Finite Group Databasebeta

← #13155799

Subgroups — #13155799

Order 1728.

814 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C211yesmaximal, characteristic, complemented
2C2×#88811C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic, complemented
4C3×#63909C211yesmaximal, characteristic
5#11722039C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591584—31nomaximal
9#10591171—31nomaximal
10#10590603—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#627—21no—
16C2×C3×#626—21no—
17C2×C3×D3×Dic3—21no—
18C3×D3×Dic6—22no—
19C3×D3×Dic6—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C32×C4⋊Dic3C2211yescharacteristic
23#88811—22no—
24#88793—22no—
25C3×#3722—21no—
26C2×C3×D3×Q8—31no—
27C22×C3×Dic6—31no—
28C2×D3×Dic6D311yescharacteristic, complemented
29C2×#3755—31no—
30C3×#930—31no—
31C3×#895—31no—
32#63909D311yescharacteristic
33C2×#3717—31no—
34C2×#3716—31no—
35C2×#3715—31no—
36C2×#3715—31no—
37C2×#3715—31no—
38C2×#3715—61no—
39#63657—31no—
40#63637—31no—
41C3×#835—31no—
42C3×#810—31no—
43#63565—31no—
44#63565—31no—
45#63565—61no—
46#63565D311yescharacteristic, complemented
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yescomplemented
65C3×#627—24no—
66C3×#626—42no—
67C3×D3×Dic3—24no—
68C3⋊(C3⋊(C3⋊C8))—21no—
69C3×C3⋊(C3⋊C8)—21no—
70#5227—91no—
71#5040—91no—
72#4806—91no—
73C2×C32×Q8—31no—
74C22×C32×C4—31no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×#67—61no—
78C22×C3×Dic3—61no—
79C3×D3×Q8—62no—
80C3×D3×Q8—62no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×D12—61no—
85C2×C3×C4×D3—21no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×C3×Dic6—21no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—62no—
93C2×C3×Dic6—61no—
94C2×C3×Dic6—61no—
95C2×C3×Dic6—62no—
96C2×C3×Dic6—61no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#627—21no—
100C2×#626—21no—
101C2×D3×Dic3—21no—
102D3×Dic6—22no—
103D3×Dic6—22no—
104C32×C4⋊C4—31no—
105#3755—62no—
106C2×C3⋊(C3⋊C8)—31no—
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—61no—
110C2×C3⋊(C3⋊C8)—61no—
111C2×C3⋊(C3⋊C8)—61no—
112C2×C3⋊(C3⋊C8)—121no—
113C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
114C3×(C2×D3)⋊C4—61no—
115C3×(C2×D3)⋊C4—61no—
116C3×C4⋊Dic3—21no—
117C3×C4⋊Dic3C2×D311yescharacteristic
118C3×C4⋊Dic3C2×D311yescharacteristic
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—61no—
122#3722—21no—
123#3717—62no—
124#3716—62no—
125#3715—62no—
126#3715—64no—
127#3715—62no—
128#3715—62no—
129C22×C33C2×D411yescharacteristic
130C2×C3×C3⋊D3—41no—
131C2×C3×C3⋊D3—41no—
132C2×C3×C3⋊D3—41no—
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3SD1612yescomplemented
136C33×C4C2×D411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3×C3⋊Dic3—22no—
139C32×Dic3—22no—
140C32×Dic3—21no—
141C32×Dic3—41no—
142C22×C3×Q8—91no—
143C2×D3×Q8—31no—
144C22×Dic6—31no—
145C3×#121—91no—
146C2×#215—91no—
147C2×#215—91no—
148C2×#215—91no—
149C2×#215—181no—
150C2×#214—91no—
151C2×#214—91no—
152#930—31no—
153C2×#204—91no—
154#895—31no—
155C3×#103—91no—
156#835—31no—
157#835—91no—
158#830—91no—
159#830—91no—
160#830—181no—
161(C2×D3)⋊C8—91no—
162#810—31no—
163#810—91no—
164#810—91no—
165#810—181no—
166C23×C32—31no—
167C22×C3⋊D3—21no—
