Finite Group Databasebeta

← #13178765

Subgroups — #13178765

Order 1728.

1254 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C3×#64155C212yesmaximal, complemented
3#11723701C211yesmaximal, characteristic
4#11723700C212yesmaximal, complemented
5#11723690C212yesmaximal, complemented
6#11723686C211yesmaximal, characteristic, complemented
7#11723684C212yesmaximal, complemented
8#11723683C212yesmaximal, complemented
9#11723675C212yesmaximal, complemented
10#10595437—32nomaximal
11#10595436—31nomaximal
12C22×#1113C2211yescharacteristic
13C2×C33×Q8C2211yescharacteristic, complemented
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2212yes—
16C2×#6295C2212yes—
17C2×#6294C2212yes—
18C2×#6294C2211yescharacteristic, complemented
19C3×#3752C2212yes—
20C3×#3752C2212yes—
21C3×#3752C2212yes—
22C3×#3752C2212yescomplemented
23C3×C4⋊(C3⋊Dic3)C2212yescomplemented
24C3×#3750C2212yescomplemented
25C3×#3750C2212yescomplemented
26C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
27#88815C2212yescomplemented
28#88814C2211yescharacteristic
29#88814C2212yes—
30#88814C2212yes—
31#88814C2211yescharacteristic, complemented
32#88813C2212yescomplemented
33#64156D311yescharacteristic, complemented
34#64155D312yescomplemented
35C3×#947—32no—
36C3×#947—61no—
37C3×#946—32no—
38C3×#946—32no—
39C3×#946—62no—
40#63923—31no—
41#63922—32no—
42#63921—32no—
43D3×(C2×D3)⋊C4—32no—
44D3×(C2×D3)⋊C4—32no—
45#63904—32no—
46#63897—31no—
47#63891—32no—
48#63890—32no—
49#63884—32no—
50#63881—32no—
51#63879—32no—
52#63878—32no—
53#63877—32no—
54#63861—32no—
55#63858—32no—
56#63854—32no—
57#63848—32no—
58#63836—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C2×#1113—24no—
63C2×#1113—41no—
64C33×Q8—41no—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4⋊(C3⋊D3)—41no—
72C3×C4×C3⋊D3—42no—
73#6295—24no—
74#6294—24no—
75#6294—41no—
76#5679—91no—
77#5637—92no—
78C22×D32—31no—
79C22×D32—32no—
80C2×C32×Q8—22no—
81C2×C32×Q8—21no—
82C2×C32×Q8—41no—
83C2×C32×Q8C2×D311yescharacteristic, complemented
84C2×C32×Q8C2×D312yescomplemented
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D312yes—
87C2×C3×D12—32no—
88C2×C3×D12—61no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—62no—
92C2×#626—32no—
93C2×#626—32no—
94C2×#625—32no—
95C2×#624—32no—
96C2×#624—31no—
97C2×D3×Dic3—32no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100#3752C2×D312yes—
101#3752C2×D312yes—
102#3752C2×D312yes—
103#3752C2×D312yescomplemented
104C4⋊(C3⋊Dic3)C2×D312yescomplemented
105#3750C2×D312yescomplemented
106#3750C2×D312yescomplemented
107C4×C3⋊Dic3C2×D311yescharacteristic, complemented
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—62no—
117C3×(C2×D3)⋊C4—62no—
118C3×(C2×D3)⋊C4—62no—
119C3×(C2×D3)⋊C4—62no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—62no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—62no—
128C3×Dic3⋊C4—62no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—61no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134#3722—32no—
135#3722—31no—
136#3722—32no—
137#3722—31no—
138(C2×D3)⋊Dic3—32no—
