Finite Group Databasebeta

← #13177580

Subgroups — #13177580

Order 1728.

2660 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723636C212yesmaximal, complemented
5#11723632C212yesmaximal, complemented
6#11723609C212yesmaximal, complemented
7C2×#88803C212yesmaximal, complemented
8#11723532C211yesmaximal, characteristic, complemented
9C4×#6292C211yesmaximal, characteristic, complemented
10#11723439C211yesmaximal, characteristic, complemented
11#11723431C211yesmaximal, characteristic, complemented
12#10596025—31nomaximal
13#10595246—34nomaximal
14C23×C3×C3⋊D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yescomplemented
17C22×C3⋊(C3⋊Dic3)C2212yescomplemented
18#89078—24no—
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
20C2×#6308C2211yescharacteristic, complemented
21C2×C3×#638C2212yes—
22C3×D4×C3⋊D3—24no—
23C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
24C2×C3×C4×C3⋊D3C2211yescharacteristic
25C22×C32×Dic3C2212yescomplemented
26C2×#6292C2211yescharacteristic
27C2×#6292C2212yescomplemented
28C32×#217C2212yescomplemented
29C32×C4⋊Dic3C2211yescharacteristic, complemented
30C32×C4×Dic3C2211yescharacteristic, complemented
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
32Dic3⋊(C3⋊Dic3)C2212yescomplemented
33#88804C2211yescharacteristic, complemented
34#88803C2211yescharacteristic
35#88803C2212yes—
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38#88803C2212yescomplemented
39C2×#3830—31no—
40C2×#3830—34no—
41C2×#3830—64no—
42C2×D4×C3⋊D3D311yescharacteristic, complemented
43C2×C3×D3×D4—34no—
44C32×C4×D4—31no—
45#64139—32no—
46C2×#3752—32no—
47#64106—31no—
48#64104—31no—
49#64095—32no—
50#64091—32no—
51#64084—31no—
52C4×C4⋊(C3⋊D3)—31no—
53C3×D4×Dic3D314yescomplemented
54#63978—38no—
55#63977—38no—
56#63962—38no—
57C2×(C2×D3)⋊Dic3—38no—
58#63916—34no—
59C4×#626—34no—
60#63860—34no—
61#63857—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C3×C3⋊D3—24no—
65C22×C3×C3⋊D3—24no—
66C22×C3×C3⋊D3—42no—
67C22×C3×C3⋊D3C2312yes—
68C22×C3×C3⋊D3C2311yescharacteristic
69C33×D4D414yescomplemented
70C2×C33×C4C2311yescharacteristic
71#6312—24no—
72C2×C3⋊(C3⋊Dic3)—24no—
73C2×C3⋊(C3⋊Dic3)C2312yes—
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—22no—
76#6308—22no—
77C3×#638—24no—
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4⋊(C3⋊D3)—22no—
80C3×C4×C3⋊D3—22no—
81C2×C32×Dic3—42no—
82C2×C32×Dic3C2311yescharacteristic
83C2×C32×Dic3C2312yes—
84C2×C32×Dic3C2311yescharacteristic
85#6292—22no—
86#6292—42no—
87#5806—91no—
88#5554—94no—
89C23×C3⋊D3C2×D312yes—
90C23×C3×D3—38no—
91C2×C32×D4—24no—
92C2×C32×D4C2×D314yes—
93C2×C32×D4C2×D311yescharacteristic
94C22×C32×C4—32no—
95C2×#638—32no—
96C2×#638—38no—
97C2×#638—68no—
98C2×#638C2×D312yes—
99C22×C3⋊Dic3—32no—
100C22×C3⋊Dic3—38no—
101C22×C3⋊Dic3—68no—
102#3830—632no—
103#3830—616no—
104#3830—64no—
