Finite Group Databasebeta

← #13176542

Subgroups — #13176542

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
2C2×#89020C211yesmaximal, characteristic, complemented
3#11723936C211yesmaximal, characteristic, complemented
4C3×#64101C211yesmaximal, characteristic, complemented
5C32×#895C211yesmaximal, characteristic, complemented
6#11723542C212yesmaximal, complemented
7#11723540C212yesmaximal, complemented
8#11723531C211yesmaximal, characteristic, complemented
9#11723524C212yesmaximal, complemented
10#11723516C211yesmaximal, characteristic, complemented
11#11723419C212yesmaximal, complemented
12#10596078—31nomaximal
13#10594628—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4⋊(C3⋊(C3⋊D3))C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C32×D12C2211yescharacteristic
20C2×C32×D12C2212yescomplemented
21C2×C32×C4×D3C2211yescharacteristic, complemented
22C2×#6293C2211yescharacteristic, complemented
23C2×#6292C2211yescharacteristic
24C2×#6291C2212yes—
25C2×#6291C2211yescharacteristic, complemented
26C2×#6291C2212yescomplemented
27C2×#6291C2212yescomplemented
28C2×#6290C2211yescharacteristic
29C3⋊D3×D12—24no—
30#89020—24no—
31C33×C4⋊C4C2211yescharacteristic, complemented
32#88859C2212yescomplemented
33C3×#3752C2212yes—
34C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
35C3×C4×C3⋊Dic3C2212yescomplemented
36C32×(C2×D3)⋊C4C2212yescomplemented
37#88804C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C2×#3828—31no—
41C2×D3×D12—34no—
42C2×#3798—34no—
43C32×#121—31no—
44#64149—32no—
45#64147—31no—
46#64146—32no—
47#64139—31no—
48#64104—31no—
49#64104—34no—
50#64104—64no—
51#64101D311yescharacteristic, complemented
52#64094—32no—
53#64091—32no—
54C3×#895D314yescomplemented
55C3×#892—34no—
56#63923—38no—
57#63922—38no—
58#63916—34no—
59#63911—38no—
60#63909—34no—
61#63852—38no—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2312yes—
66C22×C32×D3C2311yescharacteristic
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×D3×C3⋊D3—42no—
70C2×C33×C4C2312yes—
71C2×C33×C4C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C4⋊(C3⋊(C3⋊D3))—22no—
74C4⋊(C3⋊(C3⋊D3))—42no—
75C4×C3⋊(C3⋊D3)—22no—
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×C4×C3⋊D3D414yescomplemented
79C2×C32×Dic3C2311yescharacteristic
80C32×D12—22no—
81C32×D12—42no—
82C32×C4×D3—22no—
83#6293—22no—
84#6292—22no—
85#6291—22no—
86#6291—24no—
87#6291—42no—
88#6291—42no—
89#6290—22no—
90#5607—94no—
91#5525—91no—
92C23×C3⋊D3—32no—
93C22×D32—38no—
94C2×C32×D4—31no—
95C2×C32×D4—32no—
96C22×C32×C4—31no—
97C2×#638—31no—
98C2×#638—32no—
99C2×#638—31no—
100C2×#638—32no—
101C2×#638—32no—
102D4×C3⋊D3—64no—
103#3828—64no—
104C2×C4⋊(C3⋊D3)—31no—
105C2×C4⋊(C3⋊D3)—31no—
106C2×C4⋊(C3⋊D3)—34no—
107C2×C4⋊(C3⋊D3)—32no—
108C2×C4⋊(C3⋊D3)—38no—
109C2×C4⋊(C3⋊D3)—64no—
110C2×C4⋊(C3⋊D3)—68no—
111C2×C4×C3⋊D3—31no—
112C2×C4×C3⋊D3—34no—
