Finite Group Databasebeta

← #13168258

Subgroups — #13168258

Order 1728.

1726 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89040C211yesmaximal, characteristic, complemented
2C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
3#11723449C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63857C212yesmaximal, complemented
6#11722982C212yesmaximal, complemented
7#11722981C212yesmaximal, complemented
8#11722908C212yesmaximal, complemented
9#11722904C212yesmaximal
10#10594925—31nomaximal
11D3×#897—32nomaximal
12C22×D3×C3⋊D3C2211yescharacteristic
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×#634C2212yescomplemented
16C2×C32×C4×D3C2211yescharacteristic
17C2×#6296C2212yescomplemented
18C2×#6294C2212yes—
19#89040—24no—
20C2×#6289C2211yescharacteristic
21C2×Dic3×C3⋊D3C2211yescharacteristic
22C2×D3×C3⋊Dic3C2211yescharacteristic
23C4×D3×C3⋊D3—24no—
24C2×C3×#624C2212yes—
25C32×C4⋊Dic3C2212yescomplemented
26Dic3⋊(C3⋊Dic3)C2214yescomplemented
27#88804C2214yes—
28C3×#3724C2212yescomplemented
29C3×Dic3⋊Dic3C2214yescomplemented
30C3×(C2×D3)⋊Dic3C2214yes—
31C22×C4×C3⋊D3—31no—
32C2×C4×D32—32no—
33C2×C4×D32—61no—
34C2×#3797—31no—
35C2×D3×Dic6—32no—
36#64106—32no—
37C3×#924—32no—
38C3×#897—32no—
39C3×D3×C4⋊C4—32no—
40#63982—32no—
41#63974—32no—
42D3×(C2×D3)⋊C4—34no—
43#63900D311yescharacteristic, complemented
44#63895—34no—
45#63893—34no—
46D3×C4⋊Dic3—34no—
47#63883—32no—
48#63871—32no—
49#63871—32no—
50#63871—62no—
51#63865—32no—
52#63857D312yescomplemented
53C22×C3⋊(C3⋊D3)C2311yescharacteristic
54C22×C3×C3⋊D3C2311yescharacteristic
55C22×C32×D3C2311yescharacteristic
56C2×D3×C3⋊D3—22no—
57C2×D3×C3⋊D3—22no—
58C2×D3×C3⋊D3Q814yescomplemented
59C2×C33×C4C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
61C4×C3⋊(C3⋊D3)—22no—
62C2×C3×C3⋊Dic3C2314yes—
63C2×C3×C3⋊Dic3C2311yescharacteristic
64C3×C4×C3⋊D3D414yes—
65C3×#634—24no—
66C2×C32×Dic3C2314yes—
67C2×C32×Dic3C2311yescharacteristic
68C32×C4×D3—22no—
69#6296—24no—
70#6294—24no—
71#6289—22no—
72Dic3×C3⋊D3—22no—
73D3×C3⋊Dic3—22no—
74C3×#624—24no—
75D3×C22⋊Q8—92no—
76C2×#897—91no—
77C23×C3⋊D3—31no—
78C22×D32—32no—
79C22×D32—61no—
80C22×C32×C4—31no—
81C22×C3⋊Dic3—31no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—32no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—61no—
87C2×C4×C3⋊D3—64no—
88C2×C4×C3⋊D3—62no—
89C2×C4×C3⋊D3—121no—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×#634C2×D312yescomplemented
92C2×C3×C4×D3—21no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3—34no—
95C2×C3×C4×D3—61no—
96C2×C3×C4×D3C2×D312yes—
97C2×C3×Dic6—32no—
98C2×C3×Dic6—32no—
99C2×C3×Dic6—62no—
100C2×#627—34no—
101C2×#627—32no—
102C2×#624—32no—
103C2×#624—32no—
104C2×#624—61no—
105C2×#624C2×D312yes—
106C2×D3×Dic3—32no—
107C2×D3×Dic3—34no—
108C2×D3×Dic3—32no—
109C2×D3×Dic3—61no—
110C2×D3×Dic3—61no—
