Finite Group Databasebeta

← #13175500

Subgroups — #13175500

Order 1728.

2524 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6292C211yesmaximal, characteristic, complemented
2C2×#89031C211yesmaximal, characteristic, complemented
3#11723923C211yesmaximal, characteristic, complemented
4C3×#64091C211yesmaximal, characteristic, complemented
5C32×#877C211yesmaximal, characteristic, complemented
6#11723639C211yesmaximal, characteristic, complemented
7#11723636C211yesmaximal, characteristic, complemented
8#11723517C212yesmaximal, complemented
9#11723486C212yesmaximal, complemented
10#11723482C212yesmaximal, complemented
11#11723416C212yesmaximal, complemented
12#10595996—31nomaximal
13#10594080—34nomaximal
14C23×C3⋊(C3⋊D3)C2211yescharacteristic
15C23×C3×C3⋊D3C2211yescharacteristic
16C2×#6312C2211yescharacteristic
17C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
18C2×C3×#638C2211yescharacteristic
19C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
20C22×C32×Dic3C2211yescharacteristic
21C2×#6293C2211yescharacteristic, complemented
22C2×#6292—24no—
23C2×#6292C2212yes—
24C2×#6292C2212yes—
25C2×#6292C2211yescharacteristic, complemented
26C2×#6292C2212yescomplemented
27C2×#6292C2212yescomplemented
28C2×#6289C2211yescharacteristic, complemented
29#89031—24no—
30C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
31C33×C22⋊C4C2211yescharacteristic
32#88859C2212yescomplemented
33C3×#3752C2212yescomplemented
34C32×#217C2211yescharacteristic
35C32×Dic3⋊C4C2212yescomplemented
36C32×C4×Dic3C2212yescomplemented
37#88804C2212yescomplemented
38#88803C2212yescomplemented
39C2×#3828—31no—
40C22×C4⋊(C3⋊D3)—31no—
41C22×#626—34no—
42C2×#3804—34no—
43C32×#117—31no—
44#64104—32no—
45#64104—32no—
46#64091—31no—
47#64091—31no—
48#64091—34no—
49#64091—31no—
50#64091—64no—
51#64091D311yescharacteristic, complemented
52#64084—32no—
53#64083—32no—
54C3×#882—34no—
55C3×#877D314yescomplemented
56#63981—34no—
57#63977—34no—
58#63911—38no—
59#63910—38no—
60#63886—38no—
61#63852—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)—22no—
64C22×C3⋊(C3⋊D3)—42no—
65C22×C3⋊(C3⋊D3)C2312yes—
66C22×C3⋊(C3⋊D3)C2311yescharacteristic
67C22×C3×C3⋊D3—22no—
68C22×C3×C3⋊D3—42no—
69C22×C3×C3⋊D3C2312yes—
70C22×C3×C3⋊D3C2311yescharacteristic
71C2×C33×C4C2312yes—
72#6312—22no—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C4⋊(C3⋊(C3⋊D3))—42no—
75C3×#638—22no—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)—42no—
78C2×C32×Dic3D414yescomplemented
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C2×C32×Dic3C2311yescharacteristic
82#6293—22no—
83#6292—22no—
84#6292—24no—
85#6292—24no—
86#6292—42no—
87#6292—42no—
88#6289—22no—
89Dic3×C3⋊D3—22no—
90#5592—91no—
91#5525—94no—
92C23×C3⋊D3—31no—
93C23×C3⋊D3—34no—
94C23×C3⋊D3—64no—
95C23×C3⋊D3C2×D311yescharacteristic
96C23×C3×D3—34no—
97C22×C32×C4—31no—
98C2×#638—31no—
