Finite Group Databasebeta

← #13156702

Subgroups — #13156702

Order 1728.

1666 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic, complemented
3C2×#88804C212yesmaximal
4C2×C3×Dic32C211yesmaximal, characteristic, complemented
5C3×#63579C212yesmaximal, complemented
6#10591260—31nomaximal
7#10591233—32nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×#6289—22no—
14C2×#6289—22no—
15C2×#6289C414yescomplemented
16#88813—22no—
17#88804C418yescomplemented
18C3×Dic32—22no—
19C22×C4×C3⋊D3—31no—
20C22×#624—32no—
21C22×#624—61no—
22C2×#3752—32no—
23C2×C3×C4×Dic3—32no—
24C2×#3722—32no—
25C2×#3722—32no—
26C2×#3722—62no—
27C2×Dic32—32no—
28C2×Dic32—31no—
29C2×Dic32D311yescharacteristic, complemented
30C3×C3⋊#95—32no—
31C3×C3⋊#95—32no—
32#63579D312yescomplemented
33C23×C33C2311yescharacteristic
34C22×C3⋊(C3⋊D3)—21no—
35C22×C3⋊(C3⋊D3)—21no—
36C22×C3⋊(C3⋊D3)—22no—
37C22×C3⋊(C3⋊D3)C2×C412yes—
38C22×C3⋊(C3⋊D3)C2×C414yes—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3C2×C412yes—
42C2×C3×C3⋊Dic3C2×C414yescomplemented
43C2×C3×C3⋊Dic3D412yes—
44C2×C3×C3⋊Dic3D412yes—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3C2×C412yes—
50C2×C32×Dic3C2×C414yescomplemented
51#6289—28no—
52C4×(C2×D3)⋊C4—91no—
53#4666—92no—
54C23×C3⋊D3—32no—
55C23×C3⋊D3—31no—
56C23×C3⋊D3—61no—
57C23×C3⋊D3—62no—
58C23×C3⋊D3—121no—
59C22×C32×C4—32no—
60C22×C32×C4—31no—
61C22×C3⋊Dic3C2×D312yes—
62C22×C3⋊Dic3C2×D311yescharacteristic
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—62no—
65C2×C4×C3⋊D3—62no—
66C22×C3×Dic3—21no—
67C22×C3×Dic3—22no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3—62no—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C2×D312yes—
75C22×C3×Dic3C2×D312yes—
76C2×#624—38no—
77C2×#624—64no—
78C2×#624—64no—
79C2×#624—64no—
80C2×#624—64no—
81C2×#624—64no—
82#3752—38no—
83C3×C4×Dic3—64no—
84#3722—38no—
85#3722—38no—
86#3722—68no—
87Dic32—22no—
88Dic32—64no—
89Dic32—62no—
90C22×C33C22×C411yescharacteristic
91C22×C33C22×C412yes—
92C22×C33C2×D411yescharacteristic
93C22×C33C2×D411yescharacteristic
94C22×C33#4112yes—
95C2×C3⋊(C3⋊D3)—22no—
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—24no—
98C2×C3⋊(C3⋊D3)—24no—
99C2×C3⋊(C3⋊D3)C4214yescomplemented
100C3×C3⋊Dic3—24no—
101C3×C3⋊Dic3C22⋊C414yescomplemented
102C32×Dic3—22no—
103C32×Dic3—24no—
104C22×C4×D3—92no—
105C22×C4×D3—91no—
106C22×C4×D3—181no—
107C2×C3×C42—91no—
108C2×(C2×D3)⋊C4—92no—
109C2×(C2×D3)⋊C4—92no—
110C2×(C2×D3)⋊C4—92no—
111C2×(C2×D3)⋊C4—182no—
112C2×C4×Dic3—32no—
113C2×C4×Dic3—92no—
114C2×C4×Dic3—91no—
115C3×#95—92no—
116C3⋊#95—32no—
117C3⋊#95—32no—
118C23×C32—21no—
119C23×C32—22no—
120C23×C32—41no—
121C23×C32C22×D311yescharacteristic
122C23×C32C22×D312yes—
123C22×C3⋊D3—34no—
124C22×C3⋊D3—34no—
125C22×C3⋊D3—34no—
126C22×C3⋊D3—34no—
127C22×C3⋊D3—32no—
128C22×C3⋊D3—64no—
129C22×C3⋊D3—62no—
130C22×C3⋊D3—64no—
