Finite Group Databasebeta

← #13178313

Subgroups — #13178313

Order 1728.

1134 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#64098C212yesmaximal, complemented
3#11723689C211yesmaximal, characteristic
4#11723682C211yesmaximal, characteristic
5#11723682C211yesmaximal, characteristic
6#11723682C211yesmaximal, characteristic
7#11723682C211yesmaximal, characteristic
8C4×#6294C211yesmaximal, characteristic
9#11723679C212yesmaximal, complemented
10#11723669C212yesmaximal, complemented
11#11723669C212yesmaximal, complemented
12#10594399—31nomaximal
13D3×#889—32nomaximal
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×#6294C2211yescharacteristic
18C2×#6294C2211yescharacteristic
19C2×#6294C2211yescharacteristic
20C2×#6294C2211yescharacteristic
21C33×C4⋊C4C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27C3×#3750C2212yes—
28C3×#3750C2211yescharacteristic
29C3×#3750C2211yescharacteristic
30C3×C4×C3⋊Dic3C2212yes—
31#88815C2212yes—
32#88815C2212yes—
33#88815C2212yes—
34#88815C2212yes—
35#88815C2212yes—
36#88813C2212yes—
37C4⋊C4×C3⋊D3D311yescharacteristic
38#64098D312yescomplemented
39C3×D3×C4⋊C4—32no—
40C3×D3×C4⋊C4—61no—
41C3×#889—32no—
42C3×#889—32no—
43C3×#889—62no—
44#63900—31no—
45D3×C4⋊Dic3—32no—
46#63883—31no—
47#63883—31no—
48#63883—31no—
49#63883—31no—
50C4×#624—31no—
51D3×Dic3⋊C4—32no—
52D3×Dic3⋊C4—32no—
53D3×Dic3⋊C4—32no—
54D3×Dic3⋊C4—32no—
55C4×D3×Dic3—32no—
56#63870—32no—
57#63867—32no—
58#63867—32no—
59#63847—32no—
60#63847—32no—
61#63846—32no—
62#63846—32no—
63#63845—32no—
64#63845—32no—
65C22×C3×C3⋊D3C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4×C3⋊D3—22no—
77C3×C4×C3⋊D3—22no—
78C3×C4×C3⋊D3—22no—
79#6294—22no—
80#6294—22no—
81#6294—22no—
82#6294Q814yes—
83D3×#125—92no—
84C2×#889—91no—
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—61no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—61no—
94C2×#624—31no—
95C2×#624—31no—
96C2×#624—31no—
97C2×#624—31no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C32×C4⋊C4—22no—
103C32×C4⋊C4—21no—
104C32×C4⋊C4—41no—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C4⋊C4C2×D312yes—
107C4⋊(C3⋊Dic3)C2×D312yes—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750C2×D312yes—
110#3750C2×D312yes—
111#3750C2×D312yes—
112#3750C2×D312yes—
113#3750C2×D311yescharacteristic
114#3750C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D312yes—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—62no—
120C3×C4⋊Dic3—61no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—62no—
132C3×Dic3⋊C4—62no—
133C3×Dic3⋊C4—62no—
134C3×Dic3⋊C4—62no—
135C3×Dic3⋊C4—61no—
136C3×Dic3⋊C4—61no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139C3×C4×Dic3—62no—
140#3724—32no—
141#3724—32no—
142#3724—32no—
143#3724—32no—
144#3724—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151Dic3⋊Dic3—32no—
152Dic3⋊Dic3—32no—
