Finite Group Databasebeta

← #13161755

Subgroups — #13161755

Order 1728.

838 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic, complemented
2C2×C3×#3744C211yesmaximal, characteristic, complemented
3#11723696C211yesmaximal, characteristic, complemented
4C2×#88810C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7#11722076C211yesmaximal, characteristic, complemented
8#10590968—31nomaximal
9#10590964—31nomaximal
10#10590823—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C2×#6295—21no—
16C2×#6295—21no—
17C2×#6294—21no—
18#89041—22no—
19#89041—22no—
20C2×C33×C8C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3744—22no—
24#88814—21no—
25#88810—22no—
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29C2×#3744D311yescharacteristic, complemented
30C2×C3×#204—31no—
31C2×C3×#204—31no—
32C2×C3×#204—61no—
33#63916—31no—
34#63909—31no—
35#63908—31no—
36C2×#3714—31no—
37C2×#3714—31no—
38C2×#3713—31no—
39#63648D311yescharacteristic, complemented
40#63647D311yescharacteristic, complemented
41C3×#821—31no—
42C3×#821—31no—
43C3×#821—61no—
44C3×(C2×D3)⋊C8—31no—
45C3×(C2×D3)⋊C8—31no—
46C3×(C2×D3)⋊C8—61no—
47#63567—31no—
48#63566—31no—
49#63565—31no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—21no—
52C22×C3×C3⋊D3D411yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)D412yescomplemented
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C3×#634—21no—
64C3×#634D412yescomplemented
65#6295—24no—
66#6295—42no—
67#6294—24no—
68C33×C8—21no—
69C3×C3⋊(C3⋊C8)—21no—
70#5172—91no—
71#4781—91no—
72#4734—91no—
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—61no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×#626—61no—
90C2×#626—61no—
91C2×#626—61no—
92C2×#626—61no—
93C2×#625—61no—
94C2×#625—61no—
95C2×#624—61no—
96C2×D3×Dic3—61no—
97C2×D3×Dic3—61no—
98#3799—62no—
99#3799—62no—
100#3796—62no—
101#3796—62no—
102#3795—62no—
103#3795—62no—
104C2×C32×C8—21no—
105C2×C32×C8—21no—
106C2×C32×C8—21no—
107C2×C32×C8—41no—
108C2×C32×C8C2×D311yescharacteristic
109C2×C32×C8C2×D311yescharacteristic
110C2×C32×C8C2×D311yescharacteristic
111C4⋊(C3⋊Dic3)C2×D311yescharacteristic
112C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
113#3744—22no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—61no—
120C3×#204—62no—
121C3×#204—62no—
122C3×#204—64no—
123#3722—61no—
124(C2×D3)⋊Dic3—61no—
125(C2×D3)⋊Dic3—61no—
126#3714—62no—
127#3714—62no—
128#3713—62no—
129C22×C33C2×D411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—41no—
135C2×C3×C3⋊D3—41no—
136C2×C3×C3⋊D3—41no—
137C2×C3×C3⋊D3—41no—
138C2×C3×C3⋊D3—41no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—41no—
144C2×#239—91no—
145C2×#237—91no—
146C2×#235—91no—
147C2×C3×SD16—91no—
148#938—91no—
149C2×#213—91no—
150C2×#213—91no—
151#930—91no—
152C2×D24—91no—
153C2×#204—31no—
154C2×#204—31no—
155C2×#204—61no—
156#895—91no—
157C3×#102—91no—
158C3×C22⋊C8—91no—
159#835—91no—
160#821—31no—
161#821—31no—
162#821—61no—
163(C2×D3)⋊C8—31no—
164(C2×D3)⋊C8—31no—
165(C2×D3)⋊C8—61no—
166#816—91no—
167#810—91no—
168C22×C3⋊D3—21no—
169C22×C3⋊D3—21no—
170C22×C3⋊D3D1211yescharacteristic
171C22×C3×D3—31no—
172C22×C3×D3—31no—
173C22×C3×D3—61no—
174C22×C3×D3—61no—
