Finite Group Databasebeta

← #13155987

Subgroups — #13155987

Order 1728.

708 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic, complemented
2Dic3×#634C211yesmaximal, characteristic, complemented
3C3×#63911C211yesmaximal, characteristic
4C2×C3×#3716C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591438—31nomaximal
9#10590799—31nomaximal
10#10590377—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88811—22no—
21#88807—21no—
22Dic3×C3⋊Dic3—21no—
23C3×#3716—22no—
24C4×#634—31no—
25C3×#940—31no—
26C2×C3×#214—31no—
27C2×C3×#204—31no—
28C3×#874—31no—
29#63911D311yescharacteristic
30Dic3×Dic6—31no—
31Dic3×Dic6—31no—
32Dic3×Dic6—61no—
33C2×#3717—31no—
34C2×#3716—31no—
35C2×#3716D311yescharacteristic, complemented
36C2×#3715—31no—
37#63659—31no—
38#63648—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63563—31no—
43#63563—31no—
44#63563—31no—
45#63563—31no—
46#63563—61no—
47#63563—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×#634—21no—
56C3×#634D412yescomplemented
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C3×#626—42no—
62C3×#626—42no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5163—91no—
66#5075—91no—
67#4848—91no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—61no—
78C2×C3×D12—61no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×#626—21no—
85C2×#626—21no—
86C2×C32×C8—31no—
87C32×C42—31no—
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—61no—
92C4⋊(C3⋊Dic3)—61no—
93C4⋊(C3⋊Dic3)—61no—
94C4⋊(C3⋊Dic3)—121no—
95#3750—61no—
96C4×C3⋊Dic3—61no—
97C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
98C3×#214—62no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D311yescharacteristic
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109Dic3⋊Dic3—62no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—61no—
112Dic32—62no—
113Dic32—61no—
114Dic32—61no—
115#3717—62no—
116#3716—22no—
117#3716—62no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3⋊(C3⋊Dic3)—41no—
129C3×C3⋊Dic3—22no—
130C32×Dic3—21no—
131C32×Dic3SD1612yescomplemented
132C2×C3×SD16—91no—
133C3×#127—91no—
134Q8×Dic3—91no—
135C2×#215—91no—
136#940—31no—
137C2×#214—31no—
138C2×#214—91no—
139C2×#204—31no—
140C2×#204—91no—
141#874—31no—
142C4×Dic6—91no—
143C4×Dic6—91no—
144C4×Dic6—181no—
145C3×C4⋊C8—91no—
146#832—91no—
147#832—91no—
148#832—91no—
149#832—181no—
150#821—91no—
151#821—91no—
152#821—91no—
153#821—181no—
154Dic3⋊C8—31no—
155C4⋊(C3⋊C8)—31no—
156C23×C32—61no—
157C22×C3⋊D3—21no—
158C22×C3×D3—21no—
159C22×C3×D3—21no—
160C22×C3×D3—41no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×D4—61no—
165C32×D4—61no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—41no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—121no—
181C2×C3⋊Dic3—121no—
182C2×C3⋊Dic3—122no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185#634—21no—
186#634#6712yescomplemented
187C3×#67—122no—
188C3×#67—122no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3D1211yescharacteristic
195C2×C3×Dic3D1211yescharacteristic
196C2×C3×Dic3#6711yescharacteristic
197C2×C3×Dic3#6711yescharacteristic
198C3×D12—22no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—41no—
204C3×D12—62no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—122no—
211C3×D12—122no—
212C3×Dic6—32no—
213C3×Dic6—32no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218#626—42no—
219#626—42no—
220C32×C8—61no—
221C3⋊(C3⋊C8)—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225C3×C3⋊C8—61no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—121no—
228#460—271noSylow
229C2×C33#12111yescharacteristic
230C2×C33C2×SD1611yescharacteristic
231C2×C33AGL(1,ℤ/8)11yescharacteristic
232C3×C3⋊D3—81no—
233C32×D3—81no—
234C2×C3×Q8—91no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C2×C3×D4—91no—