153Dic3⋊Dic3—32no—
154Dic3⋊Dic3—32no—
155Dic32—32no—
156Dic32—32no—
157Dic32—32no—
158C22×C33C2411yescharacteristic
159C2×C3×C3⋊D3C2×Q812yes—
160C2×C3×C3⋊D3#4112yes—
161C2×C3×C3⋊D3#4112yes—
162C33×C4—21no—
163C33×C4—21no—
164C33×C4—21no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—21no—
170C3×C3⋊Dic3—21no—
171C3×C3⋊Dic3C2×Q814yes—
172C3×#125—92no—
173C2×C3×C4⋊C4—91no—
174C2×C4⋊Dic3—91no—
175C2×Dic3⋊C4—91no—
176C2×Dic3⋊C4—91no—
177C2×Dic3⋊C4—91no—
178C2×Dic3⋊C4—91no—
179C2×C4×Dic3—91no—
180D3×C4⋊C4—32no—
181D3×C4⋊C4—61no—
182D3×C4⋊C4—92no—
183D3×C4⋊C4—92no—
184D3×C4⋊C4—92no—
185D3×C4⋊C4—92no—
186D3×C4⋊C4—92no—
187#890—92no—
188#890—92no—
189#889—32no—
190#889—32no—
191#889—62no—
192#889—92no—
193#889—92no—
194#889—92no—
195#889—92no—
196#889—92no—
197#889—92no—
198C42×D3—92no—
199#870—92no—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3×D3—32no—
202C22×C3×D3—61no—
203C2×C32×C4—21no—
204C2×C32×C4—22no—
205C2×C32×C4—22no—
206C2×C32×C4—22no—
207C2×C32×C4—21no—
208C2×C32×C4—21no—
209C2×C32×C4—41no—
210C2×C32×C4—41no—
211C2×C32×C4—41no—
212C2×C32×C4C22×D311yescharacteristic
213C2×C32×C4C22×D312yes—
214C2×C32×C4C22×D311yescharacteristic
215C2×C32×C4C22×D311yescharacteristic
216C2×C32×C4C22×D312yes—
217C2×C32×C4C22×D312yes—
218C2×C3⋊Dic3C22×D311yescharacteristic
219C2×C3⋊Dic3C22×D312yes—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C2×C3⋊Dic3C22×D311yescharacteristic
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C4×C3⋊D3—22no—
226C4×C3⋊D3—22no—
227C4×C3⋊D3—22no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—62no—
240C2×C3×Dic3—61no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—62no—
243C2×C3×Dic3—62no—
244C2×C3×Dic3—61no—
245C2×C3×Dic3—61no—
246C3×C4×D3—64no—
247C3×C4×D3—64no—
248C3×C4×D3—64no—
249C3×C4×D3—64no—
250C3×C4×D3—122no—
251C3×C4×D3—122no—
252#624—34no—
253#624—62no—
254#624—62no—
255#624—62no—
256D3×Dic3—38no—
257D3×Dic3—64no—
258D3×Dic3—64no—
259D3×Dic3—64no—
260C2×#125—271noSylow
261C2×C33C22×Q811yescharacteristic
262C2×C33C2×#4111yescharacteristic
263C2×C33C2×#4111yescharacteristic
264C3×C3⋊D3#12514yescomplemented
265C22×C3×C4—91no—
266C22×C3×C4—91no—
267C22×C3×C4—91no—
268C22×Dic3—91no—
269C22×Dic3—91no—
270C22×Dic3—91no—
271C22×Dic3—91no—
272C2×C4×D3—32no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—61no—
276C2×C4×D3—61no—
277C2×C4×D3—61no—
278C2×C4×D3—92no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3—92no—
282C3×C4⋊C4—22no—
283C3×C4⋊C4—21no—
284C3×C4⋊C4—41no—
285C3×C4⋊C4—91no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—92no—
289C3×C4⋊C4—92no—
290C3×C4⋊C4—92no—
291C3×C4⋊C4—92no—
292C3×C4⋊C4—92no—
293C3×C4⋊C4D3211yescharacteristic
294C3×C4⋊C4D3212yes—
295C3×C42—92no—
296C4⋊Dic3—32no—
297C4⋊Dic3—32no—
298C4⋊Dic3—32no—
299C4⋊Dic3—62no—
300C4⋊Dic3—61no—
301C4⋊Dic3—92no—
302C4⋊Dic3—92no—
303C4⋊Dic3—92no—
304C4⋊Dic3—92no—