105D4×C3⋊D3—24no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4⋊(C3⋊D3)—32no—
108C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
109C2×C4×C3⋊D3—31no—
110C2×C4×C3⋊D3—34no—
111C2×C4×C3⋊D3—64no—
112C2×C4×C3⋊D3C2×D311yescharacteristic
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—64no—
116C2×C3×#67—38no—
117C22×C3×Dic3C2×D318yescomplemented
118C3×D3×D4—616no—
119C2×C3×D12—34no—
120C2×C3×C4×D3—34no—
121C2×#626—34no—
122C2×#626—38no—
123C32×C4⋊C4—31no—
124C32×C22⋊C4—32no—
125C32×C42—31no—
126#3752—31no—
127#3752—32no—
128#3752—32no—
129#3752—32no—
130#3752—32no—
131#3752—31no—
132C4⋊(C3⋊Dic3)—31no—
133C4⋊(C3⋊Dic3)—32no—
134C3×#217C2×D318yescomplemented
135C3×C4⋊Dic3C2×D314yescomplemented
136C3×C4×Dic3C2×D314yescomplemented
137Dic3⋊Dic3—38no—
138Dic3⋊Dic3—34no—
139#3722—34no—
140(C2×D3)⋊Dic3—38no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144(C2×D3)⋊Dic3—38no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2×D414yes—
148C22×C33C2411yescharacteristic
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—21no—
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—21no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—48no—
158C2×C3×C3⋊D3#4112yescomplemented
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—22no—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—21no—
163C32×Dic3—21no—
164C32×Dic3—22no—
165C32×Dic3—21no—
166C22×C3×D4—91no—
167C2×#237—94no—
168C2×#237—91no—
169C2×#237—184no—
170C2×D3×D4—34no—
171C3×C4×D4—34no—
172#953—92no—
173C2×#217—92no—
174#939—92no—
175#938—91no—
176#936—91no—
177#935—92no—
178D4×Dic3C3⋊D311yescharacteristic, complemented
179#930—98no—
180C4×#67—91no—
181C2×(C2×D3)⋊C4—98no—
182#924—91no—
183#897—94no—
184#895—94no—
185#886—98no—
186#882—98no—
187#875—94no—
188C4×D12—94no—
189C23×C32—28no—
190C23×C32C22×D318yes—
191C23×C32C22×D312yes—
192C22×C3⋊D3—24no—
193C22×C3⋊D3—24no—
194C22×C3⋊D3—31no—
195C22×C3⋊D3—34no—
196C22×C3⋊D3—42no—
197C22×C3⋊D3—64no—
198C22×C3⋊D3C22×D312yes—
199C22×C3⋊D3C22×D311yescharacteristic
200C22×C3×D3—34no—
201C22×C3×D3—38no—
202C22×C3×D3—616no—
203C22×C3×D3—616no—
204C22×C3×D3—128no—
205C32×D4—216no—
206C32×D4D12116yescomplemented
207C32×D4#6714yescomplemented
208C2×C32×C4—24no—
209C2×C32×C4—31no—
210C2×C32×C4—32no—
211C2×C32×C4—31no—
212C2×C32×C4—122no—
213C2×C32×C4C22×D314yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—24no—
216#638—632no—
217#638—616no—
218#638—64no—
219C2×C3⋊Dic3—32no—
220C2×C3⋊Dic3—31no—
221C2×C3⋊Dic3—38no—
222C2×C3⋊Dic3—34no—
223C2×C3⋊Dic3—68no—
224C2×C3⋊Dic3—64no—
225C2×C3⋊Dic3—632no—
226C2×C3⋊Dic3—616no—
227C2×C3⋊Dic3—64no—
228C2×C3⋊Dic3C22×D311yescharacteristic
229C4⋊(C3⋊D3)—22no—
230C4⋊(C3⋊D3)—62no—
231C4⋊(C3⋊D3)—122no—
232C4×C3⋊D3—22no—
233C4×C3⋊D3—616no—
234C4×C3⋊D3—68no—
235C4×C3⋊D3—62no—
236#634—616no—
237#634—68no—
238#634—62no—
239C3×#67—616no—
240C2×C3×Dic3—34no—
