Finite Group Databasebeta

← #13176555

Subgroups — #13176555

Order 1728.

2380 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
1C2×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723934C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C32×#897C211yesmaximal, characteristic, complemented
6#11723541C212yesmaximal, complemented
7#11723539C212yesmaximal, complemented
8#11723532C212yesmaximal, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10#11723504C212yesmaximal, complemented
11#11723436C211yesmaximal, characteristic, complemented
12#10596101—31nomaximal
13#10594633—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2212yescomplemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
19C2×C3×C4×C3⋊D3C2211yescharacteristic, complemented
20C2×C32×C4×D3C2211yescharacteristic
21C2×C32×Dic6C2211yescharacteristic, complemented
22C2×#6292C2212yes—
23C2×#6292C2212yescomplemented
24C2×#6290C2211yescharacteristic
25C2×#6289C2212yescomplemented
26D3×C4⋊(C3⋊D3)—24no—
27#89018—24no—
28C33×C4⋊C4C2211yescharacteristic, complemented
29#88859C2212yescomplemented
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31C3×#3752C2212yescomplemented
32C32×(C2×D3)⋊C4C2212yes—
33C32×C4⋊Dic3C2212yescomplemented
34#88806C2211yescharacteristic, complemented
35#88803C2211yescharacteristic, complemented
36#88803C2212yescomplemented
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
39C2×#3832—31no—
40C22×C4⋊(C3⋊D3)—31no—
41C2×D3×D12—34no—
42C2×#3799—34no—
43C32×C22⋊Q8—31no—
44#64156—31no—
45C4×#638—31no—
46#64104—32no—
47#64104D311yescharacteristic, complemented
48#64103—32no—
49#64102—31no—
50#64102—34no—
51#64102—64no—
52#64091—32no—
53C22⋊C4×C3⋊D3—32no—
54C3×#897D314yescomplemented
55C3×#895—34no—
56#63922—38no—
57#63921—38no—
58#63916—38no—
59C4×#625—34no—
60#63863—34no—
61#63854—38no—
62C22×C3⋊(C3⋊D3)C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C22×C32×D3C2311yescharacteristic
66C2×D3×C3⋊D3—24no—
67C2×D3×C3⋊D3—24no—
68C2×D3×C3⋊D3—42no—
69C2×C33×C4C2312yes—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C4⋊(C3⋊(C3⋊D3))D414yescomplemented
74C4×C3⋊(C3⋊D3)—42no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×C4⋊(C3⋊D3)—22no—
77C3×C4⋊(C3⋊D3)—42no—
78C3×C4×C3⋊D3—22no—
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C32×C4×D3—22no—
82C32×Dic6—22no—
83#6292—24no—
84#6292—24no—
85#6290—22no—
86#6289—24no—
87#5630—94no—
88#5612—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×Q8—31no—
92C22×C32×C4—31no—
93C2×#638—31no—
94#3832—64no—
95C2×C4⋊(C3⋊D3)—31no—
96C2×C4⋊(C3⋊D3)—34no—
97C2×C4⋊(C3⋊D3)—32no—
98C2×C4⋊(C3⋊D3)—32no—
99C2×C4⋊(C3⋊D3)—64no—
100C2×C4⋊(C3⋊D3)—64no—
101C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
102C2×C4⋊(C3⋊D3)C2×D312yescomplemented
103C2×C4×C3⋊D3—32no—