113C2×C4×C3⋊D3—64no—
114C2×C4×C3⋊D3C2×D311yescharacteristic
115C2×#634—31no—
116C2×C3×D12C2×D314yes—
117C2×C3×D12C2×D318yescomplemented
118C2×C3×C4×D3—34no—
119C2×C3×C4×D3C2×D314yescomplemented
120C2×#627—34no—
121C2×#626—34no—
122C2×#626—38no—
123C2×#626—34no—
124C2×#626—38no—
125C2×#626—38no—
126C2×#625—34no—
127D3×D12—616no—
128#3798—616no—
129C32×C4⋊C4—24no—
130C32×C4⋊C4C2×D314yescomplemented
131C32×C4⋊C4C2×D311yescharacteristic, complemented
132C32×C22⋊C4—32no—
133#3757—32no—
134#3752—32no—
135#3752—38no—
136#3752—32no—
137#3752—68no—
138#3752C2×D312yes—
139C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
140C4×C3⋊Dic3C2×D312yescomplemented
141C3×(C2×D3)⋊C4—38no—
142C3×(C2×D3)⋊C4C2×D318yescomplemented
143C3×C4⋊Dic3—34no—
144C3×C4×Dic3—38no—
145#3722—38no—
146(C2×D3)⋊Dic3—38no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3⋊(C3⋊D3)—22no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3×C3⋊D3—21no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3C2×D412yes—
157C2×C32×D3—21no—
158C2×C32×D3—22no—
159C2×C32×D3—21no—
160C2×C32×D3—22no—
161C2×C32×D3—21no—
162C2×C32×D3—22no—
163D3×C3⋊D3—48no—
164C33×C4—22no—
165C33×C4C2×D412yes—
166C3⋊(C3⋊Dic3)—21no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3C2×D412yes—
169C32×Dic3—21no—
170C2×D3×D4—94no—
171C2×#235—94no—
172C2×#235—91no—
173C22×D12—91no—
174C3×#121—34no—
175C3×#117—91no—
176#940—98no—
177#938—94no—
178#937—98no—
179#930—94no—
180#895—91no—
181#895—91no—
182#895—94no—
183#895—91no—
184#895—184no—
185#895C3⋊D311yescharacteristic, complemented
186#892—34no—
187#885—98no—
188#882—98no—
189#882—92no—
190#882—92no—
191#875—92no—
192#874—92no—
193C23×C32—32no—
194C23×C32—31no—
195C22×C3⋊D3—32no—
196C22×C3⋊D3—31no—
197C22×C3⋊D3—38no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—68no—
201C22×C3⋊D3—64no—
202C22×C3⋊D3—122no—
203C22×C3⋊D3—122no—
204C22×C3⋊D3C22×D311yescharacteristic
205C22×C3×D3—34no—
206C22×C3×D3C22×D318yes—
207C22×C3×D3C22×D314yes—
208C2×D32—616no—
209C2×D32—128no—
210C2×D32—128no—
211C32×D4—62no—
212C32×D4—122no—
213C2×C32×C4—24no—
214C2×C32×C4—28no—
215C2×C32×C4—31no—
216C2×C32×C4—62no—
217C2×C32×C4C22×D318yes—
218C2×C32×C4C22×D314yes—
219C2×C32×C4C22×D312yes—
220C2×C32×C4C22×D311yescharacteristic
221#638—62no—
222#638—62no—
223#638—64no—
224#638—122no—
225#638—122no—
226C2×C3⋊Dic3—31no—
227C2×C3⋊Dic3—34no—
228C2×C3⋊Dic3—64no—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C4⋊(C3⋊D3)—62no—
232C4⋊(C3⋊D3)—62no—
233C4⋊(C3⋊D3)—68no—
234C4⋊(C3⋊D3)—128no—
235C4⋊(C3⋊D3)—122no—
236C4⋊(C3⋊D3)—128no—
237C4⋊(C3⋊D3)—1216no—
238C4×C3⋊D3—62no—
239C4×C3⋊D3—68no—
240C4×C3⋊D3—128no—
241C4×C3⋊D3D1214yescomplemented
242#634—62no—
243C2×C3×Dic3—34no—
244C2×C3×Dic3—38no—
245C2×C3×Dic3C22×D314yes—
246C3×D12—28no—
247C3×D12—48no—
248C3×C4×D3—28no—
249C3×C4×D3—316no—
250#627—68no—
251#626—68no—
252#626—68no—
253#626—616no—
