Finite Group Databasebeta

← #13170791

Subgroups — #13170791

Order 1728.

2030 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic
2C2×#88949C212yesmaximal, complemented
3#11723501C212yesmaximal, complemented
4C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
5C3×#63900C211yesmaximal, characteristic, complemented
6C3×#63894C212yesmaximal, complemented
7#11722938C212yesmaximal, complemented
8#11722931C212yesmaximal, complemented
9#11722905C212yesmaximal, complemented
10D3×#893—32nomaximal
11C4×#3802—31nomaximal
12C22×#1113C2212yes—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6295C2212yes—
16C4⋊#1113C418yescomplemented
17C2×Dic3×C3⋊D3C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C2×Dic3×C3⋊D3C2214yescomplemented
20C2×C3×#624C2212yes—
21C2×C3×#624C2214yescomplemented
22#88949—28no—
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
24C32×C4⋊Dic3C2212yescomplemented
25C32×C4×Dic3C2211yescharacteristic, complemented
26#88803C2214yescomplemented
27Dic3×C3⋊Dic3C2211yescharacteristic, complemented
28Dic3×C3⋊Dic3C2211yescharacteristic, complemented
29C3×#3724C2212yescomplemented
30C3×#3722C2214yescomplemented
31C2×#3802—31no—
32C2×D3×D12—32no—
33C2×C4×D32—32no—
34C2×C4×D32—34no—
35#64102—32no—
36C42×C3⋊D3—31no—
37C3×#893—32no—
38C3×D3×C4⋊C4—32no—
39C3×C4×D12—32no—
40#63921—32no—
41D3×(C2×D3)⋊C4—34no—
42#63904—34no—
43#63902—32no—
44C4×#625—32no—
45#63900D311yescharacteristic, complemented
46#63898—32no—
47#63898—32no—
48#63898—62no—
49#63894D312yescomplemented
50#63881—34no—
51C4×D3×Dic3—32no—
52C4×D3×Dic3—61no—
53C22×C3×C3⋊D3C2314yes—
54C22×C3×C3⋊D3C2311yescharacteristic
55C2×#1113—24no—
56C2×#1113—24no—
57C2×#1113C2×C418yescomplemented
58C2×C33×C4C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C3×C4⋊(C3⋊D3)C2×C418yescomplemented
63C3×C4×C3⋊D3C2×C414yescomplemented
64C2×C32×Dic3C2314yes—
65C2×C32×Dic3C2311yescharacteristic
66C2×C32×Dic3C2311yescharacteristic
67#6295C2×C418yescomplemented
68Dic3×C3⋊D3—22no—
69Dic3×C3⋊D3—28no—
70Dic3×C3⋊D3D414yescomplemented
71C3×#624—24no—
72C3×#624—28no—
73C4×D3×D4—92no—
74C2×#893—91no—
75C22×D32—34no—
76C22×D32—32no—
77C2×C4⋊(C3⋊D3)C2×D312yes—
78C2×C4×C3⋊D3—31no—
79C2×C4×C3⋊D3—34no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C3×D12—32no—
83C2×C3×D12—32no—
84C2×C3×D12—62no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—34no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—34no—
89C2×C3×C4×D3—61no—
90C2×#626—34no—
91C2×#625—32no—
92C2×#624C2×D312yes—
93C2×#624C2×D314yescomplemented
94C2×D3×Dic3—32no—
95C2×D3×Dic3—34no—
96C2×D3×Dic3—34no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—61no—
99C2×D3×Dic3—61no—
100C2×D3×Dic3—64no—
101#3802—38no—
102D3×D12—316no—
103C4×D32—616no—
104C4×D32—68no—
105C32×C4⋊C4—32no—
106C32×C42—31no—