111C4×D32—68no—
112C4×D32—68no—
113#3797—64no—
114D3×Dic6—68no—
115C32×C4⋊C4—32no—
116#3752—34no—
117C4⋊(C3⋊Dic3)—34no—
118C3×#217—34no—
119C3×(C2×D3)⋊C4—34no—
120C3×C4⋊Dic3—22no—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3C2×D312yescomplemented
123C3×C4⋊Dic3C2×D312yescomplemented
124C3×Dic3⋊C4—34no—
125C3×Dic3⋊C4—34no—
126#3724C2×D312yescomplemented
127Dic3⋊Dic3—34no—
128Dic3⋊Dic3—34no—
129Dic3⋊Dic3—64no—
130Dic3⋊Dic3C2×D314yescomplemented
131#3722—34no—
132#3722—34no—
133#3722—64no—
134(C2×D3)⋊Dic3C2×D314yes—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C2×C3⋊(C3⋊D3)C2×Q812yescomplemented
139C2×C3×C3⋊D3C2×D412yes—
140C2×C3×C3⋊D3C2×Q812yes—
141C2×C3×C3⋊D3#4112yes—
142C2×C32×D3—21no—
143C2×C32×D3—21no—
144C2×C32×D3C2×Q812yescomplemented
145D3×C3⋊D3—28no—
146C33×C4C2×D412yes—
147C3⋊(C3⋊Dic3)—21no—
148C3×C3⋊Dic3—24no—
149C3×C3⋊Dic3C2×D412yes—
150C32×Dic3—21no—
151C32×Dic3—24no—
152C2×D3×Q8—92no—
153C22×C4×D3—92no—
154C22×C4×D3—91no—
155C22×C4×D3—181no—
156C22×Dic6—91no—
157C3×C22⋊Q8—92no—
158C2×C3×C4⋊C4—91no—
159#946—92no—
160C2×(C2×D3)⋊C4—92no—
161C2×C4⋊Dic3—92no—
162#924—32no—
163#897—32no—
164#897—92no—
165#897—92no—
166#897—92no—
167#897—92no—
168#897—182no—
169#896—94no—
170D3×C4⋊C4—32no—
171D3×C4⋊C4—94no—
172D3×C22⋊C4—94no—
173#878—94no—
174C23×C32—31no—
175C22×C3⋊D3—32no—
176C22×C3⋊D3—31no—
177C22×C3⋊D3—34no—
178C22×C3⋊D3—61no—
179C22×C3⋊D3—62no—
180C22×C3⋊D3—62no—
181C22×C3⋊D3—62no—
182C22×C3⋊D3—121no—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—21no—
185C22×C3×D3—32no—
186C22×C3×D3—61no—
187C22×C3×D3C22×D312yes—
188C2×D32—38no—
189C2×D32—64no—
190C2×D32—64no—
191C2×D32—64no—
192C2×D32—64no—
193C2×D32—64no—
194C2×C32×C4—22no—
195C2×C32×C4—21no—
196C2×C32×C4—31no—
197C2×C32×C4—34no—
198C2×C32×C4—41no—
199C2×C32×C4—62no—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D312yes—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—32no—
204C2×C3⋊Dic3—61no—
205C2×C3⋊Dic3—62no—
206C2×C3⋊Dic3—62no—
207C2×C3⋊Dic3—121no—
208C2×C3⋊Dic3C22×D314yes—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C4×C3⋊D3—62no—
211C4×C3⋊D3—62no—
212C4×C3⋊D3—62no—
213C4×C3⋊D3—64no—
214C4×C3⋊D3—64no—
215C4×C3⋊D3—124no—
216C4×C3⋊D3—124no—
217C4×C3⋊D3D1214yes—
218#634—24no—
219C2×C3×Dic3—21no—
220C2×C3×Dic3—24no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3C22×D314yes—
227C2×C3×Dic3C22×D314yes—
228C2×C3×Dic3C22×D312yes—
229C3×C4×D3—24no—
230C3×C4×D3—24no—
231C3×C4×D3—38no—
232C3×C4×D3—64no—
233C3×C4×D3—68no—
234C3×Dic6—64no—
235C3×Dic6—64no—
236C3×Dic6—124no—
237#627—68no—
238#627—64no—
239#624—24no—
240#624—64no—
241#624—64no—
242#624—64no—
243D3×Dic3—64no—
244D3×Dic3—64no—
245D3×Dic3—64no—
246D3×Dic3—68no—
247D3×Dic3—64no—
248C2×C22⋊Q8—271noSylow
249C2×C33C22×D411yescharacteristic
250C2×C33C22×Q811yescharacteristic
