Finite Group Databasebeta

← #13168467

Subgroups — #13168467

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
1#11723679C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3#11723409C212yesmaximal, complemented
4C3×#63884C212yesmaximal
5C3×#63838C211yesmaximal, characteristic, complemented
6#11723020C212yesmaximal, complemented
7#11722976C211yesmaximal, characteristic, complemented
8#11722976C211yesmaximal, characteristic, complemented
9#11722917C211yesmaximal, characteristic, complemented
10#11722913C212yesmaximal, complemented
11D3×#3724C211yesmaximal, characteristic, complemented
12#10595120—31nomaximal
13#10594420—32nomaximal
14C2×C32×D12C2211yescharacteristic
15C2×D3×C3⋊Dic3C2211yescharacteristic
16C2×D3×C3⋊Dic3C2211yescharacteristic
17C2×C3×#627C2211yescharacteristic
18C2×C3×D3×Dic3C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×#3750C2212yes—
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C32×Dic3⋊C4C2212yes—
23#88815C2212yes—
24#88815C2211yescharacteristic
25#88815C2211yescharacteristic
26#88807C2212yes—
27#88807C2212yes—
28#88806C2212yes—
29(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
30(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
31Dic3×C3⋊Dic3C2212yes—
32C3×#3724C2211yescharacteristic
33C3×(C2×D3)⋊Dic3C2212yes—
34C3×(C2×D3)⋊Dic3C2212yes—
35C3×(C2×D3)⋊Dic3C2212yes—
36C3×Dic32C2212yes—
37D4×C3⋊Dic3—31no—
38#64098—32no—
39C3×#935—32no—
40C3×#894—32no—
41C3×#887—32no—
42#63982—31no—
43#63982—31no—
44C2×#3724—31no—
45#63957—31no—
46#63956—32no—
47#63955—32no—
48Dic3×D12—32no—
49Dic3×D12—61no—
50#63887—32no—
51#63887—32no—
52#63884D312yes—
53#63878—32no—
54#63877—32no—
55D3×Dic3⋊C4—32no—
56#63870—31no—
57#63867—32no—
58#63859—32no—
59#63859—32no—
60#63855—32no—
61#63845—32no—
62#63845—32no—
63#63845—62no—
64#63838D311yescharacteristic, complemented
65C22×C32×D3C2311yescharacteristic
66C22×C32×D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2312yes—
76C32×D12—22no—
77D3×C3⋊Dic3—22no—
78D3×C3⋊Dic3—22no—
79C3×#627—22no—
80C3×D3×Dic3—24no—
81#5650—92no—
82#5581—91no—
83C2×C32×D4—31no—
84C22×C3⋊Dic3—31no—
85C22×C3⋊Dic3—31no—
86C22×C3×Dic3—32no—
87C2×C3×D12—21no—
88C2×C3×D12C2×D312yes—
89C2×C3×C4×D3—32no—
90C2×C3×Dic6—32no—
91C2×#627C2×D311yescharacteristic
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—61no—
95C2×D3×Dic3—61no—
96C2×D3×Dic3C2×D312yes—
97C32×C4⋊C4—32no—
98#3757—31no—
99#3757—31no—
100C4⋊(C3⋊Dic3)—32no—
101C4⋊(C3⋊Dic3)—31no—
102C4⋊(C3⋊Dic3)—62no—
103C4⋊(C3⋊Dic3)—61no—
104C4⋊(C3⋊Dic3)—121no—
105#3750—32no—
106#3750—32no—
107#3750—32no—
108#3750C2×D312yes—
109C4×C3⋊Dic3—32no—
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×Dic3⋊C4—22no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—62no—
122C3×Dic3⋊C4C2×D312yes—
123C3×Dic3⋊C4C2×D312yes—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—61no—
128#3724—32no—
129#3724—31no—
130#3724—31no—
131#3724—32no—
132#3724—32no—
133#3724—62no—
134#3724C2×D311yescharacteristic
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—62no—
