Finite Group Databasebeta

← #13155385

Subgroups — #13155385

Order 1728.

830 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3799C211yesmaximal, characteristic
2C2×#88786C211yesmaximal, characteristic, complemented
3C3×D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11722075C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591550—31nomaximal
9#10591087—31nomaximal
10#10590609—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#624—21no—
16C2×C3×D3×Dic3—21no—
17C3×#3799—22no—
18C3×#3799C414yescomplemented
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4⋊Dic3C2211yescharacteristic
23#88786—22no—
24C3×Dic3⋊Dic3—21no—
25C2×C3×#239—31no—
26C2×C3×#235—31no—
27C2×#3799D311yescharacteristic
28C2×#3748—31no—
29C2×C3×C4⋊Dic3—31no—
30C3×D3×C4⋊C4—31no—
31D3×C4⋊Dic3D311yescharacteristic, complemented
32C2×#3711—31no—
33C2×#3711—31no—
34C2×#3711—61no—
35#63662—31no—
36#63636—31no—
37C3×#835—31no—
38C3×#810—31no—
39#63565—31no—
40#63565—31no—
41#63565—61no—
42#63565D311yescharacteristic, complemented
43#63564—31no—
44#63563—31no—
45#63562—31no—
46#63562—31no—
47#63562—31no—
48#63562—61no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C32×C4×D3C2×C412yes—
59C32×C4×D3D412yes—
60C32×Dic6—21no—
61C32×Dic6C2×C412yescomplemented
62C3×#626—24no—
63C3×#624—24no—
64C3×D3×Dic3—42no—
65C3⋊(C3⋊(C3⋊C8))—21no—
66C3×C3⋊(C3⋊C8)—21no—
67#5232—91no—
68#4966—91no—
69#4803—91no—
70C2×C32×Q8—31no—
71C22×C32×C4—31no—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
73C2×C3×#67—61no—
74C22×C3×Dic3—61no—
75C3×#239—34no—
76C3×#239—62no—
77C3×#235—34no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3—61no—
85C2×C3×C4×D3—61no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×C3×Dic6—21no—
90C2×C3×Dic6C2×D311yescharacteristic
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#626—21no—
93C2×#624—21no—
94C2×D3×Dic3—21no—
95#3799—22no—
96#3799Dic314yescomplemented
97C32×C4⋊C4—31no—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99#3748—62no—
100C2×C3⋊(C3⋊C8)—31no—
101C2×C3⋊(C3⋊C8)—31no—
102C2×C3⋊(C3⋊C8)—31no—
103C2×C3⋊(C3⋊C8)—61no—
104C2×C3⋊(C3⋊C8)—61no—
105C2×C3⋊(C3⋊C8)—61no—
106C2×C3⋊(C3⋊C8)—121no—
107C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
108C3×C4⋊Dic3—21no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—61no—
112C3×C4⋊Dic3—61no—
113C3×C4⋊Dic3C2×D311yescharacteristic
114C3×C4⋊Dic3C2×D311yescharacteristic
115C3×Dic3⋊C4—61no—
116C2×C3×C3⋊C8—31no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—61no—
119Dic3⋊Dic3—21no—
120#3711—64no—
121#3711—62no—
122#3711—62no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—41no—
127C2×C32×D3—21no—
128C2×C32×D3—21no—
129C2×C32×D3C22⋊C412yes—
130C33×C4C22×C411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—41no—
133C32×Dic3—22no—
134C32×Dic3—41no—
135C32×Dic3C22⋊C412yes—
136C2×C3×#41—91no—
137C2×#239—31no—
138C2×#235—31no—
139C2×C3×C4⋊C4—91no—
140C2×C4⋊Dic3—31no—
141C2×#208—91no—
142C2×#208—91no—
143C2×#208—181no—
144C2×ΓL(1,𝔽25)—91no—
145D3×C4⋊C4—31no—
146C3×#103—91no—
147#835—31no—
148#835—91no—
149#835—91no—
150#835—181no—
151#832—91no—