251C2×C33C2×#4111yescharacteristic
252C3⋊(C3⋊D3)—22no—
253C3×C3⋊D3C22⋊Q814yescomplemented
254C32×D3—22no—
255C23×D3—92no—
256C23×D3—91no—
257C23×D3—181no—
258C2×C3×Q8—92no—
259C22×C3×C4—32no—
260C22×C3×C4—61no—
261C22×C3×C4—92no—
262C22×C3×C4—91no—
263C22×Dic3—92no—
264C22×Dic3—92no—
265C22×Dic3—91no—
266C22×Dic3—181no—
267D3×Q8—188no—
268C2×C4×D3—32no—
269C2×C4×D3—34no—
270C2×C4×D3—61no—
271C2×C4×D3—92no—
272C2×C4×D3—94no—
273C2×C4×D3—92no—
274C2×C4×D3—91no—
275C2×C4×D3—91no—
276C2×C4×D3—92no—
277C2×C4×D3—91no—
278C2×C4×D3—181no—
279C2×C4×D3—181no—
280C2×C4×D3—181no—
281C2×C4×D3—182no—
282C2×C4×D3—184no—
283C2×C4×D3—188no—
284C2×C4×D3—188no—
285C2×C4×D3—361no—
286C2×C4×D3D3211yescharacteristic
287C2×Dic6—32no—
288C2×Dic6—32no—
289C2×Dic6—62no—
290C2×Dic6—94no—
291C2×Dic6—92no—
292C2×Dic6—184no—
293C3×C4⋊C4—32no—
294C3×C4⋊C4—32no—
295C3×C4⋊C4—62no—
296C3×C4⋊C4—94no—
297C3×C4⋊C4—92no—
298C3×C22⋊C4—94no—
299#217—34no—
300(C2×D3)⋊C4—34no—
301(C2×D3)⋊C4—94no—
302(C2×D3)⋊C4—94no—
303(C2×D3)⋊C4—94no—
304(C2×D3)⋊C4—184no—
305C4⋊Dic3—34no—
306C4⋊Dic3—94no—
307C4⋊Dic3—94no—
308C4⋊Dic3—184no—
309C4⋊Dic3D3212yescomplemented
310Dic3⋊C4—34no—
311Dic3⋊C4—34no—
312Dic3⋊C4—94no—
313C22×C32—22no—
314C22×C32—21no—
315C22×C32—32no—
316C22×C32—41no—
317C22×C32—61no—
318C22×C32—61no—
319C22×C32C23×D311yescharacteristic
320C22×C32C23×D312yes—
321C2×C3⋊D3—34no—
322C2×C3⋊D3—32no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—64no—
325C2×C3⋊D3—61no—
326C2×C3⋊D3—61no—
327C2×C3⋊D3—68no—
328C2×C3⋊D3—62no—
329C2×C3⋊D3—62no—
330C2×C3⋊D3—62no—
331C2×C3⋊D3—62no—
332C2×C3⋊D3—122no—
333C2×C3⋊D3—122no—
334C2×C3⋊D3—122no—
335C2×C3⋊D3—122no—
336C2×C3⋊D3—122no—
337C2×C3⋊D3C2×D1212yes—
338C2×C3⋊D3#23712yes—
339C2×C3⋊D3D3×Q812yes—
340C2×C3×D3—22no—
341C2×C3×D3—22no—
342C2×C3×D3—22no—
343C2×C3×D3—22no—
344C2×C3×D3—22no—
345C2×C3×D3—34no—
346C2×C3×D3—34no—
347C2×C3×D3—34no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3D3×Q814yescomplemented
352D32—616no—
353D32—616no—
354C32×C4—24no—
355C32×C4—22no—
356C32×C4—42no—
357C32×C4—61no—
358C32×C4—64no—
359C32×C4C2×D1212yes—
360C32×C4D3×D414yes—
361C3⋊Dic3—24no—
362C3⋊Dic3—61no—
363C3⋊Dic3—62no—
364C3⋊Dic3—62no—
365C3⋊Dic3—122no—
366C3⋊Dic3—122no—
367C3⋊Dic3C2×D1212yes—
368C3×Dic3—22no—
369C3×Dic3—22no—
370C3×Dic3—24no—
371C3×Dic3—24no—
372C3×Dic3—34no—
373C3×Dic3—44no—
374C3×Dic3—62no—
375C3×Dic3—64no—
376C3×Dic3—64no—
377C3×Dic3—124no—
378C22×Q8—271no—
379C23×C4—271no—
380C22⋊Q8—92no—
381C22⋊Q8—272no—
382C22⋊Q8—272no—
383C22⋊Q8—272no—
384C2×C4⋊C4—91no—
385C2×C4⋊C4—272no—
386C2×C22⋊C4—272no—
387C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
388C23×C3—32no—
389C23×C3—61no—
390C23×C3—91no—
391C22×D3—32no—
392C22×D3—61no—
393C22×D3—98no—
394C22×D3—91no—
395C22×D3—92no—
396C22×D3—94no—
397C22×D3—184no—
398C22×D3—181no—
399C22×D3—182no—