107#3752—34no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—32no—
110C4×C3⋊Dic3—31no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—61no—
113C4×C3⋊Dic3—62no—
114C4×C3⋊Dic3—121no—
115C3×(C2×D3)⋊C4—34no—
116C3×(C2×D3)⋊C4—34no—
117C3×C4⋊Dic3—22no—
118C3×C4⋊Dic3C2×D312yescomplemented
119C3×C4⋊Dic3C2×D312yescomplemented
120C3×Dic3⋊C4—34no—
121C3×C4×Dic3—21no—
122C3×C4×Dic3C2×D312yescomplemented
123#3724C2×D312yescomplemented
124#3722C2×D314yescomplemented
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—34no—
127(C2×D3)⋊Dic3—64no—
128Dic32—32no—
129Dic32—32no—
130Dic32—61no—
131Dic32—61no—
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—24no—
134C2×C3×C3⋊D3—24no—
135C2×C3×C3⋊D3C22×C412yes—
136C2×C3×C3⋊D3C22×C418yescomplemented
137C2×C3×C3⋊D3C2×D412yes—
138C2×C3×C3⋊D3#4112yes—
139#1113—216no—
140C33×C4C22×C412yescomplemented
141C3⋊(C3⋊Dic3)—21no—
142C3⋊(C3⋊Dic3)C2×D412yescomplemented
143C3×C3⋊Dic3C22×C412yescomplemented
144C32×Dic3—24no—
145C32×Dic3—21no—
146C32×Dic3C2×D412yescomplemented
147C2×D3×D4—92no—
148C22×D12—91no—
149C22×C4×D3—92no—
150C22×C4×D3—94no—
151C3×C4×D4—92no—
152C2×C3×C4⋊C4—91no—
153D4×Dic3—92no—
154C4×#67—94no—
155C2×(C2×D3)⋊C4—92no—
156C2×C4×Dic3—91no—
157#893—32no—
158#893—92no—
159#893—92no—
160#893—92no—
161#893—92no—
162#893—182no—
163D3×C4⋊C4—32no—
164#67⋊C4—94no—
165D3×C22⋊C4—94no—
166C4×D12—32no—
167C42×D3—92no—
168C42×D3—181no—
169C22×C3⋊D3C22×D314yes—
170C22×C3⋊D3C22×D311yescharacteristic
171C22×C3×D3—32no—
172C22×C3×D3—34no—
173C22×C3×D3—34no—
174C22×C3×D3—61no—
175C22×C3×D3—64no—
176C2×D32—38no—
177C2×D32—316no—
178C2×D32—68no—
179C2×D32—68no—
180C2×D32—64no—
181C2×D32—64no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—31no—
185C2×C32×C4—34no—
186C2×C32×C4—31no—
187C2×C32×C4—41no—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—32no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—61no—
196C2×C3⋊Dic3—62no—
197C2×C3⋊Dic3—62no—
198C2×C3⋊Dic3—121no—
199C2×C3⋊Dic3—121no—
200C2×C3⋊Dic3C22×D311yescharacteristic
201C4⋊(C3⋊D3)C4×D318yescomplemented
202C4×C3⋊D3—34no—
203C4×C3⋊D3—62no—
204C4×C3⋊D3—68no—
205C4×C3⋊D3C4×D314yescomplemented
206C2×C3×Dic3—21no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—24no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3C22×D312yes—
212C2×C3×Dic3C22×D314yes—
213C2×C3×Dic3C22×D314yes—
214C2×C3×Dic3C22×D312yes—
215C3×D12—38no—
216C3×D12—38no—
217C3×D12—68no—
218C3×C4×D3—38no—
219C3×C4×D3—64no—
220C3×C4×D3—68no—
221C3×C4×D3—68no—
222C3×C4×D3—68no—
223#626—316no—
224#625—38no—
225#624—24no—
226#624—28no—
227D3×Dic3—38no—
228D3×Dic3—64no—
229D3×Dic3—68no—
230D3×Dic3—68no—
231D3×Dic3—64no—
232D3×Dic3—64no—
233D3×Dic3—128no—
234C2×C4×D4—271noSylow
235C2×C33C23×C411yescharacteristic
236C2×C33C22×D411yescharacteristic
237C2×C33C2×#4111yescharacteristic
238C3×C3⋊D3—28no—