139(C2×D3)⋊Dic3—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142(C2×D3)⋊Dic3—32no—
143(C2×D3)⋊Dic3—32no—
144Dic32—32no—
145Dic32—32no—
146Dic32—32no—
147C22×C33C2411yescharacteristic
148C2×C3×C3⋊D3—22no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3#4114yes—
155#1113—44no—
156C33×C4—22no—
157C33×C4—21no—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3—22no—
162C3×#124—91no—
163C3×C22⋊Q8—92no—
164#947—32no—
165#947—61no—
166#946—32no—
167#946—32no—
168#946—62no—
169#937—91no—
170#935—91no—
171C4×#67—92no—
172C2×(C2×D3)⋊C4—92no—
173#898—92no—
174#898—92no—
175#894—92no—
176#894—92no—
177#893—92no—
178#892—92no—
179#886—92no—
180#883—92no—
181#882—92no—
182#67⋊C4—92no—
183D3×C22⋊C4—92no—
184D3×C22⋊C4—92no—
185#879—92no—
186#877—92no—
187#872—92no—
188C22×C3⋊D3C22×D311yescharacteristic
189C22×C3⋊D3C22×D311yescharacteristic
190C22×C3⋊D3C22×D312yes—
191C22×C3×D3—32no—
192C22×C3×D3—32no—
193C22×C3×D3—32no—
194C22×C3×D3—32no—
195C22×C3×D3—62no—
196C22×C3×D3—61no—
197C22×C3×D3—61no—
198C2×D32—64no—
199C2×D32—64no—
200C2×D32—64no—
201C2×D32—121no—
202C2×D32—122no—
203C32×Q8—42no—
204C32×Q8—41no—
205C32×Q8—44no—
206C32×Q8—42no—
207C32×Q8—44no—
208C2×C32×C4—21no—
209C2×C32×C4—22no—
210C2×C32×C4—22no—
211C2×C32×C4—22no—
212C2×C32×C4—22no—
213C2×C32×C4—41no—
214C2×C32×C4—42no—
215C2×C32×C4C22×D312yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C32×C4C22×D312yes—
218C2×C32×C4C22×D312yes—
219C2×C32×C4C22×D312yes—
220C2×C3⋊Dic3C22×D312yes—
221C2×C3⋊Dic3C22×D312yes—
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D312yes—
224C4⋊(C3⋊D3)—41no—
225C4×C3⋊D3—42no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
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—62no—
235C2×C3×Dic3—62no—
236C2×C3×Dic3—62no—
237C2×C3×Dic3—62no—
238C3×D12—122no—
239C3×D12—122no—
240C3×C4×D3—122no—
241C3×C4×D3—122no—
242C3×C4×D3—124no—
243#626—64no—
244#626—64no—
245#625—64no—
246#624—64no—
247#624—121no—
248D3×Dic3—64no—
249D3×Dic3—64no—
250D3×Dic3—122no—
251#551—271noSylow
252C2×C33C2×#4112yes—
253C2×C33H(5,𝔽2)11yescharacteristic
254C3×C3⋊D3—24no—
255C3×C3⋊D3—41no—
256C3×C3⋊D3—41no—
257C23×D3—91no—
258C23×D3—92no—
259C2×C3×Q8—22no—
260C2×C3×Q8—21no—
261C2×C3×Q8—41no—
262C2×C3×Q8D3211yescharacteristic, complemented
263C2×C3×Q8D3212yescomplemented
264C2×C3×D4—91no—
265C22×C3×C4—92no—
266C2×#67—92no—
267C2×#67—92no—
268C22×Dic3—92no—
269C22×Dic3—91no—
270C2×D12—32no—
271C2×D12—61no—
272C2×D12—92no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—62no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—92no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C42—91no—
287#217—92no—
288#217—91no—
289#217—92no—
290#217—91no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—62no—
301(C2×D3)⋊C4—62no—
302(C2×D3)⋊C4—62no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309C4⋊Dic3—32no—
310C4⋊Dic3—32no—
311C4⋊Dic3—62no—
312C4⋊Dic3—92no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—62no—
318Dic3⋊C4—62no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321C4×Dic3—32no—
322C4×Dic3—61no—
323C4×Dic3—92no—
324C4×Dic3—92no—
325C4×Dic3—92no—
