Finite Group Databasebeta

← #13176553

Subgroups — #13176553

Order 1728.

2300 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
2C2×#89019C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic, complemented
5C32×#897C211yesmaximal, characteristic, complemented
6#11723539C212yesmaximal
7#11723503C212yesmaximal, complemented
8#11723502C212yesmaximal, complemented
9#11723500C212yesmaximal, complemented
10#11723435C211yesmaximal, characteristic, complemented
11#11723425C211yesmaximal, characteristic, complemented
12C3⋊D3×C22⋊Q8—31nomaximal
13#10594631—34nomaximal
14C22×D3×C3⋊D3C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C2×C32×C4×D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6293C2212yescomplemented
22C2×#6289C2212yes—
23C2×#6289C2211yescharacteristic
24C2×Dic3×C3⋊D3C2211yescharacteristic
25C2×D3×C3⋊Dic3C2211yescharacteristic
26C4×D3×C3⋊D3—24no—
27#89019—24no—
28C33×C4⋊C4C2211yescharacteristic, complemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30#88857C2212yescomplemented
31C3×#3752C2212yes—
32C3×C4⋊(C3⋊Dic3)C2212yescomplemented
33C32×(C2×D3)⋊C4C2212yes—
34C32×C4⋊Dic3C2212yescomplemented
35#88806C2212yescomplemented
36#88806C2212yescomplemented
37#88803C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×Q8×C3⋊D3—31no—
40C22×C4×C3⋊D3—31no—
41C2×C4×D32—34no—
42C2×#3797—34no—
43C32×C22⋊Q8—31no—
44#64155—31no—
45#64133—31no—
46#64106D311yescharacteristic, complemented
47#64105—32no—
48C4⋊C4×C3⋊D3—31no—
49C4⋊C4×C3⋊D3—32no—
50C4⋊C4×C3⋊D3—34no—
51C4⋊C4×C3⋊D3—64no—
52C22⋊C4×C3⋊D3—32no—
53#64087—32no—
54C3×#897—34no—
55C3×#897D314yescomplemented
56#63921—38no—
57#63900—38no—
58#63899—38no—
59#63899—38no—
60#63856—34no—
61#63856—34no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C2×D3×C3⋊D3—22no—
66C2×D3×C3⋊D3—22no—
67C2×D3×C3⋊D3Q814yescomplemented
68C2×C33×C4C2312yes—
69C2×C33×C4C2311yescharacteristic
70C2×C3⋊(C3⋊Dic3)C2312yes—
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C4×C3⋊(C3⋊D3)—24no—
73C4×C3⋊(C3⋊D3)D414yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4×C3⋊D3—22no—
77C3×#634—22no—
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2312yes—
80C32×C4×D3—22no—
81C32×Dic6—22no—
82#6293—24no—
83#6289—24no—
84#6289—22no—
85Dic3×C3⋊D3—22no—
86D3×C3⋊Dic3—22no—
87D3×C22⋊Q8—94no—
88D3×C22⋊Q8—91no—
89C23×C3⋊D3—31no—
90C22×D32—34no—
91C2×C32×Q8—31no—
92C22×C32×C4—31no—
93C22×C3⋊Dic3—31no—
94Q8×C3⋊D3—64no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—34no—
97C2×C4×C3⋊D3—31no—
98C2×C4×C3⋊D3—32no—
99C2×C4×C3⋊D3—38no—
100C2×C4×C3⋊D3—32no—
101C2×C4×C3⋊D3—31no—
102C2×C4×C3⋊D3—64no—
103C2×C4×C3⋊D3—64no—
104C2×C4×C3⋊D3—68no—
105C2×C4×C3⋊D3C2×D311yescharacteristic
106C2×#634—32no—
107C2×#634C2×D311yescharacteristic, complemented