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—38no—
106C2×C4×C3⋊D3—68no—
107C2×C4×C3⋊D3C2×D311yescharacteristic, complemented
108C2×C3×D12—34no—
109C2×C3×D12—38no—
110C2×C3×C4×D3—34no—
111C2×C3×C4×D3C2×D314yes—
112C2×C3×Dic6C2×D314yescomplemented
113C2×#626—38no—
114C2×#626—38no—
115C2×#625—34no—
116C2×#624—38no—
117D3×D12—616no—
118#3799—616no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4—32no—
121C32×C4⋊C4C2×D314yescomplemented
122C32×C4⋊C4C2×D311yescharacteristic, complemented
123C32×C22⋊C4—32no—
124#3757—31no—
125#3752—32no—
126#3752—32no—
127#3752—31no—
128#3752—32no—
129#3752—38no—
130#3752—68no—
131#3752C2×D312yescomplemented
132#3750—31no—
133C4×C3⋊Dic3—31no—
134C4×C3⋊Dic3—34no—
135C4×C3⋊Dic3—64no—
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4C2×D318yes—
138C3×C4⋊Dic3C2×D318yescomplemented
139#3724—34no—
140(C2×D3)⋊Dic3—34no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)—22no—
146C2×C3⋊(C3⋊D3)—22no—
147C2×C3⋊(C3⋊D3)C2×D414yes—
148C2×C3×C3⋊D3—21no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3—21no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—21no—
153C2×C3×C3⋊D3—22no—
154C2×C32×D3—21no—
155C2×C32×D3—21no—
156C2×C32×D3#4112yescomplemented
157D3×C3⋊D3—48no—
158C33×C4—22no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—21no—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—21no—
163C32×Dic3—22no—
164C32×Dic3—21no—
165C2×#239—94no—
166C2×D3×D4—91no—
167C2×#235—91no—
168C22×D12—94no—
169C3×C22⋊Q8—34no—
170C3×#121—91no—
171#947—94no—
172#938—92no—
173#937—92no—
174#936—91no—
175C4×#67—94no—
176C4×#67—91no—
177#897C3⋊D311yescharacteristic, complemented
178#895—34no—
179#895—98no—
180#894—98no—
181#893—94no—
182#893—91no—
183#893—184no—
184#883—92no—
185#882—98no—
186D3×C22⋊C4—98no—
187D3×C22⋊C4—92no—
188C23×C32—31no—
189C22×C3⋊D3—32no—
190C22×C3⋊D3—38no—
191C22×C3⋊D3—68no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—122no—
195C22×C3⋊D3C22×D312yes—
196C22×C3⋊D3C22×D311yescharacteristic
197C22×C3×D3—38no—
198C22×C3×D3—34no—
199C22×C3×D3C22×D314yes—
200C2×D32—616no—
201C2×D32—616no—
202C2×D32—128no—
203C32×Q8—62no—
204C2×C32×C4—24no—
205C2×C32×C4—28no—
206C2×C32×C4—31no—
207C2×C32×C4—32no—
208C2×C32×C4—62no—
209C2×C32×C4C22×D318yes—
210C2×C32×C4C22×D314yes—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D311yescharacteristic
213#638—62no—
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—34no—
217C2×C3⋊Dic3—34no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C4⋊(C3⋊D3)—22no—
222C4⋊(C3⋊D3)—34no—
223C4⋊(C3⋊D3)—316no—
224C4⋊(C3⋊D3)—42no—
225C4⋊(C3⋊D3)—616no—
226C4⋊(C3⋊D3)—64no—
227C4⋊(C3⋊D3)—64no—
228C4×C3⋊D3—22no—
229C4×C3⋊D3—64no—
230C4×C3⋊D3—122no—
231C4×C3⋊D3—128no—
232C4×C3⋊D3—1216no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3C22×D318yes—