254#626—128no—
255#626—128no—
256#625—68no—
257#535—271noSylow
258C2×C33C22×D411yescharacteristic
259C2×C33H(5,𝔽2)11yescharacteristic
260C2×C33H(5,𝔽2)11yescharacteristic
261C3⋊(C3⋊D3)—41no—
262C3⋊(C3⋊D3)—42no—
263C3×C3⋊D3—22no—
264C32×D3—42no—
265C32×D3—41no—
266C23×D3—98no—
267C23×D3—92no—
268C2×C3×D4—34no—
269C2×C3×D4—38no—
270C22×C3×C4—34no—
271C22×C3×C4—91no—
272C2×#67—98no—
273C2×#67—94no—
274C2×#67—98no—
275C2×#67—98no—
276C2×#67—94no—
277C2×#67—91no—
278C2×#67—91no—
279D3×D4—1816no—
280#235—184no—
281#235—1816no—
282C2×D12—94no—
283C2×D12—91no—
284C2×D12—92no—
285C2×D12—94no—
286C2×D12—91no—
287C2×D12—92no—
288C2×D12—92no—
289C2×D12—98no—
290C2×D12—92no—
291C2×D12—184no—
292C2×D12—188no—
293C2×D12—184no—
294C2×D12C2×C3⋊D311yescharacteristic
295C2×D12C2×C3⋊D312yescomplemented
296C2×C4×D3—34no—
297C2×C4×D3—94no—
298C2×C4×D3—91no—
299C2×C4×D3—184no—
300C2×C4×D3C2×C3⋊D311yescharacteristic, complemented
301C2×Dic6—94no—
302C2×Dic6—91no—
303C3×C4⋊C4—24no—
304C3×C4⋊C4—91no—
305C3×C4⋊C4D3214yescomplemented
306C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
307C3×C22⋊C4—38no—
308C3×C22⋊C4—92no—
309C3×C42—92no—
310#217—98no—
311(C2×D3)⋊C4—38no—
312(C2×D3)⋊C4—98no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—98no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317(C2×D3)⋊C4—188no—
318(C2×D3)⋊C4C2×C3⋊D312yescomplemented
319C4⋊Dic3—34no—
320C4×Dic3—38no—
321C22×C32—24no—
322C22×C32—61no—
323C22×C32—62no—
324C22×C32—61no—
325C22×C32—62no—
326C22×C32—61no—
327C22×C32—62no—
328C22×C32C23×D314yes—
329C22×C32C23×D311yescharacteristic
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3⋊D3—68no—
333C2×C3⋊D3—61no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—64no—
336C2×C3⋊D3—61no—
337C2×C3⋊D3—62no—
338C2×C3⋊D3—61no—
339C2×C3⋊D3—62no—
340C2×C3⋊D3—64no—
341C2×C3⋊D3—68no—
342C2×C3⋊D3—68no—
343C2×C3⋊D3—64no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—124no—
346C2×C3⋊D3—124no—
347C2×C3⋊D3—128no—
348C2×C3⋊D3—128no—
349C2×C3⋊D3—128no—
350C2×C3⋊D3—128no—
351C2×C3⋊D3C2×D1212yes—
352C2×C3×D3—24no—
353C2×C3×D3—28no—
354C2×C3×D3—24no—
355C2×C3×D3—28no—
356C2×C3×D3—24no—
357C2×C3×D3—28no—
358C2×C3×D3—38no—
359C2×C3×D3—64no—
360C2×C3×D3—64no—
361D32—1232no—
362C32×C4—22no—
363C32×C4—28no—
364C32×C4—28no—
365C32×C4—48no—
366C32×C4—61no—
367C32×C4C2×D1212yes—
368C32×C4D3×D418yes—
369C3⋊Dic3—22no—
370C3⋊Dic3—61no—
371C3⋊Dic3—64no—
372C3⋊Dic3—124no—
373C3⋊Dic3C2×D1212yes—
374C3×Dic3—24no—
375C3×Dic3—38no—
376C3×Dic3—68no—
377C2×#41—271no—
378C22×D4—271no—
379#127—272no—
380#124—272no—
381#121—31no—
382#121—271no—
383#121—271no—
384#121—271no—
385#120—272no—
386#120—272no—
387#117—91no—
388C33#53511yescharacteristic, complemented, Sylow
389C23×C3—38no—
390C23×C3—34no—
391C23×C3—91no—
392C22×D3—34no—
393C22×D3—98no—
394C22×D3—94no—
395C22×D3—92no—
396C22×D3—91no—
397C22×D3—184no—
