Finite Group Databasebeta

← #13167997

Subgroups — #13167997

Order 1728.

1216 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#11723671C211yesmaximal, characteristic, complemented
2#11723491C212yesmaximal, complemented
3#11723435C211yesmaximal, characteristic, complemented
4C3×#63921C211yesmaximal, characteristic
5C3×#63887C212yesmaximal, complemented
6#11722980C212yesmaximal, complemented
7#11722978C212yesmaximal, complemented
8#11722972C212yesmaximal, complemented
9D3×(C2×D3)⋊Dic3C212yesmaximal
10#10595434—31nomaximal
11#10594170—32nomaximal
12C22×C3×D32C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×#6296C2211yescharacteristic, complemented
16C2×#6293C2212yescomplemented
17C2×D3×C3⋊Dic3C2212yes—
18C2×C3×#624C2211yescharacteristic
19C2×C3×D3×Dic3C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
21C3×#3750C2212yescomplemented
22C32×(C2×D3)⋊C4C2212yes—
23#88815C2211yescharacteristic, complemented
24#88814C2212yes—
25#88807C2212yescomplemented
26#88806C2212yescomplemented
27#88803C2212yes—
28(C2×D3)⋊(C3⋊Dic3)C2212yes—
29C3×Dic3⋊Dic3C2212yescomplemented
30C3×Dic3⋊Dic3C2212yescomplemented
31C3×(C2×D3)⋊Dic3C2212yes—
32C3×(C2×D3)⋊Dic3C2212yes—
33#64155—31no—
34#64087—32no—
35C3×#946—32no—
36C3×D3×C22⋊C4—32no—
37C3×#878—32no—
38#63982—32no—
39D3×#217—32no—
40#63964—32no—
41C2×(C2×D3)⋊Dic3—32no—
42#63921D311yescharacteristic
43D3×(C2×D3)⋊C4—32no—
44#63899—32no—
45#63899—32no—
46#63896—32no—
47#63895—32no—
48#63893—32no—
49#63893—32no—
50#63893—62no—
51#63887D312yescomplemented
52#63865—31no—
53#63857—32no—
54#63856—32no—
55#63856—61no—
56#63851—32no—
57#63851—32no—
58#63851—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C22×C32×D3C2312yes—
61C2×C3×D32—41no—
62C2×C3×D32—41no—
63C2×C3×D32Q814yescomplemented
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C3×C4×C3⋊D3—41no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C32×Dic6—41no—
73#6296—41no—
74#6293—42no—
75D3×C3⋊Dic3—42no—
76C3×#624—41no—
77C3×D3×Dic3—42no—
78#5634—92no—
79#5483—91no—
80C23×C3×D3—32no—
81C22×D32C2×D311yescharacteristic
82C2×C32×Q8—31no—
83C22×C3⋊Dic3—32no—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×#634—32no—
86C22×C3×Dic3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—61no—
91C2×C3×Dic6—21no—
92C2×C3×Dic6C2×D312yescomplemented
93C2×#627—31no—
94C2×#627—32no—
95C2×#627—32no—
96C2×#627—32no—
97C2×#627—62no—
98C2×#624C2×D311yescharacteristic
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—62no—
102C2×D3×Dic3C2×D312yes—
103C32×C22⋊C4—32no—
104#3757—32no—
105#3752—32no—
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—61no—
109C4⋊(C3⋊Dic3)—62no—
110C4⋊(C3⋊Dic3)—121no—
111#3750—32no—
112#3750—32no—
113#3750C2×D312yescomplemented
114C3×#217—32no—
115C3×#217—32no—
116C3×(C2×D3)⋊C4—22no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4C2×D312yes—
120C3×(C2×D3)⋊C4C2×D312yes—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—62no—
128#3724—31no—
129#3724—32no—
130#3724—32no—
131#3724—62no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—62no—
136Dic3⋊Dic3C2×D312yescomplemented
137Dic3⋊Dic3C2×D312yescomplemented
138#3722—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—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—62no—
146(C2×D3)⋊Dic3—62no—
147(C2×D3)⋊Dic3C2×D312yes—