131C22×C3⋊D3—64no—
132C22×C3⋊D3—64no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—62no—
138C22×C3⋊D3—62no—
139C22×C3⋊D3—62no—
140C22×C3⋊D3—62no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—122no—
144C22×C3⋊D3—124no—
145C22×C3⋊D3—122no—
146C22×C3⋊D3—122no—
147C22×C3⋊D3—122no—
148C22×C3⋊D3—122no—
149C22×C3⋊D3—122no—
150C22×C3⋊D3—122no—
151C22×C3⋊D3—124no—
152C2×C32×C4—32no—
153C2×C32×C4—34no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4—62no—
157C2×C32×C4—62no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3C4×D312yes—
161C2×C3⋊Dic3C4×D314yescomplemented
162C2×C3⋊Dic3D1212yes—
163C2×C3⋊Dic3#6712yes—
164C4×C3⋊D3—68no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—34no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—42no—
181C2×C3×Dic3—42no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—122no—
191C2×C3×Dic3—122no—
192C2×C3×Dic3C4×D314yes—
193C2×C3×Dic3C4×D314yescomplemented
194C2×C3×Dic3C4×D314yescomplemented
195#624—616no—
196#624—616no—
197C4×C22⋊C4—271noSylow
198C2×C33C2×C4211yescharacteristic
199C2×C33C2×C22⋊C411yescharacteristic
200C2×C33#11711yescharacteristic
201C2×C33C4×D412yes—
202C2×C33C4×D412yes—
203C3⋊(C3⋊D3)—24no—
204C23×D3—91no—
205C23×D3—92no—
206C23×D3—181no—
207C23×D3—182no—
208C23×D3—361no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—32no—
212C22×C3×C4—61no—
213C22×C3×C4—62no—
214C22×C3×C4—91no—
215C22×C3×C4—92no—
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—61no—
220C22×Dic3—62no—
221C22×Dic3D3211yescharacteristic
222C22×Dic3D3212yes—
223C2×C4×D3—98no—
224C2×C4×D3—94no—
225C2×C4×D3—184no—
226C2×C4×D3—182no—
227C2×C4×D3—182no—
228C2×C4×D3—184no—
229C2×C4×D3—184no—
230C2×C4×D3—184no—
231C2×C4×D3—184no—
232C3×C42—182no—
233(C2×D3)⋊C4—98no—
234(C2×D3)⋊C4—98no—
235(C2×D3)⋊C4—98no—
236(C2×D3)⋊C4—188no—
237C4×Dic3—64no—
238C4×Dic3—184no—
239C4×Dic3—182no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—42no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—42no—
257C22×C32C2×C4×D311yescharacteristic
258C22×C32C2×C4×D312yes—
259C22×C32C2×C4×D312yes—
260C22×C32C2×C4×D312yes—
261C22×C32C2×C4×D312yes—
262C22×C32C2×D1211yescharacteristic
263C22×C32#23512yes—
264C22×C32D3×D412yes—
265C22×C32D3×D412yes—
266C22×C32#23712yes—
267C22×C32#23912yes—
268C22×C32C2×#6711yescharacteristic
269C2×C3⋊D3—34no—
270C2×C3⋊D3—38no—
271C2×C3⋊D3—64no—
272C2×C3⋊D3—68no—
273C2×C3⋊D3—62no—
274C2×C3⋊D3—64no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—64no—
277C2×C3⋊D3—64no—
278C2×C3⋊D3—64no—
279C2×C3⋊D3—68no—
280C2×C3⋊D3—68no—
281C2×C3⋊D3—64no—
282C2×C3⋊D3—64no—
283C2×C3⋊D3—64no—
284C2×C3⋊D3—64no—
285C2×C3⋊D3—64no—
286C2×C3⋊D3—64no—
287C2×C3⋊D3—124no—
288C2×C3⋊D3—124no—
289C2×C3⋊D3—124no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—124no—
292C2×C3⋊D3—124no—
293C2×C3⋊D3—124no—
294C2×C3⋊D3—124no—