168C22×C3×D3—21no—
169C22×C3×D3—61no—
170C22×C3×D3—61no—
171C22×C3×D3—121no—
172C22×C3×D3#6711yescharacteristic
173C22×C3×D3#6711yescharacteristic
174C32×Q8—32no—
175C32×Q8—61no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—31no—
180C2×C32×C4—41no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4—61no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—21no—
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—21no—
191#634—21no—
192#634#6712yescomplemented
193C3×#67—122no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3—64no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—121no—
205C2×C3×Dic3#6711yescharacteristic
206C2×C3×Dic3#6711yescharacteristic
207C3×D12—62no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—121no—
213C3×D12—122no—
214C3×C4×D3—21no—
215C3×C4×D3—21no—
216C3×C4×D3—21no—
217C3×C4×D3—21no—
218C3×C4×D3—41no—
219C3×C4×D3—41no—
220C3×C4×D3—64no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—22no—
224C3×Dic6—32no—
225C3×Dic6—32no—
226C3×Dic6—41no—
227C3×Dic6—62no—
228C3×Dic6—64no—
229C3×Dic6—64no—
230C3×Dic6—61no—
231C3×Dic6—61no—
232C3×Dic6—121no—
233C3×Dic6D1212yescomplemented
234C3×Dic6#6712yescomplemented
235#627—24no—
236#626—42no—
237D3×Dic3—24no—
238C3⋊(C3⋊C8)—21no—
239C3⋊(C3⋊C8)—61no—
240C3⋊(C3⋊C8)—62no—
241C3⋊(C3⋊C8)—61no—
242C3⋊(C3⋊C8)—61no—
243C3⋊(C3⋊C8)—121no—
244C3⋊(C3⋊C8)—121no—
245C3⋊(C3⋊C8)—122no—
246C3×C3⋊C8—61no—
247C3×C3⋊C8—61no—
248C3×C3⋊C8—121no—
249#448—271noSylow
250C2×C33#12011yescharacteristic
251C2×C33C2×SD1611yescharacteristic
252C2×C33#13711yescharacteristic
253C3×C3⋊D3—81no—
254C32×D3—22no—
255C2×C3×Q8—31no—
256C2×C3×Q8—31no—
257C2×C3×Q8—61no—
258C2×C3×Q8—91no—
259C2×C3×Q8—181no—
260C2×C3×Q8—182no—
261C2×C3×Q8—182no—
262C2×C3×D4—91no—
263C2×C3×D4—181no—
264C22×C3×C4—31no—
265C22×C3×C4—31no—
266C22×C3×C4—61no—
267C22×C3×C4—181no—
268C2×#67—61no—
269C22×Dic3—61no—
270D3×Q8—62no—
271D3×Q8—62no—
272C2×D12—31no—
273C2×D12—31no—
274C2×D12—61no—
275C2×D12—61no—
276C2×C4×D3—61no—
277C2×C4×D3D3211yescharacteristic
278C2×Dic6—31no—
279C2×Dic6—31no—
280C2×Dic6—61no—
281C2×Dic6—62no—
282C2×Dic6—61no—
283C2×Dic6—61no—
284C2×Dic6—62no—
285C2×Dic6D3211yescharacteristic
286C2×C3×C8—91no—
287C3×C4⋊C4—31no—
288C3×C4⋊C4—31no—
289C3×C4⋊C4—61no—
290C3×C22⋊C4—181no—
291#215—182no—
292#215—182no—
293#215—182no—
294#215—184no—
295#214—182no—
296#214—182no—
297(C2×D3)⋊C4—61no—
298(C2×D3)⋊C4—61no—
299C4⋊Dic3D3211yescharacteristic
300C2×C3⋊C8—31no—
301C2×C3⋊C8—31no—
302C2×C3⋊C8—61no—
303C2×C3⋊C8—91no—
304C2×C3⋊C8—91no—
305C2×C3⋊C8—91no—
306C2×C3⋊C8—181no—
307C2×C3⋊C8—181no—
308C2×C3⋊C8—181no—
309C2×C3⋊C8—361no—
310#204—182no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—32no—
315C22×C32—41no—
316C22×C32—61no—
317C22×C32—61no—
318C22×C32D3×D411yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C22×C32C2×#6711yescharacteristic
321C2×C3⋊D3—41no—
322C2×C3⋊D3—41no—
323C2×C3⋊D3—41no—
324C2×C3×D3—21no—
325C2×C3×D3—21no—
326C2×C3×D3—21no—
327C2×C3×D3—21no—
328C2×C3×D3—22no—
329C2×C3×D3—41no—
330C2×C3×D3—41no—
331C2×C3×D3—122no—
332C2×C3×D3—121no—
333C2×C3×D3—121no—
334C2×C3×D3—121no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C2×C3×D3—121no—
338C2×C3×D3—241no—