152#832—91no—
153#821—91no—
154#810—31no—
155#810—91no—
156#809—91no—
157#809—91no—
158#809—91no—
159#809—181no—
160C23×C32—31no—
161C22×C3⋊D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—121no—
166C22×C3×D3#6711yescharacteristic
167C22×C3×D3#6711yescharacteristic
168C32×Q8—32no—
169C32×Q8—61no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—31no—
174C2×C32×C4—32no—
175C2×C32×C4—32no—
176C2×C32×C4—41no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C3⋊Dic3—21no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)C2×Dic312yescomplemented
185C3×#67—64no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—121no—
196C2×C3×Dic3—122no—
197C2×C3×Dic3#6711yescharacteristic
198C2×C3×Dic3#6711yescharacteristic
199C3×D12—32no—
200C3×D12—32no—
201C3×D12—62no—
202C3×D12—64no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—121no—
206C3×C4×D3—22no—
207C3×C4×D3—22no—
208C3×C4×D3—64no—
209C3×C4×D3—64no—
210C3×C4×D3—122no—
211C3×C4×D3C4×D312yes—
212C3×C4×D3D1212yes—
213C3×C4×D3C2×Dic312yes—
214C3×C4×D3#6712yes—
215C3×Dic6—21no—
216C3×Dic6—21no—
217C3×Dic6—22no—
218C3×Dic6—41no—
219C3×Dic6C4×D312yescomplemented
220C3×Dic6C2×Dic312yescomplemented
221#626—24no—
222#624—24no—
223D3×Dic3—42no—
224C3⋊(C3⋊C8)—21no—
225C3⋊(C3⋊C8)—61no—
226C3⋊(C3⋊C8)—62no—
227C3⋊(C3⋊C8)—61no—
228C3⋊(C3⋊C8)—61no—
229C3⋊(C3⋊C8)—121no—
230C3⋊(C3⋊C8)—121no—
231C3⋊(C3⋊C8)—122no—
232C3×C3⋊C8—62no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235#417—271noSylow
236C2×C33C2×C22⋊C411yescharacteristic
237C2×C33AGL(1,ℤ/8)11yescharacteristic
238C2×C33#13711yescharacteristic
239C3×C3⋊D3—42no—
240C32×D3—22no—
241C3×#41—94no—
242C3×#41—182no—
243C2×C3×Q8—31no—
244C2×C3×Q8—31no—
245C2×C3×Q8—61no—
246C2×C3×D4—91no—
247C2×C3×D4—181no—
248C22×C3×C4—31no—
249C22×C3×C4—31no—
250C22×C3×C4—61no—
251C22×C3×C4—181no—
252C22×C3×C4—181no—
253C2×#67—61no—
254C22×Dic3—61no—
255#239—34no—
256#239—62no—
257#235—34no—
258#235—62no—
259C2×D12—31no—
260C2×D12—31no—
261C2×D12—61no—
262C2×D12—61no—
263C2×C4×D3—61no—
264C2×C4×D3—61no—
265C2×C4×D3—61no—
266C2×C4×D3D3211yescharacteristic
267C2×Dic6D3211yescharacteristic
268C2×C3×C8—91no—
269C3×C4⋊C4—31no—
270C3×C4⋊C4—31no—
271C3×C4⋊C4—61no—
272C3×C4⋊C4—91no—
273C3×C4⋊C4—181no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—61no—
277C4⋊Dic3—61no—
278C4⋊Dic3D3211yescharacteristic
279Dic3⋊C4—61no—
280#208—182no—
281#208—182no—
282#208—184no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—61no—
286C2×C3⋊C8—91no—
287C2×C3⋊C8—91no—
288C2×C3⋊C8—91no—
289C2×C3⋊C8—181no—
290C2×C3⋊C8—181no—
291C2×C3⋊C8—181no—
292C2×C3⋊C8—361no—
293ΓL(1,𝔽25)—182no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—32no—
298C22×C32—41no—
299C22×C32—61no—
300C22×C32—61no—
301C22×C32D3×D411yescharacteristic
302C22×C32C2×#6711yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C2×C3⋊D3—22no—
305C2×C3⋊D3—41no—
306C2×C3⋊D3—41no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—21no—
310C2×C3×D3—21no—
311C2×C3×D3—22no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—122no—
317C2×C3×D3—121no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—241no—
