Finite Group Databasebeta

← #13147893

Subgroups — #13147893

Order 1728.

1231 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2#11721718C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#11721160C212yesmaximal
6#11721120C212yesmaximal, complemented
7#11721119C212yesmaximal, complemented
8C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
9#11721105C211yesmaximal, characteristic, complemented
10#11721097C211yesmaximal, characteristic, complemented
11#11721076C212yesmaximal, complemented
12#10593992—31nomaximal
13D3×#872—31nomaximal
14C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#6199C2211yescharacteristic
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
23C4×((C3⋊D3)⋊C3)⋊C2C418yescomplemented
24C42×H(3,𝔽3)C2211yescharacteristic, complemented
25#88542C2212yes—
26#88540C2212yescomplemented
27C4×#1084C2211yescharacteristic, complemented
28#3752⋊C3C2212yes—
29#3750⋊C3C2212yescomplemented
30C4×(C3⋊Dic3)⋊C3C2212yescomplemented
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
32C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
33#88451C2212yes—
34#88450C2212yes—
35#88449C2212yescomplemented
36#88448C2212yescomplemented
37#88447C2212yescomplemented
38C2×C4×D32—31no—
39C2×C4×D32—31no—
40#64081D311yescharacteristic, complemented
41C42×C3⋊D3D311yescharacteristic, complemented
42C3×#872—31no—
43C3×#872—31no—
44C3×#872—31no—
45C3×#872—61no—
46C3×C42×D3—31no—
47#63921—32no—
48D3×(C2×D3)⋊C4—32no—
49#63883—32no—
50C4×#624—31no—
51#63878—32no—
52#63878—32no—
53#63877—32no—
54D3×Dic3⋊C4—32no—
55C4×D3×Dic3—31no—
56#63874—31no—
57#63855—31no—
58#63855—31no—
59#63855—31no—
60#63836—32no—
61C22×#272C2311yescharacteristic
62C22×(C3⋊D3)⋊C3C2311yescharacteristic
63C22×(C3⋊D3)⋊C3C2311yescharacteristic
64C2×((C3⋊D3)⋊C3)⋊C2—24no—
65C2×((C3⋊D3)⋊C3)⋊C2C2×C414yes—
66C2×C4×H(3,𝔽3)C2312yes—
67C2×C4×H(3,𝔽3)C2311yescharacteristic
68C2×#1084C2312yes—
69C2×#1084C2311yescharacteristic
70C4×#272C2×C414yescomplemented
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C4×(C3⋊D3)⋊C3—24no—
76C4×(C3⋊D3)⋊C3C2×C414yes—
77C4×(C3⋊D3)⋊C3C2×C414yescomplemented
78#6199C2×C414yescomplemented
79((C3⋊D3)⋊C3)⋊C4—24no—
80((C3⋊D3)⋊C3)⋊C4C2×C414yes—
81((C3⋊D3)⋊C3)⋊C4C2×C414yescomplemented
82D3×#117—91no—
83D3×#117—91no—
84C2×#872—91no—
85C22×D32—31no—
86C22×D32—31no—
87C2×C4×C3⋊D3C2×D312yes—
88C2×C4×C3⋊D3C2×D311yescharacteristic
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96C2×#624—31no—
97C2×#624—31no—
98C2×#624—32no—
99C2×D3×Dic3—31no—
100C2×D3×Dic3—31no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—32no—
104C4×D32—38no—
105C4×D32—38no—
106C32×C42—21no—
107C32×C42C2×D311yescharacteristic, complemented
108C32×C42C2×D311yescharacteristic, complemented
109#3752C2×D312yes—
110#3750C2×D312yescomplemented
111C4×C3⋊Dic3C2×D311yescharacteristic, complemented
112C4×C3⋊Dic3C2×D312yescomplemented
113C4×C3⋊Dic3C2×D311yescharacteristic, complemented
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—62no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—62no—
122C3×C4×Dic3—32no—
123C3×C4×Dic3—31no—
124C3×C4×Dic3—31no—
125C3×C4×Dic3—31no—
126C3×C4×Dic3—31no—
127C3×C4×Dic3—61no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132#3722—32no—