339C2×C3×D3—241no—
340C2×C3×D3#21412yescomplemented
341C2×C3×D3#21512yescomplemented
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—41no—
349C32×C4—41no—
350C32×C4—61no—
351C32×C4—62no—
352C32×C4—121no—
353C32×C4C2×D1211yescharacteristic
354C32×C4D3×D411yescharacteristic
355C32×C4D3×D411yescharacteristic
356C32×C4D3×D411yescharacteristic
357C32×C4C2×#6711yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C3⋊Dic3—22no—
360C3×Dic3—22no—
361C3×Dic3—21no—
362C3×Dic3—22no—
363C3×Dic3—21no—
364C3×Dic3—42no—
365C3×Dic3—41no—
366C3×Dic3—41no—
367C3×Dic3—41no—
368C3×Dic3—62no—
369C3×Dic3—62no—
370C3×Dic3—81no—
371C3×Dic3—122no—
372C22×Q8—91no—
373C2×SD16—271no—
374C2×SD16—271no—
375#121—91no—
376#103—91no—
377#103—271no—
378C22⋊C8—271no—
379C33#44811yescharacteristic, complemented, Sylow
380C23×C3—31no—
381C23×C3—31no—
382C23×C3—61no—
383C23×C3—181no—
384C22×D3—61no—
385C22×D3—61no—
386C22×D3—121no—
387C22×D3#62511yescharacteristic
388C3×Q8—32no—
389C3×Q8—32no—
390C3×Q8—62no—
391C3×Q8—61no—
392C3×Q8—61no—
393C3×Q8—92no—
394C3×Q8—121no—
395C3×Q8—184no—
396C3×Q8—181no—
397C3×D4—181no—
398C3×D4—181no—
399C3×D4—362no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—31no—
404C2×C3×C4—31no—
405C2×C3×C4—41no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—61no—
412C2×C3×C4—61no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—121no—
416C2×C3×C4—121no—
417C2×C3×C4—121no—
418C2×C3×C4—121no—
419C2×C3×C4—184no—
420C2×C3×C4—181no—
421C2×C3×C4C2×D3211yescharacteristic
422C2×C3×C4C2×D3211yescharacteristic
423C2×C3×C4C2×D3211yescharacteristic
424#67—122no—
425C2×Dic3—21no—
426C2×Dic3—21no—
427C2×Dic3—64no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—121no—
431C2×Dic3#62511yescharacteristic
432D12—62no—
433D12—61no—
434D12—61no—
435D12—61no—
436D12—61no—
437D12—121no—
438D12—122no—
439C4×D3—21no—
440C4×D3—21no—
441C4×D3—64no—
442Dic6—21no—
443Dic6—32no—
444Dic6—32no—
445Dic6—62no—
446Dic6—64no—
447Dic6—64no—
448Dic6—61no—
449Dic6—61no—
450Dic6—121no—
451Dic6#62612yescomplemented
452C3×C8—181no—
453C3⋊C8—61no—
454C3⋊C8—61no—
455C3⋊C8—121no—
456C3⋊C8—182no—
457C3⋊C8—181no—
458C3⋊C8—181no—
459C3⋊C8—181no—
460C3⋊C8—361no—
461C3⋊C8—361no—
462C3⋊C8—362no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—41no—
473C2×C32—41no—
474C2×C32—41no—
475C2×C32—62no—
476C2×C32#88211yescharacteristic
477C2×C32D3×SD1611yescharacteristic
478C2×C32#91811yescharacteristic
479C2×C32C2×#21411yescharacteristic
480C2×C32#93711yescharacteristic
481C2×C32C2×#21511yescharacteristic
482C2×C32#94211yescharacteristic
483C2×C32#95111yescharacteristic
484C2×C32#95311yescharacteristic
485C3⋊D3—81no—
486C3×D3—22no—
487C3×D3—22no—
488C3×D3—42no—
489C3×D3—242no—
490C3×D3—241no—
491C3×D3—241no—
492C2×Q8—31no—
493C2×Q8—91no—
494C2×Q8—181no—
495C2×Q8—182no—
496C2×Q8—182no—
497C2×D4—91no—
498C2×D4—181no—
499C22×C4—31no—
500C22×C4—181no—
501SD16—542no—
502SD16—542no—
503C2×C8—91no—
504C2×C8—271no—
505C4⋊C4—31no—
506C22⋊C4—181no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—32no—
511C22×C3—32no—
512C22×C3—41no—
513C22×C3—62no—
514C22×C3—61no—
515C22×C3—61no—
516C22×C3—61no—
517C22×C3—61no—
518C22×C3—121no—
519C22×C3—121no—
520C22×C3—361no—
521C22×C3—361no—
