Finite Group Databasebeta

← #13169436

Subgroups — #13169436

Order 1728.

1144 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723689C211yesmaximal, characteristic, complemented
2#11723507C211yesmaximal, characteristic
3#11723402C212yesmaximal, complemented
4C3×#63893C212yesmaximal, complemented
5C3×#63865C211yesmaximal, characteristic, complemented
6#11723032C211yesmaximal, characteristic, complemented
7#11722979C212yesmaximal, complemented
8#11722976C211yesmaximal, characteristic, complemented
9#11722941C211yesmaximal, characteristic, complemented
10#11722936C212yesmaximal
11D3×#3724C211yesmaximal, characteristic, complemented
12#10594535—32nomaximal
13#10594168—31nomaximal
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6294C2212yes—
16C2×#6290C2211yescharacteristic
17C2×D3×C3⋊Dic3C2211yescharacteristic
18C2×C3×#627C2211yescharacteristic
19C2×C3×#627C2212yes—
20C2×C3×D3×Dic3C2212yes—
21#88857C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C32×(C2×D3)⋊C4C2211yescharacteristic
24C32×Dic3⋊C4C2212yes—
25#88815C2211yescharacteristic
26#88815C2211yescharacteristic
27#88807C2212yes—
28#88806C2212yes—
29#88806C2212yes—
30#88803C2211yescharacteristic
31Dic3×C3⋊Dic3C2211yescharacteristic
32Dic3×C3⋊Dic3C2212yes—
33C3×#3724C2211yescharacteristic
34C3×Dic3⋊Dic3C2212yes—
35C3×#3722C2212yes—
36C3×(C2×D3)⋊Dic3C2212yes—
37#64098—32no—
38#64090—31no—
39C3×#897—32no—
40C3×#896—32no—
41C3×#878—32no—
42#63982—31no—
43C2×#3724—31no—
44#63952—32no—
45#63902—32no—
46#63900—31no—
47#63899—32no—
48#63893D312yescomplemented
49D3×C4⋊Dic3—32no—
50#63888—32no—
51#63887—32no—
52#63887—32no—
53#63879—32no—
54#63879—32no—
55#63879—61no—
56D3×Dic3⋊C4—32no—
57#63873—31no—
58#63865D311yescharacteristic, complemented
59#63847—32no—
60#63847—32no—
61#63847—62no—
62#63843—32no—
63#63839—32no—
64#63838—31no—
65C22×C3×C3⋊D3C2311yescharacteristic
66C22×C32×D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C2×C32×Dic3C2312yes—
77#6294—42no—
78#6290Q814yescomplemented
79D3×C3⋊Dic3—22no—
80C3×#627—22no—
81C3×#627—42no—
82C3×D3×Dic3—42no—
83#5636—92no—
84#5635—91no—
85C2×#638—31no—
86C22×C3⋊Dic3—31no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C22×C3×Dic3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—61no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—32no—
94C2×C3×Dic6—32no—
95C2×#627C2×D311yescharacteristic
96C2×#627C2×D312yes—
97C2×#625—32no—
98C2×#625—61no—
99C2×#624—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—32no—
103C2×D3×Dic3—61no—
104C2×D3×Dic3C2×D312yes—
105C32×C4⋊C4—32no—
106C32×C22⋊C4—31no—
107#3752—31no—
108C4⋊(C3⋊Dic3)C2×D312yes—