239C3×C3⋊D3C4×D414yescomplemented
240C23×D3—94no—
241C23×D3—92no—
242C2×C3×D4—92no—
243C22×C3×C4—92no—
244C22×C3×C4—94no—
245C22×C3×C4—91no—
246C2×#67—94no—
247C22×Dic3—91no—
248C22×Dic3—94no—
249C22×Dic3—91no—
250D3×D4—916no—
251C2×D12—32no—
252C2×D12—32no—
253C2×D12—62no—
254C2×D12—98no—
255C2×D12—92no—
256C2×C4×D3—32no—
257C2×C4×D3—34no—
258C2×C4×D3—34no—
259C2×C4×D3—34no—
260C2×C4×D3—61no—
261C2×C4×D3—92no—
262C2×C4×D3—92no—
263C2×C4×D3—94no—
264C2×C4×D3—94no—
265C2×C4×D3—181no—
266C2×C4×D3—181no—
267C2×C4×D3—184no—
268C2×C4×D3—1816no—
269C2×C4×D3—188no—
270C3×C4⋊C4—32no—
271C3×C4⋊C4—32no—
272C3×C4⋊C4—62no—
273C3×C4⋊C4—92no—
274C3×C22⋊C4—94no—
275C3×C42—32no—
276C3×C42—61no—
277#217—94no—
278(C2×D3)⋊C4—34no—
279(C2×D3)⋊C4—34no—
280(C2×D3)⋊C4—94no—
281(C2×D3)⋊C4—94no—
282(C2×D3)⋊C4—184no—
283C4⋊Dic3D3212yescomplemented
284Dic3⋊C4—34no—
285C4×Dic3—92no—
286C4×Dic3—92no—
287C4×Dic3—92no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—181no—
292C4×Dic3—181no—
293C4×Dic3—181no—
294C4×Dic3—182no—
295C4×Dic3—361no—
296C4×Dic3D3211yescharacteristic, complemented
297C22×C32—21no—
298C22×C32—22no—
299C22×C32—41no—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D312yes—
302C2×C3⋊D3—24no—
303C2×C3⋊D3—24no—
304C2×C3⋊D3C2×C4×D312yes—
305C2×C3⋊D3C2×C4×D318yescomplemented
306C2×C3⋊D3D3×D412yes—
307C2×C3⋊D3#23912yes—
308C2×C3×D3—34no—
309C2×C3×D3—34no—
310C2×C3×D3—34no—
311C2×C3×D3—38no—
312C2×C3×D3—38no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—68no—
317C2×C3×D3—64no—
318C2×C3×D3—64no—
319C2×C3×D3—64no—
320C2×C3×D3—64no—
321C2×C3×D3—124no—
322C2×C3×D3—124no—
323D32—632no—
324D32—616no—
325C32×C4—22no—
326C32×C4—24no—
327C32×C4—32no—
328C32×C4—42no—
329C32×C4—61no—
330C32×C4—64no—
331C32×C4C2×C4×D314yescomplemented
332C32×C4C2×C4×D312yescomplemented
333C3⋊Dic3—34no—
334C3⋊Dic3—32no—
335C3⋊Dic3—62no—
336C3⋊Dic3—64no—
337C3⋊Dic3—61no—
338C3⋊Dic3—62no—
339C3⋊Dic3—62no—
340C3⋊Dic3—122no—
341C3⋊Dic3—122no—
342C3⋊Dic3—122no—
343C3⋊Dic3C2×C4×D312yescomplemented
344C3×Dic3—22no—
345C3×Dic3—24no—
346C3×Dic3—24no—
347C3×Dic3—22no—
348C3×Dic3—22no—
349C3×Dic3—34no—
350C3×Dic3—44no—
351C3×Dic3—62no—
352C3×Dic3D3×D414yescomplemented
353C22×D4—271no—
354C23×C4—272no—
355C4×D4—92no—
356C4×D4—272no—
357C4×D4—272no—
358C4×D4—272no—
359C2×C4⋊C4—91no—
360C2×C22⋊C4—272no—
361C2×C42—271no—
362C33C2×C4×D411yescharacteristic, complemented, Sylow
363C23×C3—91no—
364C23×C3—94no—
365C22×D3—32no—
366C22×D3—34no—
367C22×D3—34no—
368C22×D3—61no—
369C22×D3—64no—
370C22×D3—98no—
371C22×D3—916no—
372C22×D3—188no—
373C22×D3—188no—
374C22×D3—184no—
375C22×D3—184no—
376C3×D4—98no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—32no—
380C2×C3×C4—32no—
381C2×C3×C4—34no—
382C2×C3×C4—34no—
383C2×C3×C4—41no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—64no—