305C4⋊Dic3—92no—
306Dic3⋊C4—32no—
307Dic3⋊C4—32no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—62no—
317Dic3⋊C4—62no—
318Dic3⋊C4—62no—
319Dic3⋊C4—62no—
320Dic3⋊C4—61no—
321Dic3⋊C4—61no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328Dic3⋊C4—92no—
329Dic3⋊C4—92no—
330Dic3⋊C4—92no—
331Dic3⋊C4—92no—
332C4×Dic3—32no—
333C4×Dic3—32no—
334C4×Dic3—62no—
335C4×Dic3—92no—
336C4×Dic3—92no—
337C4×Dic3—92no—
338C22×C32—22no—
339C22×C32—21no—
340C22×C32—41no—
341C22×C32C23×D311yescharacteristic
342C22×C32C23×D312yes—
343C2×C3⋊D3#23512yes—
344C2×C3⋊D3#23712yes—
345C2×C3⋊D3D3×Q812yes—
346C2×C3×D3—34no—
347C2×C3×D3—34no—
348C2×C3×D3—34no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C32×C4—21no—
353C32×C4—22no—
354C32×C4—21no—
355C32×C4—21no—
356C32×C4—22no—
357C32×C4—22no—
358C32×C4—22no—
359C32×C4—42no—
360C32×C4—42no—
361C32×C4—41no—
362C32×C4—41no—
363C32×C4—42no—
364C32×C4—42no—
365C32×C4—81no—
366C32×C4—81no—
367C3⋊Dic3—21no—
368C3⋊Dic3—21no—
369C3⋊Dic3—21no—
370C3⋊Dic3—22no—
371C3⋊Dic3—22no—
372C3⋊Dic3—22no—
373C3⋊Dic3D3×Q814yes—
374C3×Dic3—34no—
375C3×Dic3—34no—
376C3×Dic3—64no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—121no—
388C3×Dic3—121no—
389C3×Dic3—122no—
390C3×Dic3—122no—
391C3×Dic3—122no—
392#125—92no—
393#125—272no—
394#125—272no—
395#125—272no—
396C2×C4⋊C4—91no—
397C2×C4⋊C4—271no—
398C2×C4⋊C4—271no—
399C2×C4⋊C4—271no—
400C2×C4⋊C4—271no—
401C2×C4⋊C4—271no—
402C2×C42—271no—
403C33C2×#12511yescharacteristic, complemented, Sylow
404C23×C3—91no—
405C22×D3—32no—
406C22×D3—61no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—92no—
417C2×C3×C4—91no—
418C2×C3×C4—92no—
419C2×C3×C4—91no—
420C2×C3×C4—91no—
421C2×C3×C4—92no—
422C2×C3×C4—92no—
423C2×C3×C4—182no—
424C2×C3×C4—182no—
425C2×C3×C4—182no—
426C2×C3×C4C2×D3211yescharacteristic
427C2×C3×C4C2×D3212yes—
428C2×C3×C4C2×D3211yescharacteristic
429C2×C3×C4C2×D3211yescharacteristic
430C2×C3×C4C2×D3212yes—
431C2×C3×C4C2×D3212yes—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—32no—
443C2×Dic3—62no—
444C2×Dic3—61no—
445C2×Dic3—62no—
446C2×Dic3—62no—
447C2×Dic3—62no—
448C2×Dic3—61no—
449C2×Dic3—61no—
450C2×Dic3—94no—
451C2×Dic3—182no—
452C2×Dic3—182no—
453C2×Dic3—182no—
454C4×D3—64no—
455C4×D3—64no—
456C4×D3—64no—
457C4×D3—64no—
458C4×D3—98no—
459C4×D3—122no—
460C4×D3—122no—
461C4×D3—184no—
462C4×D3—184no—
463C4×D3—184no—
464C2×C32—22no—
465C2×C32—21no—
466C2×C32—22no—
467C2×C32—21no—
468C2×C32—22no—
469C2×C32—21no—
470C2×C32—41no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32C2×#23511yescharacteristic
474C2×C32C2×#23711yescharacteristic
475C2×C32C2×D3×Q812yes—
476C2×C32C2×D3×Q811yescharacteristic
477C2×C32D3×#4112yes—
478C2×C32D3×#4112yes—
479C3⋊D3#88914yescomplemented
480C3×D3—38no—
481C3×D3—64no—
482C22×C4—91no—
483C22×C4—91no—
484C22×C4—91no—
485C22×C4—271no—
486C22×C4—271no—
487C22×C4—271no—
