Finite Group Databasebeta

← #13178321

Subgroups — #13178321

Order 1728.

1088 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64102C211yesmaximal, characteristic
2C3×#64098C212yesmaximal, complemented
3#11723689C211yesmaximal, characteristic, complemented
4#11723686C212yesmaximal
5#11723683C212yesmaximal, complemented
6#11723679C211yesmaximal, characteristic, complemented
7#11723675C212yesmaximal, complemented
8#11723673C211yesmaximal, characteristic, complemented
9#11723671C211yesmaximal, characteristic, complemented
10#11723671C211yesmaximal, characteristic, complemented
11#11723665C211yesmaximal, characteristic, complemented
12#10594476—31nomaximal
13#10594410—32nomaximal
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×#6296C2211yescharacteristic
18C2×#6294C2212yes—
19C33×C4⋊C4C2211yescharacteristic
20C3×#3752C2211yescharacteristic
21C3×#3752C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29#88815C2211yescharacteristic
30#88815C2211yescharacteristic
31#88815C2211yescharacteristic
32#88815C2212yes—
33#88814C2212yes—
34#88814C2212yes—
35#88814C2212yes—
36#88813C2212yes—
37#64102D311yescharacteristic
38#64098D312yescomplemented
39C3×#893—32no—
40C3×#893—61no—
41C3×#889—32no—
42C3×#889—32no—
43C3×#889—62no—
44#63900—31no—
45#63897—32no—
46D3×C4⋊Dic3—32no—
47#63884—32no—
48#63884—32no—
49#63878—32no—
50#63877—32no—
51#63874—31no—
52#63870—31no—
53#63867—32no—
54#63865—31no—
55#63865—31no—
56#63861—32no—
57#63858—32no—
58#63855—32no—
59#63851—32no—
60#63851—32no—
61#63848—32no—
62#63838—31no—
63#63837—32no—
64#63836—32no—
65C22×C3×C3⋊D3C2311yescharacteristic
66C22×C3×C3⋊D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4⋊(C3⋊D3)—22no—
77C3×C4×C3⋊D3—22no—
78C3×C4×C3⋊D3—22no—
79#6296—22no—
80#6294—24no—
81#5693—92no—
82#5541—91no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C3×D12—32no—
87C2×C3×D12—61no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—61no—
91C2×C3×C4×D3—61no—
92C2×#627—32no—
93C2×#627—31no—
94C2×#624—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C32×C4⋊C4—22no—
98C32×C4⋊C4—21no—
99C32×C4⋊C4—41no—
100C32×C4⋊C4C2×D311yescharacteristic
101C32×C4⋊C4C2×D312yes—
102#3752C2×D311yescharacteristic
103#3752C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D312yes—
105#3750C2×D312yes—
106#3750C2×D312yes—
107#3750C2×D312yes—
108#3750C2×D312yes—
109C4×C3⋊Dic3C2×D311yescharacteristic
110C4×C3⋊Dic3C2×D312yes—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—61no—
114C3×(C2×D3)⋊C4—61no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—62no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—62no—
127C3×Dic3⋊C4—62no—
128C3×Dic3⋊C4—62no—
129C3×Dic3⋊C4—62no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—62no—
134C3×C4×Dic3—61no—
135#3724—31no—
136#3724—32no—
137#3724—31no—
138#3724—31no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144#3722—32no—
145#3722—32no—
146#3722—32no—
147(C2×D3)⋊Dic3—32no—
148(C2×D3)⋊Dic3—32no—
149(C2×D3)⋊Dic3—32no—
150(C2×D3)⋊Dic3—32no—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—32no—
153Dic32—32no—
154Dic32—32no—
155Dic32—32no—
156C22×C33C2411yescharacteristic
157C2×C3×C3⋊D3—21no—
158C2×C3×C3⋊D3—21no—
159C2×C3×C3⋊D3—21no—
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3—21no—
162C2×C3×C3⋊D3#4112yes—
163C33×C4—21no—
164C33×C4—21no—
165C33×C4—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—22no—
170C3×C3⋊Dic3—21no—
171C3×C3⋊Dic3#4112yes—
172C3×#125—92no—
173C3×C4×D4—91no—
174#935—92no—
175#926—91no—
176C2×C4⋊Dic3—91no—
177#924—91no—
178#924—91no—
179#898—92no—
180#898—92no—
181#896—92no—
182#896—92no—
183#894—92no—
184#894—92no—
185#893—32no—
186#893—61no—
187#892—92no—
188D3×C4⋊C4—92no—
189#889—32no—
190#889—32no—
191#889—62no—
192#889—92no—
193#887—92no—
194#879—92no—
195#877—92no—
196#872—92no—
197#872—92no—
198#870—91no—
199C4×Dic6—91no—
200C22×C3⋊D3C22×D311yescharacteristic
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3×D3—32no—
203C22×C3×D3—32no—