99C2×#638—34no—
100C2×#638—64no—
101C2×#638C2×D311yescharacteristic
102#3828—64no—
103C2×C4⋊(C3⋊D3)—31no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4⋊(C3⋊D3)—32no—
106C2×C4⋊(C3⋊D3)—32no—
107C2×C4⋊(C3⋊D3)—32no—
108C2×C4⋊(C3⋊D3)—32no—
109C2×C4⋊(C3⋊D3)—38no—
110C2×C4⋊(C3⋊D3)—64no—
111C2×C4⋊(C3⋊D3)—68no—
112C2×C4⋊(C3⋊D3)C2×D312yescomplemented
113C2×C4×C3⋊D3—31no—
114C2×C4×C3⋊D3—31no—
115C2×#634—31no—
116C2×C3×#67—34no—
117C22×C3×Dic3C2×D314yes—
118C2×C3×D12—38no—
119C2×#627—34no—
120C2×#626—34no—
121C2×#626—38no—
122C2×#626—38no—
123C2×#626—38no—
124C2×#626—38no—
125C2×#626—616no—
126C2×#624—34no—
127#3804—616no—
128C2×D3×Dic3—34no—
129C32×C4⋊C4—32no—
130C32×C22⋊C4—24no—
131C32×C22⋊C4—31no—
132C32×C22⋊C4C2×D314yes—
133C32×C22⋊C4C2×D311yescharacteristic
134C32×C42—32no—
135#3752—32no—
136#3752—32no—
137#3752—38no—
138#3752—32no—
139#3752—68no—
140#3752C2×D312yescomplemented
141C3×#217C2×D314yes—
142C3×(C2×D3)⋊C4—38no—
143C3×Dic3⋊C4C2×D318yescomplemented
144C3×C4×Dic3C2×D318yescomplemented
145#3722—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33—22no—
148C22×C33C2×D412yes—
149C22×C33C2411yescharacteristic
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—22no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3⋊(C3⋊D3)—22no—
155C2×C3⋊(C3⋊D3)—44no—
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—22no—
161C2×C3×C3⋊D3—44no—
162C33×C4—22no—
163C3⋊(C3⋊Dic3)—21no—
164C3×C3⋊Dic3—21no—
165C32×Dic3—22no—
166C32×Dic3C2×D412yes—
167C32×Dic3C2×D412yes—
168C22×#67—91no—
169C2×#237—91no—
170C2×#235—94no—
171C22×D12—94no—
172C3×#120—91no—
173C3×#117—34no—
174#953—91no—
175#940—92no—
176#939—92no—
177#937—91no—
178#895—98no—
179#895—98no—
180#885—92no—
181#884—92no—
182#882—34no—
183#882—94no—
184#882—94no—
185#882—94no—
186#882—91no—
187#882—184no—
188#877C3⋊D311yescharacteristic, complemented
189#875—98no—
190#874—98no—
191C23×C32—24no—
192C23×C32C22×D314yes—
193C23×C32C22×D311yescharacteristic
194C22×C3⋊D3—22no—
195C22×C3⋊D3—32no—
196C22×C3⋊D3—31no—
197C22×C3⋊D3—38no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—42no—
200C22×C3⋊D3—68no—
201C22×C3⋊D3—64no—
202C22×C3⋊D3—62no—
203C22×C3⋊D3—68no—
204C22×C3⋊D3—128no—
205C22×C3⋊D3—122no—
206C22×C3⋊D3—128no—
207C22×C3⋊D3—1216no—
208C22×C3⋊D3C22×D312yes—
209C22×C3⋊D3C22×D311yescharacteristic
210C22×C3×D3—38no—
211C22×C3×D3—34no—
212C22×C3×D3—68no—
213C22×C3×D3—128no—
214C2×C32×C4—28no—
215C2×C32×C4—34no—
216C2×C32×C4—31no—
217C2×C32×C4—31no—
218C2×C32×C4—32no—
219C2×C32×C4C22×D318yes—
220C2×C32×C4C22×D312yes—
221#638—22no—
222#638—62no—
223#638—68no—
224#638—128no—
225C2×C3⋊Dic3—31no—
226C2×C3⋊Dic3—34no—
227C2×C3⋊Dic3—64no—
228C2×C3⋊Dic3C22×D311yescharacteristic
229C4⋊(C3⋊D3)—42no—
230C4⋊(C3⋊D3)—64no—
231C4⋊(C3⋊D3)—64no—
232C4⋊(C3⋊D3)—62no—
233C4⋊(C3⋊D3)—122no—
234C4⋊(C3⋊D3)—128no—
235C4⋊(C3⋊D3)—1216no—
236C4⋊(C3⋊D3)—122no—
237C4⋊(C3⋊D3)—122no—