109#3750—31no—
110#3750—32no—
111#3750—32no—
112#3750—32no—
113#3750—32no—
114#3750—62no—
115#3750—61no—
116#3750—121no—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—32no—
119C3×#217—32no—
120C3×(C2×D3)⋊C4—21no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4C2×D312yes—
125C3×C4⋊Dic3—32no—
126C3×C4⋊Dic3—32no—
127C3×C4⋊Dic3—32no—
128C3×C4⋊Dic3—62no—
129C3×Dic3⋊C4—22no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4C2×D312yes—
133C3×Dic3⋊C4C2×D312yes—
134#3724—31no—
135#3724—31no—
136#3724—32no—
137#3724—32no—
138#3724—32no—
139#3724—32no—
140#3724—62no—
141#3724—62no—
142#3724C2×D311yescharacteristic
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—62no—
148Dic3⋊Dic3C2×D312yes—
149#3722C2×D312yes—
150(C2×D3)⋊Dic3—32no—
151(C2×D3)⋊Dic3—61no—
152(C2×D3)⋊Dic3C2×D312yes—
153Dic32—32no—
154Dic32—32no—
155Dic32—32no—
156Dic32—62no—
157Dic32—61no—
158C22×C33C2411yescharacteristic
159C2×C3×C3⋊D3—21no—
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3C2×Q812yes—
162C2×C32×D3—21no—
163C2×C32×D3—21no—
164C2×C32×D3C2×Q812yes—
165C33×C4—21no—
166C3⋊(C3⋊Dic3)—21no—
167C3⋊(C3⋊Dic3)C2×Q812yes—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—22no—
170C3×C3⋊Dic3#4112yes—
171C32×Dic3—22no—
172C32×Dic3—22no—
173C32×Dic3—21no—
174C3×C22⋊Q8—92no—
175C3×C22⋊Q8—91no—
176#946—92no—
177#946—91no—
178Q8×Dic3—91no—
179#944—92no—
180C4×#67—92no—
181C4×#67—91no—
182C2×C4⋊Dic3—91no—
183C2×Dic3⋊C4—92no—
184#897—32no—
185#896—32no—
186#896—92no—
187D3×C4⋊C4—92no—
188D3×C4⋊C4—91no—
189#889—92no—
190#889—92no—
191#889—92no—
192#889—182no—
193#888—91no—
194#887—92no—
195#67⋊C4—92no—
196#67⋊C4—92no—
197#67⋊C4—92no—
198#67⋊C4—181no—
199#878—32no—
200#878—92no—
201#878—92no—
202C23×C32—31no—
203C22×C3⋊D3C22×D311yescharacteristic
204C22×C3×D3—21no—
205C22×C3×D3—32no—
206C22×C3×D3—61no—
207C22×C3×D3C22×D312yes—
208C2×C32×C4—22no—
209C2×C32×C4—21no—
210C2×C32×C4—31no—
211C2×C32×C4—32no—
212C2×C32×C4—32no—
213C2×C32×C4—41no—
214C2×C32×C4C22×D311yescharacteristic
215C2×C32×C4C22×D312yes—
216#638—34no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—31no—
221C2×C3⋊Dic3—62no—
222C2×C3⋊Dic3—61no—
223C2×C3⋊Dic3—61no—
224C2×C3⋊Dic3—62no—
225C2×C3⋊Dic3—62no—
226C2×C3⋊Dic3—121no—
227C2×C3⋊Dic3—121no—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C4×C3⋊D3—22no—
232C2×C3×Dic3—21no—
233C2×C3×Dic3—22no—
234C2×C3×Dic3—22no—
235C2×C3×Dic3—32no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—61no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—62no—
243C2×C3×Dic3—122no—
244C2×C3×Dic3C22×D312yes—
245C2×C3×Dic3C22×D312yes—
246C2×C3×Dic3C22×D312yes—
247C2×C3×Dic3C22×D312yes—