108C2×C3×C4×D3—34no—
109C2×C3×C4×D3C2×D314yes—
110C2×C3×Dic6—34no—
111C2×C3×Dic6C2×D314yescomplemented
112C2×#627—38no—
113C2×#624—38no—
114C2×#624—34no—
115C2×D3×Dic3—34no—
116C2×D3×Dic3—34no—
117C4×D32—616no—
118#3797—616no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4—32no—
121C32×C4⋊C4C2×D314yescomplemented
122C32×C4⋊C4C2×D311yescharacteristic, complemented
123C32×C22⋊C4—32no—
124#3757—32no—
125#3752—32no—
126#3752C2×D312yes—
127C4⋊(C3⋊Dic3)—31no—
128C4⋊(C3⋊Dic3)—34no—
129C4⋊(C3⋊Dic3)—64no—
130C4⋊(C3⋊Dic3)C2×D312yescomplemented
131#3750—32no—
132#3750—38no—
133#3750—32no—
134#3750—32no—
135#3750—68no—
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4C2×D318yes—
138C3×C4⋊Dic3—38no—
139C3×C4⋊Dic3C2×D318yescomplemented
140#3724—38no—
141#3724—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)C2×D412yes—
146C2×C3⋊(C3⋊D3)C2×Q812yes—
147C2×C3⋊(C3⋊D3)#4112yes—
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—21no—
150C2×C3×C3⋊D3C2×Q812yescomplemented
151C2×C32×D3—21no—
152C2×C32×D3—21no—
153C2×C32×D3C2×Q812yescomplemented
154D3×C3⋊D3—28no—
155C33×C4—22no—
156C33×C4C2×D412yes—
157C3⋊(C3⋊Dic3)—22no—
158C3⋊(C3⋊Dic3)C2×D412yes—
159C3×C3⋊Dic3—21no—
160C3×C3⋊Dic3—22no—
161C32×Dic3—21no—
162C32×Dic3—22no—
163C2×D3×Q8—94no—
164C2×D3×Q8—91no—
165C22×C4×D3—94no—
166C22×C4×D3—91no—
167C3×C22⋊Q8—34no—
168C3×C22⋊Q8—91no—
169#946—94no—
170#946—91no—
171#924—94no—
172#924—91no—
173#897—34no—
174#897C3⋊D311yescharacteristic, complemented
175#896—98no—
176#896—92no—
177D3×C4⋊C4—98no—
178D3×C4⋊C4—94no—
179D3×C4⋊C4—91no—
180D3×C4⋊C4—92no—
181D3×C4⋊C4—184no—
182D3×C22⋊C4—98no—
183D3×C22⋊C4—92no—
184#878—98no—
185#878—92no—
186C23×C32—31no—
187C22×C3⋊D3—31no—
188C22×C3⋊D3—34no—
189C22×C3⋊D3—34no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—62no—
192C22×C3⋊D3—62no—
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3×D3—34no—
195C22×C3×D3C22×D314yes—
196C2×D32—316no—
197C2×D32—68no—
198C2×D32—68no—
199C32×Q8—62no—
200C2×C32×C4—24no—
201C2×C32×C4—28no—
202C2×C32×C4—31no—
203C2×C32×C4—32no—
204C2×C32×C4—62no—
205C2×C32×C4C22×D318yes—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D312yes—
208C2×C32×C4C22×D311yescharacteristic
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—32no—
211C2×C3⋊Dic3—34no—
212C2×C3⋊Dic3—38no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—68no—
215C2×C3⋊Dic3—62no—
216C2×C3⋊Dic3C22×D311yescharacteristic
217C2×C3⋊Dic3C22×D312yes—
218C4×C3⋊D3—22no—
219C4×C3⋊D3—34no—
220C4×C3⋊D3—316no—
221C4×C3⋊D3—616no—
222C4×C3⋊D3—62no—
223C4×C3⋊D3—64no—
224C4×C3⋊D3—62no—
225C4×C3⋊D3—64no—
226C4×C3⋊D3—616no—
227C4×C3⋊D3—1216no—
228#634—22no—
229#634—64no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—34no—
232C2×C3×Dic3C22×D318yes—
