Finite Group Databasebeta

← #13167966

Subgroups — #13167966

Order 1728.

1762 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2#11723689C211yesmaximal, characteristic, complemented
3#11723413C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63841C212yesmaximal, complemented
6#11722918C212yesmaximal, complemented
7#11722917C212yesmaximal, complemented
8#11722904C212yesmaximal
9Dic3×#624C212yesmaximal, complemented
10#10594969—31nomaximal
11D3×#892—32nomaximal
12C22×D3×C3⋊D3C2211yescharacteristic
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C4×D3C2211yescharacteristic
16C2×#6294C2212yes—
17C2×#6289C2211yescharacteristic
18C2×Dic3×C3⋊D3C2211yescharacteristic
19C2×D3×C3⋊Dic3C2211yescharacteristic
20C4×D3×C3⋊D3C418yescomplemented
21C2×C3×#624C2212yes—
22C3×C4⋊(C3⋊Dic3)C2212yescomplemented
23C32×C4⋊Dic3C2212yescomplemented
24#88815C2212yescomplemented
25#88804C2214yes—
26Dic3×C3⋊Dic3C2214yescomplemented
27C3×#3724C2212yescomplemented
28C3×(C2×D3)⋊Dic3C2214yes—
29C3×Dic32C2214yescomplemented
30C22×C4×C3⋊D3—31no—
31C2×C4×D32—32no—
32C2×C4×D32—61no—
33#64101—32no—
34C3×#926—32no—
35C3×#892—32no—
36C3×D3×C4⋊C4—32no—
37#63974—32no—
38#63957—32no—
39D3×(C2×D3)⋊C4—34no—
40#63900—31no—
41#63900D311yescharacteristic, complemented
42D3×C4⋊Dic3—32no—
43C4×#624—32no—
44#63878—34no—
45C4×D3×Dic3—34no—
46#63874—32no—
47#63849—32no—
48#63849—32no—
49#63849—62no—
50#63841D312yescomplemented
51#63836—34no—
52C22×C3⋊(C3⋊D3)C2311yescharacteristic
53C22×C3×C3⋊D3C2311yescharacteristic
54C22×C32×D3C2311yescharacteristic
55C2×D3×C3⋊D3—22no—
56C2×D3×C3⋊D3—22no—
57C2×D3×C3⋊D3C2×C414yes—
58C2×C33×C4C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C4×C3⋊(C3⋊D3)C2×C414yescomplemented
61C2×C3×C3⋊Dic3C2314yes—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4×C3⋊D3C2×C414yes—
64C2×C32×Dic3C2314yes—
65C2×C32×Dic3C2311yescharacteristic
66C32×C4×D3C2×C414yescomplemented
67#6294—24no—
68#6289C2×C414yescomplemented
69Dic3×C3⋊D3C2×C414yes—
70D3×C3⋊Dic3C2×C414yescomplemented
71C3×#624—24no—
72D3×#117—92no—
73C2×#892—91no—
74C23×C3⋊D3—31no—
75C22×D32—32no—
76C22×D32—61no—
77C22×C32×C4—31no—
78C22×C3⋊Dic3—31no—
79C2×C4×C3⋊D3—31no—
80C2×C4×C3⋊D3—32no—
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—38no—
84C2×C4×C3⋊D3—61no—
85C2×C4×C3⋊D3—62no—
86C2×C4×C3⋊D3—121no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×C3×C4×D3—21no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—34no—
91C2×C3×C4×D3—61no—
92C2×C3×C4×D3C2×D312yes—
93C2×#624—32no—
94C2×#624—32no—
95C2×#624—61no—
96C2×#624C2×D312yes—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—34no—
100C2×D3×Dic3—61no—
101C2×D3×Dic3—61no—
102C4×D32—316no—
103C4×D32—68no—
104C32×C4⋊C4—32no—
105#3752—34no—
106C4⋊(C3⋊Dic3)C2×D312yescomplemented
107C4×C3⋊Dic3—34no—