326C22×C32—22no—
327C22×C32—21no—
328C22×C32—41no—
329C22×C32C23×D312yes—
330C22×C32C23×D311yescharacteristic
331C2×C3⋊D3—22no—
332C2×C3⋊D3—22no—
333C2×C3⋊D3—21no—
334C2×C3⋊D3—21no—
335C2×C3⋊D3—22no—
336C2×C3⋊D3—22no—
337C2×C3⋊D3#23914yes—
338C2×C3×D3—34no—
339C2×C3×D3—34no—
340C2×C3×D3—64no—
341C2×C3×D3—62no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—62no—
345C2×C3×D3—62no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3—122no—
352C2×C3×D3—122no—
353C2×C3×D3—121no—
354C2×C3×D3—121no—
355C2×C3×D3—122no—
356C2×C3×D3—122no—
357D32—124no—
358D32—128no—
359C32×C4—22no—
360C32×C4—21no—
361C32×C4—24no—
362C32×C4—22no—
363C32×C4—22no—
364C32×C4—22no—
365C32×C4—22no—
366C32×C4—42no—
367C32×C4—42no—
368C32×C4—42no—
369C32×C4—44no—
370C32×C4—42no—
371C3⋊Dic3—22no—
372C3⋊Dic3—22no—
373C3⋊Dic3—22no—
374C3⋊Dic3—22no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—122no—
384C3×Dic3—122no—
385C3×Dic3—122no—
386C3×Dic3—122no—
387#126—272no—
388#124—91no—
389#123—272no—
390#123—271no—
391C22⋊Q8—92no—
392#120—271no—
393C4×D4—272no—
394#117—272no—
395C2×C22⋊C4—272no—
396C33#55111yescharacteristic, complemented, Sylow
397C23×C3—92no—
398C23×C3—91no—
399C23×C3—91no—
400C22×D3—32no—
401C22×D3—32no—
402C22×D3—32no—
403C22×D3—32no—
404C22×D3—62no—
405C22×D3—61no—
406C22×D3—61no—
407C22×D3—184no—
408C22×D3—184no—
409C22×D3—184no—
410C22×D3—361no—
411C22×D3—362no—
412C3×Q8—41no—
413C3×Q8—42no—
414C3×Q8—44no—
415C3×Q8—42no—
416C3×Q8—44no—
417C3×D4—361no—
418C2×C3×C4—21no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—41no—
424C2×C3×C4—42no—
425C2×C3×C4—92no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—92no—
429C2×C3×C4—362no—
430C2×C3×C4C2×D3212yes—
431C2×C3×C4C2×D3211yescharacteristic
432C2×C3×C4C2×D3212yes—
433C2×C3×C4C2×D3212yes—
434C2×C3×C4C2×D3212yes—
435#67—184no—
436#67—184no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—32no—
443C2×Dic3—32no—
444C2×Dic3—32no—
445C2×Dic3—62no—
446C2×Dic3—62no—
447C2×Dic3—62no—
448C2×Dic3—62no—
449C2×Dic3—184no—
450C2×Dic3—361no—
451D12—122no—
452D12—122no—
453D12—184no—
454C4×D3—122no—
455C4×D3—122no—
456C4×D3—124no—
457C4×D3—184no—
458C4×D3—184no—
459C4×D3—362no—
460C2×C32—22no—
461C2×C32—22no—
462C2×C32—22no—
463C2×C32—22no—
464C2×C32—21no—
465C2×C32—42no—
466C2×C32—41no—
467C2×C32#100412yes—
468C2×C32#100411yescharacteristic
469C2×C32C2×#23912yes—
470C2×C32D3×#4112yes—
471C2×C32D3×#4112yes—
472C3⋊D3—24no—
473C3⋊D3—41no—
474C3⋊D3—41no—
475C3×D3—64no—
476C3×D3—64no—
477C3×D3—124no—
478C3×D3—122no—
479C3×D3—122no—
480C3×D3—241no—
481C3×D3—241no—
482C24—271no—
483C2×Q8#111311yescharacteristic, complemented
484C2×D4—91no—
485C2×D4—272no—
486C22×C4—92no—
487C22×C4—272no—
488C22×C4—271no—
489C4⋊C4—92no—
490C4⋊C4—92no—
491C4⋊C4—92no—
492C4⋊C4—272no—
493C22⋊C4—92no—
494C22⋊C4—92no—
495C22⋊C4—92no—
496C22⋊C4—92no—
497C22⋊C4—272no—
498C22⋊C4—272no—
499C22⋊C4—271no—
500C22⋊C4—271no—
501C42—91no—
502C42—272no—
503C22×C3—22no—
504C22×C3—21no—
505C22×C3—41no—
506C22×C3—94no—
507C22×C3—182no—
508C22×C3—182no—
509C22×C3—182no—
510C22×C3—182no—
511C22×C3—181no—
512C22×C3—181no—