248C2×C3×Dic3C22×D312yes—
249C3×C4×D3—64no—
250C3×C4×D3—122no—
251C3×C4×D3—122no—
252C3×Dic6—64no—
253C3×Dic6—122no—
254C3×Dic6—122no—
255#627—22no—
256#627—42no—
257#625—38no—
258#625—64no—
259#624—122no—
260D3×Dic3—42no—
261D3×Dic3—64no—
262D3×Dic3—122no—
263D3×Dic3—122no—
264D3×Dic3—122no—
265#554—271noSylow
266C2×C33C22×Q811yescharacteristic
267C2×C33C2×#4111yescharacteristic
268C2×C33H(5,𝔽2)11yescharacteristic
269C3×C3⋊D3—22no—
270C32×D3—22no—
271C2×C3×Q8—91no—
272C2×C3×Q8—92no—
273C22×C3×C4—92no—
274C22×C3×C4—91no—
275C2×#67—92no—
276C2×#67—91no—
277C2×#67—181no—
278C22×Dic3—32no—
279C22×Dic3—92no—
280C22×Dic3—92no—
281C22×Dic3—181no—
282C2×C4×D3—32no—
283C2×C4×D3—32no—
284C2×C4×D3—61no—
285C2×C4×D3—91no—
286C2×C4×D3—92no—
287C2×C4×D3—92no—
288C2×Dic6—32no—
289C2×Dic6—32no—
290C2×Dic6—32no—
291C3×C4⋊C4—32no—
292C3×C4⋊C4—32no—
293C3×C4⋊C4—62no—
294C3×C4⋊C4—92no—
295C3×C4⋊C4—91no—
296C3×C4⋊C4—92no—
297C3×C22⋊C4—32no—
298C3×C22⋊C4—61no—
299C3×C22⋊C4—92no—
300C3×C22⋊C4—92no—
301#217—32no—
302#217—91no—
303(C2×D3)⋊C4—32no—
304(C2×D3)⋊C4—32no—
305(C2×D3)⋊C4—32no—
306(C2×D3)⋊C4—92no—
307(C2×D3)⋊C4—181no—
308(C2×D3)⋊C4D3211yescharacteristic
309C4⋊Dic3—32no—
310C4⋊Dic3—32no—
311C4⋊Dic3—32no—
312C4⋊Dic3—62no—
313C4⋊Dic3—91no—
314C4⋊Dic3—91no—
315C4⋊Dic3—92no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—91no—
328Dic3⋊C4—92no—
329Dic3⋊C4—92no—
330Dic3⋊C4—182no—
331Dic3⋊C4—182no—
332Dic3⋊C4—182no—
333Dic3⋊C4—181no—
334Dic3⋊C4—182no—
335Dic3⋊C4—361no—
336Dic3⋊C4D3212yes—
337C4×Dic3—92no—
338C4×Dic3—92no—
339C4×Dic3—92no—
340C4×Dic3—92no—
341C4×Dic3—91no—
342C4×Dic3—181no—
343C4×Dic3—182no—
344C22×C32—22no—
345C22×C32—21no—
346C22×C32—32no—
347C22×C32—41no—
348C22×C32—61no—
349C22×C32—61no—
350C22×C32C23×D311yescharacteristic
351C22×C32C23×D312yes—
352C2×C3⋊D3—21no—
353C2×C3⋊D3—21no—
354C2×C3⋊D3D3×Q812yes—
355C2×C3×D3—22no—
356C2×C3×D3—22no—
357C2×C3×D3—22no—
358C2×C3×D3—34no—
359C2×C3×D3—41no—
360C2×C3×D3—41no—
361C2×C3×D3—62no—
362C2×C3×D3—62no—
363C2×C3×D3—62no—
364C2×C3×D3—121no—
365C2×C3×D3—121no—
366C2×C3×D3D3×Q814yes—
367C32×C4—21no—
368C32×C4—22no—
369C32×C4—41no—
370C32×C4—42no—
371C32×C4—62no—
372C32×C4—62no—
373C32×C4—61no—
374C32×C4—81no—
375C3⋊Dic3—22no—
376C3⋊Dic3—22no—
377C3⋊Dic3—34no—
378C3⋊Dic3—32no—
379C3⋊Dic3—62no—
380C3⋊Dic3—64no—
381C3⋊Dic3—61no—
382C3⋊Dic3—62no—
383C3⋊Dic3—122no—
384C3⋊Dic3—121no—
385C3⋊Dic3—122no—
386C3⋊Dic3—241no—
387C3⋊Dic3#23712yes—
388C3×Dic3—22no—
389C3×Dic3—22no—
390C3×Dic3—22no—
391C3×Dic3—22no—
392C3×Dic3—22no—
393C3×Dic3—34no—
394C3×Dic3—42no—
395C3×Dic3—42no—
396C3×Dic3—41no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—62no—