108C3×#217—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×C4⋊Dic3—22no—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—62no—
114C3×C4⋊Dic3C2×D312yescomplemented
115C3×C4⋊Dic3C2×D312yescomplemented
116C3×Dic3⋊C4—34no—
117C3×C4×Dic3—34no—
118C3×C4×Dic3—34no—
119#3724—32no—
120#3724C2×D312yescomplemented
121Dic3⋊Dic3—34no—
122#3722—34no—
123#3722—34no—
124#3722—64no—
125(C2×D3)⋊Dic3C2×D314yes—
126Dic32—34no—
127Dic32—34no—
128Dic32—64no—
129Dic32C2×D314yescomplemented
130C22×C33C2411yescharacteristic
131C2×C3⋊(C3⋊D3)—21no—
132C2×C3⋊(C3⋊D3)—21no—
133C2×C3⋊(C3⋊D3)C22×C412yes—
134C2×C3×C3⋊D3C22×C412yes—
135C2×C3×C3⋊D3#4112yes—
136C2×C3×C3⋊D3#4112yes—
137C2×C32×D3—21no—
138C2×C32×D3—21no—
139C2×C32×D3C22×C412yes—
140D3×C3⋊D3—28no—
141C33×C4C22×C412yes—
142C3⋊(C3⋊Dic3)C22×C412yes—
143C3×C3⋊Dic3—24no—
144C3×C3⋊Dic3C22×C412yes—
145C32×Dic3—24no—
146C32×Dic3C22×C412yes—
147C22×C4×D3—92no—
148C22×C4×D3—91no—
149C22×C4×D3—181no—
150C3×#117—92no—
151C2×C3×C4⋊C4—91no—
152C2×(C2×D3)⋊C4—92no—
153#926—32no—
154C2×C4⋊Dic3—91no—
155C2×C4×Dic3—92no—
156#892—32no—
157#892—92no—
158#892—92no—
159#892—92no—
160#892—92no—
161#892—92no—
162#892—182no—
163D3×C4⋊C4—32no—
164D3×C4⋊C4—92no—
165D3×C22⋊C4—94no—
166#877—94no—
167#872—94no—
168C42×D3—94no—
169C23×C32—31no—
170C22×C3⋊D3—32no—
171C22×C3⋊D3—31no—
172C22×C3⋊D3—34no—
173C22×C3⋊D3—61no—
174C22×C3⋊D3—62no—
175C22×C3⋊D3—62no—
176C22×C3⋊D3—62no—
177C22×C3⋊D3—121no—
178C22×C3⋊D3C22×D311yescharacteristic
179C22×C3×D3—21no—
180C22×C3×D3—32no—
181C22×C3×D3—61no—
182C22×C3×D3C22×D312yes—
183C2×D32—38no—
184C2×D32—64no—
185C2×D32—64no—
186C2×D32—64no—
187C2×D32—64no—
188C2×D32—64no—
189C2×C32×C4—22no—
190C2×C32×C4—21no—
191C2×C32×C4—31no—
192C2×C32×C4—34no—
193C2×C32×C4—34no—
194C2×C32×C4—41no—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D312yes—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—34no—
200C2×C3⋊Dic3—62no—
201C2×C3⋊Dic3—61no—
202C2×C3⋊Dic3—121no—
203C2×C3⋊Dic3C22×D314yes—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C4×C3⋊D3—34no—
206C4×C3⋊D3—38no—
207C4×C3⋊D3—34no—
208C4×C3⋊D3—34no—
209C4×C3⋊D3—64no—
210C4×C3⋊D3—68no—
211C4×C3⋊D3—124no—
212C4×C3⋊D3C4×D314yes—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—24no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—34no—
217C2×C3×Dic3—34no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—64no—
220C2×C3×Dic3C22×D314yes—
221C2×C3×Dic3C22×D314yes—
222C2×C3×Dic3C22×D312yes—
223C3×C4×D3—24no—
224C3×C4×D3—38no—
225C3×C4×D3—64no—
226C3×C4×D3—68no—
227C3×C4×D3C4×D318yescomplemented
228#624—24no—
229#624—38no—
230#624—64no—
231#624—64no—
232D3×Dic3—38no—
233D3×Dic3—38no—
234D3×Dic3—64no—
235D3×Dic3—64no—
236D3×Dic3—68no—
237C2×#117—271noSylow
238C2×C33C23×C411yescharacteristic
239C2×C33C2×#4111yescharacteristic