322C2×C3×D3—241no—
323C2×C3×D3(C2×D3)⋊C412yes—
324C2×C3×D3#21712yes—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—32no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—62no—
335C32×C4—121no—
336C32×C4C2×C4×D311yescharacteristic
337C32×C4C2×C4×D311yescharacteristic
338C32×C4C2×D1211yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4C22×Dic311yescharacteristic
341C32×C4C2×#6711yescharacteristic
342C3⋊Dic3—41no—
343C3×Dic3—22no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—41no—
348C3×Dic3—41no—
349C3×Dic3—81no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—241no—
353C3×Dic3(C2×D3)⋊C412yes—
354C3×Dic3#21712yes—
355C2×#41—91no—
356C2×M16—271no—
357C2×C4⋊C4—91no—
358#103—91no—
359#103—271no—
360#102—271no—
361#102—271no—
362C33#41711yescharacteristic, complemented, Sylow
363C23×C3—31no—
364C23×C3—31no—
365C23×C3—61no—
366C23×C3—181no—
367C22×D3—61no—
368C22×D3—61no—
369C22×D3—121no—
370C22×D3#62511yescharacteristic
371C3×Q8—32no—
372C3×Q8—32no—
373C3×Q8—62no—
374C3×Q8—61no—
375C3×Q8—61no—
376C3×Q8—121no—
377C3×D4—92no—
378C3×D4—184no—
379C3×D4—181no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—32no—
389C2×C3×C4—41no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—121no—
398C2×C3×C4—121no—
399C2×C3×C4—184no—
400C2×C3×C4—181no—
401C2×C3×C4—362no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405#67—64no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—121no—
411C2×Dic3—122no—
412C2×Dic3#62511yescharacteristic
413D12—32no—
414D12—32no—
415D12—62no—
416D12—64no—
417D12—61no—
418D12—61no—
419D12—121no—
420C4×D3—64no—
421C4×D3—64no—
422C4×D3—122no—
423C4×D3D3×Dic312yes—
424C4×D3#62612yes—
425Dic6—21no—
426Dic6D3×Dic312yescomplemented
427C3×C8—181no—
428C3⋊C8—62no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C3⋊C8—182no—
432C3⋊C8—181no—
433C3⋊C8—181no—
434C3⋊C8—181no—
435C3⋊C8—361no—
436C3⋊C8—361no—
437C3⋊C8—362no—
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—21no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—41no—
450C2×C32—62no—
451C2×C32D3×C22⋊C411yescharacteristic
452C2×C32#91411yescharacteristic
453C2×C32#91811yescharacteristic
454C2×C32C2×(C2×D3)⋊C411yescharacteristic
455C2×C32#93211yescharacteristic
456C2×C32#94211yescharacteristic
457C2×C32#94911yescharacteristic
458C2×C32#95111yescharacteristic
459C2×C32C2×#21711yescharacteristic
460C3⋊D3—42no—
461C3×D3—22no—
462C3×D3—22no—
463C3×D3—42no—
464C3×D3—122no—
465C3×D3—122no—
466C3×D3—242no—
467#41—94no—
468#41—182no—
469C2×Q8—31no—
470C2×D4—91no—
471C2×D4—181no—
472C22×C4—31no—
473C22×C4—181no—
474C22×C4—181no—
475M16—542no—
476C2×C8—91no—
477C2×C8—271no—
478C4⋊C4—31no—
479C4⋊C4—91no—
480C4⋊C4—181no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—32no—
485C22×C3—32no—
486C22×C3—41no—
487C22×C3—62no—
488C22×C3—61no—
489C22×C3—61no—
490C22×C3—61no—
491C22×C3—61no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3—182no—
495C22×C3—361no—
496C22×C3—361no—
497C22×C3C2×#62511yescharacteristic
498C22×C3D3×#6711yescharacteristic
499C22×C3D3×#6711yescharacteristic
500C2×D3—21no—