401C3×Dic3—62no—
402C3×Dic3—122no—
403C3×Dic3—122no—
404C4⋊Q8—271no—
405#125—272no—
406C22⋊Q8—92no—
407C22⋊Q8—91no—
408C22⋊Q8—271no—
409C22⋊Q8—272no—
410C4×Q8—271no—
411C4×D4—271no—
412C4×D4—272no—
413C2×C4⋊C4—271no—
414C2×C4⋊C4—271no—
415C33#55411yescharacteristic, complemented, Sylow
416C23×C3—32no—
417C23×C3—61no—
418C23×C3—91no—
419C22×D3—32no—
420C22×D3—61no—
421C22×D3C2×D3211yescharacteristic
422C3×Q8—182no—
423C3×Q8—362no—
424C2×C3×C4—22no—
425C2×C3×C4—21no—
426C2×C3×C4—32no—
427C2×C3×C4—32no—
428C2×C3×C4—32no—
429C2×C3×C4—32no—
430C2×C3×C4—32no—
431C2×C3×C4—41no—
432C2×C3×C4—62no—
433C2×C3×C4—61no—
434C2×C3×C4—62no—
435C2×C3×C4—91no—
436C2×C3×C4—92no—
437C2×C3×C4—92no—
438C2×C3×C4—182no—
439C2×C3×C4—362no—
440C2×C3×C4C2×D3212yes—
441C2×C3×C4C2×D3211yescharacteristic
442#67—98no—
443#67—94no—
444#67—184no—
445C2×Dic3—32no—
446C2×Dic3—32no—
447C2×Dic3—32no—
448C2×Dic3—32no—
449C2×Dic3—32no—
450C2×Dic3—61no—
451C2×Dic3—62no—
452C2×Dic3—62no—
453C2×Dic3—91no—
454C2×Dic3—92no—
455C2×Dic3—91no—
456C2×Dic3—92no—
457C2×Dic3—122no—
458C2×Dic3—182no—
459C2×Dic3—181no—
460C2×Dic3—182no—
461C2×Dic3—181no—
462C2×Dic3—184no—
463C2×Dic3—361no—
464C2×Dic3—361no—
465C2×Dic3—362no—
466C2×Dic3—362no—
467C2×Dic3C2×D3212yes—
468C2×Dic3C2×D3211yescharacteristic
469C2×Dic3C2×D3212yes—
470C4×D3—64no—
471C4×D3—122no—
472C4×D3—122no—
473C4×D3—182no—
474C4×D3—362no—
475C4×D3—362no—
476Dic6—64no—
477Dic6—122no—
478Dic6—122no—
479C2×C32—21no—
480C2×C32—22no—
481C2×C32—22no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—22no—
485C2×C32—41no—
486C2×C32—41no—
487C2×C32—41no—
488C2×C32—62no—
489C2×C32C2×#23711yescharacteristic
490C2×C32#100412yes—
491C2×C32C2×D3×Q812yes—
492C2×C32C2×D3×Q811yescharacteristic
493C2×C32D3×#4112yes—
494C2×C32#100911yescharacteristic
495C3⋊D3—22no—
496C3×D3—24no—
497C3×D3—42no—
498C3×D3—64no—
499C3×D3—122no—
500C2×Q8—91no—
501C2×Q8—92no—
502C2×D4—271no—
503C22×C4—92no—
504C22×C4—91no—
505C22×C4—272no—
506C22×C4—271no—
507C4⋊C4—32no—
508C4⋊C4—91no—
509C4⋊C4—92no—
510C4⋊C4—92no—
511C4⋊C4—272no—
512C4⋊C4—272no—
513C4⋊C4—272no—
514C4⋊C4—271no—
515C4⋊C4—271no—
516C4⋊C4—271no—
517C22⋊C4—31no—
518C22⋊C4—92no—
519C22⋊C4—92no—
520C22⋊C4—271no—
521C42—272no—
522C42—271no—
523C22×C3—22no—
524C22×C3—21no—
525C22×C3—34no—
526C22×C3—41no—
527C22×C3—62no—
528C22×C3—62no—
529C22×C3—62no—
530C22×C3—92no—
531C22×C3—121no—
532C22×C3—121no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3C22×D3211yescharacteristic
536C22×C3C22×D3212yes—
537C2×D3—21no—
538C2×D3—21no—
539C2×D3—34no—
540C2×D3—62no—
541C2×D3—62no—
542C2×D3—62no—
543C2×D3—121no—