238C4×C3⋊D3—62no—
239C4×C3⋊D3—62no—
240#634—62no—
241C3×#67—68no—
242C2×C3×Dic3—34no—
243C2×C3×Dic3D12116yescomplemented
244C2×C3×Dic3C22×D318yes—
245C2×C3×Dic3C22×D314yes—
246C2×C3×Dic3C22×D314yes—
247C3×D12—128no—
248#627—68no—
249#626—68no—
250#626—616no—
251#626—616no—
252#626—128no—
253#626—128no—
254#624—68no—
255D3×Dic3—68no—
256#535—271noSylow
257C2×C33—22no—
258C2×C33C22×D411yescharacteristic
259C2×C33H(5,𝔽2)12yes—
260C3⋊(C3⋊D3)—42no—
261C3⋊(C3⋊D3)—41no—
262C3×C3⋊D3—41no—
263C3×C3⋊D3—42no—
264C24×C3—91no—
265C23×D3—34no—
266C23×D3—94no—
267C23×D3—91no—
268C23×D3—184no—
269C2×C3×D4—91no—
270C2×C3×D4—92no—
271C22×C3×C4—34no—
272C2×#67—34no—
273C2×#67—94no—
274C2×#67—91no—
275C2×#67—92no—
276C2×#67—92no—
277C2×#67—91no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—184no—
281C2×#67—184no—
282C22×Dic3—91no—
283C22×Dic3C2×C3⋊D311yescharacteristic
284#237—184no—
285#235—1816no—
286C2×D12—38no—
287C2×D12—94no—
288C2×D12—98no—
289C2×D12—98no—
290C2×D12—98no—
291C2×D12—98no—
292C2×D12—98no—
293C2×D12—92no—
294C2×D12—188no—
295C2×D12—1816no—
296C2×C4×D3—94no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×Dic6—94no—
300C2×Dic6—91no—
301C3×C4⋊C4—38no—
302C3×C22⋊C4—24no—
303C3×C22⋊C4—34no—
304C3×C22⋊C4—92no—
305C3×C22⋊C4D3214yes—
306C3×C22⋊C4C2×C3⋊D311yescharacteristic
307C3×C42—38no—
308#217—92no—
309#217C2×C3⋊D311yescharacteristic
310(C2×D3)⋊C4—38no—
311(C2×D3)⋊C4—98no—
312(C2×D3)⋊C4—98no—
313(C2×D3)⋊C4—98no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—188no—
317Dic3⋊C4C2×C3⋊D312yescomplemented
318C4×Dic3C2×C3⋊D312yescomplemented
319C22×C32—22no—
320C22×C32—24no—
321C22×C32—28no—
322C22×C32—28no—
323C22×C32—48no—
324C22×C32C2×D1218yes—
325C22×C32D3×D412yes—
326C22×C32C23×D314yes—
327C22×C32C23×D311yescharacteristic
328C2×C3⋊D3—21no—
329C2×C3⋊D3—22no—
330C2×C3⋊D3—22no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—22no—
333C2×C3⋊D3—44no—
334C2×C3⋊D3—616no—
335C2×C3⋊D3—61no—
336C2×C3⋊D3—62no—
337C2×C3⋊D3—62no—
338C2×C3⋊D3—62no—
339C2×C3⋊D3—62no—
340C2×C3⋊D3—68no—
341C2×C3⋊D3—64no—
342C2×C3⋊D3—68no—
343C2×C3⋊D3—68no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—128no—
346C2×C3⋊D3—128no—
347C2×C3⋊D3—128no—
348C2×C3⋊D3—124no—
349C2×C3⋊D3—124no—
350C2×C3⋊D3—1216no—
351C2×C3⋊D3—1232no—
352C2×C3×D3—64no—
353C2×C3×D3—68no—
354C2×C3×D3—68no—
355C2×C3×D3—68no—
356C2×C3×D3—68no—
357C2×C3×D3—1216no—
358C32×C4—22no—
359C32×C4—28no—
360C32×C4—32no—
361C32×C4—32no—
362C32×C4—48no—
363C32×C4—62no—
364C3⋊Dic3—21no—
365C3⋊Dic3—61no—
366C3⋊Dic3—64no—
367C3⋊Dic3—124no—
368C3×Dic3—28no—
369C3×Dic3—64no—
370C3×Dic3C2×D1218yes—
371C3×Dic3C2×D1218yes—
372C2×#41—271no—
373C22×D4—271no—
374#127—272no—
375#124—272no—
376#121—272no—
377#121—272no—
378#120—91no—
379#120—271no—
380#120—271no—
381#120—271no—
382#117—31no—
383C33#53511yescharacteristic, complemented, Sylow
384C23×C3—24no—