144Dic3⋊Dic3—62no—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—61no—
148(C2×D3)⋊Dic3—61no—
149(C2×D3)⋊Dic3C2×D312yes—
150(C2×D3)⋊Dic3C2×D312yes—
151(C2×D3)⋊Dic3C2×D312yes—
152Dic32—32no—
153Dic32—32no—
154Dic32—62no—
155Dic32C2×D312yes—
156C22×C33C2411yescharacteristic
157C2×C32×D3—21no—
158C2×C32×D3—21no—
159C2×C32×D3—21no—
160C2×C32×D3—21no—
161C2×C32×D3—21no—
162C2×C32×D3#4112yes—
163C33×C4—21no—
164C3⋊(C3⋊Dic3)—21no—
165C3⋊(C3⋊Dic3)—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—21no—
169C3×C3⋊Dic3#4112yes—
170C32×Dic3—22no—
171C32×Dic3—22no—
172C3×#123—92no—
173C3×C4×Q8—91no—
174#946—91no—
175#946—91no—
176#935—32no—
177D4×Dic3—92no—
178D4×Dic3—181no—
179#926—92no—
180C2×Dic3⋊C4—92no—
181#894—32no—
182#892—91no—
183D3×C4⋊C4—91no—
184#890—92no—
185#889—92no—
186#889—92no—
187#889—92no—
188#889—182no—
189#887—32no—
190#879—92no—
191#879—92no—
192#878—92no—
193#878—92no—
194#877—92no—
195#877—92no—
196#876—92no—
197C4×D12—91no—
198#872—92no—
199#870—91no—
200C23×C32—31no—
201C23×C32—31no—
202C22×C3×D3—21no—
203C22×C3×D3—21no—
204C22×C3×D3C22×D312yes—
205C22×C3×D3C22×D312yes—
206C32×D4—62no—
207C2×C32×C4—21no—
208C2×C32×C4—22no—
209C2×C32×C4—32no—
210C2×C32×C4—32no—
211C2×C32×C4—41no—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C22×D311yescharacteristic
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—32no—
216C2×C3⋊Dic3—32no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—61no—
220C2×C3⋊Dic3—61no—
221C2×C3⋊Dic3—62no—
222C2×C3⋊Dic3—62no—
223C2×C3⋊Dic3—62no—
224C2×C3⋊Dic3—121no—
225C2×C3⋊Dic3—121no—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D311yescharacteristic
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—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—62no—
239C2×C3×Dic3—61no—
240C2×C3×Dic3—61no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—64no—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C2×C3×Dic3C22×D312yes—
246C2×C3×Dic3C22×D312yes—
247C3×D12—24no—
248C3×D12—24no—
249C3×C4×D3—64no—
250C3×Dic6—64no—
251#627—22no—
252D3×Dic3—24no—
253D3×Dic3—64no—
254D3×Dic3—64no—
255D3×Dic3—122no—
256D3×Dic3—122no—
257#552—271noSylow
258C2×C33C2×#4111yescharacteristic
259C2×C33C2×#4111yescharacteristic
260C2×C332−1+411yescharacteristic
261C32×D3—22no—
262C32×D3—41no—
263C2×C3×Q8—91no—
264C2×C3×D4—32no—
265C2×C3×D4—61no—
266C22×C3×C4—92no—
267C22×Dic3—32no—
268C22×Dic3—92no—
269C22×Dic3—92no—
270C22×Dic3—181no—
271C22×Dic3—181no—
272C2×D12D3211yescharacteristic
273C2×C4×D3—32no—
274C2×C4×D3—91no—
275C2×C4×D3—91no—
276C2×Dic6—32no—
277C3×C4⋊C4—32no—
278C3×C4⋊C4—32no—
279C3×C4⋊C4—62no—
280C3×C4⋊C4—91no—
281C3×C4⋊C4—92no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C42—92no—
286C3×C42—91no—
287#217—32no—
288#217—32no—
289#217—32no—
290#217—92no—
291#217—92no—
292#217—181no—
293#217—181no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—32no—
296(C2×D3)⋊C4—32no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—91no—
299C4⋊Dic3—91no—
300C4⋊Dic3—91no—
301C4⋊Dic3—92no—
302C4⋊Dic3—91no—
303C4⋊Dic3—92no—
304C4⋊Dic3—92no—
305C4⋊Dic3—92no—
306C4⋊Dic3—182no—