544C2×D3—121no—
545C2×D3#379712yes—
546C3×C4—22no—
547C3×C4—21no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—62no—
551C3×C4—62no—
552C3×C4—62no—
553C3×C4—62no—
554C3×C4—62no—
555C3×C4—81no—
556C3×C4—92no—
557C3×C4—122no—
558C3×C4—121no—
559C3×C4—122no—
560C3×C4—182no—
561C3×C4—182no—
562Dic3—22no—
563Dic3—22no—
564Dic3—21no—
565Dic3—34no—
566Dic3—62no—
567Dic3—62no—
568Dic3—62no—
569Dic3—62no—
570Dic3—62no—
571Dic3—92no—
572Dic3—94no—
573Dic3—122no—
574Dic3—122no—
575Dic3—182no—
576Dic3—184no—
577Dic3—182no—
578Dic3—181no—
579Dic3—362no—
580Dic3—361no—
581Dic3—362no—
582Dic3—721no—
583C32—21no—
584C32—22no—
585C32—41no—
586C32#563511yescharacteristic, complemented
587C32#563612yescomplemented
588C23—31no—
589C23—91no—
590Q8—182no—
591Q8—362no—
592D4—274no—
593C2×C4—31no—
594C2×C4—32no—
595C2×C4—32no—
596C2×C4—91no—
597C2×C4—92no—
598C2×C4—92no—
599C2×C4—182no—
600C2×C4—271no—
601C2×C4—271no—
602C2×C4—362no—
603C2×C4—542no—
604C2×C4—1082no—
605C2×C4D3311yescharacteristic
606C2×C3—21no—
607C2×C3—22no—
608C2×C3—22no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—22no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—64no—
616C2×C3—122no—
617C2×C3—182no—
618C2×C3C2×#379711yescharacteristic
619C2×C3#6430712yes—
620C2×C3D3×#23712yes—
621C2×C3#6431211yescharacteristic
622C2×C3#6431311yescharacteristic
623C2×C3D32×Q812yes—
624D3—22no—
625D3—64no—
626D3—122no—
627C22—32no—
628C22—61no—
629C22—61no—
630C22—92no—
631C22—181no—
632C22—181no—
633C22C2×D3311yescharacteristic
634C4—21no—
635C4—62no—
636C4—61no—
637C4—62no—
638C4—92no—
639C4—182no—
640C4—182no—
641C4—272no—
642C4—541no—
643C3—21no—
644C3—22no—
645C3—41no—
646C3#1059416811yescharacteristic, complemented
647C3#1059453512yescomplemented
648C2—62no—
649C2—182no—
650C2D3×#379711yescharacteristic
651C2#1172438011yescharacteristic
652C2#1172440811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×#6290C22
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
1. #11723689C2semidirect
3. #11723402C2semidirect
4. C3×#63893C2semidirect
5. C3×#63865C2semidirect
6. #11723032C2semidirect
7. #11722979C2semidirect
8. #11722976C2semidirect
9. #11722941C2semidirect
11. D3×#3724C2semidirect
48. #63893D3semidirect
58. #63865D3semidirect
78. #6290Q8semidirect
415. C33#554semidirect
586. C32#5635semidirect
587. C32#5636semidirect
646. C3#10594168semidirect
647. C3#10594535semidirect

Maximal subgroups

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

subgroupindex
#117236892
#117235072
#117234022
C3×#638932
C3×#638652
#117230322
#117229792
#117229762
#117229412
#117229362
D3×#37242
#105945353
#105941683