148(C2×D3)⋊Dic3C2×D312yes—
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3C2×Q812yes—
153C2×C32×D3—22no—
154C2×C32×D3—22no—
155C2×C32×D3C2×Q814yes—
156C3×D32—44no—
157C33×C4—21no—
158C3⋊(C3⋊Dic3)—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—22no—
161C3×C3⋊Dic3—21no—
162C32×Dic3—22no—
163C32×Dic3—22no—
164C3×C22⋊Q8—92no—
165C2×C3×C22⋊C4—91no—
166C2×#217—92no—
167#946—32no—
168#946—92no—
169#946—181no—
170C2×(C2×D3)⋊C4—92no—
171#924—91no—
172#924—92no—
173#924—92no—
174#924—91no—
175#897—92no—
176#897—92no—
177#896—92no—
178#896—92no—
179#896—92no—
180#896—92no—
181D3×C22⋊C4—32no—
182D3×C22⋊C4—92no—
183#878—32no—
184#878—92no—
185#878—92no—
186#878—92no—
187#878—92no—
188#878—92no—
189#878—182no—
190C23×C32—32no—
191C22×C3⋊D3C22×D311yescharacteristic
192C22×C3×D3—22no—
193C22×C3×D3—32no—
194C22×C3×D3—38no—
195C22×C3×D3—61no—
196C22×C3×D3—122no—
197C22×C3×D3—122no—
198C22×C3×D3C22×D312yes—
199C22×C3×D3C22×D312yes—
200C2×D32—41no—
201C2×D32—41no—
202C2×D32Dic614yescomplemented
203C32×Q8—121no—
204C2×C32×C4—21no—
205C2×C32×C4—22no—
206C2×C32×C4—32no—
207C2×C32×C4—32no—
208C2×C32×C4—41no—
209C2×C32×C4C22×D311yescharacteristic
210C2×C32×C4C22×D312yes—
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—62no—
215C2×C3⋊Dic3—62no—
216C2×C3⋊Dic3—62no—
217C2×C3⋊Dic3—122no—
218C2×C3⋊Dic3—122no—
219C2×C3⋊Dic3C22×D312yes—
220C2×C3⋊Dic3C22×D312yes—
221C2×C3⋊Dic3C22×D311yescharacteristic
222C4×C3⋊D3—41no—
223#634—122no—
224C2×C3×Dic3—22no—
225C2×C3×Dic3—22no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—62no—
232C2×C3×Dic3—62no—
233C2×C3×Dic3—61no—
234C2×C3×Dic3—122no—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C22×D312yes—
238C2×C3×Dic3C22×D312yes—
239C3×C4×D3—122no—
240C3×C4×D3—122no—
241C3×C4×D3—122no—
242C3×C4×D3—122no—
243C3×Dic6—42no—
244C3×Dic6—42no—
245#627—122no—
246#627—122no—
247#627—124no—
248#627—121no—
249#627—122no—
250#624—41no—
251D3×Dic3—42no—
252D3×Dic3—122no—
253D3×Dic3—122no—
254D3×Dic3—124no—
255#543—271noSylow
256C2×C33C22×Q811yescharacteristic
257C2×C33H(5,𝔽2)11yescharacteristic
258C2×C33H(5,𝔽2)11yescharacteristic
259C3×C3⋊D3—22no—
260C32×D3—24no—
261C24×C3—91no—
262C23×D3—32no—
263C2×C3×Q8—32no—
264C2×C3×Q8—61no—
265C22×C3×C4—91no—
266C22×C3×C4—92no—
267C22×C3×C4—91no—
268C22×Dic3—32no—
269C22×Dic3—92no—
270C22×Dic3—92no—
271C22×Dic3—182no—
272C2×C4×D3—32no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—61no—
276C2×C4×D3—92no—
277C2×Dic6—91no—
278C2×Dic6—92no—
279C2×Dic6—92no—
280C2×Dic6—92no—
281C2×Dic6—92no—
282C2×Dic6—182no—
283C2×Dic6D3211yescharacteristic, complemented
284C3×C4⋊C4—92no—
285C3×C4⋊C4—92no—
286C3×C4⋊C4—92no—
287C3×C22⋊C4—32no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—62no—
290C3×C22⋊C4—92no—
291C3×C22⋊C4—92no—
292#217—32no—
293#217—32no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—92no—
298#217—182no—
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—32no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—92no—
305(C2×D3)⋊C4—92no—
306(C2×D3)⋊C4—182no—
307(C2×D3)⋊C4D3212yes—
308C4⋊Dic3—32no—
309C4⋊Dic3—32no—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—92no—
313C4⋊Dic3—92no—
314C4⋊Dic3—182no—
315C4⋊Dic3—181no—
316C4⋊Dic3—361no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—32no—
320Dic3⋊C4—32no—