240C2×C33C2×#4111yescharacteristic
241C3⋊(C3⋊D3)—22no—
242C3×C3⋊D3#11714yescomplemented
243C32×D3—22no—
244C23×D3—92no—
245C23×D3—91no—
246C23×D3—181no—
247C22×C3×C4—32no—
248C22×C3×C4—61no—
249C22×C3×C4—92no—
250C22×C3×C4—91no—
251C22×Dic3—92no—
252C22×Dic3—92no—
253C22×Dic3—91no—
254C22×Dic3—181no—
255C2×C4×D3—32no—
256C2×C4×D3—34no—
257C2×C4×D3—61no—
258C2×C4×D3—916no—
259C2×C4×D3—92no—
260C2×C4×D3—92no—
261C2×C4×D3—91no—
262C2×C4×D3—92no—
263C2×C4×D3—91no—
264C2×C4×D3—91no—
265C2×C4×D3—98no—
266C2×C4×D3—94no—
267C2×C4×D3—188no—
268C2×C4×D3—181no—
269C2×C4×D3—182no—
270C2×C4×D3—181no—
271C2×C4×D3—181no—
272C2×C4×D3—361no—
273C2×C4×D3D3211yescharacteristic
274C3×C4⋊C4—32no—
275C3×C4⋊C4—32no—
276C3×C4⋊C4—62no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C22⋊C4—94no—
280C3×C42—94no—
281#217—34no—
282(C2×D3)⋊C4—34no—
283(C2×D3)⋊C4—94no—
284(C2×D3)⋊C4—94no—
285(C2×D3)⋊C4—94no—
286(C2×D3)⋊C4—184no—
287C4⋊Dic3—32no—
288C4⋊Dic3—32no—
289C4⋊Dic3—62no—
290C4⋊Dic3—92no—
291C4⋊Dic3D3212yescomplemented
292Dic3⋊C4—34no—
293Dic3⋊C4—94no—
294C4×Dic3—34no—
295C4×Dic3—34no—
296C4×Dic3—94no—
297C4×Dic3—94no—
298C4×Dic3—94no—
299C4×Dic3—184no—
300C22×C32—22no—
301C22×C32—21no—
302C22×C32—32no—
303C22×C32—41no—
304C22×C32—61no—
305C22×C32—61no—
306C22×C32C23×D311yescharacteristic
307C22×C32C23×D312yes—
308C2×C3⋊D3—34no—
309C2×C3⋊D3—32no—
310C2×C3⋊D3—62no—
311C2×C3⋊D3—64no—
312C2×C3⋊D3—61no—
313C2×C3⋊D3—61no—
314C2×C3⋊D3—68no—
315C2×C3⋊D3—62no—
316C2×C3⋊D3—62no—
317C2×C3⋊D3—62no—
318C2×C3⋊D3—62no—
319C2×C3⋊D3—122no—
320C2×C3⋊D3—122no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—122no—
323C2×C3⋊D3—122no—
324C2×C3⋊D3C2×C4×D312yes—
325C2×C3⋊D3#23712yes—
326C2×C3⋊D3#23912yes—
327C2×C3×D3—22no—
328C2×C3×D3—22no—
329C2×C3×D3—22no—
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—34no—
333C2×C3×D3—34no—
334C2×C3×D3—34no—
335C2×C3×D3—62no—
336C2×C3×D3—62no—
337C2×C3×D3—62no—
338C2×C3×D3C2×C4×D314yes—
339D32—616no—
340D32—616no—
341C32×C4—24no—
342C32×C4—22no—
343C32×C4—32no—
344C32×C4—42no—
345C32×C4—64no—
346C32×C4C2×C4×D312yes—
347C32×C4C2×C4×D314yes—
348C3⋊Dic3—24no—
349C3⋊Dic3—34no—
350C3⋊Dic3—32no—
351C3⋊Dic3—62no—
352C3⋊Dic3—64no—
353C3⋊Dic3—122no—
354C3⋊Dic3C2×C4×D312yes—
355C3×Dic3—24no—
356C3×Dic3—24no—
357C3×Dic3—22no—
358C3×Dic3—34no—
359C3×Dic3—44no—
360C3×Dic3—62no—
361C3×Dic3—64no—
362C3×Dic3—64no—
363C3×Dic3—124no—
364C3×Dic3C2×C4×D314yes—
365C23×C4—271no—
366#117—92no—
367#117—272no—
368#117—272no—
369#117—272no—
370C2×C4⋊C4—91no—
371C2×C4⋊C4—271no—
372C2×C22⋊C4—272no—
373C2×C42—272no—
374C33C2×#11711yescharacteristic, complemented, Sylow
375C23×C3—32no—
376C23×C3—61no—
377C23×C3—91no—
378C22×D3—32no—
379C22×D3—61no—
380C22×D3—98no—
381C22×D3—92no—
382C22×D3—91no—
383C22×D3—94no—