233C2×C3×Dic3C22×D314yes—
234C3×C4×D3—28no—
235C3×C4×D3—68no—
236C3×Dic6—28no—
237C3×Dic6—68no—
238#627—616no—
239#624—616no—
240#624—68no—
241D3×Dic3—68no—
242D3×Dic3—68no—
243C2×C22⋊Q8—271noSylow
244C2×C33C22×D411yescharacteristic
245C2×C33C22×Q811yescharacteristic
246C2×C33C2×#4111yescharacteristic
247C3⋊(C3⋊D3)C22⋊Q814yescomplemented
248C3×C3⋊D3—22no—
249C32×D3—22no—
250C23×D3—94no—
251C23×D3—91no—
252C2×C3×Q8—34no—
253C2×C3×Q8—91no—
254C22×C3×C4—34no—
255C22×C3×C4—91no—
256C22×Dic3—94no—
257C22×Dic3—91no—
258D3×Q8—184no—
259D3×Q8—1816no—
260C2×C4×D3—34no—
261C2×C4×D3—94no—
262C2×C4×D3—91no—
263C2×C4×D3—98no—
264C2×C4×D3—94no—
265C2×C4×D3—94no—
266C2×C4×D3—91no—
267C2×C4×D3—98no—
268C2×C4×D3—92no—
269C2×C4×D3—92no—
270C2×C4×D3—91no—
271C2×C4×D3—184no—
272C2×C4×D3—188no—
273C2×C4×D3—184no—
274C2×C4×D3—1816no—
275C2×C4×D3C2×C3⋊D311yescharacteristic
276C2×Dic6—34no—
277C2×Dic6—98no—
278C2×Dic6—92no—
279C2×Dic6C2×C3⋊D311yescharacteristic, complemented
280C3×C4⋊C4—24no—
281C3×C4⋊C4—38no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4D3214yescomplemented
284C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
285C3×C22⋊C4—38no—
286C3×C22⋊C4—92no—
287#217—98no—
288#217—92no—
289(C2×D3)⋊C4—38no—
290(C2×D3)⋊C4—98no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4C2×C3⋊D312yes—
293C4⋊Dic3—38no—
294C4⋊Dic3—94no—
295C4⋊Dic3—91no—
296C4⋊Dic3—184no—
297C4⋊Dic3C2×C3⋊D312yescomplemented
298Dic3⋊C4—98no—
299Dic3⋊C4—98no—
300Dic3⋊C4—98no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—188no—
305C22×C32—24no—
306C22×C32—32no—
307C22×C32—61no—
308C22×C32—61no—
309C22×C32C23×D314yes—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3⋊D3—32no—
314C2×C3⋊D3—32no—
315C2×C3⋊D3—32no—
316C2×C3⋊D3—38no—
317C2×C3⋊D3—38no—
318C2×C3⋊D3—38no—
319C2×C3⋊D3—68no—
320C2×C3⋊D3—68no—
321C2×C3⋊D3—68no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3D3×Q812yescomplemented
324C2×C3×D3—24no—
325C2×C3×D3—24no—
326C2×C3×D3—38no—
327C2×C3×D3—64no—
328C2×C3×D3—64no—
329C2×C3×D3D3×Q818yescomplemented
330D32—632no—
331C32×C4—22no—
332C32×C4—28no—
333C32×C4—28no—
334C32×C4—48no—
335C32×C4—62no—
336C32×C4—61no—
337C32×C4D3×D418yes—
338C32×C4D3×D412yes—
339C3⋊Dic3—22no—
340C3⋊Dic3—21no—
341C3⋊Dic3—32no—
342C3⋊Dic3—38no—
343C3⋊Dic3—68no—
344C3⋊Dic3—62no—
345C3⋊Dic3—68no—
346C3⋊Dic3—128no—
347C3×Dic3—24no—
348C3×Dic3—28no—
349C3×Dic3—64no—
350C3×Dic3—68no—
351C22×Q8—271no—
352C23×C4—271no—
353C22⋊Q8—31no—
354C22⋊Q8—91no—
355C22⋊Q8—272no—
356C22⋊Q8—272no—
357C22⋊Q8—271no—
358C22⋊Q8—271no—
359C2×C4⋊C4—272no—
360C2×C4⋊C4—271no—
361C2×C22⋊C4—272no—
362C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
363C23×C3—34no—
364C23×C3—91no—
365C22×D3—34no—
366C22×D3—916no—
367C22×D3—94no—