307C4⋊Dic3—181no—
308C4⋊Dic3—361no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—62no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—182no—
325Dic3⋊C4—182no—
326Dic3⋊C4—182no—
327Dic3⋊C4D3212yes—
328C4×Dic3—32no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—61no—
332C4×Dic3—92no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—182no—
336C22×C32—22no—
337C22×C32—21no—
338C22×C32—32no—
339C22×C32—41no—
340C22×C32—61no—
341C22×C32—61no—
342C22×C32—61no—
343C22×C32—61no—
344C22×C32—61no—
345C22×C32C23×D311yescharacteristic
346C22×C32C23×D312yes—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—22no—
353C2×C3×D3—22no—
354C2×C3×D3—41no—
355C2×C3×D3—41no—
356C2×C3×D3—41no—
357C2×C3×D3—41no—
358C2×C3×D3#23714yes—
359C32×C4—21no—
360C32×C4—22no—
361C32×C4—22no—
362C32×C4—42no—
363C32×C4—42no—
364C32×C4—62no—
365C32×C4—62no—
366C3⋊Dic3—22no—
367C3⋊Dic3—22no—
368C3⋊Dic3—21no—
369C3⋊Dic3—61no—
370C3⋊Dic3—61no—
371C3⋊Dic3—62no—
372C3⋊Dic3—62no—
373C3⋊Dic3—121no—
374C3⋊Dic3—121no—
375C3⋊Dic3—122no—
376C3⋊Dic3—122no—
377C3⋊Dic3—241no—
378C3⋊Dic3—241no—
379C3⋊Dic3#23512yes—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—22no—
384C3×Dic3—34no—
385C3×Dic3—42no—
386C3×Dic3—42no—
387C3×Dic3—62no—
388C3×Dic3—64no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—62no—
393C3×Dic3—62no—
394C3×Dic3—121no—
395C3×Dic3—122no—
396#126—272no—
397#125—271no—
398#125—272no—
399#123—92no—
400C22⋊Q8—271no—
401C22⋊Q8—271no—
402C4×Q8—91no—
403C4×D4—271no—
404#117—272no—
405#117—271no—
406C2×C4⋊C4—271no—
407C33#55211yescharacteristic, complemented, Sylow
408C23×C3—32no—
409C23×C3—32no—
410C23×C3—61no—
411C23×C3—61no—
412C22×D3C2×D3211yescharacteristic
413C22×D3C2×D3211yescharacteristic
414C3×Q8—182no—
415C3×D4—64no—
416C3×D4—64no—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—32no—
420C2×C3×C4—32no—
421C2×C3×C4—32no—
422C2×C3×C4—32no—
423C2×C3×C4—41no—
424C2×C3×C4—62no—
425C2×C3×C4—62no—
426C2×C3×C4—91no—
427C2×C3×C4—92no—
428C2×C3×C4—92no—
429C2×C3×C4—91no—
430C2×C3×C4—184no—
431C2×C3×C4C2×D3212yes—
432C2×C3×C4C2×D3211yescharacteristic
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—61no—
440C2×Dic3—62no—
441C2×Dic3—61no—
442C2×Dic3—62no—
443C2×Dic3—64no—
444C2×Dic3—91no—
445C2×Dic3—92no—
446C2×Dic3—91no—
447C2×Dic3—92no—
448C2×Dic3—182no—
449C2×Dic3—182no—
450C2×Dic3—181no—
451C2×Dic3—181no—
452C2×Dic3—184no—
453C2×Dic3—184no—
454C2×Dic3—361no—
455C2×Dic3—361no—
456C2×Dic3—362no—
457C2×Dic3—362no—
458C2×Dic3C2×D3212yes—
459C2×Dic3C2×D3212yes—
460D12—22no—
461C4×D3—64no—
462C4×D3—182no—
463C4×D3—182no—
464Dic6—64no—
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×C32—62no—
475C2×C32—121no—
476C2×C32C2×#23511yescharacteristic
477C2×C32C2×#23712yes—
478C2×C32#100711yescharacteristic
479C2×C32D3×#4112yes—
480C2×C32D3×#4111yescharacteristic
481C2×C32#101012yes—
482C3×D3—24no—
483C3×D3—42no—
484C3×D3—42no—
485C3×D3—42no—
486C2×Q8—91no—
487C2×D4—31no—
488C22×C4—92no—
489C22×C4—271no—
490C22×C4—271no—
491C4⋊C4—32no—
492C4⋊C4—91no—
493C4⋊C4—92no—
494C4⋊C4—272no—
495C4⋊C4—272no—