522C22×C3—361no—
523C22×C3C2×#62511yescharacteristic
524C22×C3D3×#6711yescharacteristic
525C22×C3D3×#6711yescharacteristic
526C2×D3—21no—
527C2×D3—21no—
528C2×D3—122no—
529C2×D3—121no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—121no—
535C2×D3—241no—
536C2×D3—241no—
537C2×D3#371512yescomplemented
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—61no—
547C3×C4—62no—
548C3×C4—62no—
549C3×C4—61no—
550C3×C4—121no—
551C3×C4—122no—
552C3×C4—121no—
553C3×C4—121no—
554C3×C4—182no—
555C3×C4—241no—
556C3×C4D3×D1211yescharacteristic
557C3×C4#380211yescharacteristic
558C3×C4C2×#62611yescharacteristic
559C3×C4D3×#6711yescharacteristic
560C3×C4D3×#6711yescharacteristic
561C3×C4D3×#6711yescharacteristic
562Dic3—21no—
563Dic3—22no—
564Dic3—41no—
565Dic3—62no—
566Dic3—62no—
567Dic3—122no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#480611yescharacteristic, complemented
573C32#504011yescharacteristic, complemented
574C32#522711yescharacteristic, complemented
575C23—31no—
576C23—181no—
577Q8—32no—
578Q8—61no—
579Q8—92no—
580Q8—184no—
581Q8—181no—
582D4—181no—
583D4—181no—
584D4—362no—
585C2×C4—31no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—184no—
591C2×C4—181no—
592C2×C4D3311yescharacteristic
593C8—181no—
594C8—541no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—62no—
608C2×C3—62no—
609C2×C3—122no—
610C2×C3—721no—
611C2×C3C2×#371511yescharacteristic
612C2×C3#6382711yescharacteristic
613C2×C3#6392211yescharacteristic
614C2×C3D3×#21411yescharacteristic
615C2×C3D3×#21511yescharacteristic
616C2×C3#6394311yescharacteristic
617C2×C3#6394811yescharacteristic
618C2×C3#6397611yescharacteristic
619C2×C3#6397711yescharacteristic
620D3—22no—
621D3—242no—
622D3—241no—
623D3—241no—
624C22—32no—
625C22—61no—
626C22—61no—
627C22—361no—
628C22—361no—
629C22—361no—
630C22D3×#62511yescharacteristic
631C4—61no—
632C4—62no—
633C4—121no—
634C4—182no—
635C4D3×#62611yescharacteristic
636C4#8895311yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059060311yescharacteristic, complemented
642C3#1059117111yescharacteristic, complemented
643C3#1059158411yescharacteristic, complemented
644C2—62no—
645C2—721no—
646C2D3×#371511yescharacteristic
647C2#1172288311yescharacteristic
648C2#1172294711yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypocenterC33#448
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×C3×D3×Dic6C2semidirect
2. C2×#88811C2semidirect
3. C2×#88793C2semidirect
5. #11722039C2semidirect
6. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
7. C3×#63565C2semidirect
28. C2×D3×Dic6D3semidirect
46. #63565D3semidirect
57. C3×#634D4semidirect
64. C32×Dic6D4semidirect
135. C2×C32×D3SD16semidirect
192. #634#67semidirect
233. C3×Dic6D12semidirect
234. C3×Dic6#67semidirect
340. C2×C3×D3#214semidirect
341. C2×C3×D3#215semidirect
379. C33#448semidirect
451. Dic6#626semidirect
537. C2×D3#3715semidirect
572. C32#4806semidirect
573. C32#5040semidirect
574. C32#5227semidirect
641. C3#10590603semidirect
642. C3#10591171semidirect
643. C3#10591584semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C2×#888112
C2×#887932
C3×#639092
#117220392
(C2×D3)⋊(C3⋊(C3⋊C8))2
C3×#635652
#105915843
#105911713
#105906033