Finite Group Databasebeta

← #13155839

Subgroups — #13155839

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×#88792C211yesmaximal, characteristic, complemented
3C3×#63916C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic, complemented
5#11722033C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8#10591519—31nomaximal
9#10590963—31nomaximal
10#10590615—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×#6295—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041—22no—
19C2×C3×#626—21no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88792—22no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3714—22no—
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29C2×#3754—31no—
30C3×#938—31no—
31C2×C3×#213—31no—
32C2×C3×D24—31no—
33C3×#895—31no—
34#63916D311yescharacteristic, complemented
35C2×#3715—31no—
36C2×#3715—31no—
37C2×#3715—61no—
38C2×#3714D311yescharacteristic, complemented
39#63647—31no—
40#63636—31no—
41C3×#816—31no—
42C3×#810—31no—
43#63567D311yescharacteristic, complemented
44#63562—31no—
45#63562—31no—
46#63562—61no—
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C22×C32×D3—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)D412yescomplemented
58C3×C4×C3⋊D3—21no—
59C3×C4×C3⋊D3—21no—
60C3×#634—21no—
61C3×#634D412yescomplemented
62C2×C32×Dic3—21no—
63C32×D12—21no—
64C32×D12—21no—
65#6295—24no—
66#6294—24no—
67C3×#626—42no—
68C3⋊(C3⋊(C3⋊C8))—21no—
69C32×C3⋊C8—21no—
70#5158—91no—
71#5037—91no—
72#4810—91no—
73C2×C32×D4—31no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×#67—61no—
78C2×C3×D12—21no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—61no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—61no—
91C2×#626—21no—
92C2×#626—61no—
93C2×#626—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—31no—
105C32×C4⋊C4—31no—
106#3754—62no—
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—31no—
110C2×C3⋊(C3⋊C8)—61no—
111C2×C3⋊(C3⋊C8)—61no—
112C2×C3⋊(C3⋊C8)—61no—
113C2×C3⋊(C3⋊C8)—121no—
114C3×#217—61no—
115C3×#213—62no—
116C3×(C2×D3)⋊C4—61no—
117C3×C4⋊Dic3—21no—
118C3×C4⋊Dic3C2×D311yescharacteristic
119C3×C4⋊Dic3C2×D311yescharacteristic
120C2×C3×C3⋊C8—21no—
121C2×C3×C3⋊C8C2×D311yescharacteristic
122C2×C3×C3⋊C8C2×D311yescharacteristic
123C3×D24—62no—
124(C2×D3)⋊Dic3—21no—
125#3715—62no—
126#3715—62no—
127#3715—64no—
128#3714—22no—
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×C32×D3—41no—
137C2×C32×D3—41no—
138C2×C32×D3—41no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—22no—
143C32×Dic3—41no—
144C2×#239—91no—
145C2×#237—91no—
146C2×#235—91no—
147C2×C3×D8—91no—
148C3×#121—91no—
149C2×#215—91no—
150#938—31no—
151C2×#214—91no—
152C2×#214—91no—
153C2×#214—181no—
154C2×#213—31no—
155C2×D24—31no—
156#895—31no—
157C3×#103—91no—
158#832—91no—
159#830—91no—
160(C2×D3)⋊C8—91no—
161(C2×D3)⋊C8—91no—
162(C2×D3)⋊C8—181no—
163#816—31no—
164#810—31no—
165#809—91no—
166#809—91no—
167#809—181no—
168C23×C32—61no—
169C22×C3⋊D3—21no—
170C22×C3⋊D3#6711yescharacteristic
171C22×C3×D3—21no—
172C22×C3×D3—21no—
173C22×C3×D3—31no—
174C22×C3×D3—31no—
