Finite Group Databasebeta

← #13177573

Subgroups — #13177573

Order 1728.

2948 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723639C212yesmaximal, complemented
6#11723628C212yesmaximal, complemented
7#11723627C212yesmaximal, complemented
8#11723623C212yesmaximal, complemented
9C4×#6292C211yesmaximal, characteristic, complemented
10#11723484C211yesmaximal, characteristic, complemented
11#11723456C211yesmaximal, characteristic, complemented
12#10596147—31nomaximal
13#10595243—34nomaximal
14C23×C3⋊(C3⋊D3)C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yes—
17D4×C3⋊(C3⋊D3)—24no—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×#638C2212yescomplemented
21C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C32×Dic6C2211yescharacteristic, complemented
23C2×#6292C2211yescharacteristic
24C2×#6292C2212yescomplemented
25C2×#6289C2212yescomplemented
26C2×#6289C2212yescomplemented
27#89018—24no—
28C3×#3757C2212yescomplemented
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
30C32×#217C2212yescomplemented
31C32×#217C2212yescomplemented
32C32×C4×Dic3C2211yescharacteristic, complemented
33#88807C2212yescomplemented
34Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
35#88804C2211yescharacteristic
36#88804C2212yes—
37#88804C2212yescomplemented
38#88803C2211yescharacteristic, complemented
39C2×#3832—31no—
40C2×D4×C3⋊D3—31no—
41C2×D4×C3⋊D3—34no—
42C2×D4×C3⋊D3—64no—
43C2×#3799—34no—
44C32×#124—31no—
45#64155—31no—
46#64147D311yescharacteristic, complemented
47#64093—32no—
48#64092—32no—
49#64091—32no—
50C22⋊C4×C3⋊D3—32no—
51C4×C4⋊(C3⋊D3)—31no—
52C4×C4⋊(C3⋊D3)—31no—
53C3×#938—34no—
54C3×#936D314yescomplemented
55#63981—38no—
56#63975—38no—
57#63974—38no—
58#63958—38no—
59C4×#626—34no—
60#63894—34no—
61#63871—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)—24no—
64C22×C3⋊(C3⋊D3)—24no—
65C22×C3⋊(C3⋊D3)—42no—
66C22×C3⋊(C3⋊D3)C2312yes—
67C22×C3⋊(C3⋊D3)C2311yescharacteristic
68C22×C3×C3⋊D3C2311yescharacteristic
69C33×D4—22no—
70C2×C33×C4C2311yescharacteristic
71#6312—24no—
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C4⋊(C3⋊(C3⋊D3))D414yescomplemented
74C4×C3⋊(C3⋊D3)—22no—
75C3×#638—42no—
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×C4×C3⋊D3—22no—
79C2×C32×Dic3C2311yescharacteristic
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2311yescharacteristic
82C32×Dic6—22no—
83#6292—24no—
84#6292—22no—
85#6289—24no—
86#6289—42no—
87#5676—94no—
88#5612—91no—
89C23×C3⋊D3—32no—
90C23×C3⋊D3—38no—
91C23×C3⋊D3—68no—
92C2×C32×Q8—31no—
93C2×C32×D4—24no—
94C2×C32×D4C2×D314yes—
95C2×C32×D4C2×D311yescharacteristic
96C2×#638—32no—
97C2×#638—38no—
98C2×#638—68no—
99C2×#638C2×D312yescomplemented
100#3832—64no—
101D4×C3⋊D3—64no—
102D4×C3⋊D3—616no—
103D4×C3⋊D3—632no—
104C2×C4⋊(C3⋊D3)—31no—
105C2×C4⋊(C3⋊D3)—34no—
106C2×C4⋊(C3⋊D3)—31no—
107C2×C4⋊(C3⋊D3)—32no—
108C2×C4⋊(C3⋊D3)—64no—