295C2×C3⋊D3—124no—
296C2×C3⋊D3—128no—
297C2×C3⋊D3—128no—
298C32×C4—62no—
299C32×C4—64no—
300C3⋊Dic3—24no—
301C3⋊Dic3(C2×D3)⋊C414yescomplemented
302C3×Dic3—24no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—24no—
306C3×Dic3—38no—
307C3×Dic3—44no—
308C3×Dic3—64no—
309C3×Dic3—64no—
310C3×Dic3—64no—
311C3×Dic3—124no—
312C23×C4—271no—
313C2×C22⋊C4—272no—
314C2×C42—91no—
315C2×C42—271no—
316#95—92no—
317C33C4×C22⋊C411yescharacteristic, complemented, Sylow
318C23×C3—21no—
319C23×C3—22no—
320C23×C3—41no—
321C23×C3C2×D3211yescharacteristic
322C23×C3C2×D3212yes—
323C22×D3—94no—
324C22×D3—94no—
325C22×D3—94no—
326C22×D3—94no—
327C22×D3—92no—
328C22×D3—182no—
329C22×D3—184no—
330C22×D3—184no—
331C22×D3—184no—
332C22×D3—184no—
333C22×D3—182no—
334C22×D3—182no—
335C22×D3—184no—
336C22×D3—182no—
337C22×D3—182no—
338C22×D3—182no—
339C22×D3—182no—
340C22×D3—182no—
341C22×D3—181no—
342C22×D3—181no—
343C22×D3—364no—
344C22×D3—362no—
345C22×D3—362no—
346C22×D3—362no—
347C22×D3—362no—
348C22×D3—362no—
349C22×D3—362no—
350C22×D3—362no—
351C22×D3—364no—
352C2×C3×C4—34no—
353C2×C3×C4—34no—
354C2×C3×C4—34no—
355C2×C3×C4—62no—
356C2×C3×C4—64no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4—92no—
368C2×C3×C4—94no—
369C2×C3×C4—122no—
370C2×C3×C4—122no—
371C2×C3×C4—182no—
372C2×C3×C4—182no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—34no—
381C2×Dic3—34no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—122no—
391C2×Dic3—122no—
392C2×Dic3#62412yes—
393C2×Dic3#62414yescomplemented
394C4×D3—188no—
395C4×D3—1816no—
396C4×D3—1816no—
397C2×C32—22no—
398C2×C32—21no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—22no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—42no—
414C2×C32C42×D312yes—
415C2×C32C42×D311yescharacteristic
416C2×C32#87211yescharacteristic
417C2×C32C4×D1212yes—
418C2×C32D3×C22⋊C412yes—
419C2×C32#67⋊C412yes—
420C2×C32#67⋊C412yes—
421C2×C32#89212yes—
422C2×C32#89312yes—
423C2×C32#89312yes—
424C2×C32C2×(C2×D3)⋊C411yescharacteristic
425C2×C32C4×#6712yes—
426C3⋊D3—64no—
427C3⋊D3—68no—
428C3⋊D3—68no—
429C3⋊D3—128no—
430C3⋊D3—128no—
431C24—271no—
432C22×C4—31no—
433C22×C4—32no—
434C22×C4—91no—
435C22×C4—92no—
436C22×C4—274no—
437C22×C4—542no—
438C22×C4—542no—
439C22⋊C4—278no—
440C42—182no—
441C42—542no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—41no—
455C22×C3—42no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3#379612yes—
460C22×C3#379912yes—
461C22×C3C4×D3212yes—
462C22×C3C4×D3212yes—
463C22×C3C4×D3212yes—
464C22×C3D3×D1212yes—
465C22×C3#380412yes—
466C22×C3C2×#62411yescharacteristic
467C22×C3C2×#62412yes—
468C22×C3D3×#6712yes—
469C22×C3#381111yescharacteristic
470C22×C3#381111yescharacteristic
471C2×D3—98no—
472C2×D3—94no—
473C2×D3—184no—
474C2×D3—188no—
475C2×D3—184no—
476C2×D3—184no—
477C2×D3—188no—
478C2×D3—188no—
479C2×D3—184no—
480C2×D3—182no—