513C22×C3C22×D3211yescharacteristic
514C22×C3C22×D3212yes—
515C2×D3—34no—
516C2×D3—34no—
517C2×D3—64no—
518C2×D3—62no—
519C2×D3—62no—
520C2×D3—62no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—122no—
529C2×D3—122no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—122no—
533C2×D3—122no—
534C2×D3—364no—
535C2×D3—368no—
536C3×C4—24no—
537C3×C4—22no—
538C3×C4—22no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—21no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4—44no—
547C3×C4—42no—
548C3×C4—182no—
549C3×C4—182no—
550C3×C4—182no—
551C3×C4—182no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—62no—
558Dic3—62no—
559Dic3—62no—
560Dic3—122no—
561Dic3—122no—
562Dic3—122no—
563Dic3—122no—
564C32—21no—
565C32—22no—
566C32—41no—
567C32#563712yescomplemented
568C32#567911yescharacteristic, complemented
569C23—92no—
570C23—91no—
571C23—91no—
572C23—544no—
573C23—1081no—
574Q8—41no—
575D4—361no—
576D4—544no—
577C2×C4—92no—
578C2×C4—92no—
579C2×C4—92no—
580C2×C4—92no—
581C2×C4—362no—
582C2×C4—544no—
583C2×C4—1081no—
584C2×C4C2×#111311yescharacteristic
585C2×C4C2×#111312yes—
586C2×C3—22no—
587C2×C3—22no—
588C2×C3—22no—
589C2×C3—22no—
590C2×C3—21no—
591C2×C3—42no—
592C2×C3—41no—
593C2×C3—184no—
594C2×C3—361no—
595C2×C3—361no—
596C2×C3D3×#23912yes—
597C2×C3D3×#23912yes—
598C2×C3#6431912yes—
599C2×C3#6433011yescharacteristic
600C2×C3#6433012yes—
601D3—64no—
602D3—64no—
603D3—124no—
604D3—122no—
605D3—122no—
606D3—241no—
607D3—241no—
608C22—94no—
609C22—182no—
610C22—182no—
611C22—182no—
612C22—182no—
613C22—181no—
614C22—181no—
615C22—1084no—
616C22C22×#111311yescharacteristic
617C4—22no—
618C4—21no—
619C4—182no—
620C4—182no—
621C4—182no—
622C4—182no—
623C3—22no—
624C3—21no—
625C3—41no—
626C3#1059543611yescharacteristic, complemented
627C3#1059543712yescomplemented
628C2—184no—
629C2—361no—
630C2—361no—
631C2#1172462612yes—
632C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radical#11723701C2
hypercenterC2×Q8#1113
hypocenterC33#551
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C3×#64155C2semidirect
4. #11723700C2semidirect
5. #11723690C2semidirect
6. #11723686C2semidirect
7. #11723684C2semidirect
8. #11723683C2semidirect
9. #11723675C2semidirect
13. C2×C33×Q8C22semidirect
18. C2×#6294C22semidirect
22. C3×#3752C22semidirect
23. C3×C4⋊(C3⋊Dic3)C22semidirect
24. C3×#3750C22semidirect
25. C3×#3750C22semidirect
26. C3×C4×C3⋊Dic3C22semidirect
27. #88815C22semidirect
31. #88814C22semidirect
32. #88813C22semidirect
33. #64156D3semidirect
34. #64155D3semidirect
83. C2×C32×Q8C2×D3semidirect
84. C2×C32×Q8C2×D3semidirect
103. #3752C2×D3semidirect
104. C4⋊(C3⋊Dic3)C2×D3semidirect
105. #3750C2×D3semidirect
106. #3750C2×D3semidirect
107. C4×C3⋊Dic3C2×D3semidirect
262. C2×C3×Q8D32semidirect
263. C2×C3×Q8D32semidirect
396. C33#551semidirect
483. C2×Q8#1113semidirect
567. C32#5637semidirect
568. C32#5679semidirect
626. C3#10595436semidirect
627. C3#10595437semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C3×#641552
#117237012
#117237002
#117236902
#117236862
#117236842
#117236832
#117236752
#105954373
#105954363