488C22×C4—271no—
489C4⋊C4—91no—
490C4⋊C4—91no—
491C4⋊C4—91no—
492C4⋊C4—92no—
493C4⋊C4—92no—
494C4⋊C4—92no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C4⋊C4—272no—
498C4⋊C4—272no—
499C4⋊C4—272no—
500C4⋊C4—272no—
501C4⋊C4—272no—
502C4⋊C4#111311yescharacteristic
503C42—92no—
504C42—272no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—41no—
508C22×C3—92no—
509C22×C3—92no—
510C22×C3—92no—
511C22×C3C22×D3212yes—
512C22×C3C22×D3211yescharacteristic
513C2×D3—34no—
514C2×D3—34no—
515C2×D3—34no—
516C2×D3—62no—
517C2×D3—62no—
518C2×D3—62no—
519C3×C4—22no—
520C3×C4—22no—
521C3×C4—21no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—42no—
529C3×C4—42no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—94no—
535C3×C4—182no—
536C3×C4—182no—
537C3×C4—181no—
538C3×C4—182no—
539C3×C4—181no—
540C3×C4—181no—
541Dic3—34no—
542Dic3—34no—
543Dic3—64no—
544Dic3—62no—
545Dic3—62no—
546Dic3—62no—
547Dic3—62no—
548Dic3—62no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—121no—
555Dic3—121no—
556Dic3—122no—
557Dic3—122no—
558Dic3—122no—
559C32—22no—
560C32—21no—
561C32—41no—
562C32C2×#88911yescharacteristic, complemented
563C32D3×#12512yescomplemented
564C23—91no—
565C2×C4—92no—
566C2×C4—92no—
567C2×C4—91no—
568C2×C4—92no—
569C2×C4—92no—
570C2×C4—91no—
571C2×C4—91no—
572C2×C4—182no—
573C2×C4—182no—
574C2×C4—182no—
575C2×C4—274no—
576C2×C4—542no—
577C2×C4—542no—
578C2×C4—542no—
579C2×C4C2×#111311yescharacteristic
580C2×C4C2×#111311yescharacteristic
581C2×C4C2×#111311yescharacteristic
582C2×C3—21no—
583C2×C3—22no—
584C2×C3—21no—
585C2×C3—22no—
586C2×C3—22no—
587C2×C3—21no—
588C2×C3—41no—
589C2×C3—41no—
590C2×C3—41no—
591C2×C3—94no—
592C2×C3D3×#23512yes—
593C2×C3#6430611yescharacteristic
594C2×C3D3×#23712yes—
595C2×C3#6431211yescharacteristic
596C2×C3D32×Q812yes—
597C2×C3D32×Q811yescharacteristic
598D3—38no—
599D3—64no—
600C22—92no—
601C22—92no—
602C22—92no—
603C22C22×#111311yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—94no—
608C4—181no—
609C4—182no—
610C4—182no—
611C4—181no—
612C4—181no—
613C4—182no—
614C3—21no—
615C3—22no—
616C3—41no—
617C3D3×#88912yescomplemented
618C3#1059439911yescharacteristic, complemented
619C2—94no—
620C2#1172461811yescharacteristic
621C2#1172462211yescharacteristic
622C2Q8×#111311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC4⋊C4#1113
hypocenterC33C2×#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64098C2semidirect
9. #11723679C2semidirect
10. #11723669C2semidirect
11. #11723669C2semidirect
38. #64098D3semidirect
264. C3×C3⋊D3#125semidirect
403. C33C2×#125semidirect
479. C3⋊D3#889semidirect
562. C32C2×#889semidirect
563. C32D3×#125semidirect
617. C3D3×#889semidirect
618. C3#10594399semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊C4×C3⋊D32
C3×#640982
#117236892
#117236822
C4×#62942
#117236792
#117236692
#105943993
D3×#8893