204C22×C3×D3—61no—
205C22×C3×D3—61no—
206C2×C32×C4—21no—
207C2×C32×C4—22no—
208C2×C32×C4—22no—
209C2×C32×C4—22no—
210C2×C32×C4—21no—
211C2×C32×C4—21no—
212C2×C32×C4—41no—
213C2×C32×C4—41no—
214C2×C32×C4—41no—
215C2×C32×C4C22×D311yescharacteristic
216C2×C32×C4C22×D312yes—
217C2×C32×C4C22×D312yes—
218C2×C32×C4C22×D311yescharacteristic
219C2×C32×C4C22×D311yescharacteristic
220C2×C32×C4C22×D312yes—
221C2×C3⋊Dic3C22×D311yescharacteristic
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C4⋊(C3⋊D3)—22no—
228C4×C3⋊D3—22no—
229C4×C3⋊D3—22no—
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—32no—
240C2×C3×Dic3—62no—
241C2×C3×Dic3—61no—
242C2×C3×Dic3—62no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—62no—
246C3×D12—64no—
247C3×D12—64no—
248C3×C4×D3—64no—
249C3×C4×D3—64no—
250C3×C4×D3—122no—
251C3×C4×D3—122no—
252#627—64no—
253#627—62no—
254#624—64no—
255D3×Dic3—64no—
256D3×Dic3—64no—
257#552—271noSylow
258C2×C33C2×#4111yescharacteristic
259C2×C33C2×#4111yescharacteristic
260C2×C332−1+411yescharacteristic
261C3×C3⋊D3—22no—
262C3×C3⋊D3—41no—
263C2×C3×D4—91no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C22×Dic3—92no—
267C2×D12—32no—
268C2×D12—61no—
269C2×C4×D3—32no—
270C2×C4×D3—32no—
271C2×C4×D3—61no—
272C2×C4×D3—61no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×Dic6—91no—
276C2×Dic6—92no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—21no—
279C3×C4⋊C4—41no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—92no—
282C3×C4⋊C4—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—92no—
285C3×C4⋊C4D3211yescharacteristic
286C3×C4⋊C4D3212yes—
287C3×C22⋊C4—91no—
288C3×C22⋊C4—91no—
289C3×C42—91no—
290C3×C42—92no—
291#217—92no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—61no—
297(C2×D3)⋊C4—61no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304C4⋊Dic3—32no—
305C4⋊Dic3—32no—
306C4⋊Dic3—62no—
307C4⋊Dic3—91no—
308C4⋊Dic3—91no—
309C4⋊Dic3—91no—
310C4⋊Dic3—92no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—62no—
320Dic3⋊C4—62no—
321Dic3⋊C4—62no—
322Dic3⋊C4—62no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328C4×Dic3—32no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—62no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—92no—
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—41no—
339C22×C32C23×D311yescharacteristic
340C22×C32C23×D312yes—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—21no—
343C2×C3⋊D3—21no—
344C2×C3⋊D3—21no—
345C2×C3⋊D3—21no—
346C2×C3⋊D3#23512yes—
347C2×C3×D3—34no—
348C2×C3×D3—62no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C2×C3×D3—62no—
353C2×C3×D3—62no—
354C2×C3×D3—62no—
355C2×C3×D3—121no—
356C2×C3×D3—121no—
357C2×C3×D3—121no—
358C2×C3×D3—121no—
359C32×C4—21no—
360C32×C4—22no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—22no—
364C32×C4—22no—
365C32×C4—22no—
366C32×C4—42no—
367C32×C4—42no—
368C32×C4—41no—
369C32×C4—41no—
370C32×C4—42no—
371C32×C4—42no—
372C32×C4—81no—
373C32×C4—81no—
374C3⋊Dic3—22no—
375C3⋊Dic3—22no—
376C3⋊Dic3—22no—
377C3⋊Dic3—22no—
378C3⋊Dic3—21no—
379C3⋊Dic3#23712yes—
380C3×Dic3—34no—
381C3×Dic3—62no—
382C3×Dic3—64no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—122no—
393C3×Dic3—122no—
394C3×Dic3—122no—
395C3×Dic3—121no—
396#126—272no—
397#125—92no—
398#125—271no—
399#123—272no—
400C22⋊Q8—271no—
401C22⋊Q8—271no—
402C4×Q8—271no—
403C4×D4—91no—
404#117—271no—
405#117—272no—
406C2×C4⋊C4—271no—
407C33#55211yescharacteristic, complemented, Sylow
408C23×C3—91no—
409C23×C3—91no—
410C22×D3—32no—
411C22×D3—32no—
412C22×D3—61no—
413C22×D3—61no—
414C3×D4—182no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—92no—
429C2×C3×C4—92no—
430C2×C3×C4—182no—
431C2×C3×C4—182no—
432C2×C3×C4C2×D3211yescharacteristic
433C2×C3×C4C2×D3212yes—
434C2×C3×C4C2×D3212yes—
435C2×C3×C4C2×D3211yescharacteristic
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3212yes—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—32no—
441C2×Dic3—32no—
442C2×Dic3—32no—