400C22×D3—182no—
401C22×D3—182no—
402C22×D3—184no—
403C22×D3—184no—
404C22×D3—184no—
405C22×D3—184no—
406C22×D3—361no—
407C22×D3C2×D3211yescharacteristic
408C3×Q8—184no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—32no—
412C2×C3×C4—34no—
413C2×C3×C4—34no—
414C2×C3×C4—41no—
415C2×C3×C4—61no—
416C2×C3×C4—64no—
417C2×C3×C4—64no—
418C2×C3×C4—64no—
419C2×C3×C4—91no—
420C2×C3×C4—94no—
421C2×C3×C4—94no—
422C2×C3×C4—184no—
423C2×C3×C4C2×D3212yes—
424C2×C3×C4C2×D3211yescharacteristic
425C2×Dic3—32no—
426C2×Dic3—34no—
427C2×Dic3—34no—
428C2×Dic3—61no—
429C2×Dic3—64no—
430C2×Dic3—92no—
431C2×Dic3—91no—
432C2×Dic3—181no—
433C2×Dic3—182no—
434C2×Dic3—182no—
435C2×Dic3—184no—
436C2×Dic3—184no—
437C2×Dic3—184no—
438C2×Dic3—361no—
439C2×Dic3C2×D3214yes—
440C2×Dic3C2×D3211yescharacteristic
441C4×D3—22no—
442C4×D3—38no—
443C4×D3—64no—
444C4×D3—68no—
445C4×D3—182no—
446C4×D3—184no—
447C4×D3—184no—
448C4×D3—188no—
449C4×D3—184no—
450C4×D3—184no—
451C4×D3—184no—
452C4×D3—182no—
453C4×D3—184no—
454C4×D3—182no—
455C4×D3—364no—
456C4×D3—364no—
457Dic6—64no—
458Dic6—64no—
459Dic6—124no—
460Dic6—188no—
461Dic6—184no—
462C2×C32—21no—
463C2×C32—22no—
464C2×C32—21no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—21no—
468C2×C32—41no—
469C2×C32—41no—
470C2×C32—41no—
471C2×C32—62no—
472C2×C32C22×D1211yescharacteristic
473C2×C32C2×D3×D412yes—
474C2×C32C2×#23711yescharacteristic
475C2×C32C2×D3×Q812yes—
476C2×C32C2×D3×Q811yescharacteristic
477C2×C32D3×#4112yes—
478C3⋊D3—62no—
479C3⋊D3—64no—
480C3⋊D3—64no—
481C3⋊D3—124no—
482C3⋊D3—124no—
483C3⋊D3#89714yescomplemented
484C3×D3—24no—
485C3×D3—24no—
486C3×D3—38no—
487C3×D3—64no—
488C24—271no—
489C2×Q8—92no—
490C2×Q8—272no—
491C2×Q8—544no—
492C22×C4—31no—
493C22×C4—92no—
494C22×C4—91no—
495C22×C4—271no—
496C22×C4—271no—
497C22×C4—272no—
498C22×C4—271no—
499C22×C4—271no—
500C22×C4—544no—
501C4⋊C4—32no—
502C4⋊C4—92no—
503C4⋊C4—94no—
504C4⋊C4—274no—
505C22⋊C4—94no—
506C22⋊C4—274no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—34no—
510C22×C3—41no—
511C22×C3—62no—
512C22×C3—62no—
513C22×C3—62no—
514C22×C3—62no—
515C22×C3—62no—
516C22×C3—92no—
517C22×C3—92no—
518C22×C3—92no—
519C22×C3C22×D3211yescharacteristic
520C22×C3C22×D3212yes—
521C2×D3—21no—
522C2×D3—21no—
523C2×D3—34no—
524C2×D3—34no—
525C2×D3—34no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—62no—
529C2×D3—92no—
530C2×D3—94no—
531C2×D3—182no—
532C2×D3—184no—
533C2×D3—188no—
534C2×D3—1816no—
535C2×D3—1816no—
536C2×D3—182no—
537C2×D3—182no—
538C2×D3—182no—
539C2×D3—181no—
540C2×D3—182no—
541C2×D3—181no—
542C2×D3—362no—
543C2×D3—362no—
544C2×D3—362no—
545C2×D3—362no—
546C2×D3—362no—
547C2×D3#379712yescomplemented
548C3×C4—24no—
549C3×C4—22no—
550C3×C4—42no—
551C3×C4—62no—
552C3×C4—62no—
553C3×C4—64no—
554C3×C4—64no—
555C3×C4—92no—
556C3×C4—124no—
557C3×C4—184no—
