Finite Group Databasebeta

← #13168218

Subgroups — #13168218

Order 1728.

1156 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723685C211yesmaximal, characteristic
2#11723483C212yesmaximal, complemented
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×#63888C212yesmaximal, complemented
5C3×#63870C211yesmaximal, characteristic, complemented
6#11723037C211yesmaximal, characteristic, complemented
7#11723018C211yesmaximal, characteristic, complemented
8#11722984C212yesmaximal, complemented
9#11722938C211yesmaximal, characteristic
10#11722931C211yesmaximal, characteristic
11#11722908C212yesmaximal, complemented
12#10595407—31nomaximal
13#10594396—32nomaximal
14C2×C32×Dic6C2211yescharacteristic
15C2×#6295C2211yescharacteristic
16C2×#6294C2212yes—
17C2×#6293C2212yes—
18C2×Dic3×C3⋊D3C2212yes—
19C2×C3×#624C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C3×#3752C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×Dic3⋊C4C2212yes—
24#88815C2211yescharacteristic
25Dic3⋊(C3⋊Dic3)C2212yes—
26Dic3⋊(C3⋊Dic3)C2211yescharacteristic
27Dic3⋊(C3⋊Dic3)C2211yescharacteristic
28#88803C2212yes—
29Dic3×C3⋊Dic3C2211yescharacteristic
30Dic3×C3⋊Dic3C2211yescharacteristic
31C3×#3724C2211yescharacteristic
32C3×#3724C2212yes—
33C3×#3724C2211yescharacteristic
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic3⋊Dic3C2212yes—
36C3×#3722C2212yes—
37Q8×C3⋊Dic3—31no—
38#64106—32no—
39C3×#946—32no—
40C3×#896—32no—
41C3×#889—32no—
42#63919—31no—
43#63904—32no—
44#63902—31no—
45#63902—31no—
46C4×#625—31no—
47#63896—32no—
48#63895—32no—
49#63893—32no—
50#63893—32no—
51#63893—32no—
52#63893—62no—
53D3×C4⋊Dic3—32no—
54D3×C4⋊Dic3—32no—
55#63888D312yescomplemented
56#63883—32no—
57#63881—32no—
58#63880—32no—
59#63870—31no—
60#63870D311yescharacteristic, complemented
61#63845—32no—
62Dic3×Dic6—32no—
63Dic3×Dic6—61no—
64#63840—32no—
65C22×C3×C3⋊D3C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76C32×Dic6—22no—
77#6295Q814yescomplemented
78#6294—42no—
79#6293—42no—
80Dic3×C3⋊D3—42no—
81C3×#624—24no—
82#5690—91no—
83#5635—92no—
84C2×C32×Q8—31no—
85C2×C4×C3⋊D3—32no—
86C2×#634—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—21no—
90C2×C3×Dic6C2×D312yes—
91C2×#627—32no—
92C2×#627—32no—
93C2×#627—62no—
94C2×#626—32no—
95C2×#625—31no—
96C2×#624—32no—
97C2×#624C2×D312yes—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—62no—
103C32×C4⋊C4—32no—
104#3752—32no—
105#3752C2×D312yes—
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—32no—
109C4⋊(C3⋊Dic3)—31no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—61no—
112C4⋊(C3⋊Dic3)—62no—
113C4⋊(C3⋊Dic3)—121no—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—62no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—32no—
124C3×Dic3⋊C4—22no—
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⋊C4C2×D312yes—
132C3×Dic3⋊C4C2×D312yes—
133C3×C4×Dic3—32no—
134C3×C4×Dic3—61no—
135#3724—31no—
136#3724C2×D312yes—
137#3724C2×D311yescharacteristic
138#3724C2×D311yescharacteristic
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—61no—
