Finite Group Databasebeta

← #13148534

Subgroups — #13148534

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
1C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2#11721733C211yesmaximal, characteristic, complemented
3C22⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
4#64086⋊C3C211yesmaximal, characteristic, complemented
5#11721310C211yesmaximal, characteristic
6#11721298C211yesmaximal, characteristic, complemented
7#11721296C211yesmaximal, characteristic, complemented
8#11721121C212yesmaximal, complemented
9#11721120C212yesmaximal, complemented
10#11721119C212yesmaximal, complemented
11C4×((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
12#10594052—31nomaximal
13D3×#877—31nomaximal
14C23×(C3⋊D3)⋊C3C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C418yescomplemented
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
21C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
22C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
23C2×((C3⋊D3)⋊C3)⋊C4C2211yescharacteristic
24C22⋊C4×H(3,𝔽3)C2211yescharacteristic
25#88552C2211yescharacteristic
26#88540C2212yescomplemented
27C4×#1084C2212yescomplemented
28#3757⋊C3C2211yescharacteristic
29#3757⋊C3C2211yescharacteristic
30#3752⋊C3C2212yes—
31#3750⋊C3C2212yescomplemented
32C4×(C3⋊Dic3)⋊C3C2212yescomplemented
33#88450C2212yes—
34#88449C2212yescomplemented
35#88448C2212yescomplemented
36#88447C2212yescomplemented
37#88447C2212yescomplemented
38C22×#624—31no—
39C22×D3×Dic3—31no—
40C22⋊C4×C3⋊D3D311yescharacteristic
41#64086D311yescharacteristic, complemented
42C3×D3×C22⋊C4—31no—
43C3×#877—31no—
44C3×#877—31no—
45C3×#877—31no—
46C3×#877—61no—
47#63974—31no—
48D3×#217—31no—
49#63957—31no—
50#63955—31no—
51#63955—31no—
52#63955—31no—
53#63883—32no—
54C4×#624—32no—
55#63878—32no—
56#63877—32no—
57D3×Dic3⋊C4—32no—
58C4×D3×Dic3—32no—
59#63836—32no—
60#63836—32no—
61C23×H(3,𝔽3)C2311yescharacteristic
62C22×(C3⋊D3)⋊C3—24no—
63C22×(C3⋊D3)⋊C3C2×C414yes—
64C22×(C3⋊D3)⋊C3C2311yescharacteristic
65C22×(C3⋊D3)⋊C3C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2312yes—
67C2×#1084C2×C414yescomplemented
68C2×#1084C2312yes—
69C2×#1084C2311yescharacteristic
70C2×#1084C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
75C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
76C4×(C3⋊D3)⋊C3—24no—
77((C3⋊D3)⋊C3)⋊C4—24no—
78((C3⋊D3)⋊C3)⋊C4C2×C414yes—
79((C3⋊D3)⋊C3)⋊C4C2×C414yes—
80((C3⋊D3)⋊C3)⋊C4C2×C414yescomplemented
81((C3⋊D3)⋊C3)⋊C4C2×C414yescomplemented
82D3×#117—91no—
83D3×#117—91no—
84C2×#877—91no—
85C23×C3⋊D3C2×D311yescharacteristic
86C23×C3×D3—31no—
87C22×C3⋊Dic3C2×D311yescharacteristic
88C2×C4×C3⋊D3C2×D312yes—
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C22×C3×Dic3—31no—
92C22×C3×Dic3—61no—
93C2×C3×C4×D3—32no—
94C2×#624—38no—
95C2×#624—31no—
96C2×#624—31no—
97C2×#624—31no—
98C2×#624—31no—
99C2×#624—32no—
100C2×D3×Dic3—38no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—31no—
104C2×D3×Dic3—31no—
105C2×D3×Dic3—32no—
106C32×C22⋊C4—21no—
107C32×C22⋊C4C2×D311yescharacteristic
108C32×C22⋊C4C2×D311yescharacteristic
109#3757C2×D311yescharacteristic
110#3757C2×D311yescharacteristic
111#3752C2×D312yes—