133(C2×D3)⋊Dic3—32no—
134(C2×D3)⋊Dic3—32no—
135(C2×D3)⋊Dic3—32no—
136Dic32—32no—
137Dic32—32no—
138C22×H(3,𝔽3)C2411yescharacteristic
139C2×#272—22no—
140C2×#272C22×C412yes—
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3C22×C412yes—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C2×(C3⋊D3)⋊C3#4114yes—
145((C3⋊D3)⋊C3)⋊C2—28no—
146C4×H(3,𝔽3)—22no—
147C4×H(3,𝔽3)C22×C412yes—
148#1084—22no—
149#1084C22×C412yes—
150(C3⋊Dic3)⋊C3—22no—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3C22×C412yes—
153(C3⋊Dic3)⋊C3C22×C412yes—
154C22×C4×D3—91no—
155C22×C4×D3—91no—
156C22×C4×D3—91no—
157C3×#117—91no—
158C3×#117—91no—
159C2×C3×C42—91no—
160C2×(C2×D3)⋊C4—92no—
161#926—91no—
162#926—91no—
163C2×Dic3⋊C4—92no—
164C2×C4×Dic3—91no—
165#892—92no—
166#892—92no—
167D3×C4⋊C4—92no—
168D3×C4⋊C4—92no—
169D3×C22⋊C4—92no—
170D3×C22⋊C4—92no—
171#877—92no—
172#877—92no—
173#872—31no—
174#872—31no—
175#872—31no—
176#872—61no—
177#872—91no—
178#872—91no—
179#872—91no—
180#872—92no—
181#872—92no—
182#872—91no—
183#872—91no—
184C42×D3—31no—
185C42×D3—91no—
186C42×D3—91no—
187C42×D3—91no—
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3×D3—31no—
191C22×C3×D3—31no—
192C22×C3×D3—31no—
193C22×C3×D3—31no—
194C22×C3×D3—61no—
195C2×D32—34no—
196C2×D32—34no—
197C2×D32—64no—
198C2×D32—64no—
199C2×C32×C4—21no—
200C2×C32×C4—22no—
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D312yes—
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D312yes—
207C2×C3⋊Dic3C22×D311yescharacteristic
208C2×C3⋊Dic3C22×D311yescharacteristic
209C4×C3⋊D3—24no—
210C4×C3⋊D3C4×D314yes—
211C4×C3⋊D3C4×D314yescomplemented
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—62no—
222C3×C4×D3—34no—
223C3×C4×D3—34no—
224C3×C4×D3—34no—
225C3×C4×D3—34no—
226C3×C4×D3—64no—
227C3×C4×D3—64no—
228#624—34no—
229#624—34no—
230#624—64no—
231D3×Dic3—34no—
232D3×Dic3—34no—
233D3×Dic3—34no—
234D3×Dic3—34no—
235D3×Dic3—64no—
236C2×#117—271noSylow
237C2×H(3,𝔽3)C23×C411yescharacteristic
238C2×H(3,𝔽3)C2×#4112yes—
239#272—22no—
240(C3⋊D3)⋊C3—22no—
241(C3⋊D3)⋊C3#11714yescomplemented
242C23×D3—91no—
243C23×D3—91no—
244C23×D3—91no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C22×C3×C4—92no—
248C22×C3×C4—91no—
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—92no—
253C2×C4×D3—31no—
254C2×C4×D3—31no—
255C2×C4×D3—31no—
256C2×C4×D3—31no—
257C2×C4×D3—32no—
258C2×C4×D3—61no—
259C2×C4×D3—91no—
260C2×C4×D3—91no—
261C2×C4×D3—92no—
262C2×C4×D3—98no—
263C2×C4×D3—98no—
264C2×C4×D3—98no—
265C2×C4×D3—91no—
266C2×C4×D3—91no—
267C2×C4×D3—91no—
268C2×C4×D3—91no—
269C2×C4×D3—91no—
270C2×C4×D3—91no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—92no—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—61no—
280C3×C42—91no—
281C3×C42—92no—
282C3×C42—91no—
283C3×C42—91no—
284C3×C42D3211yescharacteristic
285#217—92no—
286#217—92no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—62no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296C4⋊Dic3—92no—
297C4⋊Dic3—92no—
298Dic3⋊C4—32no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—62no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C4×Dic3—31no—
308C4×Dic3—31no—
309C4×Dic3—31no—
310C4×Dic3—32no—
311C4×Dic3—31no—
312C4×Dic3—61no—
313C4×Dic3—91no—