398C22×D3—188no—
399C22×D3—184no—
400C22×D3—1816no—
401C22×D3—362no—
402C22×D3—362no—
403C22×D3—368no—
404C22×D3—368no—
405C22×D3C22×C3⋊D312yes—
406C22×D3C22×C3⋊D311yescharacteristic
407C3×D4—68no—
408C3×D4—128no—
409C2×C3×C4—24no—
410C2×C3×C4—28no—
411C2×C3×C4—34no—
412C2×C3×C4—68no—
413C2×C3×C4—91no—
414C2×C3×C4—94no—
415C2×C3×C4—92no—
416C2×C3×C4C2×D3218yes—
417C2×C3×C4C2×D3214yes—
418C2×C3×C4C22×C3⋊D312yes—
419C2×C3×C4C22×C3⋊D311yescharacteristic
420#67—182no—
421#67—188no—
422#67—1816no—
423#67—182no—
424#67—188no—
425#67—368no—
426#67—368no—
427C2×Dic3—34no—
428C2×Dic3—38no—
429C2×Dic3—94no—
430C2×Dic3—91no—
431C2×Dic3—184no—
432C2×Dic3C22×C3⋊D311yescharacteristic
433D12—22no—
434D12—42no—
435D12—188no—
436D12—184no—
437D12—182no—
438D12—188no—
439D12—182no—
440D12—368no—
441D12—368no—
442D12—362no—
443D12—3616no—
444D12—362no—
445D12—362no—
446C4×D3—22no—
447C4×D3—316no—
448C4×D3—188no—
449C4×D3—182no—
450C4×D3—368no—
451Dic6—182no—
452Dic6—188no—
453C2×C32—24no—
454C2×C32—24no—
455C2×C32—24no—
456C2×C32—122no—
457C2×C32—121no—
458C2×C32C22×D1211yescharacteristic
459C2×C32C2×D3×D414yes—
460C2×C32#100414yes—
461C2×C32#100914yes—
462C2×C32#100911yescharacteristic
463C2×C32#100911yescharacteristic
464C3⋊D3—22no—
465C3⋊D3—128no—
466C3⋊D3—121no—
467C3⋊D3—122no—
468C3⋊D3—124no—
469C3⋊D3—128no—
470C3⋊D3—248no—
471C3×D3—48no—
472C3×D3—44no—
473C3×D3—68no—
474C24—272no—
475#41—544no—
476C2×Q8—271no—
477C2×D4—31no—
478C2×D4—32no—
479C2×D4—271no—
480C2×D4—272no—
481C2×D4—271no—
482C2×D4—272no—
483C2×D4—272no—
484C2×D4—271no—
485C2×D4—271no—
486C2×D4—272no—
487C2×D4—544no—
488C22×C4—31no—
489C22×C4—91no—
490C22×C4—271no—
491C4⋊C4—91no—
492C4⋊C4D3×C3⋊D311yescharacteristic, complemented
493C22⋊C4—32no—
494C22⋊C4—92no—
495C22⋊C4—272no—
496C22⋊C4—272no—
497C22⋊C4—272no—
498C42—92no—
499C22×C3—24no—
500C22×C3—64no—
501C22×C3—68no—
502C22×C3—64no—
503C22×C3—68no—
504C22×C3—64no—
505C22×C3—68no—
506C22×C3—92no—
507C22×C3—181no—
508C22×C3—181no—
509C22×C3C22×D3214yes—
510C22×C3C23×C3⋊D311yescharacteristic
511C2×D3—21no—
512C2×D3—22no—
513C2×D3—21no—
514C2×D3—22no—
515C2×D3—21no—
516C2×D3—22no—
517C2×D3—38no—
518C2×D3—64no—
519C2×D3—64no—
520C2×D3—188no—
521C2×D3—184no—
522C2×D3—188no—
523C2×D3—184no—
524C2×D3—188no—
525C2×D3—184no—
526C2×D3—188no—
527C2×D3—181no—
528C2×D3—182no—
529C2×D3—181no—
530C2×D3—182no—
531C2×D3—181no—
532C2×D3—182no—
533C2×D3—364no—
534C2×D3—364no—
535C2×D3—368no—
536C2×D3—368no—
537C2×D3—368no—
538C2×D3—368no—
539C2×D3—3632no—
540C3×C4—28no—
541C3×C4—22no—
542C3×C4—28no—
543C3×C4—48no—
544C3×C4—64no—
545C3×C4—92no—
546C3×C4—182no—
547C3×C4D3×D1218yes—
548C3×C4D4×C3⋊D312yes—
549Dic3—21no—
550Dic3—38no—
551Dic3—68no—
552Dic3—184no—
553Dic3—181no—
554Dic3—364no—
555C32—24no—
556C32#552511yescharacteristic, complemented
557C32#560714yescomplemented
558C23—32no—
559C23—31no—
560C23—91no—