112#3750C2×D312yescomplemented
113C4×C3⋊Dic3C2×D312yescomplemented
114C3×#217—31no—
115C3×#217—31no—
116C3×#217—31no—
117C3×#217—31no—
118C3×#217—61no—
119C3×(C2×D3)⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—62no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—62no—
128#3724—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131Dic3⋊Dic3—32no—
132#3722—32no—
133(C2×D3)⋊Dic3—32no—
134Dic32—32no—
135Dic32—32no—
136Dic32—32no—
137Dic32—32no—
138C22×H(3,𝔽3)—22no—
139C22×H(3,𝔽3)C22×C412yes—
140C22×H(3,𝔽3)C2411yescharacteristic
141C2×(C3⋊D3)⋊C3—22no—
142C2×(C3⋊D3)⋊C3—28no—
143C2×(C3⋊D3)⋊C3C22×C412yes—
144C2×(C3⋊D3)⋊C3C22×C412yes—
145C2×(C3⋊D3)⋊C3#4114yes—
146C4×H(3,𝔽3)—22no—
147#1084—22no—
148#1084C22×C412yes—
149#1084C22×C412yes—
150(C3⋊Dic3)⋊C3—22no—
151(C3⋊Dic3)⋊C3—22no—
152(C3⋊Dic3)⋊C3C22×C412yes—
153(C3⋊Dic3)⋊C3C22×C412yes—
154C23×Dic3—91no—
155C22×C4×D3—91no—
156C22×C4×D3—91no—
157C3×#117—91no—
158C3×#117—91no—
159C2×C3×C22⋊C4—91no—
160C2×#217—91no—
161#926—91no—
162#926—91no—
163C2×Dic3⋊C4—92no—
164C2×C4×Dic3—92no—
165#892—92no—
166#892—92no—
167D3×C4⋊C4—92no—
168D3×C4⋊C4—92no—
169D3×C22⋊C4—31no—
170D3×C22⋊C4—91no—
171D3×C22⋊C4—91no—
172D3×C22⋊C4—91no—
173#877—31no—
174#877—31no—
175#877—31no—
176#877—61no—
177#877—91no—
178#877—92no—
179#877—92no—
180#877—91no—
181#877—91no—
182#877—91no—
183#877—91no—
184#872—92no—
185#872—92no—
186C42×D3—92no—
187C42×D3—92no—
188C23×C32—21no—
189C23×C32C22×D311yescharacteristic
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—24no—
192C22×C3⋊D3C4×D314yes—
193C22×C3⋊D3C22×D311yescharacteristic
194C22×C3⋊D3C22×D311yescharacteristic
195C22×C3×D3—31no—
196C22×C3×D3—34no—
197C22×C3×D3—31no—
198C22×C3×D3—64no—
199C2×C32×C4—22no—
200C2×C32×C4C22×D312yes—
201C2×C32×C4C22×D312yes—
202C2×C3⋊Dic3C4×D314yescomplemented
203C2×C3⋊Dic3C22×D312yes—
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D311yescharacteristic
207C4×C3⋊D3—24no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—31no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—31no—
215C2×C3×Dic3—31no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—62no—
225C3×C4×D3—64no—
226#624—34no—
227#624—34no—
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)—22no—
238C2×H(3,𝔽3)C23×C411yescharacteristic
239C2×H(3,𝔽3)C2×#4112yes—
240(C3⋊D3)⋊C3—22no—
241(C3⋊D3)⋊C3#11714yescomplemented
242C24×C3—91no—
243C23×D3—31no—
244C23×D3—91no—
245C22×C3×C4—92no—
246C22×C3×C4—91no—
247C22×C3×C4—91no—
248C22×Dic3—31no—
249C22×Dic3—31no—
250C22×Dic3—31no—
251C22×Dic3—61no—
252C22×Dic3—91no—
253C22×Dic3—91no—
254C22×Dic3—91no—
255C22×Dic3—91no—
256C22×Dic3—91no—
257C22×Dic3—98no—
258C22×Dic3—92no—
259C2×C4×D3—32no—
260C2×C4×D3—92no—
261C2×C4×D3—91no—
262C2×C4×D3—91no—
263C2×C4×D3—91no—
264C2×C4×D3—91no—
265C2×C4×D3—98no—
266C2×C4×D3—91no—
267C2×C4×D3—91no—
268C2×C4×D3—91no—
269C2×C4×D3—91no—
270C2×C4×D3—98no—
271C2×C4×D3—92no—
272C2×C4×D3—92no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C22⋊C4—31no—
276C3×C22⋊C4—31no—
277C3×C22⋊C4—61no—
278C3×C22⋊C4—92no—
279C3×C22⋊C4—91no—
280C3×C22⋊C4—91no—
281C3×C22⋊C4—91no—
282C3×C22⋊C4D3211yescharacteristic
283C3×C42—92no—
284C3×C42—92no—