239C2×C3×D4—181no—
240C2×C3×D4—181no—
241C2×#67—61no—
242C2×#67—61no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—61no—
246C2×D12—61no—
247C2×D12—61no—
248C2×D12D3211yescharacteristic
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C3×SD16—182no—
253C2×C3×C8—31no—
254C2×C3×C8—31no—
255C2×C3×C8—61no—
256C2×C3×C8—91no—
257C3×C42—31no—
258C3×C42—31no—
259C3×C42—61no—
260#215—182no—
261#214—62no—
262#214—182no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—181no—
267C4⋊Dic3—181no—
268C4⋊Dic3—181no—
269C4⋊Dic3—181no—
270C4⋊Dic3—361no—
271Dic3⋊C4—181no—
272Dic3⋊C4—181no—
273Dic3⋊C4—182no—
274C4×Dic3—181no—
275C4×Dic3—181no—
276C4×Dic3—182no—
277C4×Dic3—181no—
278C4×Dic3D3211yescharacteristic
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8D3211yescharacteristic
283#204—62no—
284#204—182no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—41no—
289C22×C32—121no—
290C22×C32—121no—
291C22×C32—121no—
292C22×C32C2×D1211yescharacteristic
293C22×C32D3×D411yescharacteristic
294C22×C32C2×#6711yescharacteristic
295C2×C3⋊D3—41no—
296C2×C3⋊D3—41no—
297C2×C3⋊D3—41no—
298C2×C3×D3—42no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—81no—
306C2×C3×D3—81no—
307C2×C3×D3—122no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—241no—
315C2×C3×D3—241no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—32no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—61no—
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4#23711yescharacteristic
329C32×C4#23711yescharacteristic
330C32×C4#23911yescharacteristic
331C32×C4C2×#6711yescharacteristic
332C3⋊Dic3—22no—
333C3⋊Dic3—121no—
334C3⋊Dic3—122no—
335C3⋊Dic3—121no—
336C3⋊Dic3—121no—
337C3⋊Dic3—241no—
338C3⋊Dic3—241no—
339C3⋊Dic3—242no—
340C3×Dic3—21no—
341C3×Dic3—21no—
342C3×Dic3—22no—
343C3×Dic3—41no—
344C3×Dic3—64no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3#20412yescomplemented
348C3×Dic3#21412yescomplemented
349C2×SD16—91no—
350C2×SD16—271no—
351#127—91no—
352C4×Q8—271no—
353C4⋊C8—91no—
354#102—271no—
355#102—271no—
356C33#46011yescharacteristic, complemented, Sylow
357C23×C3—61no—
358C23×C3—61no—
359C23×C3—121no—
360C23×C3—181no—
361C22×D3—21no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3—121no—
365C3×Q8—92no—
366C3×Q8—181no—
367C3×D4—62no—
368C3×D4—61no—
369C3×D4—61no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—121no—
373C3×D4—181no—
374C3×D4—181no—
375C3×D4—362no—
376C3×D4—362no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—31no—
381C2×C3×C4—31no—
382C2×C3×C4—31no—
383C2×C3×C4—31no—
384C2×C3×C4—41no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—181no—
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391#67—122no—
392#67—122no—
393C2×Dic3—62no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—182no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—361no—
401C2×Dic3—361no—
402C2×Dic3—362no—
403C2×Dic3#62611yescharacteristic
404C2×Dic3#62611yescharacteristic
405D12—21no—
406D12—21no—
407D12—62no—
408D12—61no—
409D12—61no—
410D12—61no—
411D12—61no—
412D12—121no—
413D12—122no—
414D12—122no—
415Dic6—32no—
416Dic6—32no—
417Dic6—62no—
418Dic6—62no—
419Dic6—61no—
420Dic6—61no—
421C3×C8—61no—
422C3×C8—61no—
423C3×C8—121no—
424C3×C8—181no—
425C3⋊C8—21no—
426C3⋊C8—61no—
427C3⋊C8—61no—
428C3⋊C8—121no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—41no—
439C2×C32—41no—
440C2×C32—41no—
441C2×C32—241no—
442C2×C32#89511yescharacteristic
443C2×C32C2×#20411yescharacteristic
444C2×C32#90811yescharacteristic
445C2×C32D3×SD1611yescharacteristic
446C2×C32#91411yescharacteristic
447C2×C32#93211yescharacteristic
448C2×C32C2×#21411yescharacteristic
449C2×C32#93811yescharacteristic
450C2×C32#93911yescharacteristic