561C23—272no—
562C23—271no—
563C23—544no—
564C23—1082no—
565C23—1082no—
566Q8—542no—
567D4—62no—
568D4—122no—
569D4—542no—
570D4—544no—
571D4—542no—
572D4—542no—
573D4—542no—
574D4—1082no—
575D4—1082no—
576D4—1082no—
577C2×C4—31no—
578C2×C4—62no—
579C2×C4—91no—
580C2×C4—94no—
581C2×C4—92no—
582C2×C4—271no—
583C2×C4—542no—
584C2×C4C2×D3×C3⋊D312yes—
585C2×C4C2×D3×C3⋊D311yescharacteristic
586C2×C3—24no—
587C2×C3—24no—
588C2×C3—24no—
589C2×C3—128no—
590C2×C3—124no—
591C2×C3—182no—
592C2×C3C2×D3×D1214yes—
593C2×C3#6431414yes—
594C2×C3#6432014yes—
595C2×C3C2×D4×C3⋊D311yescharacteristic
596C2×C3#6436111yescharacteristic
597C2×C3#6436611yescharacteristic
598D3—42no—
599D3—41no—
600D3—68no—
601D3—368no—
602D3—368no—
603D3—364no—
604D3—362no—
605D3—361no—
606D3—728no—
607C22—61no—
608C22—62no—
609C22—61no—
610C22—62no—
611C22—61no—
612C22—62no—
613C22—92no—
614C22—181no—
615C22—181no—
616C22—541no—
617C22—542no—
618C22—541no—
619C22—542no—
620C22—541no—
621C22—542no—
622C22—1088no—
623C22C22×D3×C3⋊D311yescharacteristic
624C4—22no—
625C4—61no—
626C4—92no—
627C4—182no—
628C4—541no—
629C4C3⋊D3×D1212yes—
630C3—24no—
631C3#1059462814yescomplemented
632C3#1059607811yescharacteristic, complemented
633C2—122no—
634C2—121no—
635C2—182no—
636C2—1082no—
637C2—1081no—
638C2C2×C3⋊D3×D1211yescharacteristic
639C2#1172458811yescharacteristic
640C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. C2×#89020C2semidirect
3. #11723936C2semidirect
4. C3×#64101C2semidirect
5. C32×#895C2semidirect
6. #11723542C2semidirect
7. #11723540C2semidirect
8. #11723531C2semidirect
9. #11723524C2semidirect
10. #11723516C2semidirect
11. #11723419C2semidirect
16. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
22. C2×#6293C22semidirect
25. C2×#6291C22semidirect
26. C2×#6291C22semidirect
27. C2×#6291C22semidirect
31. C33×C4⋊C4C22semidirect
32. #88859C22semidirect
34. C3×C4⋊(C3⋊Dic3)C22semidirect
35. C3×C4×C3⋊Dic3C22semidirect
36. C32×(C2×D3)⋊C4C22semidirect
37. #88804C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
51. #64101D3semidirect
54. C3×#895D3semidirect
78. C3×C4×C3⋊D3D4semidirect
117. C2×C3×D12C2×D3semidirect
119. C2×C3×C4×D3C2×D3semidirect
130. C32×C4⋊C4C2×D3semidirect
131. C32×C4⋊C4C2×D3semidirect
139. C4⋊(C3⋊Dic3)C2×D3semidirect
140. C4×C3⋊Dic3C2×D3semidirect
142. C3×(C2×D3)⋊C4C2×D3semidirect
185. #895C3⋊D3semidirect
241. C4×C3⋊D3D12semidirect
295. C2×D12C2×C3⋊D3semidirect
300. C2×C4×D3C2×C3⋊D3semidirect
305. C3×C4⋊C4D32semidirect
306. C3×C4⋊C4C2×C3⋊D3semidirect
318. (C2×D3)⋊C4C2×C3⋊D3semidirect
388. C33#535semidirect
492. C4⋊C4D3×C3⋊D3semidirect
556. C32#5525semidirect
557. C32#5607semidirect
631. C3#10594628semidirect
632. C3#10596078semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×#890202
#117239362
C3×#641012
C32×#8952
#117235422
#117235402
#117235312
#117235242
#117235162
#117234192
#105960783
#105946283