145Dic3⋊Dic3—61no—
146Dic3⋊Dic3—62no—
147Dic3⋊Dic3C2×D312yes—
148Dic3⋊Dic3C2×D312yes—
149#3722C2×D312yes—
150(C2×D3)⋊Dic3—32no—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—62no—
153Dic32—32no—
154Dic32—32no—
155Dic32—61no—
156Dic32—61no—
157C22×C33C2411yescharacteristic
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3C2×Q814yes—
161C33×C4—21no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)—21no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3C2×Q812yes—
167C32×Dic3—21no—
168C32×Dic3—22no—
169C32×Dic3—22no—
170C32×Dic3#4112yes—
171C3×#125—91no—
172C3×C22⋊Q8—92no—
173#946—32no—
174Q8×Dic3—92no—
175Q8×Dic3—181no—
176C4×#67—92no—
177C2×C4⋊Dic3—92no—
178#897—92no—
179#897—92no—
180#897—92no—
181#897—182no—
182#896—32no—
183#896—92no—
184#893—91no—
185#893—91no—
186D3×C4⋊C4—92no—
187D3×C4⋊C4—92no—
188#890—91no—
189#889—32no—
190#889—92no—
191#888—91no—
192#888—92no—
193#67⋊C4—92no—
194#67⋊C4—92no—
195#878—92no—
196#878—92no—
197C4×D12—91no—
198C4×Dic6—91no—
199C22×C3⋊D3C22×D312yes—
200C22×C3×D3—32no—
201C22×C3×D3—32no—
202C22×C3×D3—62no—
203C32×Q8—62no—
204C2×C32×C4—21no—
205C2×C32×C4—22no—
206C2×C32×C4—31no—
207C2×C32×C4—32no—
208C2×C32×C4—32no—
209C2×C32×C4—31no—
210C2×C32×C4—41no—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D311yescharacteristic
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—61no—
218C2×C3⋊Dic3—61no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3—62no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3—121no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C2×C3⋊Dic3C22×D311yescharacteristic
226C4×C3⋊D3—122no—
227#634—122no—
228C2×C3×Dic3—21no—
229C2×C3×Dic3—21no—
230C2×C3×Dic3—22no—
231C2×C3×Dic3—22no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—61no—
238C2×C3×Dic3—62no—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C22×D312yes—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C3×C4×D3—64no—
246C3×C4×D3—64no—
247C3×Dic6—24no—
248C3×Dic6—24no—
249#627—122no—
250#627—122no—
251#627—124no—
252#626—38no—
253#625—34no—
254#624—24no—
255#624—122no—
256D3×Dic3—122no—
257D3×Dic3—122no—
258D3×Dic3—124no—
259D3×Dic3—122no—
260D3×Dic3—122no—
261#554—271noSylow
262C2×C33C22×Q811yescharacteristic
263C2×C33C2×#4111yescharacteristic
264C2×C33H(5,𝔽2)11yescharacteristic
265C3×C3⋊D3—24no—
266C2×C3×Q8—32no—
267C2×C3×Q8—61no—
268C22×C3×C4—92no—
269C2×#67—92no—
270C22×Dic3—92no—
271C22×Dic3—92no—
272C2×D12—91no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—92no—
276C2×C4×D3—92no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3—182no—
280C2×Dic6—92no—
281C2×Dic6—92no—
282C2×Dic6—92no—
283C2×Dic6—182no—
284C2×Dic6D3211yescharacteristic
285C3×C4⋊C4—32no—
286C3×C4⋊C4—32no—
287C3×C4⋊C4—62no—
288C3×C4⋊C4—92no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—92no—
291C3×C4⋊C4—91no—
292C3×C4⋊C4—92no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4—92no—
295C3×C42—91no—
296#217—92no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—32no—
301(C2×D3)⋊C4—62no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—182no—