285#217—31no—
286#217—31no—
287#217—31no—
288#217—31no—
289#217—61no—
290#217—91no—
291#217—91no—
292#217—92no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—92no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297C4⋊Dic3—92no—
298C4⋊Dic3—92no—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—62no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308C4×Dic3—32no—
309C4×Dic3—32no—
310C4×Dic3—32no—
311C4×Dic3—62no—
312C4×Dic3—92no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—92no—
316C4×Dic3—92no—
317C4×Dic3—92no—
318C4×Dic3—92no—
319C22×C32—22no—
320C22×C32—22no—
321C22×C32—21no—
322C22×C32—22no—
323C22×C32—42no—
324C22×C32C2×C4×D312yes—
325C22×C32C2×C4×D312yes—
326C22×C32C23×D311yescharacteristic
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—22no—
329C2×C3⋊D3—28no—
330C2×C3⋊D3C2×C4×D312yes—
331C2×C3⋊D3C2×C4×D312yes—
332C2×C3⋊D3#23714yes—
333C2×C3×D3—32no—
334C2×C3×D3—34no—
335C2×C3×D3—32no—
336C2×C3×D3—68no—
337C2×C3×D3—62no—
338C32×C4—22no—
339C32×C4—22no—
340C32×C4—42no—
341C3⋊Dic3—22no—
342C3⋊Dic3—22no—
343C3⋊Dic3C2×C4×D312yes—
344C3⋊Dic3C2×C4×D312yes—
345C3×Dic3—32no—
346C3×Dic3—32no—
347C3×Dic3—32no—
348C3×Dic3—32no—
349C3×Dic3—32no—
350C3×Dic3—32no—
351C3×Dic3—62no—
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—272no—
362#117—272no—
363#117—271no—
364#117—271no—
365C2×C4⋊C4—272no—
366C2×C22⋊C4—91no—
367C2×C22⋊C4—271no—
368C2×C42—272no—
369H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
370C23×C3—31no—
371C23×C3—31no—
372C23×C3—61no—
373C23×C3—91no—
374C23×C3—91no—
375C23×C3—94no—
376C23×C3—184no—
377C23×C3C2×D3211yescharacteristic
378C22×D3—31no—
379C22×D3—31no—
380C22×D3—34no—
381C22×D3—64no—
382C22×D3—91no—
383C22×D3—94no—
384C22×D3—91no—
385C22×D3—184no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—62no—
389C2×C3×C4—91no—
390C2×C3×C4—91no—
391C2×C3×C4—94no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—94no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—92no—
398C2×C3×C4—184no—
399C2×C3×C4C2×D3212yes—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—34no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—64no—
416C2×Dic3—62no—
417C2×Dic3—94no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—92no—
421C2×Dic3—92no—
422C2×Dic3—94no—
423C2×Dic3—94no—
424C2×Dic3—94no—
425C2×Dic3—94no—
426C2×Dic3—184no—
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—22no—
440C2×C32—22no—
441C2×C32—21no—
442C2×C32—22no—
443C2×C32—42no—
444C2×C32C22×C4×D311yescharacteristic
445C2×C32C22×C4×D311yescharacteristic
446C2×C32C2×#23712yes—
447C2×C32D3×#4112yes—
448C3⋊D3—22no—
449C3⋊D3#87714yescomplemented
450C3×D3—34no—
451C3×D3—62no—
452C24—91no—
453C22×C4—91no—
454C22×C4—91no—
455C22×C4—92no—
456C22×C4—271no—
457C22×C4—271no—
458C22×C4—271no—
459C22×C4—271no—
460C22×C4—278no—
461C22×C4—272no—
462C4⋊C4—92no—
463C4⋊C4—92no—
464C4⋊C4—272no—
465C4⋊C4—272no—
466C22⋊C4—91no—
467C22⋊C4—91no—
468C22⋊C4—91no—
469C22⋊C4—92no—
470C22⋊C4—272no—
471C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
472C42—92no—
473C42—92no—
474C42—272no—
475C42—272no—
476C22×C3—22no—
477C22×C3—31no—
478C22×C3—31no—
479C22×C3—32no—
480C22×C3—32no—