481C2×D3—184no—
482C2×D3—184no—
483C2×D3—184no—
484C2×D3—184no—
485C2×D3—184no—
486C2×D3—184no—
487C2×D3—184no—
488C2×D3—182no—
489C2×D3—364no—
490C2×D3—364no—
491C2×D3—364no—
492C2×D3—364no—
493C2×D3—364no—
494C2×D3—364no—
495C2×D3—364no—
496C2×D3—364no—
497C2×D3—364no—
498C2×D3—368no—
499C2×D3—368no—
500C3×C4—64no—
501C3×C4—64no—
502C3×C4—64no—
503C3×C4—64no—
504C3×C4—94no—
505C3×C4—124no—
506C3×C4—184no—
507Dic3—24no—
508Dic3—22no—
509Dic3—38no—
510Dic3—64no—
511Dic3—64no—
512Dic3—64no—
513Dic3—124no—
514C32—22no—
515C32—21no—
516C32—41no—
517C32#466612yescomplemented
518C32C4×(C2×D3)⋊C411yescharacteristic, complemented
519C23—274no—
520C23—272no—
521C23—541no—
522C23—541no—
523C23—542no—
524C23D3311yescharacteristic
525C2×C4—34no—
526C2×C4—32no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—94no—
535C2×C4—182no—
536C2×C4—182no—
537C2×C4—548no—
538C2×C3—21no—
539C2×C3—22no—
540C2×C3—21no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—22no—
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—42no—
554C2×C3—42no—
555C2×C3#6383612yes—
556C2×C3#6384911yescharacteristic
557C2×C3#6388012yes—
558C2×C3#6388112yes—
559C2×C3#6388112yes—
560C2×C3C4×#62412yes—
561C2×C3C4×#62411yescharacteristic
562C2×C3#6389412yes—
563C2×C3#6390412yes—
564C2×C3D3×(C2×D3)⋊C412yes—
565C2×C3#6395212yes—
566C2×C3#6397411yescharacteristic
567D3—188no—
568D3—188no—
569D3—184no—
570D3—368no—
571D3—368no—
572C22—274no—
573C22—542no—
574C22—542no—
575C22—544no—
576C22—544no—
577C22D3×#62411yescharacteristic
578C22D3×#62412yes—
579C22#8895211yescharacteristic
580C22#8895511yescharacteristic
581C22#8896212yes—
582C4—62no—
583C4—64no—
584C4—94no—
585C4—184no—
586C3—21no—
587C3—22no—
588C3—41no—
589C3#1059123312yescomplemented
590C3#1059126011yescharacteristic, complemented
591C2—544no—
592C2#1172290011yescharacteristic
593C2#1172290411yescharacteristic
594C2#1172292011yescharacteristic
595C2#1172293512yes—
596C2#1172293712yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC2×C33C2×C42
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypocenterC33C4×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88813C2semidirect
4. C2×C3×Dic32C2semidirect
5. C3×#63579C2semidirect
15. C2×#6289C4semidirect
17. #88804C4semidirect
29. C2×Dic32D3semidirect
32. #63579D3semidirect
42. C2×C3×C3⋊Dic3C2×C4semidirect
50. C2×C32×Dic3C2×C4semidirect
99. C2×C3⋊(C3⋊D3)C42semidirect
101. C3×C3⋊Dic3C22⋊C4semidirect
161. C2×C3⋊Dic3C4×D3semidirect
193. C2×C3×Dic3C4×D3semidirect
194. C2×C3×Dic3C4×D3semidirect
301. C3⋊Dic3(C2×D3)⋊C4semidirect
317. C33C4×C22⋊C4semidirect
393. C2×Dic3#624semidirect
517. C32#4666semidirect
518. C32C4×(C2×D3)⋊C4semidirect
589. C3#10591233semidirect
590. C3#10591260semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#888132
C2×#888042
C2×C3×Dic322
C3×#635792
#105912603
#105912333