305C4⋊Dic3—32no—
306C4⋊Dic3—32no—
307C4⋊Dic3—91no—
308C4⋊Dic3—92no—
309C4⋊Dic3—92no—
310C4⋊Dic3—92no—
311C4⋊Dic3—92no—
312C4⋊Dic3—91no—
313C4⋊Dic3—92no—
314C4⋊Dic3—182no—
315C4⋊Dic3—181no—
316C4⋊Dic3—181no—
317C4⋊Dic3—182no—
318C4⋊Dic3—181no—
319C4⋊Dic3—361no—
320Dic3⋊C4—32no—
321Dic3⋊C4—32no—
322Dic3⋊C4—32no—
323Dic3⋊C4—32no—
324Dic3⋊C4—32no—
325Dic3⋊C4—32no—
326Dic3⋊C4—92no—
327Dic3⋊C4—91no—
328Dic3⋊C4—91no—
329Dic3⋊C4—92no—
330Dic3⋊C4D3212yes—
331C4×Dic3—32no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—91no—
336C4×Dic3—91no—
337C4×Dic3—181no—
338C4×Dic3—181no—
339C22×C32—21no—
340C22×C32—22no—
341C22×C32—41no—
342C22×C32C23×D312yes—
343C22×C32C23×D311yescharacteristic
344C2×C3⋊D3—22no—
345C2×C3⋊D3—22no—
346C2×C3⋊D3D3×Q814yes—
347C2×C3×D3—34no—
348C2×C3×D3—34no—
349C2×C3×D3—64no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—122no—
355C2×C3×D3—122no—
356C32×C4—21no—
357C32×C4—22no—
358C32×C4—22no—
359C32×C4—32no—
360C32×C4—42no—
361C32×C4—42no—
362C32×C4—61no—
363C32×C4—62no—
364C32×C4—62no—
365C3⋊Dic3—21no—
366C3⋊Dic3—22no—
367C3⋊Dic3—61no—
368C3⋊Dic3—62no—
369C3⋊Dic3—61no—
370C3⋊Dic3—62no—
371C3⋊Dic3—122no—
372C3⋊Dic3—122no—
373C3⋊Dic3—121no—
374C3⋊Dic3—121no—
375C3⋊Dic3—241no—
376C3⋊Dic3—241no—
377C3⋊Dic3D3×Q812yes—
378C3×Dic3—24no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—22no—
384C3×Dic3—22no—
385C3×Dic3—22no—
386C3×Dic3—34no—
387C3×Dic3—42no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—122no—
393C3×Dic3—121no—
394C3×Dic3#23714yes—
395C4⋊Q8—271no—
396#125—91no—
397#125—271no—
398C22⋊Q8—92no—
399C22⋊Q8—272no—
400C22⋊Q8—272no—
401C4×Q8—271no—
402C4×D4—271no—
403C4×D4—271no—
404C4×D4—271no—
405C2×C4⋊C4—272no—
406C33#55411yescharacteristic, complemented, Sylow
407C23×C3—92no—
408C22×D3—32no—
409C22×D3—32no—
410C22×D3—62no—
411C3×Q8—64no—
412C3×Q8—64no—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—32no—
421C2×C3×C4—41no—
422C2×C3×C4—61no—
423C2×C3×C4—61no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4—92no—
429C2×C3×C4—184no—
430C2×C3×C4C2×D3212yes—
431C2×C3×C4C2×D3211yescharacteristic
432#67—98no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—61no—
438C2×Dic3—61no—
439C2×Dic3—62no—
440C2×Dic3—91no—
441C2×Dic3—92no—
442C2×Dic3—91no—
443C2×Dic3—92no—
444C2×Dic3—181no—
445C2×Dic3—181no—
446C2×Dic3—182no—
447C2×Dic3—182no—
448C2×Dic3—361no—
449C2×Dic3—361no—
450C2×Dic3—362no—
451C2×Dic3—362no—
452C2×Dic3C2×D3211yescharacteristic
453C2×Dic3C2×D3212yes—
454C2×Dic3C2×D3212yes—
455C2×Dic3C2×D3211yescharacteristic
456D12—94no—
457C4×D3—64no—
458C4×D3—64no—
459C4×D3—362no—
460C4×D3—362no—
461C4×D3—362no—
462C4×D3—362no—
463C4×D3—364no—
464Dic6—22no—
465Dic6—362no—
466Dic6—362no—
467Dic6—362no—
468Dic6—364no—
469C2×C32—22no—
470C2×C32—22no—
471C2×C32—21no—
472C2×C32—22no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32C2×#23712yes—
479C2×C32C2×D3×Q812yes—
480C2×C32C2×D3×Q811yescharacteristic
481C2×C32D3×#4111yescharacteristic
482C2×C32#100911yescharacteristic
483C2×C32#100912yes—
484C3⋊D3—24no—
485C3×D3—64no—
486C3×D3—64no—