558C3×C4D3×D1214yes—
559C3×C4#381112yes—
560Dic3—24no—
561Dic3—21no—
562Dic3—34no—
563Dic3—62no—
564Dic3—64no—
565Dic3—64no—
566Dic3—124no—
567Dic3—182no—
568Dic3—182no—
569Dic3—181no—
570Dic3—362no—
571Dic3—362no—
572C32—22no—
573C32—21no—
574C32—41no—
575C32C2×#89711yescharacteristic, complemented
576C32D3×C22⋊Q812yescomplemented
577C23—31no—
578C23—91no—
579C23—274no—
580C23—271no—
581C23—542no—
582C23—542no—
583Q8—184no—
584Q8—544no—
585C2×C4—34no—
586C2×C4—31no—
587C2×C4—62no—
588C2×C4—91no—
589C2×C4—94no—
590C2×C4—94no—
591C2×C4—184no—
592C2×C4—271no—
593C2×C4—542no—
594C2×C4—542no—
595C2×C4—542no—
596C2×C4—544no—
597C2×C4—542no—
598C2×C4D3311yescharacteristic
599C2×C3—22no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—22no—
603C2×C3—21no—
604C2×C3—22no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—41no—
608C2×C3—64no—
609C2×C3—64no—
610C2×C3—94no—
611C2×C3C2×#379711yescharacteristic
612C2×C3C2×D3×D1212yes—
613C2×C3#6430611yescharacteristic
614C2×C3D3×#23712yes—
615C2×C3D32×Q812yes—
616C2×C3C2×#381111yescharacteristic
617D3—22no—
618D3—38no—
619D3—64no—
620D3—184no—
621D3—184no—
622D3—182no—
623D3—364no—
624D3—364no—
625C22—32no—
626C22—61no—
627C22—61no—
628C22—92no—
629C22—92no—
630C22—92no—
631C22—272no—
632C22—548no—
633C22—541no—
634C22—541no—
635C22C2×D3311yescharacteristic
636C4—64no—
637C4—61no—
638C4—92no—
639C4—184no—
640C4—541no—
641C4#8895512yes—
642C3—21no—
643C3—22no—
644C3—41no—
645C3D3×#89712yescomplemented
646C3#1059492511yescharacteristic, complemented
647C2—62no—
648C2—94no—
649C2—542no—
650C2#1172437311yescharacteristic
651C2D3×#379711yescharacteristic
652C2C2×#8895511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×D3×C3⋊D3C22
hypercenterC2×C4D33
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. #11723449C2semidirect
4. C3×#63900C2semidirect
5. C3×#63857C2semidirect
6. #11722982C2semidirect
7. #11722981C2semidirect
8. #11722908C2semidirect
15. C2×C3×#634C22semidirect
17. C2×#6296C22semidirect
25. C32×C4⋊Dic3C22semidirect
26. Dic3⋊(C3⋊Dic3)C22semidirect
28. C3×#3724C22semidirect
29. C3×Dic3⋊Dic3C22semidirect
43. #63900D3semidirect
52. #63857D3semidirect
58. C2×D3×C3⋊D3Q8semidirect
91. C2×#634C2×D3semidirect
122. C3×C4⋊Dic3C2×D3semidirect
123. C3×C4⋊Dic3C2×D3semidirect
126. #3724C2×D3semidirect
130. Dic3⋊Dic3C2×D3semidirect
138. C2×C3⋊(C3⋊D3)C2×Q8semidirect
144. C2×C32×D3C2×Q8semidirect
253. C3×C3⋊D3C22⋊Q8semidirect
309. C4⋊Dic3D32semidirect
351. C2×C3×D3D3×Q8semidirect
387. C33C2×C22⋊Q8semidirect
483. C3⋊D3#897semidirect
547. C2×D3#3797semidirect
575. C32C2×#897semidirect
576. C32D3×C22⋊Q8semidirect
645. C3D3×#897semidirect
646. C3#10594925semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C4×D3×C3⋊D32
#117234492
C3×#639002
C3×#638572
#117229822
#117229812
#117229082
#117229042
#105949253
D3×#8973