175C22×C3×D3—41no—
176C22×C3×D3—61no—
177C22×C3×D3—61no—
178C22×C3×D3—61no—
179C22×C3×D3—121no—
180C32×D4—61no—
181C32×D4—61no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—61no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3#6711yescharacteristic
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)#6712yescomplemented
194C4×C3⋊D3—21no—
195C4×C3⋊D3—21no—
196#634—21no—
197#634D1212yescomplemented
198C3×#67—122no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—41no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C3×D12—22no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—32no—
214C3×D12—32no—
215C3×D12—41no—
216C3×D12—62no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—121no—
220C3×D12—122no—
221C3×C4×D3—61no—
222C3×C4×D3—61no—
223C3×C4×D3—61no—
224C3×C4×D3—61no—
225C3×C4×D3—121no—
226C3×C4×D3—121no—
227C3×Dic6—32no—
228C3×Dic6—32no—
229C3×Dic6—62no—
230C3×Dic6—61no—
231C3×Dic6—61no—
232C3×Dic6—121no—
233#626—42no—
234#626—64no—
235#626—64no—
236#625—64no—
237#624—64no—
238D3×Dic3—64no—
239D3×Dic3—64no—
240C32×C8—61no—
241C3⋊(C3⋊C8)—61no—
242C3⋊(C3⋊C8)—62no—
243C3⋊(C3⋊C8)—61no—
244C3⋊(C3⋊C8)—61no—
245C3⋊(C3⋊C8)—121no—
246C3⋊(C3⋊C8)—121no—
247C3⋊(C3⋊C8)—122no—
248C3×C3⋊C8—21no—
249C3×C3⋊C8—21no—
250C3×C3⋊C8—41no—
251#449—271noSylow
252C2×C33#12011yescharacteristic
253C2×C33#13511yescharacteristic
254C2×C33AGL(1,ℤ/8)11yescharacteristic
255C3×C3⋊D3—41no—
256C3×C3⋊D3—42no—
257C32×D3—81no—
258C2×C3×Q8—91no—
259C2×C3×D4—31no—
260C2×C3×D4—31no—
261C2×C3×D4—61no—
262C2×C3×D4—91no—
263C2×C3×D4—181no—
264C22×C3×C4—91no—
265C2×#67—61no—
266C2×#67—181no—
267C2×#67—181no—
268C22×Dic3—181no—
269#239—182no—
270#239—182no—
271#237—182no—
272#237—182no—
273#235—182no—
274#235—182no—
275C2×D12—31no—
276C2×D12—31no—
277C2×D12—61no—
278C2×D12—61no—
279C2×D12—181no—
280C2×D12D3211yescharacteristic
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×D8—182no—
290C2×C3×C8—31no—
291C2×C3×C8—31no—
292C2×C3×C8—61no—
293C3×C4⋊C4—31no—
294C3×C4⋊C4—31no—
295C3×C4⋊C4—61no—
296C3×C22⋊C4—181no—
297#217—61no—
298#215—182no—
299#214—182no—
300#214—182no—
301#214—184no—
302#213—62no—
303(C2×D3)⋊C4—61no—
304C4⋊Dic3D3211yescharacteristic
305C2×C3⋊C8—91no—
306C2×C3⋊C8—91no—
307C2×C3⋊C8—91no—
308C2×C3⋊C8—181no—
309C2×C3⋊C8—181no—
310C2×C3⋊C8—181no—
311C2×C3⋊C8—361no—
312C2×C3⋊C8D3211yescharacteristic
313D24—62no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—41no—
318C22×C32—121no—
319C22×C32—121no—
320C22×C32—121no—
321C22×C32D3×D411yescharacteristic
322C22×C32D3×D411yescharacteristic
323C22×C32C2×#6711yescharacteristic
324C2×C3⋊D3—21no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—22no—
327C2×C3⋊D3—21no—
328C2×C3⋊D3—41no—
329C2×C3⋊D3—41no—
330C2×C3×D3—42no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—41no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—62no—
345C2×C3×D3—61no—
346C2×C3×D3—81no—
347C2×C3×D3—81no—
348C2×C3×D3—121no—
349C2×C3×D3—121no—
350C2×C3×D3—122no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—121no—
354C2×C3×D3—121no—