487C3×D3—124no—
488C2×Q8—31no—
489C2×Q8—272no—
490C2×D4—271no—
491C22×C4—92no—
492C22×C4—272no—
493C22×C4—272no—
494C4⋊C4—32no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C4⋊C4—91no—
498C4⋊C4—91no—
499C4⋊C4—92no—
500C4⋊C4—271no—
501C4⋊C4—272no—
502C4⋊C4—271no—
503C4⋊C4—271no—
504C4⋊C4—271no—
505C22⋊C4—92no—
506C22⋊C4—92no—
507C22⋊C4—272no—
508C42—91no—
509C42—271no—
510C42—271no—
511C22×C3—21no—
512C22×C3—22no—
513C22×C3—41no—
514C22×C3—94no—
515C22×C3—182no—
516C22×C3—182no—
517C22×C3C22×D3212yes—
518C22×C3C22×D3211yescharacteristic
519C2×D3—34no—
520C2×D3—34no—
521C2×D3—64no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—122no—
527C2×D3—122no—
528C3×C4—22no—
529C3×C4—22no—
530C3×C4—21no—
531C3×C4—34no—
532C3×C4—42no—
533C3×C4—42no—
534C3×C4—62no—
535C3×C4—64no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—62no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—92no—
543C3×C4—122no—
544C3×C4—182no—
545C3×C4—181no—
546Dic3—22no—
547Dic3—22no—
548Dic3—21no—
549Dic3—34no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—122no—
555Dic3—121no—
556Dic3—182no—
557Dic3—182no—
558Dic3—181no—
559Dic3—181no—
560Dic3—362no—
561Dic3—362no—
562Dic3—361no—
563Dic3—361no—
564Dic3—721no—
565Dic3—721no—
566Dic3#380512yes—
567C32—21no—
568C32—22no—
569C32—41no—
570C32#563512yescomplemented
571C32#569011yescharacteristic, complemented
572C23—92no—
573Q8—62no—
574Q8—1082no—
575D4—274no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—92no—
583C2×C4—184no—
584C2×C4—271no—
585C2×C4—271no—
586C2×C4—1082no—
587C2×C4—1082no—
588C2×C4D3311yescharacteristic
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—184no—
599C2×C3#6430812yes—
600C2×C3D3×#23712yes—
601C2×C3D32×Q811yescharacteristic
602C2×C3D32×Q812yes—
603C2×C3#6432011yescharacteristic
604C2×C3C2×#380511yescharacteristic
605D3—64no—
606D3—64no—
607D3—124no—
608C22—94no—
609C22—182no—
610C22—182no—
611C22C2×D3311yescharacteristic
612C4—21no—
613C4—32no—
614C4—62no—
615C4—62no—
616C4—61no—
617C4—92no—
618C4—181no—
619C4—182no—
620C4—541no—
621C4—541no—
622C3—22no—
623C3—21no—
624C3—41no—
625C3#1059439612yescomplemented
626C3#1059540711yescharacteristic, complemented
627C2—184no—
628C2#1172437811yescharacteristic
629C2#1172439011yescharacteristic
630C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×#6295C22
hypercenterC2×C4D33
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723483C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
4. C3×#63888C2semidirect
5. C3×#63870C2semidirect
6. #11723037C2semidirect
7. #11723018C2semidirect
8. #11722984C2semidirect
11. #11722908C2semidirect
55. #63888D3semidirect
60. #63870D3semidirect
77. #6295Q8semidirect
406. C33#554semidirect
570. C32#5635semidirect
571. C32#5690semidirect
625. C3#10594396semidirect
626. C3#10595407semidirect

Maximal subgroups

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

subgroupindex
#117236852
#117234832
Dic6×C3⋊Dic32
C3×#638882
C3×#638702
#117230372
#117230182
#117229842
#117229382
#117229312
#117229082
#105954073
#105943963