109C2×C4×C3⋊D3—31no—
110C2×C4×C3⋊D3—34no—
111C2×C4×C3⋊D3—32no—
112C2×C4×C3⋊D3—32no—
113C2×C4×C3⋊D3—64no—
114C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
115C2×C3×#67—38no—
116C2×C3×C4×D3—34no—
117C2×C3×Dic6C2×D314yescomplemented
118C2×#626—38no—
119C2×#626—34no—
120C2×#624—38no—
121C2×#624—38no—
122#3799—616no—
123C32×C22⋊C4—32no—
124C32×C22⋊C4—32no—
125C32×C42—31no—
126#3757C2×D312yescomplemented
127#3752—31no—
128#3752—31no—
129#3752—32no—
130#3752—32no—
131C4⋊(C3⋊Dic3)—31no—
132C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
133#3750—32no—
134C3×#217—38no—
135C3×#217C2×D318yescomplemented
136C3×#217C2×D318yescomplemented
137C3×C4⋊Dic3—34no—
138C3×C4×Dic3C2×D314yescomplemented
139Dic3⋊Dic3—38no—
140Dic3⋊Dic3—34no—
141#3722—38no—
142#3722—38no—
143#3722—34no—
144(C2×D3)⋊Dic3—34no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33—22no—
148C22×C33C2411yescharacteristic
149C2×C3⋊(C3⋊D3)—21no—
150C2×C3⋊(C3⋊D3)—22no—
151C2×C3⋊(C3⋊D3)—21no—
152C2×C3⋊(C3⋊D3)—22no—
153C2×C3⋊(C3⋊D3)—48no—
154C2×C3⋊(C3⋊D3)C2×D414yes—
155C2×C3⋊(C3⋊D3)#4112yescomplemented
156C2×C3×C3⋊D3—21no—
157C2×C3×C3⋊D3—21no—
158C2×C3×C3⋊D3—21no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—21no—
161C3×C3⋊Dic3—21no—
162C3×C3⋊Dic3—22no—
163C32×Dic3—22no—
164C32×Dic3—21no—
165C32×Dic3—21no—
166C2×#239—94no—
167C2×D3×D4—94no—
168C2×D3×D4—91no—
169C2×D3×D4—184no—
170C2×#235—91no—
171C3×#124—34no—
172C3×#121—91no—
173#946—94no—
174#938—34no—
175#938—92no—
176#937—92no—
177#936C3⋊D311yescharacteristic, complemented
178C4×#67—91no—
179#897—91no—
180#893—91no—
181#884—98no—
182#883—98no—
183#883—92no—
184#882—98no—
185D3×C22⋊C4—98no—
186D3×C22⋊C4—92no—
187C4×D12—94no—
188C4×D12—94no—
189C23×C32—28no—
190C23×C32C22×D318yes—
191C23×C32C22×D312yes—
192C22×C3⋊D3—31no—
193C22×C3⋊D3—32no—
194C22×C3⋊D3—34no—
195C22×C3⋊D3—38no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3—68no—
198C22×C3⋊D3—64no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—616no—
201C22×C3⋊D3—632no—
202C22×C3⋊D3—616no—
203C22×C3⋊D3—1216no—
204C22×C3⋊D3—122no—
205C22×C3⋊D3—128no—
206C22×C3⋊D3—1216no—
207C22×C3⋊D3C22×D311yescharacteristic
208C22×C3×D3—34no—
209C32×Q8—62no—
210C32×D4—216no—
211C32×D4—28no—
212C32×D4—22no—
213C2×C32×C4—24no—
214C2×C32×C4—32no—
215C2×C32×C4—31no—
216C2×C32×C4—31no—
217C2×C32×C4C22×D314yes—
218C2×C32×C4C22×D311yescharacteristic
219#638—42no—
220#638—64no—
221#638—616no—
222#638—632no—
223C2×C3⋊Dic3—31no—
224C2×C3⋊Dic3—34no—
225C2×C3⋊Dic3—64no—
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D311yescharacteristic
228C4⋊(C3⋊D3)—34no—
229C4⋊(C3⋊D3)—316no—
230C4⋊(C3⋊D3)—616no—
231C4⋊(C3⋊D3)—62no—
232C4⋊(C3⋊D3)—64no—
233C4×C3⋊D3—22no—
234C4×C3⋊D3—64no—
235C4×C3⋊D3—616no—
236C4×C3⋊D3—68no—
237C4×C3⋊D3—62no—
238C4×C3⋊D3—122no—
239C3×#67—128no—
240C2×C3×Dic3—38no—
241C2×C3×Dic3—34no—
242C2×C3×Dic3C22×D314yes—