387C2×C3×C4—91no—
388C2×C3×C4—94no—
389C2×C3×C4—188no—
390C2×C3×C4—184no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3212yes—
393#67—916no—
394C2×Dic3—32no—
395C2×Dic3—61no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398C2×Dic3—94no—
399C2×Dic3—91no—
400C2×Dic3—92no—
401C2×Dic3—181no—
402C2×Dic3—181no—
403C2×Dic3—182no—
404C2×Dic3—182no—
405C2×Dic3—188no—
406C2×Dic3—182no—
407C2×Dic3—361no—
408C2×Dic3—361no—
409C2×Dic3C2×D3211yescharacteristic
410C2×Dic3C2×D3214yes—
411C2×Dic3C2×D3211yescharacteristic
412D12—38no—
413D12—38no—
414D12—68no—
415D12—98no—
416C4×D3—38no—
417C4×D3—64no—
418C4×D3—68no—
419C4×D3—68no—
420C4×D3—68no—
421C4×D3—98no—
422C4×D3—184no—
423C4×D3—188no—
424C4×D3—188no—
425C4×D3—184no—
426C4×D3—184no—
427C4×D3—368no—
428C2×C32—21no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—22no—
433C2×C32—22no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32C22×C4×D311yescharacteristic
438C2×C32C22×C4×D312yes—
439C2×C32C2×D3×D412yes—
440C2×C32C2×D3×D411yescharacteristic
441C2×C32C2×#23911yescharacteristic
442C2×C32D3×#4112yes—
443C3⋊D3—28no—
444C3⋊D3#89314yescomplemented
445C3×D3—38no—
446C3×D3—64no—
447C3×D3—68no—
448C3×D3—68no—
449C3×D3—128no—
450C24—272no—
451C2×D4—92no—
452C2×D4—278no—
453C2×D4—272no—
454C22×C4—92no—
455C22×C4—94no—
456C22×C4—91no—
457C22×C4—271no—
458C22×C4—271no—
459C22×C4—274no—
460C22×C4—548no—
461C4⋊C4—32no—
462C4⋊C4—92no—
463C22⋊C4—94no—
464C22⋊C4—274no—
465C42—31no—
466C42—271no—
467C42—271no—
468C42—271no—
469C22×C3—21no—
470C22×C3—22no—
471C22×C3—41no—
472C22×C3—92no—
473C22×C3—92no—
474C22×C3—92no—
475C22×C3—98no—
476C22×C3—184no—
477C22×C3—184no—
478C22×C3C22×D3211yescharacteristic
479C22×C3C22×D3212yes—
480C2×D3—34no—
481C2×D3—34no—
482C2×D3—34no—
483C2×D3—38no—
484C2×D3—38no—
485C2×D3—62no—
486C2×D3—62no—
487C2×D3—62no—
488C2×D3—68no—
489C2×D3—64no—
490C2×D3—64no—
491C2×D3—64no—
492C2×D3—64no—
493C2×D3—124no—
494C2×D3—124no—
495C2×D3—1832no—
496C2×D3—1816no—
497C3×C4—22no—
498C3×C4—24no—
499C3×C4—34no—
500C3×C4—42no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—64no—
504C3×C4—64no—
505C3×C4—62no—
506C3×C4—92no—
507C3×C4—124no—
508C3×C4C4×D3214yescomplemented
509C3×C4C4×D3212yescomplemented
510Dic3—21no—
511Dic3—24no—
512Dic3—34no—
513Dic3—62no—
514Dic3—92no—
515Dic3—94no—
516Dic3—182no—
517Dic3—184no—
518Dic3—182no—
519Dic3—181no—
520Dic3—182no—
521Dic3—362no—
522Dic3—362no—
523Dic3—362no—
524Dic3#380212yescomplemented
525C32—22no—
526C32—21no—
527C32—41no—
528C32C2×#89311yescharacteristic, complemented
529C32C4×D3×D412yescomplemented
530C23—91no—
531C23—94no—
532C23—278no—
533C23—544no—
534C23—544no—
535D4—98no—
536D4—278no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—34no—
540C2×C4—91no—
541C2×C4—94no—
542C2×C4—188no—
543C2×C4—184no—
544C2×C4—274no—
545C2×C4—271no—