235C2×C3×Dic3C22×D314yes—
236C3×D12—68no—
237C3×D12—128no—
238C3×C4×D3—28no—
239C3×C4×D3—68no—
240C3×Dic6—28no—
241#626—616no—
242#626—616no—
243#625—68no—
244#624—616no—
245#546—271noSylow
246C2×C33C22×D411yescharacteristic
247C2×C33C2×#4111yescharacteristic
248C2×C33H(5,𝔽2)11yescharacteristic
249C3⋊(C3⋊D3)—24no—
250C3×C3⋊D3—42no—
251C3×C3⋊D3—41no—
252C32×D3—22no—
253C23×D3—98no—
254C23×D3—92no—
255C2×C3×Q8—34no—
256C2×C3×D4—91no—
257C2×C3×D4—92no—
258C22×C3×C4—34no—
259C22×C3×C4—91no—
260C2×#67—94no—
261C2×#67—91no—
262C2×#67—92no—
263C2×#67—92no—
264#239—1816no—
265D3×D4—184no—
266#235—184no—
267C2×D12—34no—
268C2×D12—38no—
269C2×D12—94no—
270C2×D12—91no—
271C2×D12—98no—
272C2×D12—98no—
273C2×D12—184no—
274C2×D12—1816no—
275C2×C4×D3—34no—
276C2×C4×D3—98no—
277C2×C4×D3—92no—
278C2×C4×D3—98no—
279C2×C4×D3—92no—
280C2×C4×D3—188no—
281C2×C4×D3C2×C3⋊D311yescharacteristic
282C2×Dic6C2×C3⋊D311yescharacteristic, complemented
283C3×C4⋊C4—24no—
284C3×C4⋊C4—38no—
285C3×C4⋊C4D3214yescomplemented
286C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
287C3×C22⋊C4—38no—
288C3×C22⋊C4—92no—
289#217—94no—
290#217—92no—
291#217—92no—
292#217—91no—
293(C2×D3)⋊C4—38no—
294(C2×D3)⋊C4—98no—
295(C2×D3)⋊C4—98no—
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—91no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—188no—
301(C2×D3)⋊C4C2×C3⋊D312yes—
302C4⋊Dic3C2×C3⋊D312yescomplemented
303Dic3⋊C4—94no—
304Dic3⋊C4—91no—
305C4×Dic3—94no—
306C4×Dic3—91no—
307C4×Dic3—184no—
308C22×C32—24no—
309C22×C32—32no—
310C22×C32—61no—
311C22×C32—61no—
312C22×C32C23×D314yes—
313C22×C32C23×D311yescharacteristic
314C2×C3⋊D3—21no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3—22no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—34no—
321C2×C3⋊D3—316no—
322C2×C3⋊D3—616no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—62no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3—128no—
330C2×C3×D3—24no—
331C2×C3×D3—24no—
332C2×C3×D3—64no—
333C2×C3×D3—68no—
334C2×C3×D3—64no—
335C2×C3×D3—68no—
336C2×C3×D3—64no—
337C2×C3×D3—68no—
338C2×C3×D3#23918yescomplemented
339D32—1232no—
340C32×C4—22no—
341C32×C4—28no—
342C32×C4—28no—
343C32×C4—48no—
344C32×C4—62no—
345C32×C4—61no—
346C32×C4D3×D418yes—
347C32×C4D3×D412yes—
348C3⋊Dic3—21no—
349C3⋊Dic3—68no—
350C3⋊Dic3—61no—
351C3⋊Dic3—61no—
352C3⋊Dic3—64no—
353C3⋊Dic3—64no—
354C3⋊Dic3—124no—
355C3×Dic3—28no—
356C3×Dic3—24no—
357C3×Dic3—64no—
358C2×#41—271no—
359C22×D4—271no—
360#124—271no—
361#123—272no—
362C22⋊Q8—31no—
363#121—91no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C33#54611yescharacteristic, complemented, Sylow
370C23×C3—34no—
371C23×C3—92no—
372C23×C3—91no—
373C22×D3—38no—
374C22×D3—34no—
375C22×D3—98no—
376C22×D3—92no—
377C22×D3—188no—
378C22×D3—184no—