241C2×C3×Dic3—48no—
242C2×C3×Dic3C22×D314yes—
243C2×C3×Dic3C22×D314yes—
244C2×C3×Dic3C22×D318yes—
245C3×D12—68no—
246C3×C4×D3—68no—
247#626—68no—
248#626—128no—
249#546—271noSylow
250C2×C33—24no—
251C2×C33C22×D411yescharacteristic
252C2×C33C2×#4111yescharacteristic
253C2×C33H(5,𝔽2)11yescharacteristic
254C3⋊(C3⋊D3)—41no—
255C3×C3⋊D3—22no—
256C3×C3⋊D3—42no—
257C24×C3—92no—
258C23×D3—38no—
259C2×C3×D4—24no—
260C2×C3×D4—92no—
261C2×C3×D4—91no—
262C2×C3×D4—184no—
263C2×C3×D4D3214yes—
264C2×C3×D4C2×C3⋊D311yescharacteristic
265C22×C3×C4—38no—
266C22×C3×C4—91no—
267C2×#67—38no—
268C2×#67—98no—
269C2×#67—92no—
270C2×#67—91no—
271C2×#67—92no—
272C2×#67—188no—
273C22×Dic3—98no—
274C22×Dic3—92no—
275C22×Dic3—188no—
276C22×Dic3C2×C3⋊D312yescomplemented
277#237—1832no—
278#237—1816no—
279#237—184no—
280D3×D4—616no—
281C2×D12—34no—
282C2×D12—94no—
283C2×D12—98no—
284C2×C4×D3—34no—
285C2×C4×D3—94no—
286C2×C4×D3—91no—
287C2×C4×D3—184no—
288C2×Dic6—94no—
289C2×Dic6—91no—
290C2×Dic6—184no—
291C3×C4⋊C4—34no—
292C3×C22⋊C4—38no—
293C3×C42—34no—
294#217—92no—
295#217—91no—
296#217—92no—
297#217—92no—
298#217—92no—
299#217C2×C3⋊D312yescomplemented
300(C2×D3)⋊C4—94no—
301(C2×D3)⋊C4—94no—
302(C2×D3)⋊C4—98no—
303(C2×D3)⋊C4—98no—
304(C2×D3)⋊C4—98no—
305(C2×D3)⋊C4—98no—
306(C2×D3)⋊C4—91no—
307C4⋊Dic3—94no—
308C4⋊Dic3—98no—
309C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
310Dic3⋊C4—92no—
311Dic3⋊C4—91no—
312C4×Dic3C2×C3⋊D311yescharacteristic, complemented
313C22×C32—22no—
314C22×C32—22no—
315C22×C32—24no—
316C22×C32—216no—
317C22×C32—28no—
318C22×C32—28no—
319C22×C32—48no—
320C22×C32—48no—
321C22×C32C2×D12116yes—
322C22×C32C2×#6714yes—
323C22×C32C23×D314yes—
324C22×C32C23×D311yescharacteristic
325C2×C3⋊D3—21no—
326C2×C3⋊D3—21no—
327C2×C3⋊D3—22no—
328C2×C3⋊D3—22no—
329C2×C3⋊D3—22no—
330C2×C3⋊D3—48no—
331C2×C3⋊D3—68no—
332C2×C3⋊D3—64no—
333C2×C3⋊D3—61no—
334C2×C3⋊D3—61no—
335C2×C3⋊D3—61no—
336C2×C3⋊D3—64no—
337C2×C3⋊D3—64no—
338C2×C3⋊D3—124no—
339C2×C3⋊D3—124no—
340C2×C3⋊D3#23712yescomplemented
341C2×C3×D3—38no—
342C2×C3×D3—64no—
343C2×C3×D3—64no—
344C2×C3×D3—68no—
345C2×C3×D3—68no—
346C2×C3×D3—68no—
347C2×C3×D3—1232no—
348C32×C4—28no—
349C32×C4—61no—
350C32×C4—62no—
351C32×C4—61no—
352C32×C4C2×D1218yes—
353C32×C4C2×#6712yes—
354C3⋊Dic3—21no—
355C3⋊Dic3—616no—
356C3⋊Dic3—68no—
357C3⋊Dic3—68no—
358C3⋊Dic3—64no—
359C3⋊Dic3—62no—
360C3⋊Dic3—61no—
361C3×Dic3—28no—
362C3×Dic3—24no—
363C3×Dic3—24no—
364C3×Dic3—64no—
365C2×#41—271no—
366C22×D4—91no—
367#124—271no—
368#123—272no—
369C22⋊Q8—271no—
370#121—271no—
371#121—272no—
372#120—272no—
373C4×D4—31no—
374C4×D4—271no—
375C2×C22⋊C4—272no—
376C33#54611yescharacteristic, complemented, Sylow
377C23×C3—28no—
378C23×C3—91no—
379C23×C3—92no—
380C23×C3—184no—
381C23×C3—184no—
382C23×C3—362no—
383C23×C3C2×D3218yes—
384C23×C3C22×C3⋊D312yes—
385C22×D3—34no—