496C4⋊C4—272no—
497C4⋊C4—271no—
498C4⋊C4—271no—
499C4⋊C4—272no—
500C4⋊C4—271no—
501C22⋊C4—92no—
502C22⋊C4—92no—
503C22⋊C4—92no—
504C22⋊C4—271no—
505C22⋊C4—271no—
506C42—92no—
507C42—91no—
508C42—272no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—34no—
512C22×C3—41no—
513C22×C3—62no—
514C22×C3—62no—
515C22×C3—62no—
516C22×C3—62no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—121no—
521C22×C3—121no—
522C22×C3—121no—
523C22×C3—121no—
524C22×C3C22×D3212yes—
525C22×C3C22×D3211yescharacteristic
526C2×D3—21no—
527C2×D3—21no—
528C2×D3—21no—
529C2×D3—21no—
530C2×D3—21no—
531C2×D3#380512yes—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—21no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—62no—
540C3×C4—62no—
541C3×C4—92no—
542C3×C4—122no—
543C3×C4—122no—
544C3×C4—181no—
545C3×C4—182no—
546C3×C4—182no—
547Dic3—22no—
548Dic3—22no—
549Dic3—34no—
550Dic3—62no—
551Dic3—64no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—121no—
558Dic3—122no—
559Dic3—182no—
560Dic3—182no—
561Dic3—181no—
562Dic3—181no—
563Dic3—361no—
564Dic3—361no—
565Dic3—362no—
566Dic3—362no—
567Dic3—721no—
568Dic3—721no—
569C32—21no—
570C32—22no—
571C32—41no—
572C32#558111yescharacteristic, complemented
573C32#565012yescomplemented
574C23—31no—
575C23—31no—
576Q8—182no—
577D4—62no—
578C2×C4—32no—
579C2×C4—32no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—92no—
583C2×C4—92no—
584C2×C4—184no—
585C2×C4—271no—
586C2×C4—271no—
587C2×C4—542no—
588C2×C4—542no—
589C2×C4D3311yescharacteristic
590C2×C3—21no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—21no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—64no—
600C2×C3—122no—
601C2×C3—122no—
602C2×C3—122no—
603C2×C3D3×#23512yes—
604C2×C3#6430911yescharacteristic
605C2×C3D3×#23712yes—
606C2×C3#6431211yescharacteristic
607C2×C3#6431512yes—
608C2×C3C2×#380511yescharacteristic
609D3—22no—
610D3—41no—
611C22—32no—
612C22—61no—
613C22—61no—
614C22—61no—
615C22—61no—
616C22—61no—
617C22C2×D3311yescharacteristic
618C4—21no—
619C4—62no—
620C4—62no—
621C4—92no—
622C4—182no—
623C4—181no—
624C4—182no—
625C4—541no—
626C4—541no—
627C3—21no—
628C3—22no—
629C3—41no—
630C3#1059442012yescomplemented
631C3#1059512011yescharacteristic, complemented
632C2—62no—
633C2—121no—
634C2#1172438811yescharacteristic
635C2#1172439411yescharacteristic
636C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C32×D12C22
hypercenterC2×C4D33
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723679C2semidirect
3. #11723409C2semidirect
5. C3×#63838C2semidirect
6. #11723020C2semidirect
7. #11722976C2semidirect
8. #11722976C2semidirect
9. #11722917C2semidirect
10. #11722913C2semidirect
11. D3×#3724C2semidirect
64. #63838D3semidirect
407. C33#552semidirect
572. C32#5581semidirect
573. C32#5650semidirect
630. C3#10594420semidirect
631. C3#10595120semidirect

Maximal subgroups

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

subgroupindex
#117236792
D12×C3⋊Dic32
#117234092
C3×#638842
C3×#638382
#117230202
#117229762
#117229172
#117229132
D3×#37242
#105951203
#105944203