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×C4—121no—
366C32×C4C2×D1211yescharacteristic
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4D3×D411yescharacteristic
370C32×C4D3×D411yescharacteristic
371C32×C4C2×#6711yescharacteristic
372C3⋊Dic3—22no—
373C3⋊Dic3—21no—
374C3×Dic3—41no—
375C3×Dic3—41no—
376C3×Dic3—61no—
377C3×Dic3—62no—
378C3×Dic3—61no—
379C3×Dic3—62no—
380C3×Dic3—81no—
381C3×Dic3—122no—
382C3×Dic3—121no—
383C2×#41—271no—
384C2×SD16—271no—
385C2×D8—91no—
386#121—91no—
387#103—91no—
388#102—271no—
389C22⋊C8—271no—
390C33#44911yescharacteristic, complemented, Sylow
391C23×C3—61no—
392C23×C3—61no—
393C23×C3—91no—
394C23×C3—121no—
395C23×C3—181no—
396C22×D3—21no—
397C22×D3—31no—
398C22×D3—31no—
399C22×D3—61no—
400C22×D3—61no—
401C22×D3—61no—
402C22×D3—121no—
403C3×Q8—92no—
404C3×Q8—181no—
405C3×D4—62no—
406C3×D4—61no—
407C3×D4—61no—
408C3×D4—61no—
409C3×D4—61no—
410C3×D4—92no—
411C3×D4—121no—
412C3×D4—181no—
413C3×D4—362no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—21no—
417C2×C3×C4—41no—
418C2×C3×C4—61no—
419C2×C3×C4—61no—
420C2×C3×C4—91no—
421C2×C3×C4—121no—
422C2×C3×C4—181no—
423C2×C3×C4—181no—
424C2×C3×C4—181no—
425C2×C3×C4C2×D3211yescharacteristic
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3211yescharacteristic
428#67—122no—
429#67—184no—
430#67—184no—
431C2×Dic3—21no—
432C2×Dic3—31no—
433C2×Dic3—31no—
434C2×Dic3—61no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—121no—
438C2×Dic3—184no—
439D12—21no—
440D12—21no—
441D12—32no—
442D12—32no—
443D12—62no—
444D12—61no—
445D12—61no—
446D12—121no—
447D12—122no—
448D12—184no—
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—61no—
464C3×C8—61no—
465C3×C8—121no—
466C3⋊C8—21no—
467C3⋊C8—182no—
468C3⋊C8—181no—
469C3⋊C8—181no—
470C3⋊C8—181no—
471C3⋊C8—361no—
472C3⋊C8—361no—
473C3⋊C8—362no—
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—241no—
487C2×C32#88211yescharacteristic
488C2×C32#91111yescharacteristic
489C2×C32#91211yescharacteristic
490C2×C32#91411yescharacteristic
491C2×C32#91911yescharacteristic
492C2×C32#93211yescharacteristic
493C2×C32#93711yescharacteristic
494C2×C32#93711yescharacteristic
495C2×C32#95011yescharacteristic
496C3⋊D3—41no—
497C3⋊D3—42no—
498C3×D3—82no—
499C3×D3—81no—
500C3×D3—81no—
501C3×D3—122no—
502C3×D3—121no—
503C3×D3—122no—
504C3×D3—121no—
505C3×D3—122no—
506C3×D3—242no—
507#41—542no—
508#41—542no—
509C2×Q8—91no—
510C2×D4—31no—
511C2×D4—91no—
512C2×D4—181no—
513C2×D4—541no—
514C22×C4—91no—
515C22×C4—541no—
516SD16—542no—
517D8—182no—
518C2×C8—31no—
519C2×C8—271no—
520C4⋊C4—31no—
521C22⋊C4—181no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—41no—
526C22×C3—122no—
527C22×C3—121no—
528C22×C3—121no—
529C22×C3—121no—
530C22×C3—121no—
531C22×C3—121no—
532C22×C3—121no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3—182no—
536C22×C3—181no—
537C22×C3—241no—
538C22×C3—241no—
539C22×C3—361no—
540C22×C3—361no—
541C22×C3#380211yescharacteristic
542C22×C3D3×#6711yescharacteristic
543C22×C3D3×#6711yescharacteristic
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—62no—