379C22×D3—1816no—
380C22×D3—1816no—
381C22×D3—184no—
382C22×D3—362no—
383C22×D3—368no—
384C22×D3C22×C3⋊D311yescharacteristic
385C3×Q8—68no—
386C3×D4—182no—
387C3×D4—362no—
388C2×C3×C4—24no—
389C2×C3×C4—28no—
390C2×C3×C4—34no—
391C2×C3×C4—38no—
392C2×C3×C4—68no—
393C2×C3×C4—91no—
394C2×C3×C4—182no—
395C2×C3×C4C2×D3218yes—
396C2×C3×C4C2×D3214yes—
397C2×C3×C4C22×C3⋊D312yes—
398C2×C3×C4C22×C3⋊D311yescharacteristic
399#67—184no—
400#67—184no—
401#67—188no—
402#67—182no—
403C2×Dic3—34no—
404C2×Dic3—94no—
405C2×Dic3—94no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—184no—
409C2×Dic3—184no—
410C2×Dic3C22×C3⋊D311yescharacteristic
411C2×Dic3C22×C3⋊D312yes—
412D12—68no—
413D12—916no—
414D12—94no—
415D12—128no—
416D12—1816no—
417D12—1816no—
418D12—1816no—
419C4×D3—22no—
420C4×D3—68no—
421C4×D3—184no—
422C4×D3—1816no—
423C4×D3—368no—
424C4×D3—362no—
425C4×D3—3616no—
426Dic6—22no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—62no—
431C2×C32C2×D3×D414yes—
432C2×C32C2×D3×D411yescharacteristic
433C2×C32#100411yescharacteristic
434C2×C32C2×#23914yes—
435C2×C32D3×#4111yescharacteristic
436C2×C32#100914yes—
437C3⋊D3—41no—
438C3⋊D3—42no—
439C3⋊D3—64no—
440C3⋊D3—616no—
441C3⋊D3—1216no—
442C3×D3—28no—
443C3×D3—128no—
444C3×D3—124no—
445C24—272no—
446#41—544no—
447C2×Q8—31no—
448C2×D4—91no—
449C2×D4—92no—
450C2×D4—272no—
451C2×D4—272no—
452C2×D4—271no—
453C2×D4—271no—
454C2×D4—544no—
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—32no—
460C4⋊C4—271no—
461C4⋊C4D3×C3⋊D311yescharacteristic, complemented
462C22⋊C4—32no—
463C22⋊C4—92no—
464C22⋊C4—272no—
465C22⋊C4—271no—
466C22⋊C4—272no—
467C22⋊C4—271no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—24no—
471C22×C3—38no—
472C22×C3—64no—
473C22×C3—64no—
474C22×C3—181no—
475C22×C3—182no—
476C22×C3—181no—
477C22×C3—182no—
478C22×C3—181no—
479C22×C3—182no—
480C22×C3C22×D3214yes—
481C22×C3C23×C3⋊D311yescharacteristic
482C2×D3—21no—
483C2×D3—21no—
484C2×D3—64no—
485C2×D3—68no—
486C2×D3—64no—
487C2×D3—68no—
488C2×D3—64no—
489C2×D3—68no—
490C2×D3—916no—
491C2×D3—94no—
492C2×D3—1816no—
493C2×D3—188no—
494C2×D3—188no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—368no—
498C2×D3—368no—
499C2×D3—368no—
500C2×D3—3632no—
501C2×D3#383212yescomplemented
502C3×C4—28no—
503C3×C4—22no—
504C3×C4—28no—
505C3×C4—48no—
506C3×C4—68no—
507C3×C4—64no—
508C3×C4—181no—
509C3×C4#380218yes—
510C3×C4D4×C3⋊D312yes—
511Dic3—22no—
512Dic3—21no—
513Dic3—64no—
514Dic3—188no—
515Dic3—184no—
516Dic3—184no—
517Dic3—181no—
518Dic3—181no—
519Dic3—364no—
520C32—24no—
521C32#561211yescharacteristic, complemented
522C32#563014yescomplemented
523C23—31no—
524C23—92no—
525C23—91no—
526C23—272no—
527C23—544no—
528C23—544no—
529C23—1082no—
530Q8—62no—
531D4—182no—
532D4—274no—
533D4—362no—
534D4—544no—