243C2×C3×Dic3C22×D314yes—
244C2×C3×Dic3C22×D318yes—
245C3×C4×D3—68no—
246C3×Dic6—28no—
247#626—616no—
248#626—68no—
249#624—616no—
250#624—128no—
251#546—271noSylow
252C2×C33—42no—
253C2×C33C22×D411yescharacteristic
254C2×C33C2×#4111yescharacteristic
255C2×C33H(5,𝔽2)11yescharacteristic
256C3⋊(C3⋊D3)—22no—
257C3⋊(C3⋊D3)—24no—
258C3×C3⋊D3—41no—
259C23×D3—98no—
260C23×D3—92no—
261C23×D3—188no—
262C2×C3×Q8—34no—
263C2×C3×D4—24no—
264C2×C3×D4—92no—
265C2×C3×D4D3214yes—
266C2×C3×D4C2×C3⋊D311yescharacteristic
267C22×C3×C4—91no—
268C2×#67—38no—
269C2×#67—98no—
270C2×#67—92no—
271C2×#67—91no—
272C2×#67—92no—
273C2×#67—188no—
274#239—1816no—
275D3×D4—1816no—
276D3×D4—184no—
277D3×D4—1832no—
278#235—184no—
279C2×D12—94no—
280C2×D12—91no—
281C2×D12—94no—
282C2×D12—98no—
283C2×D12—184no—
284C2×C4×D3—34no—
285C2×C4×D3—98no—
286C2×C4×D3—94no—
287C2×C4×D3—91no—
288C2×C4×D3—92no—
289C2×C4×D3—98no—
290C2×C4×D3—92no—
291C2×C4×D3—184no—
292C2×Dic6C2×C3⋊D311yescharacteristic, complemented
293C3×C4⋊C4—91no—
294C3×C22⋊C4—38no—
295C3×C22⋊C4—38no—
296C3×C22⋊C4—92no—
297C3×C42—34no—
298#217—38no—
299#217—91no—
300#217C2×C3⋊D312yescomplemented
301#217C2×C3⋊D312yescomplemented
302(C2×D3)⋊C4—94no—
303(C2×D3)⋊C4—98no—
304(C2×D3)⋊C4—98no—
305(C2×D3)⋊C4—94no—
306(C2×D3)⋊C4—91no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309C4⋊Dic3—34no—
310C4⋊Dic3—94no—
311C4⋊Dic3—92no—
312Dic3⋊C4—98no—
313Dic3⋊C4—91no—
314C4×Dic3C2×C3⋊D311yescharacteristic, complemented
315C22×C32—216no—
316C22×C32—28no—
317C22×C32—22no—
318C22×C32—22no—
319C22×C32—22no—
320C22×C32—24no—
321C22×C32—28no—
322C22×C32—28no—
323C22×C32—48no—
324C22×C32—48no—
325C22×C32C23×D314yes—
326C22×C32C23×D311yescharacteristic
327C2×C3⋊D3—21no—
328C2×C3⋊D3—21no—
329C2×C3⋊D3—21no—
330C2×C3⋊D3—34no—
331C2×C3⋊D3—32no—
332C2×C3⋊D3—316no—
333C2×C3⋊D3—38no—
334C2×C3⋊D3—68no—
335C2×C3⋊D3—616no—
336C2×C3⋊D3—68no—
337C2×C3⋊D3—68no—
338C2×C3⋊D3—64no—
339C2×C3⋊D3—61no—
340C2×C3⋊D3—62no—
341C2×C3⋊D3—61no—
342C2×C3⋊D3—62no—
343C2×C3⋊D3—64no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—68no—
346C2×C3⋊D3—128no—
347C2×C3⋊D3—128no—
348C2×C3⋊D3—128no—
349C2×C3⋊D3—1232no—
350C2×C3⋊D3—1264no—
351C2×C3×D3—64no—
352C2×C3×D3—64no—
353C2×C3×D3—64no—
354C32×C4—28no—
355C32×C4—62no—
356C32×C4—61no—
357C32×C4—61no—
358C32×C4D3×D418yes—
359C32×C4D3×D412yes—
360C3⋊Dic3—21no—
361C3⋊Dic3—22no—
362C3⋊Dic3—68no—
363C3⋊Dic3—64no—
364C3⋊Dic3—61no—
365C3×Dic3—28no—
366C3×Dic3—24no—
367C3×Dic3—24no—
368C3×Dic3—64no—
369C3×Dic3—68no—
370C2×#41—271no—
371C22×D4—271no—
372#124—31no—
373#123—272no—
374C22⋊Q8—271no—
375#121—91no—
376#121—272no—
377#120—272no—
378C4×D4—271no—
379C4×D4—271no—
380C2×C22⋊C4—272no—
381C33#54611yescharacteristic, complemented, Sylow
382C23×C3—28no—
383C23×C3—91no—
384C23×C3C2×D3218yes—
385C23×C3C22×C3⋊D312yes—
386C22×D3—34no—
387C22×D3—94no—
388C22×D3—98no—