548C2×D3—61no—
549C2×D3—62no—
550C2×D3—61no—
551C2×D3—61no—
552C2×D3—61no—
553C2×D3—61no—
554C2×D3—61no—
555C2×D3—62no—
556C2×D3—121no—
557C2×D3—121no—
558C2×D3—122no—
559C2×D3—121no—
560C2×D3—121no—
561C2×D3—121no—
562C2×D3—121no—
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—121no—
574C3×C4—121no—
575C3×C4—181no—
576C3×C4—182no—
577C3×C4—241no—
578C3×C4D3×D1211yescharacteristic
579C3×C4D3×D1211yescharacteristic
580C3×C4#380211yescharacteristic
581C3×C4D3×#6711yescharacteristic
582C3×C4D3×#6711yescharacteristic
583C3×C4#381111yescharacteristic
584Dic3—41no—
585Dic3—62no—
586Dic3—62no—
587Dic3—61no—
588Dic3—61no—
589Dic3—122no—
590Dic3—121no—
591C32—21no—
592C32—21no—
593C32—21no—
594C32—41no—
595C32#481011yescharacteristic, complemented
596C32#503711yescharacteristic, complemented
597C32#515811yescharacteristic, complemented
598C23—61no—
599C23—91no—
600C23—181no—
601Q8—92no—
602Q8—181no—
603D4—61no—
604D4—61no—
605D4—92no—
606D4—181no—
607D4—362no—
608D4—544no—
609C2×C4—61no—
610C2×C4—91no—
611C2×C4—181no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4—544no—
615C2×C4D3311yescharacteristic
616C8—61no—
617C8—541no—
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—242no—
631C2×C3—241no—
632C2×C3—241no—
633C2×C3—362no—
634C2×C3—361no—
635C2×C3#6379811yescharacteristic
636C2×C3#6381011yescharacteristic
637C2×C3#6392211yescharacteristic
638C2×C3#6392211yescharacteristic
639C2×C3#6393211yescharacteristic
640C2×C3#6393311yescharacteristic
641C2×C3#6393411yescharacteristic
642C2×C3#6394911yescharacteristic
643C2×C3#6398311yescharacteristic
644D3—81no—
645D3—122no—
646D3—121no—
647D3—122no—
648D3—121no—
649D3—122no—
650D3—242no—
651C22—121no—
652C22—121no—
653C22—121no—
654C22—181no—
655C22—181no—
656C22—181no—
657C22—182no—
658C22—361no—
659C22—361no—
660C22#8895311yescharacteristic
661C4—121no—
662C4—181no—
663C4—182no—
664C4#8895311yescharacteristic
665C4#8895511yescharacteristic
666C3—21no—
667C3—21no—
668C3—21no—
669C3—41no—
670C3#1059061511yescharacteristic, complemented
671C3#1059096311yescharacteristic, complemented
672C3#1059151911yescharacteristic, complemented
673C2—241no—
674C2—361no—
675C2—362no—
676C2#1172286011yescharacteristic
677C2#1172287511yescharacteristic
678C2#1172295111yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#449
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×#89041C2semidirect
2. C2×#88792C2semidirect
3. C3×#63916C2semidirect
4. C2×C3×#3714C2semidirect
5. #11722033C2semidirect
6. #11722023C2semidirect
7. C3×#63567C2semidirect
34. #63916D3semidirect
38. C2×#3714D3semidirect
43. #63567D3semidirect
57. C3×C4⋊(C3⋊D3)D4semidirect
61. C3×#634D4semidirect
193. C4⋊(C3⋊D3)#67semidirect
197. #634D12semidirect
390. C33#449semidirect
595. C32#4810semidirect
596. C32#5037semidirect
597. C32#5158semidirect
670. C3#10590615semidirect
671. C3#10590963semidirect
672. C3#10591519semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×#887922
C3×#639162
C2×C3×#37142
#117220332
#117220232
C3×#635672
#105915193
#105909633
#105906153