386C22×D3—38no—
387C22×D3—616no—
388C22×D3—616no—
389C22×D3—94no—
390C22×D3—91no—
391C22×D3—128no—
392C22×D3—184no—
393C3×D4—216no—
394C3×D4—184no—
395C3×D4—182no—
396C3×D4#626116yescomplemented
397C3×D4C4⋊(C3⋊D3)14yescomplemented
398C2×C3×C4—24no—
399C2×C3×C4—34no—
400C2×C3×C4—38no—
401C2×C3×C4—34no—
402C2×C3×C4—91no—
403C2×C3×C4—128no—
404C2×C3×C4—182no—
405C2×C3×C4C2×D3214yes—
406C2×C3×C4C22×C3⋊D311yescharacteristic
407#67—616no—
408#67—182no—
409#67—1832no—
410#67—1816no—
411#67—184no—
412#67—362no—
413C2×Dic3—34no—
414C2×Dic3—42no—
415C2×Dic3—98no—
416C2×Dic3—94no—
417C2×Dic3—92no—
418C2×Dic3—91no—
419C2×Dic3—188no—
420C2×Dic3—184no—
421C2×Dic3—1832no—
422C2×Dic3—1816no—
423C2×Dic3—184no—
424C2×Dic3C22×C3⋊D311yescharacteristic
425C2×Dic3C22×C3⋊D311yescharacteristic
426C2×Dic3C22×C3⋊D312yes—
427D12—68no—
428D12—188no—
429D12—368no—
430C4×D3—68no—
431C4×D3—1816no—
432C4×D3—188no—
433C4×D3—182no—
434Dic6—1816no—
435Dic6—188no—
436Dic6—182no—
437C2×C32—24no—
438C2×C32—24no—
439C2×C32—24no—
440C2×C32—24no—
441C2×C32—216no—
442C2×C32—416no—
443C2×C32C22×D1214yes—
444C2×C32C2×#23711yescharacteristic
445C2×C32#100414yes—
446C2×C32#100411yescharacteristic
447C2×C32D3×#4114yes—
448C2×C32C22×#6711yescharacteristic
449C3⋊D3—22no—
450C3⋊D3—42no—
451C3⋊D3—128no—
452C3⋊D3—121no—
453C3⋊D3—124no—
454C3×D3—68no—
455C3×D3—128no—
456C24—92no—
457#41—544no—
458C2×Q8—271no—
459C2×D4—92no—
460C2×D4—91no—
461C2×D4—184no—
462C2×D4—271no—
463C2×D4—272no—
464C2×D4—272no—
465C2×D4D3×C3⋊D311yescharacteristic
466C22×C4—32no—
467C22×C4—91no—
468C22×C4—272no—
469C22×C4—271no—
470C4⋊C4—31no—
471C4⋊C4—271no—
472C4⋊C4—272no—
473C22⋊C4—32no—
474C22⋊C4—271no—
475C22⋊C4—271no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C42—31no—
481C22×C3—28no—
482C22×C3—28no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—24no—
486C22×C3—216no—
487C22×C3—48no—
488C22×C3—48no—
489C22×C3—92no—
490C22×C3—181no—
491C22×C3—181no—
492C22×C3—182no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—368no—
496C22×C3C2×#626116yes—
497C22×C3C2×C4⋊(C3⋊D3)14yes—
498C22×C3C22×D3214yes—
499C22×C3C23×C3⋊D311yescharacteristic
500C2×D3—38no—
501C2×D3—64no—
502C2×D3—64no—
503C2×D3—68no—
504C2×D3—68no—
505C2×D3—68no—
506C2×D3—1232no—
507C2×D3—188no—
508C2×D3—184no—
509C2×D3—184no—
510C2×D3—184no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—181no—
514C2×D3—364no—
515C2×D3—364no—
516C3×C4—28no—
517C3×C4—64no—
518C3×C4—68no—
519C3×C4—64no—
520C3×C4—181no—
521C3×C4C2×#62618yes—
522C3×C4C2×C4⋊(C3⋊D3)12yes—
523Dic3—21no—
524Dic3—22no—
525Dic3—21no—
526Dic3—64no—
527Dic3—1816no—
528Dic3—188no—
529Dic3—188no—
530Dic3—184no—
531Dic3—182no—
532Dic3—181no—
533C32—24no—
534C32#555414yescomplemented
535C32#580611yescharacteristic, complemented
536C23—91no—
537C23—92no—
538C23—184no—
539C23—184no—
540C23—271no—
541C23—362no—
542C23C2×D3×C3⋊D312yes—