389C22×D3—91no—
390C22×D3—92no—
391C22×D3—184no—
392C22×D3—188no—
393C22×D3—1816no—
394C22×D3—184no—
395C22×D3—184no—
396C22×D3—1816no—
397C22×D3—1832no—
398C22×D3—3616no—
399C22×D3—368no—
400C22×D3—362no—
401C22×D3—3616no—
402C3×Q8—68no—
403C3×D4—216no—
404C3×D4—28no—
405C3×D4—22no—
406C3×D4—362no—
407C2×C3×C4—24no—
408C2×C3×C4—38no—
409C2×C3×C4—34no—
410C2×C3×C4—34no—
411C2×C3×C4—91no—
412C2×C3×C4—92no—
413C2×C3×C4—182no—
414C2×C3×C4C2×D3214yes—
415C2×C3×C4C22×C3⋊D311yescharacteristic
416#67—128no—
417#67—184no—
418#67—182no—
419#67—1816no—
420#67—184no—
421#67—1832no—
422C2×Dic3—38no—
423C2×Dic3—34no—
424C2×Dic3—94no—
425C2×Dic3—91no—
426C2×Dic3—184no—
427C2×Dic3C22×C3⋊D311yescharacteristic
428C2×Dic3C22×C3⋊D311yescharacteristic
429C2×Dic3C22×C3⋊D312yes—
430D12—916no—
431D12—94no—
432D12—1816no—
433D12—188no—
434D12—1816no—
435C4×D3—68no—
436C4×D3—184no—
437C4×D3—1816no—
438C4×D3—1816no—
439C4×D3—188no—
440C4×D3—182no—
441C4×D3—362no—
442C4×D3—368no—
443Dic6—22no—
444C2×C32—24no—
445C2×C32—24no—
446C2×C32—24no—
447C2×C32—416no—
448C2×C32—48no—
449C2×C32—42no—
450C2×C32C2×D3×D414yes—
451C2×C32C2×D3×D411yescharacteristic
452C2×C32#100411yescharacteristic
453C2×C32D3×#4114yes—
454C2×C32D3×#4111yescharacteristic
455C2×C32#100914yes—
456C3⋊D3—41no—
457C3⋊D3—616no—
458C3⋊D3—68no—
459C3⋊D3—64no—
460C3⋊D3—62no—
461C3⋊D3—616no—
462C3⋊D3—1216no—
463C3×D3—124no—
464C24—272no—
465#41—544no—
466C2×Q8—31no—
467C2×D4—92no—
468C2×D4—271no—
469C2×D4—272no—
470C2×D4—272no—
471C2×D4—271no—
472C2×D4—544no—
473C2×D4D3×C3⋊D311yescharacteristic
474C22×C4—91no—
475C22×C4—272no—
476C22×C4—272no—
477C22×C4—271no—
478C4⋊C4—91no—
479C4⋊C4—271no—
480C4⋊C4—272no—
481C22⋊C4—32no—
482C22⋊C4—32no—
483C22⋊C4—92no—
484C22⋊C4—271no—
485C22⋊C4—272no—
486C22⋊C4—272no—
487C22⋊C4—271no—
488C42—31no—
489C22×C3—216no—
490C22×C3—28no—
491C22×C3—28no—
492C22×C3—28no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—24no—
497C22×C3—48no—
498C22×C3—48no—
499C22×C3—181no—
500C22×C3—181no—
501C22×C3—181no—
502C22×C3C22×D3214yes—
503C22×C3C23×C3⋊D311yescharacteristic
504C2×D3—64no—
505C2×D3—64no—
506C2×D3—64no—
507C2×D3—98no—
508C2×D3—916no—
509C2×D3—94no—
510C2×D3—92no—
511C2×D3—188no—
512C2×D3—1816no—
513C2×D3—188no—
514C2×D3—188no—
515C2×D3—188no—
516C2×D3—188no—
517C2×D3—184no—
518C2×D3—184no—
519C2×D3—181no—
520C2×D3—182no—
521C2×D3—182no—
522C2×D3—181no—
523C2×D3—368no—
524C2×D3—368no—
525C2×D3—368no—
526C2×D3—3632no—
527C2×D3—3664no—
528C3×C4—28no—
529C3×C4—64no—
530C3×C4—64no—
531C3×C4—68no—
532C3×C4—181no—
533C3×C4—182no—
534C3×C4#381118yes—
535C3×C4D4×C3⋊D312yes—
536Dic3—22no—
537Dic3—21no—
538Dic3—21no—
539Dic3—68no—
540Dic3—64no—
541Dic3—188no—
542Dic3—184no—
543Dic3—181no—
544C32—24no—
545C32#561211yescharacteristic, complemented
546C32#567614yescomplemented
547C23—91no—
548C23—271no—
549C23—272no—
550C23—544no—