314C4×Dic3—92no—
315C4×Dic3—91no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C22×C32—21no—
320C22×C32C23×D311yescharacteristic
321C22×C32C23×D311yescharacteristic
322C2×C3⋊D3—22no—
323C2×C3⋊D3C2×C4×D312yes—
324C2×C3⋊D3C2×C4×D312yes—
325C2×C3⋊D3#23514yes—
326C2×C3×D3—32no—
327C2×C3×D3—34no—
328C2×C3×D3—32no—
329C2×C3×D3—32no—
330C2×C3×D3—32no—
331C2×C3×D3—62no—
332C2×C3×D3—62no—
333C2×C3×D3—62no—
334C2×C3×D3—62no—
335C2×C3×D3—122no—
336D32—68no—
337D32—68no—
338C32×C4—22no—
339C32×C4—22no—
340C32×C4—22no—
341C32×C4—42no—
342C32×C4C2×C4×D312yes—
343C32×C4C2×C4×D312yes—
344C3⋊Dic3—22no—
345C3⋊Dic3—22no—
346C3⋊Dic3C2×C4×D312yes—
347C3⋊Dic3C2×C4×D312yes—
348C3×Dic3—32no—
349C3×Dic3—32no—
350C3×Dic3—32no—
351C3×Dic3—32no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—62no—
356C3×Dic3—62no—
357C3×Dic3—122no—
358C23×C4—271no—
359#117—91no—
360#117—91no—
361#117—271no—
362#117—271no—
363#117—272no—
364#117—272no—
365C2×C4⋊C4—272no—
366C2×C22⋊C4—272no—
367C2×C42—91no—
368C2×C42—271no—
369H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
370C23×C3—91no—
371C23×C3—91no—
372C23×C3—91no—
373C22×D3—31no—
374C22×D3—31no—
375C22×D3—31no—
376C22×D3—31no—
377C22×D3—61no—
378C22×D3—91no—
379C22×D3—91no—
380C22×D3—94no—
381C22×D3—94no—
382C22×D3—94no—
383C22×D3—184no—
384C22×D3—184no—
385C22×D3—184no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—62no—
391C2×C3×C4—61no—
392C2×C3×C4—94no—
393C2×C3×C4—94no—
394C2×C3×C4—91no—
395C2×C3×C4—91no—
396C2×C3×C4—91no—
397C2×C3×C4—94no—
398C2×C3×C4—92no—
399C2×C3×C4—92no—
400C2×C3×C4—92no—
401C2×C3×C4—184no—
402C2×C3×C4C2×D3212yes—
403C2×C3×C4C2×D3211yescharacteristic
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—61no—
413C2×Dic3—62no—
414C2×Dic3—91no—
415C2×Dic3—91no—
416C2×Dic3—92no—
417C2×Dic3—92no—
418C2×Dic3—94no—
419C2×Dic3—94no—
420C2×Dic3—94no—
421C2×Dic3—184no—
422C4×D3—34no—
423C4×D3—34no—
424C4×D3—34no—
425C4×D3—34no—
426C4×D3—64no—
427C4×D3—64no—
428C4×D3—94no—
429C4×D3—94no—
430C4×D3—94no—
431C4×D3—94no—
432C4×D3—94no—
433C4×D3—94no—
434C4×D3—94no—
435C4×D3—94no—
436C4×D3—184no—
437C4×D3—184no—
438C4×D3—184no—
439C2×C32—21no—
440C2×C32—22no—
441C2×C32C22×C4×D311yescharacteristic
442C2×C32C22×C4×D311yescharacteristic
443C2×C32C2×#23512yes—
444C2×C32D3×#4112yes—
445C3⋊D3—22no—
446C3⋊D3#87214yescomplemented
447C3×D3—34no—
448C3×D3—62no—
449C3×D3—62no—
450C3×D3—62no—
451C3×D3—122no—
452C24—271no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—91no—
456C22×C4—92no—
457C22×C4—278no—
458C22×C4—271no—
459C22×C4—271no—
460C22×C4—271no—
461C22×C4—272no—
462C4⋊C4—92no—
463C4⋊C4—92no—
464C4⋊C4—272no—
465C4⋊C4—272no—
466C22⋊C4—92no—
467C22⋊C4—92no—
468C22⋊C4—272no—
469C22⋊C4—272no—
470C42—92no—
471C42—91no—
472C42—91no—
473C42—91no—
474C42—272no—
475C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
476C22×C3—31no—
477C22×C3—31no—
478C22×C3—61no—
479C22×C3—92no—
480C22×C3—94no—
481C22×C3—92no—
482C22×C3—92no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3C22×D3211yescharacteristic
486C2×D3—32no—
487C2×D3—34no—
488C2×D3—32no—
489C2×D3—32no—
490C2×D3—32no—
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—62no—
494C2×D3—62no—
495C2×D3—92no—
496C2×D3—94no—