501C2×D3—21no—
502C2×D3—62no—
503C2×D3—62no—
504C2×D3—122no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—121no—
509C2×D3—241no—
510C2×D3—241no—
511C2×D3(C2×D3)⋊Dic312yes—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—32no—
519C3×C4—32no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—62no—
523C3×C4—62no—
524C3×C4—62no—
525C3×C4—122no—
526C3×C4—121no—
527C3×C4—121no—
528C3×C4—241no—
529C3×C4—361no—
530C3×C4C4×D3211yescharacteristic
531C3×C4D3×D1211yescharacteristic
532C3×C4C2×D3×Dic311yescharacteristic
533C3×C4C2×D3×Dic311yescharacteristic
534C3×C4C2×#62611yescharacteristic
535C3×C4D3×#6711yescharacteristic
536Dic3—22no—
537Dic3—41no—
538Dic3—121no—
539Dic3—121no—
540Dic3—241no—
541Dic3(C2×D3)⋊Dic312yes—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#480311yescharacteristic, complemented
547C32#496611yescharacteristic, complemented
548C32#523211yescharacteristic, complemented
549C23—31no—
550C23—181no—
551Q8—32no—
552Q8—61no—
553D4—92no—
554D4—184no—
555D4—181no—
556C2×C4—31no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—184no—
562C2×C4—181no—
563C2×C4—362no—
564C2×C4D3311yescharacteristic
565C8—181no—
566C8—541no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—62no—
580C2×C3—62no—
581C2×C3—122no—
582C2×C3—362no—
583C2×C3#6382311yescharacteristic
584C2×C3#6382711yescharacteristic
585C2×C3D3×(C2×D3)⋊C411yescharacteristic
586C2×C3#6393411yescharacteristic
587C2×C3#6393911yescharacteristic
588C2×C3#6394311yescharacteristic
589C2×C3#6394811yescharacteristic
590C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
591C2×C3D3×#21711yescharacteristic
592D3—22no—
593D3—122no—
594D3—122no—
595D3—242no—
596C22—32no—
597C22—61no—
598C22—61no—
599C22—182no—
600C22—361no—
601C22—361no—
602C22D3×#62511yescharacteristic
603C4—32no—
604C4—62no—
605C4—121no—
606C4—361no—
607C4D32×Dic311yescharacteristic
608C4D3×#62611yescharacteristic
609C3—21no—
610C3—21no—
611C3—21no—
612C3—41no—
613C3#1059060911yescharacteristic, complemented
614C3#1059108711yescharacteristic, complemented
615C3#1059155011yescharacteristic, complemented
616C2—62no—
617C2—362no—
618C2#1172286711yescharacteristic
619C2#1172288311yescharacteristic
620C2D3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×#3799C2
hypocenterC33#417
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. C2×#88786C2semidirect
3. C3×D3×C4⋊Dic3C2semidirect
4. #11722075C2semidirect
5. #11722037C2semidirect
6. #11722033C2semidirect
7. C3×#63565C2semidirect
18. C3×#3799C4semidirect
31. D3×C4⋊Dic3D3semidirect
42. #63565D3semidirect
54. C3×C4⋊(C3⋊D3)C2×C4semidirect
61. C32×Dic6C2×C4semidirect
96. #3799Dic3semidirect
184. C4⋊(C3⋊D3)C2×Dic3semidirect
219. C3×Dic6C4×D3semidirect
220. C3×Dic6C2×Dic3semidirect
362. C33#417semidirect
426. Dic6D3×Dic3semidirect
546. C32#4803semidirect
547. C32#4966semidirect
548. C32#5232semidirect
613. C3#10590609semidirect
614. C3#10591087semidirect
615. C3#10591550semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37992
C2×#887862
C3×D3×C4⋊Dic32
#117220752
#117220372
#117220332
C3×#635652
#105915503
#105910873
#105906093