385C23×C3—92no—
386C23×C3—91no—
387C23×C3—182no—
388C23×C3—362no—
389C23×C3C2×D3214yes—
390C23×C3C22×C3⋊D311yescharacteristic
391C22×D3—38no—
392C22×D3—34no—
393C22×D3—68no—
394C22×D3—98no—
395C22×D3—94no—
396C22×D3—92no—
397C22×D3—91no—
398C22×D3—128no—
399C22×D3—188no—
400C22×D3—184no—
401C22×D3—188no—
402C22×D3—182no—
403C22×D3—368no—
404C22×D3—368no—
405C22×D3—362no—
406C22×D3—3616no—
407C3×D4—182no—
408C3×D4—362no—
409C2×C3×C4—28no—
410C2×C3×C4—34no—
411C2×C3×C4—34no—
412C2×C3×C4—316no—
413C2×C3×C4—38no—
414C2×C3×C4—91no—
415C2×C3×C4C2×D3218yes—
416C2×C3×C4C22×C3⋊D312yes—
417#67—68no—
418#67—188no—
419#67—182no—
420#67—182no—
421#67—184no—
422#67—184no—
423#67—368no—
424#67—362no—
425#67—362no—
426C2×Dic3—34no—
427C2×Dic3—94no—
428C2×Dic3—91no—
429C2×Dic3—184no—
430C2×Dic3—182no—
431C2×Dic3C4⋊(C3⋊D3)14yescomplemented
432C2×Dic3C22×C3⋊D312yes—
433C2×Dic3C22×C3⋊D311yescharacteristic
434C2×Dic3C22×C3⋊D311yescharacteristic
435D12—128no—
436D12—1816no—
437D12—1816no—
438D12—188no—
439D12—368no—
440D12—362no—
441D12—3616no—
442D12—368no—
443D12—368no—
444C4×D3—188no—
445C4×D3—188no—
446C4×D3—182no—
447Dic6—188no—
448Dic6—182no—
449C2×C32—22no—
450C2×C32—24no—
451C2×C32—28no—
452C2×C32—28no—
453C2×C32—48no—
454C2×C32C22×D1214yes—
455C2×C32C2×D3×D411yescharacteristic
456C2×C32#100412yes—
457C2×C32#100918yes—
458C3⋊D3—41no—
459C3⋊D3—42no—
460C3⋊D3—1216no—
461C3⋊D3—128no—
462C3⋊D3—121no—
463C3⋊D3—122no—
464C3⋊D3—124no—
465C3⋊D3—244no—
466C3×D3—124no—
467C3×D3—128no—
468C24—91no—
469C24—271no—
470#41—544no—
471C2×Q8—271no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—272no—
475C2×D4—272no—
476C2×D4—271no—
477C2×D4—272no—
478C2×D4—272no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—544no—
482C22×C4—31no—
483C22×C4—271no—
484C22×C4—271no—
485C4⋊C4—32no—
486C22⋊C4—31no—
487C22⋊C4—92no—
488C22⋊C4—272no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C22⋊C4D3×C3⋊D311yescharacteristic
492C42—32no—
493C22×C3—28no—
494C22×C3—22no—
495C22×C3—24no—
496C22×C3—28no—
497C22×C3—48no—
498C22×C3—181no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3—182no—
502C22×C3—182no—
503C22×C3—364no—
504C22×C3D3×D1218yes—
505C22×C3C2×C4⋊(C3⋊D3)12yes—
506C22×C3C22×D3214yes—
507C22×C3C23×C3⋊D311yescharacteristic
508C2×D3—64no—
509C2×D3—68no—
510C2×D3—68no—
511C2×D3—68no—
512C2×D3—68no—
513C2×D3—1216no—
514C2×D3—1816no—
515C2×D3—184no—
516C2×D3—188no—
517C2×D3—188no—
518C2×D3—188no—
519C2×D3—188no—
520C2×D3—181no—
521C2×D3—182no—
522C2×D3—182no—
523C2×D3—182no—
524C2×D3—182no—
525C2×D3—368no—
526C2×D3—368no—
527C2×D3—368no—
528C2×D3—364no—
529C2×D3—3616no—
530C2×D3—364no—
531C2×D3—3632no—
532C3×C4—28no—
533C3×C4—22no—
534C3×C4—38no—
535C3×C4—38no—
536C3×C4—48no—
537C3×C4—68no—
538C3×C4—181no—
539Dic3—22no—
540Dic3—64no—
541Dic3—184no—
542Dic3—181no—
543Dic3—364no—
544Dic3C2×C4⋊(C3⋊D3)12yes—
545Dic3C2×C4⋊(C3⋊D3)12yes—