321Dic3⋊C4—62no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328Dic3⋊C4—182no—
329Dic3⋊C4—182no—
330C22×C32—21no—
331C22×C32—22no—
332C22×C32—34no—
333C22×C32—41no—
334C22×C32—62no—
335C22×C32—62no—
336C22×C32C23×D311yescharacteristic
337C22×C32C23×D312yes—
338C2×C3⋊D3—21no—
339C2×C3⋊D3—21no—
340C2×C3⋊D3C2×Dic612yes—
341C2×C3×D3—22no—
342C2×C3×D3—22no—
343C2×C3×D3—22no—
344C2×C3×D3—22no—
345C2×C3×D3—24no—
346C2×C3×D3—34no—
347C2×C3×D3—42no—
348C2×C3×D3—42no—
349C2×C3×D3—62no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C2×C3×D3—121no—
353C2×C3×D3—121no—
354C2×C3×D3—128no—
355C2×C3×D3C2×Dic614yes—
356C2×C3×D3D3×Q814yes—
357D32—44no—
358C32×C4—21no—
359C32×C4—22no—
360C32×C4—22no—
361C32×C4—42no—
362C32×C4—42no—
363C32×C4—62no—
364C32×C4—62no—
365C3⋊Dic3—22no—
366C3⋊Dic3—22no—
367C3⋊Dic3—21no—
368C3⋊Dic3—62no—
369C3⋊Dic3—62no—
370C3⋊Dic3—62no—
371C3⋊Dic3—122no—
372C3⋊Dic3—122no—
373C3⋊Dic3—122no—
374C3⋊Dic3—242no—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—22no—
378C3×Dic3—22no—
379C3×Dic3—42no—
380C3×Dic3—42no—
381C3×Dic3—64no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—121no—
388C3×Dic3—122no—
389C22⋊Q8—92no—
390C22⋊Q8—272no—
391C22⋊Q8—272no—
392C22⋊Q8—272no—
393C22⋊Q8—271no—
394C22⋊Q8—272no—
395C22⋊Q8—271no—
396C2×C22⋊C4—91no—
397C2×C22⋊C4—272no—
398C33#54311yescharacteristic, complemented, Sylow
399C23×C3—32no—
400C23×C3—32no—
401C23×C3—62no—
402C23×C3—94no—
403C23×C3—91no—
404C23×C3—361no—
405C23×C3—361no—
406C22×D3—32no—
407C22×D3—38no—
408C22×D3—61no—
409C22×D3—122no—
410C22×D3—122no—
411C22×D3C2×D3212yes—
412C3×Q8—122no—
413C3×Q8—122no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—41no—
421C2×C3×C4—62no—
422C2×C3×C4—62no—
423C2×C3×C4—91no—
424C2×C3×C4—92no—
425C2×C3×C4—92no—
426C2×C3×C4—361no—
427C2×C3×C4—361no—
428C2×C3×C4—362no—
429C2×C3×C4C2×D3211yescharacteristic
430C2×C3×C4C2×D3212yes—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—62no—
437C2×Dic3—62no—
438C2×Dic3—61no—
439C2×Dic3—92no—
440C2×Dic3—92no—
441C2×Dic3—92no—
442C2×Dic3—122no—
443C2×Dic3—182no—
444C2×Dic3—182no—
445C2×Dic3—182no—
446C2×Dic3—362no—
447C2×Dic3—362no—
448C2×Dic3—362no—
449C2×Dic3—364no—
450C2×Dic3C2×D3212yes—
451C2×Dic3C2×D3212yes—
452C4×D3—122no—
453C4×D3—122no—
454C4×D3—122no—
455C4×D3—122no—
456C4×D3—362no—
457Dic6—41no—
458Dic6—362no—
459Dic6—362no—
460Dic6—361no—
461Dic6—362no—
462Dic6—362no—
463Dic6—364no—
464C2×C32—22no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—41no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32—64no—
474C2×C32C22×Dic611yescharacteristic
475C2×C32#100411yescharacteristic
476C2×C32#100412yes—
477C2×C32#100411yescharacteristic
478C2×C32C2×D3×Q812yes—
479C2×C32#100912yes—
480C3⋊D3—22no—
481C3×D3—24no—
482C3×D3—24no—
483C3×D3—44no—
484C3×D3—64no—
485C3×D3—122no—
486C24—91no—
487C2×Q8—31no—
488C2×Q8—272no—
489C2×Q8—271no—
490C22×C4—91no—
491C22×C4—92no—
492C22×C4—91no—
493C22×C4—272no—
494C4⋊C4—92no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C4⋊C4—272no—
498C4⋊C4—272no—
499C4⋊C4—271no—
500C4⋊C4—271no—
501C22⋊C4—32no—
502C22⋊C4—92no—
503C22⋊C4—92no—
504C22⋊C4—272no—
505C22⋊C4—272no—
506C22⋊C4—272no—
507C22×C3—22no—
508C22×C3—21no—
509C22×C3—34no—
510C22×C3—34no—
511C22×C3—41no—
512C22×C3—64no—
513C22×C3—62no—
514C22×C3—62no—
515C22×C3—62no—
