemented
551C32#472911yescharacteristic, complemented
552C32#512811yescharacteristic, complemented
553C23—31no—
554C23—91no—
555D4—542no—
556C2×C4—31no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—91no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4—271no—
563C2×C4—271no—
564C2×C4D3311yescharacteristic
565C8—61no—
566C8—61no—
567C8—92no—
568C8—181no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—122no—
582C2×C3—121no—
583C2×C3—121no—
584C2×C3—361no—
585C2×C3#6379111yescharacteristic
586C2×C3#6379111yescharacteristic
587C2×C3#6380311yescharacteristic
588C2×C3#6380411yescharacteristic
589C2×C3#6381711yescharacteristic
590C2×C3#6381811yescharacteristic
591C2×C3#6387911yescharacteristic
592C2×C3#6390211yescharacteristic
593C2×C3#6395211yescharacteristic
594D3—41no—
595D3—122no—
596D3—121no—
597D3—121no—
598C22—61no—
599C22—61no—
600C22—61no—
601C22—181no—
602C22—181no—
603C22—181no—
604C22D3×#62411yescharacteristic
605C4—61no—
606C4—181no—
607C4—272no—
608C4—541no—
609C4#8895311yescharacteristic
610C4#8896111yescharacteristic
611C3—21no—
612C3—21no—
613C3—21no—
614C3—41no—
615C3#1059073411yescharacteristic, complemented
616C3#1059073511yescharacteristic, complemented
617C3#1059105111yescharacteristic, complemented
618C2—121no—
619C2—361no—
620C2#1172281911yescharacteristic
621C2#1172282411yescharacteristic
622C2#1172293611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×#6290C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88809C2semidirect
3. C2×C3×#3711C2semidirect
4. #11722050C2semidirect
5. #11722018C2semidirect
6. C3×#63557C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
33. C2×#3711D3semidirect
43. #63557D3semidirect
44. (C2×D3)⋊(C3⋊C8)D3semidirect
61. C3×C3⋊(C3⋊C8)D4semidirect
130. C3⋊(C3⋊Dic3)M16semidirect
217. C3⋊(C3⋊C8)#67semidirect
363. C33#436semidirect
550. C32#4729semidirect
551. C32#4729semidirect
552. C32#5128semidirect
615. C3#10590734semidirect
616. C3#10590735semidirect
617. C3#10591051semidirect

Maximal subgroups

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

subgroupindex
C2×#888092
C4×#62902
C2×C3×#37112
#117220502
#117220182
C3×#635572
C3×(C2×D3)⋊(C3⋊C8)2
#105910513
#105907353
#105907343