368C22×D3—91no—
369C22×D3—94no—
370C22×D3—184no—
371C22×D3—182no—
372C22×D3—182no—
373C22×D3—188no—
374C22×D3—188no—
375C22×D3C22×C3⋊D311yescharacteristic
376C3×Q8—68no—
377C3×Q8—182no—
378C2×C3×C4—24no—
379C2×C3×C4—28no—
380C2×C3×C4—34no—
381C2×C3×C4—38no—
382C2×C3×C4—68no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—182no—
386C2×C3×C4C2×D3218yes—
387C2×C3×C4C2×D3214yes—
388C2×C3×C4C22×C3⋊D312yes—
389C2×C3×C4C22×C3⋊D311yescharacteristic
390C2×Dic3—34no—
391C2×Dic3—38no—
392C2×Dic3—94no—
393C2×Dic3—91no—
394C2×Dic3—98no—
395C2×Dic3—92no—
396C2×Dic3—184no—
397C2×Dic3—188no—
398C2×Dic3—182no—
399C2×Dic3—188no—
400C2×Dic3C22×C3⋊D311yescharacteristic
401C2×Dic3C22×C3⋊D312yes—
402C4×D3—22no—
403C4×D3—68no—
404C4×D3—916no—
405C4×D3—94no—
406C4×D3—1816no—
407C4×D3—182no—
408C4×D3—184no—
409C4×D3—188no—
410C4×D3—1816no—
411C4×D3—188no—
412C4×D3—182no—
413C4×D3—184no—
414C4×D3—1816no—
415C4×D3—3616no—
416Dic6—22no—
417Dic6—68no—
418Dic6—184no—
419Dic6—1816no—
420C2×C32—24no—
421C2×C32—24no—
422C2×C32—24no—
423C2×C32—62no—
424C2×C32C2×D3×D414yes—
425C2×C32C2×D3×D411yescharacteristic
426C2×C32C2×D3×Q814yes—
427C2×C32C2×D3×Q811yescharacteristic
428C2×C32D3×#4114yes—
429C2×C32D3×#4111yescharacteristic
430C3⋊D3—22no—
431C3⋊D3—34no—
432C3⋊D3—316no—
433C3⋊D3—616no—
434C3×D3—28no—
435C3×D3—68no—
436C24—271no—
437C2×Q8—31no—
438C2×Q8—91no—
439C2×Q8—272no—
440C2×Q8—544no—
441C22×C4—31no—
442C22×C4—91no—
443C22×C4—272no—
444C22×C4—271no—
445C22×C4—271no—
446C22×C4—271no—
447C22×C4—271no—
448C22×C4—272no—
449C22×C4—544no—
450C4⋊C4—32no—
451C4⋊C4—92no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—272no—
455C4⋊C4—271no—
456C4⋊C4D3×C3⋊D311yescharacteristic, complemented
457C22⋊C4—32no—
458C22⋊C4—92no—
459C22⋊C4—272no—
460C22⋊C4—272no—
461C22×C3—24no—
462C22×C3—38no—
463C22×C3—64no—
464C22×C3—64no—
465C22×C3—92no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3C22×D3214yes—
469C22×C3C23×C3⋊D311yescharacteristic
470C2×D3—21no—
471C2×D3—21no—
472C2×D3—38no—
473C2×D3—64no—
474C2×D3—64no—
475C2×D3—98no—
476C2×D3—98no—
477C2×D3—98no—
478C2×D3—92no—
479C2×D3—92no—
480C2×D3—92no—
481C2×D3—188no—
482C2×D3—188no—
483C2×D3—188no—
484C2×D3—188no—
485C2×D3—1832no—
486C2×D3Q8×C3⋊D312yescomplemented
487C3×C4—28no—
488C3×C4—22no—
489C3×C4—28no—
490C3×C4—48no—
491C3×C4—68no—
492C3×C4—64no—
493C3×C4—182no—
494C3×C4—181no—
495C3×C4#380218yes—
496C3×C4D4×C3⋊D312yes—
497Dic3—22no—
498Dic3—21no—
499Dic3—68no—
500Dic3—64no—
501Dic3—98no—
502Dic3—92no—
503Dic3—188no—
504Dic3—188no—
505Dic3—182no—
506Dic3—368no—
507C32—24no—
508C32D3×C22⋊Q814yescomplemented
509C32D3×C22⋊Q811yescharacteristic, complemented
510C23—31no—
511C23—91no—
512C23—274no—
513C23—271no—
514C23—542no—
515C23—542no—
516Q8—62no—