535D4—544no—
536D4—542no—
537C2×C4—32no—
538C2×C4—31no—
539C2×C4—62no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—271no—
543C2×C4—271no—
544C2×C4—544no—
545C2×C4—1082no—
546C2×C4C2×D3×C3⋊D312yes—
547C2×C4C2×D3×C3⋊D311yescharacteristic
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—68no—
552C2×C3—362no—
553C2×C3—361no—
554C2×C3C2×#380214yes—
555C2×C3#6431414yes—
556C2×C3D3×#23914yes—
557C2×C3C2×D4×C3⋊D311yescharacteristic
558C2×C3C2×#383211yescharacteristic
559C2×C3#6436611yescharacteristic
560D3—22no—
561D3—128no—
562D3—124no—
563D3—184no—
564D3—1816no—
565D3—3616no—
566C22—32no—
567C22—61no—
568C22—61no—
569C22—181no—
570C22—182no—
571C22—181no—
572C22—182no—
573C22—181no—
574C22—182no—
575C22—274no—
576C22—542no—
577C22—542no—
578C22—1088no—
579C22C22×D3×C3⋊D311yescharacteristic
580C4—22no—
581C4—61no—
582C4—62no—
583C4—181no—
584C4—541no—
585C4—541no—
586C4#8902612yes—
587C3—24no—
588C3#1059463314yescomplemented
589C3#1059610111yescharacteristic, complemented
590C2—62no—
591C2—362no—
592C2—361no—
593C2—544no—
594C2C2×#8902611yescharacteristic
595C2#1172458911yescharacteristic
596C2D3×#383211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#546
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×#89018C2semidirect
3. #11723934C2semidirect
4. C3×#64104C2semidirect
5. C32×#897C2semidirect
6. #11723541C2semidirect
7. #11723539C2semidirect
8. #11723532C2semidirect
9. C4×#6290C2semidirect
10. #11723504C2semidirect
11. #11723436C2semidirect
16. C2×C4×C3⋊(C3⋊D3)C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C2×C3×C4×C3⋊D3C22semidirect
21. C2×C32×Dic6C22semidirect
23. C2×#6292C22semidirect
25. C2×#6289C22semidirect
28. C33×C4⋊C4C22semidirect
29. #88859C22semidirect
30. C4×C3⋊(C3⋊Dic3)C22semidirect
31. C3×#3752C22semidirect
33. C32×C4⋊Dic3C22semidirect
34. #88806C22semidirect
35. #88803C22semidirect
36. #88803C22semidirect
37. #88803C22semidirect
47. #64104D3semidirect
54. C3×#897D3semidirect
73. C4⋊(C3⋊(C3⋊D3))D4semidirect
102. C2×C4⋊(C3⋊D3)C2×D3semidirect
107. C2×C4×C3⋊D3C2×D3semidirect
112. C2×C3×Dic6C2×D3semidirect
121. C32×C4⋊C4C2×D3semidirect
122. C32×C4⋊C4C2×D3semidirect
131. #3752C2×D3semidirect
138. C3×C4⋊Dic3C2×D3semidirect
156. C2×C32×D3#41semidirect
177. #897C3⋊D3semidirect
282. C2×Dic6C2×C3⋊D3semidirect
285. C3×C4⋊C4D32semidirect
286. C3×C4⋊C4C2×C3⋊D3semidirect
302. C4⋊Dic3C2×C3⋊D3semidirect
338. C2×C3×D3#239semidirect
369. C33#546semidirect
461. C4⋊C4D3×C3⋊D3semidirect
501. C2×D3#3832semidirect
521. C32#5612semidirect
522. C32#5630semidirect
588. C3#10594633semidirect
589. C3#10596101semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×#890182
#117239342
C3×#641042
C32×#8972
#117235412
#117235392
#117235322
C4×#62902
#117235042
#117234362
#105961013
#105946333