546C2×C4—271no—
547C2×C4—548no—
548C2×C4—542no—
549C2×C4D3311yescharacteristic
550C2×C3—21no—
551C2×C3—22no—
552C2×C3—21no—
553C2×C3—22no—
554C2×C3—21no—
555C2×C3—22no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—41no—
559C2×C3—94no—
560C2×C3—188no—
561C2×C3C2×C4×D3212yes—
562C2×C3C2×C4×D3211yescharacteristic
563C2×C3C2×#380211yescharacteristic
564C2×C3#6430611yescharacteristic
565C2×C3D32×D412yes—
566C2×C3D3×#23912yes—
567D3—38no—
568D3—64no—
569D3—68no—
570D3—68no—
571D3—128no—
572C22—92no—
573C22—92no—
574C22—92no—
575C22—98no—
576C22—184no—
577C22—184no—
578C22—5416no—
579C22C2×D3311yescharacteristic
580C4—32no—
581C4—61no—
582C4—64no—
583C4—92no—
584C4—272no—
585C4—541no—
586C4#8894912yescomplemented
587C3—21no—
588C3—22no—
589C3—41no—
590C3C4×#380211yescharacteristic, complemented
591C3D3×#89312yescomplemented
592C2—94no—
593C2—188no—
594C2#1172439111yescharacteristic
595C2D3×#380211yescharacteristic
596C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4D33
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88949C2semidirect
3. #11723501C2semidirect
4. C4×Dic3×C3⋊D3C2semidirect
5. C3×#63900C2semidirect
6. C3×#63894C2semidirect
7. #11722938C2semidirect
8. #11722931C2semidirect
9. #11722905C2semidirect
16. C4⋊#1113C4semidirect
19. C2×Dic3×C3⋊D3C22semidirect
21. C2×C3×#624C22semidirect
23. C4×C3⋊(C3⋊Dic3)C22semidirect
24. C32×C4⋊Dic3C22semidirect
25. C32×C4×Dic3C22semidirect
26. #88803C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
28. Dic3×C3⋊Dic3C22semidirect
29. C3×#3724C22semidirect
30. C3×#3722C22semidirect
45. #63900D3semidirect
49. #63894D3semidirect
57. C2×#1113C2×C4semidirect
62. C3×C4⋊(C3⋊D3)C2×C4semidirect
63. C3×C4×C3⋊D3C2×C4semidirect
67. #6295C2×C4semidirect
70. Dic3×C3⋊D3D4semidirect
93. C2×#624C2×D3semidirect
118. C3×C4⋊Dic3C2×D3semidirect
119. C3×C4⋊Dic3C2×D3semidirect
122. C3×C4×Dic3C2×D3semidirect
123. #3724C2×D3semidirect
124. #3722C2×D3semidirect
136. C2×C3×C3⋊D3C22×C4semidirect
140. C33×C4C22×C4semidirect
142. C3⋊(C3⋊Dic3)C2×D4semidirect
143. C3×C3⋊Dic3C22×C4semidirect
146. C32×Dic3C2×D4semidirect
201. C4⋊(C3⋊D3)C4×D3semidirect
205. C4×C3⋊D3C4×D3semidirect
239. C3×C3⋊D3C4×D4semidirect
283. C4⋊Dic3D32semidirect
296. C4×Dic3D32semidirect
305. C2×C3⋊D3C2×C4×D3semidirect
331. C32×C4C2×C4×D3semidirect
332. C32×C4C2×C4×D3semidirect
343. C3⋊Dic3C2×C4×D3semidirect
352. C3×Dic3D3×D4semidirect
362. C33C2×C4×D4semidirect
444. C3⋊D3#893semidirect
508. C3×C4C4×D32semidirect
509. C3×C4C4×D32semidirect
524. Dic3#3802semidirect
528. C32C2×#893semidirect
529. C32C4×D3×D4semidirect
586. C4#88949semidirect
590. C3C4×#3802semidirect
591. C3D3×#893semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#889492
#117235012
C4×Dic3×C3⋊D32
C3×#639002
C3×#638942
#117229382
#117229312
#117229052
D3×#8933
C4×#38023