516C22×C3—62no—
517C22×C3—92no—
518C22×C3—122no—
519C22×C3—122no—
520C22×C3—181no—
521C22×C3—181no—
522C22×C3—364no—
523C22×C3C22×D3212yes—
524C22×C3C22×D3211yescharacteristic
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—34no—
528C2×D3—62no—
529C2×D3—62no—
530C2×D3—62no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—128no—
534C2×D3D3×Dic614yes—
535C3×C4—22no—
536C3×C4—22no—
537C3×C4—21no—
538C3×C4—42no—
539C3×C4—42no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—62no—
543C3×C4—62no—
544C3×C4—122no—
545C3×C4—122no—
546C3×C4—182no—
547C3×C4—182no—
548C3×C4—181no—
549Dic3—22no—
550Dic3—22no—
551Dic3—64no—
552Dic3—62no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—121no—
558Dic3—122no—
559Dic3—182no—
560Dic3—182no—
561Dic3—182no—
562Dic3—362no—
563Dic3—362no—
564Dic3—362no—
565Dic3—722no—
566C32—21no—
567C32—22no—
568C32—41no—
569C32#548311yescharacteristic, complemented
570C32#563412yescomplemented
571C23—32no—
572C23—94no—
573C23—91no—
574C23—361no—
575C23—361no—
576Q8—121no—
577Q8—1082no—
578Q8—1081no—
579C2×C4—32no—
580C2×C4—32no—
581C2×C4—91no—
582C2×C4—92no—
583C2×C4—92no—
584C2×C4—272no—
585C2×C4—361no—
586C2×C4—361no—
587C2×C4—362no—
588C2×C4—1082no—
589C2×C4D3311yescharacteristic
590C2×C3—21no—
591C2×C3—22no—
592C2×C3—22no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—22no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—64no—
600C2×C3—64no—
601C2×C3—124no—
602C2×C3—182no—
603C2×C3C2×D3×Dic612yes—
604C2×C3#6431312yes—
605C2×C3#6431412yes—
606C2×C3D32×Q811yescharacteristic
607C2×C3#6432011yescharacteristic
608C2×C3#6433011yescharacteristic
609D3—24no—
610D3—64no—
611D3—122no—
612C22—34no—
613C22—62no—
614C22—62no—
615C22—92no—
616C22—181no—
617C22—181no—
618C22—364no—
619C22C2×D3311yescharacteristic
620C4—21no—
621C4—62no—
622C4—62no—
623C4—182no—
624C4—181no—
625C4—182no—
626C4—542no—
627C3—21no—
628C3—22no—
629C3—41no—
630C3#1059417012yescomplemented
631C3#1059543411yescharacteristic, complemented
632C2—64no—
633C2—182no—
634C2D32×Dic611yescharacteristic
635C2#1172438611yescharacteristic
636C2#1172441511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×D32C22
hypercenterC2×C4D33
hypocenterC33#543
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723671C2semidirect
2. #11723491C2semidirect
3. #11723435C2semidirect
5. C3×#63887C2semidirect
6. #11722980C2semidirect
7. #11722978C2semidirect
8. #11722972C2semidirect
14. C2×C32×Dic6C22semidirect
15. C2×#6296C22semidirect
16. C2×#6293C22semidirect
20. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
21. C3×#3750C22semidirect
23. #88815C22semidirect
25. #88807C22semidirect
26. #88806C22semidirect
29. C3×Dic3⋊Dic3C22semidirect
30. C3×Dic3⋊Dic3C22semidirect
51. #63887D3semidirect
63. C2×C3×D32Q8semidirect
92. C2×C3×Dic6C2×D3semidirect
113. #3750C2×D3semidirect
136. Dic3⋊Dic3C2×D3semidirect
137. Dic3⋊Dic3C2×D3semidirect
202. C2×D32Dic6semidirect
283. C2×Dic6D32semidirect
398. C33#543semidirect
569. C32#5483semidirect
570. C32#5634semidirect
630. C3#10594170semidirect
631. C3#10595434semidirect

Maximal subgroups

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

subgroupindex
#117236712
#117234912
#117234352
C3×#639212
C3×#638872
#117229802
#117229782
#117229722
D3×(C2×D3)⋊Dic32
#105954343
#105941703