175C22×C3×D3—61no—
176C22×C3×D3—61no—
177C22×C3×D3—61no—
178C22×C3×D3—121no—
179C22×C3×D3—121no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—41no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3D1211yescharacteristic
190C4⋊(C3⋊D3)—21no—
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)D1212yescomplemented
194C4×C3⋊D3—21no—
195C4×C3⋊D3—21no—
196#634—21no—
197#634#6712yescomplemented
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—121no—
206C2×C3×Dic3—121no—
207C3×D12—32no—
208C3×D12—32no—
209C3×D12—62no—
210C3×D12—62no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—121no—
219C3×C4×D3—61no—
220C3×C4×D3—61no—
221C3×C4×D3—61no—
222C3×C4×D3—61no—
223C3×C4×D3—121no—
224C3×C4×D3—121no—
225C3×Dic6—32no—
226C3×Dic6—32no—
227C3×Dic6—62no—
228C3×Dic6—61no—
229C3×Dic6—61no—
230C3×Dic6—121no—
231#626—64no—
232#626—64no—
233#626—122no—
234#626—122no—
235#625—64no—
236#625—122no—
237#624—64no—
238D3×Dic3—64no—
239D3×Dic3—64no—
240C32×C8—21no—
241C32×C8—22no—
242C32×C8—21no—
243C32×C8—21no—
244C32×C8—41no—
245C32×C8—41no—
246C32×C8—42no—
247C3⋊(C3⋊C8)—21no—
248C3×C3⋊C8—61no—
249C3×C3⋊C8—61no—
250C3×C3⋊C8—121no—
251#449—271noSylow
252C2×C33#12011yescharacteristic
253C2×C33#13511yescharacteristic
254C2×C33AGL(1,ℤ/8)11yescharacteristic
255C3×C3⋊D3—41no—
256C3×C3⋊D3—42no—
257C3×C3⋊D3—81no—
258C2×C3×Q8—91no—
259C2×C3×D4—91no—
260C2×C3×D4—91no—
261C22×C3×C4—91no—
262C2×#67—181no—
263C2×#67—181no—
264C2×#67—181no—
265C2×#67—181no—
266C22×Dic3—181no—
267#239—182no—
268#239—182no—
269#237—182no—
270#237—182no—
271#235—182no—
272#235—182no—
273C2×D12—31no—
274C2×D12—31no—
275C2×D12—31no—
276C2×D12—31no—
277C2×D12—61no—
278C2×D12—61no—
279C2×D12—181no—
280C2×D12—181no—
281C2×C4×D3—31no—
282C2×C4×D3—31no—
283C2×C4×D3—61no—
284C2×C4×D3—181no—
285C2×C4×D3—181no—
286C2×Dic6—31no—
287C2×Dic6—31no—
288C2×Dic6—61no—
289C3×SD16—182no—
290C2×C3×C8—21no—
291C2×C3×C8—21no—
292C2×C3×C8—21no—
293C2×C3×C8—41no—
294C2×C3×C8—91no—
295C2×C3×C8D3211yescharacteristic
296C2×C3×C8D3211yescharacteristic
297C2×C3×C8D3211yescharacteristic
298C3×C4⋊C4—91no—
299#217—181no—
300#213—182no—
301#213—182no—
302(C2×D3)⋊C4—181no—
303(C2×D3)⋊C4—181no—
304C4⋊Dic3—31no—
305C4⋊Dic3—31no—
306C4⋊Dic3—61no—
307C2×C3⋊C8—31no—
308C2×C3⋊C8—31no—
309C2×C3⋊C8—61no—
310D24—182no—
311#204—62no—
312#204—62no—
313#204—64no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—41no—
318C22×C32C2×D1211yescharacteristic
319C22×C32D3×D411yescharacteristic
320C22×C32D3×D411yescharacteristic
321C2×C3⋊D3—21no—
322C2×C3⋊D3—22no—
323C2×C3⋊D3—21no—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—41no—
326C2×C3⋊D3—41no—
327C2×C3⋊D3—41no—
328C2×C3⋊D3—41no—
329C2×C3⋊D3—41no—
330C2×C3×D3—62no—
331C2×C3×D3—62no—
332C2×C3×D3—61no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—61no—
336C2×C3×D3—61no—
337C2×C3×D3—62no—
338C2×C3×D3—61no—
339C2×C3×D3—121no—
340C2×C3×D3—121no—
341C2×C3×D3—122no—
342C2×C3×D3—121no—
343C2×C3×D3—121no—
344C2×C3×D3—121no—
345C2×C3×D3—121no—
346C2×C3×D3—122no—
347C2×C3×D3—121no—
348C2×C3×D3—121no—
349C2×C3×D3—121no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—241no—
354C2×C3×D3—241no—
355C2×C3×D3—241no—