517Q8—182no—
518Q8—544no—
519C2×C4—32no—
520C2×C4—31no—
521C2×C4—62no—
522C2×C4—92no—
523C2×C4—91no—
524C2×C4—182no—
525C2×C4—271no—
526C2×C4—274no—
527C2×C4—272no—
528C2×C4—544no—
529C2×C4—542no—
530C2×C4—544no—
531C2×C4—542no—
532C2×C4—542no—
533C2×C4C2×D3×C3⋊D312yes—
534C2×C4C2×D3×C3⋊D311yescharacteristic
535C2×C3—24no—
536C2×C3—24no—
537C2×C3—24no—
538C2×C3—68no—
539C2×C3—182no—
540C2×C3C2×#380214yes—
541C2×C3#6431214yes—
542C2×C3D32×Q814yes—
543C2×C3C2×D4×C3⋊D311yescharacteristic
544C2×C3C2×Q8×C3⋊D311yescharacteristic
545C2×C3#41×C3⋊D311yescharacteristic
546D3—22no—
547D3—68no—
548D3—94no—
549D3—916no—
550D3—1816no—
551C22—32no—
552C22—61no—
553C22—61no—
554C22—92no—
555C22—181no—
556C22—181no—
557C22—272no—
558C22—272no—
559C22—272no—
560C22—548no—
561C22C22×D3×C3⋊D311yescharacteristic
562C4—22no—
563C4—61no—
564C4—62no—
565C4—181no—
566C4—182no—
567C4—272no—
568C4—542no—
569C4#8902612yes—
570C3—24no—
571C3#1059463114yescomplemented
572C3C3⋊D3×C22⋊Q811yescharacteristic, complemented
573C2—62no—
574C2—182no—
575C2—274no—
576C2C2×#8902611yescharacteristic
577C2#1172458611yescharacteristic
578C2D3×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×D3×C3⋊D3C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#89019C2semidirect
3. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
4. C3×#64106C2semidirect
5. C32×#897C2semidirect
7. #11723503C2semidirect
8. #11723502C2semidirect
9. #11723500C2semidirect
10. #11723435C2semidirect
11. #11723425C2semidirect
18. C2×C3×#634C22semidirect
20. C2×C32×Dic6C22semidirect
21. C2×#6293C22semidirect
28. C33×C4⋊C4C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. #88857C22semidirect
32. C3×C4⋊(C3⋊Dic3)C22semidirect
34. C32×C4⋊Dic3C22semidirect
35. #88806C22semidirect
36. #88806C22semidirect
46. #64106D3semidirect
55. C3×#897D3semidirect
67. C2×D3×C3⋊D3Q8semidirect
107. C2×#634C2×D3semidirect
111. C2×C3×Dic6C2×D3semidirect
121. C32×C4⋊C4C2×D3semidirect
122. C32×C4⋊C4C2×D3semidirect
130. C4⋊(C3⋊Dic3)C2×D3semidirect
139. C3×C4⋊Dic3C2×D3semidirect
150. C2×C3×C3⋊D3C2×Q8semidirect
153. C2×C32×D3C2×Q8semidirect
174. #897C3⋊D3semidirect
247. C3⋊(C3⋊D3)C22⋊Q8semidirect
279. C2×Dic6C2×C3⋊D3semidirect
283. C3×C4⋊C4D32semidirect
284. C3×C4⋊C4C2×C3⋊D3semidirect
297. C4⋊Dic3C2×C3⋊D3semidirect
323. C2×C3⋊D3D3×Q8semidirect
329. C2×C3×D3D3×Q8semidirect
362. C33C2×C22⋊Q8semidirect
456. C4⋊C4D3×C3⋊D3semidirect
486. C2×D3Q8×C3⋊D3semidirect
508. C32D3×C22⋊Q8semidirect
509. C32D3×C22⋊Q8semidirect
571. C3#10594631semidirect
572. C3C3⋊D3×C22⋊Q8semidirect

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
C2×#890192
C4⋊C4×C3⋊(C3⋊D3)2
C3×#641062
C32×#8972
#117235392
#117235032
#117235022
#117235002
#117234352
#117234252
C3⋊D3×C22⋊Q83
#105946313