384C22×D3—184no—
385C22×D3—181no—
386C22×D3—182no—
387C22×D3—182no—
388C22×D3—182no—
389C22×D3—184no—
390C22×D3—184no—
391C22×D3—184no—
392C22×D3—184no—
393C22×D3—361no—
394C22×D3C2×D3211yescharacteristic
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—32no—
398C2×C3×C4—38no—
399C2×C3×C4—34no—
400C2×C3×C4—34no—
401C2×C3×C4—41no—
402C2×C3×C4—64no—
403C2×C3×C4—61no—
404C2×C3×C4—64no—
405C2×C3×C4—91no—
406C2×C3×C4—94no—
407C2×C3×C4—94no—
408C2×C3×C4—184no—
409C2×C3×C4C2×D3212yes—
410C2×C3×C4C2×D3211yescharacteristic
411C2×Dic3—32no—
412C2×Dic3—34no—
413C2×Dic3—34no—
414C2×Dic3—61no—
415C2×Dic3—64no—
416C2×Dic3—98no—
417C2×Dic3—92no—
418C2×Dic3—91no—
419C2×Dic3—94no—
420C2×Dic3—184no—
421C2×Dic3—182no—
422C2×Dic3—181no—
423C2×Dic3—184no—
424C2×Dic3—361no—
425C2×Dic3C2×D3214yes—
426C2×Dic3C2×D3211yescharacteristic
427C4×D3—38no—
428C4×D3—64no—
429C4×D3—68no—
430C4×D3—98no—
431C4×D3—98no—
432C4×D3—94no—
433C4×D3—98no—
434C4×D3—94no—
435C4×D3—94no—
436C4×D3—184no—
437C4×D3—188no—
438C4×D3—184no—
439C4×D3—184no—
440C4×D3—188no—
441C4×D3—364no—
442C4×D3#62414yescomplemented
443C2×C32—22no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—21no—
449C2×C32—41no—
450C2×C32—41no—
451C2×C32—41no—
452C2×C32—62no—
453C2×C32C22×C4×D311yescharacteristic
454C2×C32C22×C4×D312yes—
455C2×C32C2×#23711yescharacteristic
456C2×C32C2×#23911yescharacteristic
457C2×C32D3×#4112yes—
458C2×C32D3×#4112yes—
459C3⋊D3—62no—
460C3⋊D3—64no—
461C3⋊D3—64no—
462C3⋊D3—124no—
463C3⋊D3—124no—
464C3⋊D3#89214yescomplemented
465C3×D3—24no—
466C3×D3—24no—
467C3×D3—38no—
468C3×D3—64no—
469C24—271no—
470C22×C4—31no—
471C22×C4—92no—
472C22×C4—91no—
473C22×C4—271no—
474C22×C4—278no—
475C22×C4—271no—
476C22×C4—272no—
477C22×C4—271no—
478C22×C4—271no—
479C4⋊C4—32no—
480C4⋊C4—92no—
481C4⋊C4—92no—
482C4⋊C4—272no—
483C22⋊C4—94no—
484C22⋊C4—274no—
485C42—94no—
486C42—274no—
487C22×C3—22no—
488C22×C3—21no—
489C22×C3—34no—
490C22×C3—41no—
491C22×C3—62no—
492C22×C3—62no—
493C22×C3—62no—
494C22×C3—62no—
495C22×C3—62no—
496C22×C3—92no—
497C22×C3—92no—
498C22×C3—92no—
499C22×C3C22×D3212yes—
500C22×C3C22×D3211yescharacteristic
501C2×D3—21no—
502C2×D3—21no—
503C2×D3—34no—
504C2×D3—34no—
505C2×D3—34no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—94no—
510C2×D3—92no—
511C2×D3—182no—
512C2×D3—184no—
513C2×D3—188no—
514C2×D3—1816no—
515C2×D3—1816no—
516C2×D3—182no—
517C2×D3—182no—
518C2×D3—181no—
519C2×D3—181no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—362no—
523C2×D3—362no—
524C2×D3—362no—
525C2×D3—362no—
526C2×D3—362no—
527C2×D3C2×#62412yes—
528C3×C4—22no—
529C3×C4—24no—
530C3×C4—34no—
531C3×C4—42no—
532C3×C4—62no—
533C3×C4—64no—
534C3×C4—64no—
535C3×C4—92no—
536C3×C4—124no—
537C3×C4—184no—
538C3×C4C4×D3214yes—
539C3×C4C2×#62412yes—
540Dic3—24no—
541Dic3—34no—