356C2×C3×D3—241no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—41no—
364C32×C4—41no—
365C32×C4C2×D1211yescharacteristic
366C32×C4D3×D411yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4D3×D411yescharacteristic
370C32×C4C2×#6711yescharacteristic
371C3⋊Dic3—21no—
372C3⋊Dic3—22no—
373C3⋊Dic3—41no—
374C3×Dic3—62no—
375C3×Dic3—61no—
376C3×Dic3—62no—
377C3×Dic3—61no—
378C3×Dic3—122no—
379C3×Dic3—121no—
380C3×Dic3—121no—
381C3×Dic3—121no—
382C3×Dic3—241no—
383C2×#41—271no—
384C2×SD16—91no—
385C2×D8—271no—
386#121—271no—
387#103—271no—
388#102—91no—
389C22⋊C8—91no—
390C33#44911yescharacteristic, complemented, Sylow
391C23×C3—91no—
392C23×C3—181no—
393C23×C3—181no—
394C22×D3—31no—
395C22×D3—31no—
396C22×D3—61no—
397C22×D3—61no—
398C22×D3—61no—
399C22×D3—61no—
400C22×D3—61no—
401C22×D3—121no—
402C22×D3—121no—
403C3×Q8—92no—
404C3×Q8—181no—
405C3×D4—92no—
406C3×D4—181no—
407C3×D4—181no—
408C3×D4—181no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—41no—
413C2×C3×C4—91no—
414C2×C3×C4—181no—
415C2×C3×C4—181no—
416C2×C3×C4—181no—
417C2×C3×C4—181no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C2×D3211yescharacteristic
420C2×C3×C4C2×D3211yescharacteristic
421#67—184no—
422#67—184no—
423#67—362no—
424#67—362no—
425C2×Dic3—31no—
426C2×Dic3—31no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—121no—
433C2×Dic3—121no—
434C2×Dic3—184no—
435D12—32no—
436D12—32no—
437D12—62no—
438D12—62no—
439D12—61no—
440D12—61no—
441D12—61no—
442D12—61no—
443D12—61no—
444D12—61no—
445D12—121no—
446D12—121no—
447D12—184no—
448D12—362no—
449C4×D3—61no—
450C4×D3—61no—
451C4×D3—61no—
452C4×D3—61no—
453C4×D3—121no—
454C4×D3—121no—
455C4×D3—184no—
456C4×D3—184no—
457Dic6—32no—
458Dic6—32no—
459Dic6—62no—
460Dic6—61no—
461Dic6—61no—
462Dic6—121no—
463C3×C8—22no—
464C3×C8—21no—
465C3×C8—21no—
466C3×C8—21no—
467C3×C8—41no—
468C3×C8—41no—
469C3×C8—42no—
470C3×C8—181no—
471C3⋊C8—61no—
472C3⋊C8—61no—
473C3⋊C8—121no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—41no—
486C2×C32#88211yescharacteristic
487C2×C32#88211yescharacteristic
488C2×C32#90411yescharacteristic
489C2×C32#90811yescharacteristic
490C2×C32#91111yescharacteristic
491C2×C32#91411yescharacteristic
492C2×C32#91611yescharacteristic
493C2×C32#91611yescharacteristic
494C2×C32#93711yescharacteristic
495C3⋊D3—41no—
496C3⋊D3—42no—
497C3⋊D3—81no—
498C3×D3—122no—
499C3×D3—121no—
500C3×D3—122no—
501C3×D3—122no—
502C3×D3—121no—
503C3×D3—242no—
504C3×D3—242no—
505C3×D3—241no—
506C3×D3—241no—
507#41—542no—
508#41—542no—
509C2×Q8—91no—
510C2×D4—91no—
511C2×D4—91no—
512C2×D4—541no—
513C2×D4—541no—
514C22×C4—91no—
515C22×C4—541no—
516SD16—182no—
517D8—542no—
518C2×C8—91no—
519C2×C8#111311yescharacteristic
520C4⋊C4—91no—
521C22⋊C4—541no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—41no—
526C22×C3—181no—
527C22×C3—181no—
528C22×C3—182no—
529C22×C3—181no—
530C22×C3—361no—
531C22×C3—361no—
532C22×C3—361no—
533C22×C3—361no—
534C22×C3—361no—
535C22×C3D3×D1211yescharacteristic
536C22×C3D3×D1211yescharacteristic
537C22×C3#380211yescharacteristic
538C2×D3—62no—
539C2×D3—61no—
540C2×D3—61no—
541C2×D3—61no—
542C2×D3—61no—
543C2×D3—61no—
544C2×D3—62no—
545C2×D3—61no—
546C2×D3—62no—
547C2×D3—121no—