546C32—24no—
547C32#552514yescomplemented
548C32#559211yescharacteristic, complemented
549C23—92no—
550C23—91no—
551C23—182no—
552C23—272no—
553C23—271no—
554C23—362no—
555C23—542no—
556C23—1082no—
557C23C2×D3×C3⋊D311yescharacteristic
558Q8—542no—
559D4—182no—
560D4—362no—
561D4—544no—
562D4—544no—
563D4—542no—
564D4—542no—
565D4—1082no—
566D4—1082no—
567D4—1082no—
568C2×C4—31no—
569C2×C4—31no—
570C2×C4—34no—
571C2×C4—32no—
572C2×C4—91no—
573C2×C4—271no—
574C2×C4—542no—
575C2×C4—542no—
576C2×C4C2×D3×C3⋊D312yes—
577C2×C3—28no—
578C2×C3—22no—
579C2×C3—24no—
580C2×C3—28no—
581C2×C3—48no—
582C2×C3—362no—
583C2×C3—361no—
584C2×C3C2×D3×D1214yes—
585C2×C3#6431418yes—
586C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
587C2×C3#6436612yes—
588D3—128no—
589D3—124no—
590D3—3616no—
591D3—368no—
592D3—364no—
593D3—362no—
594D3—361no—
595D3—724no—
596C22—22no—
597C22—181no—
598C22—182no—
599C22—182no—
600C22—182no—
601C22—182no—
602C22—364no—
603C22—541no—
604C22—542no—
605C22—542no—
606C22—542no—
607C22—542no—
608C22—1084no—
609C22D3×C4⋊(C3⋊D3)12yes—
610C22C22×D3×C3⋊D311yescharacteristic
611C4—22no—
612C4—32no—
613C4—32no—
614C4—62no—
615C4—181no—
616C4—541no—
617C3—24no—
618C3#1059408014yescomplemented
619C3#1059599611yescharacteristic, complemented
620C2—22no—
621C2—362no—
622C2—361no—
623C2—1082no—
624C2—1081no—
625C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
626C2#1172458912yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6292C2semidirect
2. C2×#89031C2semidirect
3. #11723923C2semidirect
4. C3×#64091C2semidirect
5. C32×#877C2semidirect
6. #11723639C2semidirect
7. #11723636C2semidirect
8. #11723517C2semidirect
9. #11723486C2semidirect
10. #11723482C2semidirect
11. #11723416C2semidirect
17. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
19. C2×C3×C4⋊(C3⋊D3)C22semidirect
21. C2×#6293C22semidirect
25. C2×#6292C22semidirect
26. C2×#6292C22semidirect
27. C2×#6292C22semidirect
28. C2×#6289C22semidirect
30. C2×Dic3×C3⋊D3C22semidirect
32. #88859C22semidirect
33. C3×#3752C22semidirect
35. C32×Dic3⋊C4C22semidirect
36. C32×C4×Dic3C22semidirect
37. #88804C22semidirect
38. #88803C22semidirect
51. #64091D3semidirect
55. C3×#877D3semidirect
78. C2×C32×Dic3D4semidirect
112. C2×C4⋊(C3⋊D3)C2×D3semidirect
140. #3752C2×D3semidirect
143. C3×Dic3⋊C4C2×D3semidirect
144. C3×C4×Dic3C2×D3semidirect
188. #877C3⋊D3semidirect
243. C2×C3×Dic3D12semidirect
317. Dic3⋊C4C2×C3⋊D3semidirect
318. C4×Dic3C2×C3⋊D3semidirect
383. C33#535semidirect
431. C2×Dic3C4⋊(C3⋊D3)semidirect
547. C32#5525semidirect
548. C32#5592semidirect
618. C3#10594080semidirect
619. C3#10595996semidirect

Maximal subgroups

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

subgroupindex
C22×#62922
C2×#890312
#117239232
C3×#640912
C32×#8772
#117236392
#117236362
#117235172
#117234862
#117234822
#117234162
#105959963
#105940803