451C3⋊D3—81no—
452C3×D3—82no—
453C3×D3—81no—
454C3×D3—81no—
455C3×D3—242no—
456C3×D3—241no—
457C3×D3—241no—
458C2×Q8—91no—
459C2×D4—31no—
460C2×D4—91no—
461C2×D4—181no—
462C2×D4—181no—
463SD16—182no—
464SD16—542no—
465C2×C8—31no—
466C2×C8—91no—
467C4⋊C4—271no—
468C4⋊C4—541no—
469C42—31no—
470C42—541no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—122no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3—121no—
480C22×C3—121no—
481C22×C3—121no—
482C22×C3—241no—
483C22×C3—241no—
484C22×C3—361no—
485C22×C3—361no—
486C22×C3—361no—
487C22×C3D3×D1211yescharacteristic
488C22×C3C2×#62611yescharacteristic
489C22×C3D3×#6711yescharacteristic
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—122no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—121no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—241no—
501C2×D3—241no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—21no—
508C3×C4—32no—
509C3×C4—32no—
510C3×C4—41no—
511C3×C4—41no—
512C3×C4—62no—
513C3×C4—61no—
514C3×C4—61no—
515C3×C4—121no—
516C3×C4—182no—
517C3×C4#379611yescharacteristic
518C3×C4#379611yescharacteristic
519C3×C4#380211yescharacteristic
520C3×C4#380511yescharacteristic
521C3×C4D3×#6711yescharacteristic
522C3×C4D3×#6711yescharacteristic
523Dic3—21no—
524Dic3—64no—
525Dic3—62no—
526Dic3—62no—
527Dic3—362no—
528Dic3—361no—
529Dic3—361no—
530Dic3—361no—
531Dic3—721no—
532Dic3—721no—
533Dic3—722no—
534Dic3#371612yescomplemented
535C32—21no—
536C32—21no—
537C32—21no—
538C32—41no—
539C32#484811yescharacteristic, complemented
540C32#507511yescharacteristic, complemented
541C32#516311yescharacteristic, complemented
542C23—61no—
543C23—181no—
544Q8—92no—
545Q8—181no—
546D4—61no—
547D4—61no—
548D4—181no—
549D4—181no—
550D4—362no—
551D4—362no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—181no—
555C2×C4—541no—
556C2×C4D3311yescharacteristic
557C8—61no—
558C8—181no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—41no—
569C2×C3—41no—
570C2×C3—41no—
571C2×C3—242no—
572C2×C3—241no—
573C2×C3—241no—
574C2×C3—721no—
575C2×C3D3×#20411yescharacteristic
576C2×C3#6379411yescharacteristic
577C2×C3C2×#371611yescharacteristic
578C2×C3#6383011yescharacteristic
579C2×C3#6388611yescharacteristic
580C2×C3#6391611yescharacteristic
581C2×C3D3×#21411yescharacteristic
582C2×C3#6393411yescharacteristic
583C2×C3#6397011yescharacteristic
584D3—81no—
585D3—242no—
586D3—241no—
587D3—241no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22—361no—
592C22—361no—
593C22—361no—
594C22D3×#62611yescharacteristic
595C4—32no—
596C4—61no—
597C4—182no—
598C4—1081no—
599C4#8895311yescharacteristic
600C4#8896411yescharacteristic
601C3—21no—
602C3—21no—
603C3—21no—
604C3—41no—
605C3#1059037711yescharacteristic, complemented
606C3#1059079911yescharacteristic, complemented
607C3#1059143811yescharacteristic, complemented
608C2—241no—
609C2—721no—
610C2D3×#371611yescharacteristic
611C2#1172286811yescharacteristic
612C2#1172296211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88811C2semidirect
2. Dic3×#634C2semidirect
4. C2×C3×#3716C2semidirect
5. #11722034C2semidirect
6. #11722032C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
35. C2×#3716D3semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
56. C3×#634D4semidirect
131. C32×Dic3SD16semidirect
186. #634#67semidirect
347. C3×Dic3#204semidirect
348. C3×Dic3#214semidirect
356. C33#460semidirect
534. Dic3#3716semidirect
539. C32#4848semidirect
540. C32#5075semidirect
541. C32#5163semidirect
605. C3#10590377semidirect
606. C3#10590799semidirect
607. C3#10591438semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
Dic3×#6342
C3×#639112
C2×C3×#37162
#117220342
#117220322
C3×Dic3⋊(C3⋊C8)2
#105914383
#105907993
#105903773