548C2×D3—121no—
549C2×D3—122no—
550C2×D3—121no—
551C2×D3—121no—
552C2×D3—121no—
553C2×D3—121no—
554C2×D3—122no—
555C2×D3—121no—
556C2×D3—121no—
557C2×D3—121no—
558C2×D3—121no—
559C2×D3—121no—
560C2×D3—121no—
561C2×D3—241no—
562C2×D3—241no—
563C2×D3—241no—
564C2×D3—241no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4—21no—
570C3×C4—21no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—181no—
574C3×C4—182no—
575C3×C4—361no—
576C3×C4D3×D1211yescharacteristic
577C3×C4D3×D1211yescharacteristic
578C3×C4#380211yescharacteristic
579C3×C4D3×#6711yescharacteristic
580C3×C4D3×#6711yescharacteristic
581C3×C4#381111yescharacteristic
582Dic3—62no—
583Dic3—62no—
584Dic3—61no—
585Dic3—61no—
586Dic3—122no—
587Dic3—121no—
588Dic3—121no—
589Dic3—121no—
590Dic3—241no—
591C32—21no—
592C32—21no—
593C32—21no—
594C32—41no—
595C32#473411yescharacteristic, complemented
596C32#478111yescharacteristic, complemented
597C32#517211yescharacteristic, complemented
598C23—91no—
599C23—181no—
600C23—181no—
601Q8—92no—
602Q8—181no—
603D4—92no—
604D4—181no—
605D4—181no—
606D4—181no—
607D4—544no—
608D4—1082no—
609C2×C4—91no—
610C2×C4—181no—
611C2×C4—181no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4—544no—
615C2×C4C2×#111311yescharacteristic
616C8—21no—
617C8—181no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—41no—
628C2×C3—41no—
629C2×C3—41no—
630C2×C3—362no—
631C2×C3—361no—
632C2×C3—721no—
633C2×C3#6379411yescharacteristic
634C2×C3#6379511yescharacteristic
635C2×C3#6379811yescharacteristic
636C2×C3#6380611yescharacteristic
637C2×C3#6380611yescharacteristic
638C2×C3#6380911yescharacteristic
639C2×C3#6392211yescharacteristic
640C2×C3#6392211yescharacteristic
641C2×C3#6392311yescharacteristic
642D3—122no—
643D3—121no—
644D3—122no—
645D3—121no—
646D3—122no—
647D3—242no—
648D3—242no—
649D3—241no—
650D3—241no—
651C22—181no—
652C22—181no—
653C22—181no—
654C22—182no—
655C22—361no—
656C22—361no—
657C22—361no—
658C22—361no—
659C22—361no—
660C22C4⋊#111311yescharacteristic
661C4—181no—
662C4—182no—
663C4—361no—
664C4C4⋊#111311yescharacteristic
665C4#8904911yescharacteristic
666C3—21no—
667C3—21no—
668C3—21no—
669C3—41no—
670C3#1059082311yescharacteristic, complemented
671C3#1059096411yescharacteristic, complemented
672C3#1059096811yescharacteristic, complemented
673C2—361no—
674C2—362no—
675C2—721no—
676C2#1172364811yescharacteristic
677C2#1172365311yescharacteristic
678C2#1172370111yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C8#1113
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C2×C3×#3744C2semidirect
3. #11723696C2semidirect
4. C2×#88810C2semidirect
5. C3×#63648C2semidirect
6. C3×#63647C2semidirect
7. #11722076C2semidirect
29. C2×#3744D3semidirect
39. #63648D3semidirect
40. #63647D3semidirect
60. C3×C4⋊(C3⋊D3)D4semidirect
64. C3×#634D4semidirect
193. C4⋊(C3⋊D3)D12semidirect
197. #634#67semidirect
390. C33#449semidirect
595. C32#4734semidirect
596. C32#4781semidirect
597. C32#5172semidirect
670. C3#10590823semidirect
671. C3#10590964semidirect
672. C3#10590968semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×#37442
#117236962
C2×#888102
C3×#636482
C3×#636472
#117220762
#105909683
#105909643
#105908233