497C2×D3—92no—
498C2×D3—122no—
499C2×D3—182no—
500C2×D3—188no—
501C2×D3—188no—
502C2×D3—188no—
503C3×C4—22no—
504C3×C4—32no—
505C3×C4—32no—
506C3×C4—62no—
507C3×C4—62no—
508C3×C4—62no—
509C3×C4—92no—
510C3×C4—92no—
511C3×C4—92no—
512C3×C4—122no—
513C3×C4—182no—
514C3×C4—182no—
515C3×C4—182no—
516C3×C4C4×D3212yes—
517Dic3—32no—
518Dic3—32no—
519Dic3—32no—
520Dic3—32no—
521Dic3—62no—
522Dic3—62no—
523Dic3—62no—
524Dic3—62no—
525Dic3—62no—
526Dic3—92no—
527Dic3—92no—
528Dic3—122no—
529Dic3—182no—
530Dic3—182no—
531C32—21no—
532C32C2×#87211yescharacteristic, complemented
533C32D3×#11711yescharacteristic, complemented
534C23—91no—
535C23—91no—
536C23—91no—
537C23—274no—
538C23—544no—
539C2×C4—91no—
540C2×C4—91no—
541C2×C4—91no—
542C2×C4—94no—
543C2×C4—94no—
544C2×C4—94no—
545C2×C4—92no—
546C2×C4—92no—
547C2×C4—92no—
548C2×C4—184no—
549C2×C4—274no—
550C2×C4—274no—
551C2×C4—274no—
552C2×C4—544no—
553C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
554C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
555C2×C3—31no—
556C2×C3—31no—
557C2×C3—32no—
558C2×C3—32no—
559C2×C3—61no—
560C2×C3—62no—
561C2×C3—94no—
562C2×C3—182no—
563C2×C3—182no—
564C2×C3C2×C4×D3211yescharacteristic
565C2×C3D3×#23512yes—
566D3—34no—
567D3—62no—
568D3—62no—
569D3—62no—
570D3—94no—
571D3—122no—
572D3—182no—
573C22—92no—
574C22—94no—
575C22—92no—
576C22—92no—
577C22—182no—
578C22—182no—
579C22—548no—
580C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
581C4—22no—
582C4—92no—
583C4—92no—
584C4—92no—
585C4—182no—
586C4—182no—
587C4—182no—
588C4C4×((C3⋊D3)⋊C3)⋊C212yes—
589C3—31no—
590C3—31no—
591C3—61no—
592C3D3×#87211yescharacteristic
593C2—94no—
594C2—182no—
595C2—182no—
596C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
597C2#1172249012yes—
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
cosocleC22×H(3,𝔽3)C24
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11721718C2semidirect
3. #64081⋊C3C2semidirect
4. C42×(C3⋊D3)⋊C3C2semidirect
6. #11721120C2semidirect
7. #11721119C2semidirect
8. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
9. #11721105C2semidirect
10. #11721097C2semidirect
11. #11721076C2semidirect
23. C4×((C3⋊D3)⋊C3)⋊C2C4semidirect
24. C42×H(3,𝔽3)C22semidirect
26. #88540C22semidirect
27. C4×#1084C22semidirect
29. #3750⋊C3C22semidirect
30. C4×(C3⋊Dic3)⋊C3C22semidirect
31. C4×(C3⋊Dic3)⋊C3C22semidirect
32. C4×(C3⋊Dic3)⋊C3C22semidirect
35. #88449C22semidirect
36. #88448C22semidirect
37. #88447C22semidirect
40. #64081D3semidirect
41. C42×C3⋊D3D3semidirect
70. C4×#272C2×C4semidirect
77. C4×(C3⋊D3)⋊C3C2×C4semidirect
78. #6199C2×C4semidirect
81. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
107. C32×C42C2×D3semidirect
108. C32×C42C2×D3semidirect
110. #3750C2×D3semidirect
111. C4×C3⋊Dic3C2×D3semidirect
112. C4×C3⋊Dic3C2×D3semidirect
113. C4×C3⋊Dic3C2×D3semidirect
211. C4×C3⋊D3C4×D3semidirect
241. (C3⋊D3)⋊C3#117semidirect
369. H(3,𝔽3)C2×#117semidirect
446. C3⋊D3#872semidirect
475. C42((C3⋊D3)⋊C3)⋊C2semidirect
532. C32C2×#872semidirect
533. C32D3×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
#117217182
#64081⋊C32
C42×(C3⋊D3)⋊C32
#117211602
#117211202
#117211192
C4×((C3⋊D3)⋊C3)⋊C42
#117211052
#117210972
#117210762
#105939923
D3×#8723