481C22×C3—62no—
482C22×C3—62no—
483C22×C3—61no—
484C22×C3—62no—
485C22×C3—92no—
486C22×C3—94no—
487C22×C3—92no—
488C22×C3—122no—
489C22×C3—182no—
490C22×C3—188no—
491C22×C3C4×D3212yes—
492C22×C3C22×D3211yescharacteristic
493C2×D3—32no—
494C2×D3—34no—
495C2×D3—32no—
496C2×D3—68no—
497C2×D3—62no—
498C2×D3—92no—
499C2×D3—94no—
500C2×D3—92no—
501C2×D3—188no—
502C2×D3—182no—
503C3×C4—22no—
504C3×C4—62no—
505C3×C4—62no—
506C3×C4—92no—
507C3×C4—92no—
508C3×C4—92no—
509C3×C4—92no—
510C3×C4—122no—
511C3×C4—182no—
512C3×C4—182no—
513C3×C4—182no—
514Dic3—32no—
515Dic3—32no—
516Dic3—32no—
517Dic3—32no—
518Dic3—32no—
519Dic3—32no—
520Dic3—62no—
521Dic3—62no—
522Dic3—62no—
523Dic3—62no—
524Dic3—62no—
525Dic3—62no—
526Dic3—92no—
527Dic3—92no—
528Dic3—122no—
529Dic3—182no—
530Dic3—182no—
531C32—21no—
532C32C2×#87711yescharacteristic, complemented
533C32D3×#11711yescharacteristic, complemented
534C23—91no—
535C23—91no—
536C23—94no—
537C23—184no—
538C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
539C2×C4—91no—
540C2×C4—91no—
541C2×C4—91no—
542C2×C4—91no—
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—274no—
553C2×C4—544no—
554C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
555C2×C3—22no—
556C2×C3—32no—
557C2×C3—32no—
558C2×C3—31no—
559C2×C3—31no—
560C2×C3—62no—
561C2×C3—62no—
562C2×C3—62no—
563C2×C3—61no—
564C2×C3—94no—
565C2×C3—122no—
566C2×C3—182no—
567C2×C3C2×C4×D3211yescharacteristic
568C2×C3D3×#23712yes—
569D3—34no—
570D3—62no—
571D3—94no—
572D3—182no—
573C22—22no—
574C22—94no—
575C22—92no—
576C22—92no—
577C22—188no—
578C22—182no—
579C22C4×((C3⋊D3)⋊C3)⋊C212yes—
580C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
581C4—22no—
582C4—92no—
583C4—92no—
584C4—92no—
585C4—92no—
586C4—182no—
587C4—182no—
588C4—182no—
589C3—31no—
590C3—31no—
591C3—61no—
592C3D3×#87711yescharacteristic
593C2—22no—
594C2—94no—
595C2—182no—
596C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
597C2#1172250412yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721733C2semidirect
4. #64086⋊C3C2semidirect
6. #11721298C2semidirect
7. #11721296C2semidirect
8. #11721121C2semidirect
9. #11721120C2semidirect
10. #11721119C2semidirect
11. C4×((C3⋊D3)⋊C3)⋊C4C2semidirect
18. C2×((C3⋊D3)⋊C3)⋊C4C4semidirect
26. #88540C22semidirect
27. C4×#1084C22semidirect
31. #3750⋊C3C22semidirect
32. C4×(C3⋊Dic3)⋊C3C22semidirect
34. #88449C22semidirect
35. #88448C22semidirect
36. #88447C22semidirect
37. #88447C22semidirect
41. #64086D3semidirect
67. C2×#1084C2×C4semidirect
71. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
80. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
81. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
112. #3750C2×D3semidirect
113. C4×C3⋊Dic3C2×D3semidirect
202. C2×C3⋊Dic3C4×D3semidirect
241. (C3⋊D3)⋊C3#117semidirect
369. H(3,𝔽3)C2×#117semidirect
449. C3⋊D3#877semidirect
532. C32C2×#877semidirect
533. C32D3×#117semidirect

Maximal subgroups

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

subgroupindex
C22×((C3⋊D3)⋊C3)⋊C42
#117217332
C22⋊C4×(C3⋊D3)⋊C32
#64086⋊C32
#117213102
#117212982
#117212962
#117211212
#117211202
#117211192
C4×((C3⋊D3)⋊C3)⋊C42
#105940523
D3×#8773