542Dic3—62no—
543Dic3—64no—
544Dic3—64no—
545Dic3—94no—
546Dic3—92no—
547Dic3—124no—
548Dic3—182no—
549Dic3—184no—
550Dic3—362no—
551Dic3C2×#62412yes—
552C32—22no—
553C32—21no—
554C32—41no—
555C32C2×#89211yescharacteristic, complemented
556C32D3×#11712yescomplemented
557C23—31no—
558C23—91no—
559C23—274no—
560C23—271no—
561C23—542no—
562C23—542no—
563C2×C4—31no—
564C2×C4—34no—
565C2×C4—34no—
566C2×C4—91no—
567C2×C4—94no—
568C2×C4—94no—
569C2×C4—184no—
570C2×C4—274no—
571C2×C4—274no—
572C2×C4—274no—
573C2×C4—271no—
574C2×C4—274no—
575C2×C4—544no—
576C2×C4D3311yescharacteristic
577C2×C3—22no—
578C2×C3—21no—
579C2×C3—22no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—22no—
583C2×C3—41no—
584C2×C3—41no—
585C2×C3—41no—
586C2×C3—64no—
587C2×C3—64no—
588C2×C3—94no—
589C2×C3C2×C4×D3212yes—
590C2×C3#6430611yescharacteristic
591C2×C3#6430611yescharacteristic
592C2×C3D3×#23712yes—
593C2×C3D3×#23912yes—
594C2×C3C22×#62411yescharacteristic
595D3—22no—
596D3—38no—
597D3—64no—
598D3—184no—
599D3—184no—
600D3—182no—
601D3—364no—
602D3—364no—
603C22—32no—
604C22—61no—
605C22—61no—
606C22—92no—
607C22—92no—
608C22—92no—
609C22—272no—
610C22—548no—
611C22—541no—
612C22—541no—
613C22C2×D3311yescharacteristic
614C4—32no—
615C4—64no—
616C4—92no—
617C4—184no—
618C4—272no—
619C4D3×#62412yes—
620C3—21no—
621C3—22no—
622C3—41no—
623C3D3×#89212yescomplemented
624C3#1059496911yescharacteristic, complemented
625C2—62no—
626C2—94no—
627C2—542no—
628C2#1172437311yescharacteristic
629C2#1172439111yescharacteristic
630C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×D3×C3⋊D3C22
hypercenterC2×C4D33
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723689C2semidirect
3. #11723413C2semidirect
4. C3×#63900C2semidirect
5. C3×#63841C2semidirect
6. #11722918C2semidirect
7. #11722917C2semidirect
9. Dic3×#624C2semidirect
20. C4×D3×C3⋊D3C4semidirect
22. C3×C4⋊(C3⋊Dic3)C22semidirect
23. C32×C4⋊Dic3C22semidirect
24. #88815C22semidirect
26. Dic3×C3⋊Dic3C22semidirect
27. C3×#3724C22semidirect
29. C3×Dic32C22semidirect
41. #63900D3semidirect
50. #63841D3semidirect
60. C4×C3⋊(C3⋊D3)C2×C4semidirect
66. C32×C4×D3C2×C4semidirect
68. #6289C2×C4semidirect
70. D3×C3⋊Dic3C2×C4semidirect
106. C4⋊(C3⋊Dic3)C2×D3semidirect
114. C3×C4⋊Dic3C2×D3semidirect
115. C3×C4⋊Dic3C2×D3semidirect
120. #3724C2×D3semidirect
129. Dic32C2×D3semidirect
227. C3×C4×D3C4×D3semidirect
242. C3×C3⋊D3#117semidirect
291. C4⋊Dic3D32semidirect
374. C33C2×#117semidirect
442. C4×D3#624semidirect
464. C3⋊D3#892semidirect
555. C32C2×#892semidirect
556. C32D3×#117semidirect
623. C3D3×#892semidirect
624. C3#10594969semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×C3⋊D32
#117236892
#117234132
C3×#639002
C3×#638412
#117229182
#117229172
#117229042
Dic3×#6242
#105949693
D3×#8923