443C2×Dic3—32no—
444C2×Dic3—32no—
445C2×Dic3—32no—
446C2×Dic3—32no—
447C2×Dic3—32no—
448C2×Dic3—62no—
449C2×Dic3—61no—
450C2×Dic3—62no—
451C2×Dic3—61no—
452C2×Dic3—62no—
453C2×Dic3—62no—
454C2×Dic3—184no—
455D12—64no—
456D12—64no—
457C4×D3—64no—
458C4×D3—64no—
459C4×D3—122no—
460C4×D3—122no—
461C4×D3—184no—
462C4×D3—184no—
463Dic6—184no—
464Dic6—182no—
465C2×C32—22no—
466C2×C32—21no—
467C2×C32—22no—
468C2×C32—21no—
469C2×C32—22no—
470C2×C32—21no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32—41no—
474C2×C32C2×#23511yescharacteristic
475C2×C32C2×#23711yescharacteristic
476C2×C32#100712yes—
477C2×C32D3×#4112yes—
478C2×C32D3×#4112yes—
479C2×C32#101011yescharacteristic
480C3⋊D3—22no—
481C3⋊D3—41no—
482C3×D3—64no—
483C3×D3—122no—
484C3×D3—122no—
485C3×D3—122no—
486C2×Q8—271no—
487C2×D4—91no—
488C22×C4—91no—
489C22×C4—91no—
490C22×C4—272no—
491C4⋊C4—92no—
492C4⋊C4—92no—
493C4⋊C4—92no—
494C4⋊C4—92no—
495C4⋊C4—92no—
496C4⋊C4—271no—
497C4⋊C4—271no—
498C4⋊C4—271no—
499C4⋊C4—272no—
500C4⋊C4#111311yescharacteristic
501C22⋊C4—91no—
502C22⋊C4—91no—
503C22⋊C4—272no—
504C22⋊C4—272no—
505C22⋊C4—272no—
506C42—92no—
507C42—91no—
508C42—272no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—41no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—181no—
515C22×C3—181no—
516C22×C3—181no—
517C22×C3—181no—
518C22×C3C22×D3212yes—
519C22×C3C22×D3211yescharacteristic
520C2×D3—34no—
521C2×D3—62no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—62no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—121no—
529C2×D3—121no—
530C2×D3—121no—
531C2×D3—121no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—21no—
535C3×C4—22no—
536C3×C4—22no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—42no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4—92no—
548C3×C4—182no—
549C3×C4—182no—
550C3×C4—182no—
551C3×C4—182no—
552C3×C4—181no—
553Dic3—34no—
554Dic3—62no—
555Dic3—64no—
556Dic3—62no—
557Dic3—62no—
558Dic3—62no—
559Dic3—62no—
560Dic3—62no—
561Dic3—62no—
562Dic3—62no—
563Dic3—62no—
564Dic3—62no—
565Dic3—122no—
566Dic3—122no—
567Dic3—122no—
568Dic3—121no—
569C32—22no—
570C32—21no—
571C32—41no—
572C32#554111yescharacteristic, complemented
573C32#569312yescomplemented
574C23—91no—
575C23—91no—
576Q8—542no—
577D4—182no—
578C2×C4—91no—
579C2×C4—91no—
580C2×C4—92no—
581C2×C4—92no—
582C2×C4—92no—
583C2×C4—92no—
584C2×C4—182no—
585C2×C4—182no—
586C2×C4—544no—
587C2×C4C2×#111311yescharacteristic
588C2×C4C2×#111311yescharacteristic
589C2×C4C2×#111311yescharacteristic
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—22no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—182no—
600C2×C3—361no—
601C2×C3D3×#23512yes—
602C2×C3#6430611yescharacteristic
603C2×C3D3×#23712yes—
604C2×C3#6431211yescharacteristic
605C2×C3#6431512yes—
606C2×C3#6431611yescharacteristic
607D3—64no—
608D3—122no—
609D3—122no—
610D3—122no—
611C22—92no—
612C22—181no—
613C22—181no—
614C22—181no—
615C22—181no—
616C22—181no—
617C22C22×#111311yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—92no—
622C4—182no—
623C4—182no—
624C4—182no—
625C4—182no—
626C4—181no—
627C3—21no—
628C3—22no—
629C3—41no—
630C3#1059441012yescomplemented
631C3#1059447611yescharacteristic, complemented
632C2—182no—
633C2—361no—
634C2#1172461811yescharacteristic
635C2#1172462211yescharacteristic
636C2#1172462411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4#1113
hypocenterC33#552
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
3. #11723689C2semidirect
5. #11723683C2semidirect
6. #11723679C2semidirect
7. #11723675C2semidirect
8. #11723673C2semidirect
9. #11723671C2semidirect
10. #11723671C2semidirect
11. #11723665C2semidirect
38. #64098D3semidirect
407. C33#552semidirect
572. C32#5541semidirect
573. C32#5693semidirect
630. C3#10594410semidirect
631. C3#10594476semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#640982
#117236892
#117236862
#117236832
#117236792
#117236752
#117236732
#117236712
#117236652
#105944763
#105944103