551C23—544no—
552C23—1082no—
553C23C2×D3×C3⋊D312yes—
554Q8—62no—
555D4—22no—
556D4—274no—
557D4—362no—
558D4—544no—
559D4—542no—
560D4—544no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—32no—
564C2×C4—91no—
565C2×C4—92no—
566C2×C4—182no—
567C2×C4—271no—
568C2×C4—544no—
569C2×C4—542no—
570C2×C4—1082no—
571C2×C4C2×D3×C3⋊D311yescharacteristic
572C2×C3—24no—
573C2×C3—24no—
574C2×C3—24no—
575C2×C3—416no—
576C2×C3—42no—
577C2×C3—48no—
578C2×C3—361no—
579C2×C3#6431214yes—
580C2×C3#6431414yes—
581C2×C3C2×#381114yes—
582C2×C3C2×D4×C3⋊D311yescharacteristic
583C2×C3#41×C3⋊D311yescharacteristic
584C2×C3#6436611yescharacteristic
585D3—124no—
586D3—1816no—
587D3—182no—
588D3—184no—
589D3—188no—
590D3—1816no—
591D3—3616no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—181no—
596C22—181no—
597C22—181no—
598C22—272no—
599C22—274no—
600C22—542no—
601C22—542no—
602C22—541no—
603C22—541no—
604C22—1088no—
605C22C22×D3×C3⋊D311yescharacteristic
606C4—61no—
607C4—61no—
608C4—62no—
609C4—182no—
610C4—181no—
611C4—541no—
612C4#8903912yes—
613C3—24no—
614C3#1059524314yescomplemented
615C3#1059614711yescharacteristic, complemented
616C2—42no—
617C2—361no—
618C2—542no—
619C2—544no—
620C2#1172458611yescharacteristic
621C2#1172458911yescharacteristic
622C2C2×#8903911yescharacteristic
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×D4×C3⋊(C3⋊D3)C2semidirect
2. C2×#89018C2semidirect
3. C3×#64147C2semidirect
4. C32×#936C2semidirect
5. #11723639C2semidirect
6. #11723628C2semidirect
7. #11723627C2semidirect
8. #11723623C2semidirect
9. C4×#6292C2semidirect
10. #11723484C2semidirect
11. #11723456C2semidirect
20. C2×C3×#638C22semidirect
21. C2×C3×C4×C3⋊D3C22semidirect
22. C2×C32×Dic6C22semidirect
24. C2×#6292C22semidirect
25. C2×#6289C22semidirect
26. C2×#6289C22semidirect
28. C3×#3757C22semidirect
29. C3×C4⋊(C3⋊Dic3)C22semidirect
30. C32×#217C22semidirect
31. C32×#217C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88807C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
37. #88804C22semidirect
38. #88803C22semidirect
46. #64147D3semidirect
54. C3×#936D3semidirect
73. C4⋊(C3⋊(C3⋊D3))D4semidirect
99. C2×#638C2×D3semidirect
114. C2×C4×C3⋊D3C2×D3semidirect
117. C2×C3×Dic6C2×D3semidirect
126. #3757C2×D3semidirect
132. C4⋊(C3⋊Dic3)C2×D3semidirect
135. C3×#217C2×D3semidirect
136. C3×#217C2×D3semidirect
138. C3×C4×Dic3C2×D3semidirect
155. C2×C3⋊(C3⋊D3)#41semidirect
177. #936C3⋊D3semidirect
292. C2×Dic6C2×C3⋊D3semidirect
300. #217C2×C3⋊D3semidirect
301. #217C2×C3⋊D3semidirect
314. C4×Dic3C2×C3⋊D3semidirect
381. C33#546semidirect
545. C32#5612semidirect
546. C32#5676semidirect
614. C3#10595243semidirect
615. C3#10596147semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#890182
C3×#641472
C32×#9362
#117236392
#117236282
#117236272
#117236232
C4×#62922
#117234842
#117234562
#105961473
#105952433