543Q8—542no—
544D4—184no—
545D4—182no—
546D4—542no—
547D4—544no—
548D4—1082no—
549D4#629214yescomplemented
550C2×C4—31no—
551C2×C4—31no—
552C2×C4—32no—
553C2×C4—91no—
554C2×C4—122no—
555C2×C4—182no—
556C2×C4—271no—
557C2×C4—272no—
558C2×C4—544no—
559C2×C4—542no—
560C2×C4C2×D3×C3⋊D311yescharacteristic
561C2×C3—216no—
562C2×C3—24no—
563C2×C3—24no—
564C2×C3—24no—
565C2×C3—24no—
566C2×C3—416no—
567C2×C3—182no—
568C2×C3—362no—
569C2×C3D3×#23714yes—
570C2×C3#6431314yes—
571C2×C3C22×#62614yes—
572C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
573C2×C3#6436111yescharacteristic
574C2×C3#41×C3⋊D311yescharacteristic
575D3—68no—
576D3—128no—
577D3—368no—
578D3—364no—
579D3—361no—
580C22—22no—
581C22—22no—
582C22—92no—
583C22—181no—
584C22—181no—
585C22—182no—
586C22—182no—
587C22—182no—
588C22—368no—
589C22—541no—
590C22—541no—
591C22—541no—
592C22C2×#629214yes—
593C22C22×D3×C3⋊D311yescharacteristic
594C4—61no—
595C4—61no—
596C4—62no—
597C4—181no—
598C4—541no—
599C4—542no—
600C4C2×#629212yes—
601C3—24no—
602C3#1059524614yescomplemented
603C3#1059602511yescharacteristic, complemented
604C2—24no—
605C2—182no—
606C2—362no—
607C2—1081no—
608C2C3⋊D3×#23711yescharacteristic
609C2#1172458711yescharacteristic
610C2C22×#629211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C2×C3×D4×C3⋊D3C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723636C2semidirect
5. #11723632C2semidirect
6. #11723609C2semidirect
7. C2×#88803C2semidirect
8. #11723532C2semidirect
9. C4×#6292C2semidirect
10. #11723439C2semidirect
11. #11723431C2semidirect
16. C2×#6312C22semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
19. C2×C4×C3⋊(C3⋊D3)C22semidirect
20. C2×#6308C22semidirect
25. C22×C32×Dic3C22semidirect
27. C2×#6292C22semidirect
28. C32×#217C22semidirect
29. C32×C4⋊Dic3C22semidirect
30. C32×C4×Dic3C22semidirect
31. Dic3⋊(C3⋊Dic3)C22semidirect
32. Dic3⋊(C3⋊Dic3)C22semidirect
33. #88804C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
38. #88803C22semidirect
42. C2×D4×C3⋊D3D3semidirect
53. C3×D4×Dic3D3semidirect
69. C33×D4D4semidirect
117. C22×C3×Dic3C2×D3semidirect
134. C3×#217C2×D3semidirect
135. C3×C4⋊Dic3C2×D3semidirect
136. C3×C4×Dic3C2×D3semidirect
158. C2×C3×C3⋊D3#41semidirect
178. D4×Dic3C3⋊D3semidirect
206. C32×D4D12semidirect
207. C32×D4#67semidirect
276. C22×Dic3C2×C3⋊D3semidirect
299. #217C2×C3⋊D3semidirect
309. C4⋊Dic3C2×C3⋊D3semidirect
312. C4×Dic3C2×C3⋊D3semidirect
340. C2×C3⋊D3#237semidirect
376. C33#546semidirect
396. C3×D4#626semidirect
397. C3×D4C4⋊(C3⋊D3)semidirect
534. C32#5554semidirect
535. C32#5806semidirect
549. D4#6292semidirect
602. C3#10595246semidirect
603. C3#10596025semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×C3×D4×C3⋊D32
C32×D4×Dic32
#117236362
#117236322
#117236092
C2×#888032
#117235322
C4×#62922
#117234392
#117234312
#105960253
#105952463