Finite Group Databasebeta

← #13177392

Subgroups — #13177392

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723636C212yesmaximal, complemented
6#11723624C212yesmaximal, complemented
7#11723621C212yesmaximal, complemented
8C3⋊D3×#217C212yesmaximal, complemented
9C4×#6292C211yesmaximal, characteristic, complemented
10Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11#11723435C211yesmaximal, characteristic, complemented
12#10596147—31nomaximal
13#10595229—34nomaximal
14C23×C3×C3⋊D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2212yescomplemented
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
18C2×C3×#638C2212yes—
19C3×D4×C3⋊D3—24no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C32×Dic6C2211yescharacteristic, complemented
23C2×#6292C2211yescharacteristic
24C2×#6292C2212yescomplemented
25C2×Dic3×C3⋊D3C2212yescomplemented
26C2×Dic3×C3⋊D3C2212yescomplemented
27#89017—24no—
28#88864C2212yescomplemented
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
30C32×#217C2212yescomplemented
31C32×#217C2212yescomplemented
32C32×C4×Dic3C2211yescharacteristic, complemented
33#88806C2212yescomplemented
34Dic3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
35#88804C2211yescharacteristic, complemented
36#88803C2211yescharacteristic
37#88803C2212yes—
38#88803C2212yescomplemented
39C2×#3832—31no—
40C2×D4×C3⋊D3D311yescharacteristic, complemented
41C2×C3×D3×D4—34no—
42C2×#3796—34no—
43C32×#124—31no—
44#64155—31no—
45#64147—31no—
46#64147—34no—
47#64147—64no—
48#64093—32no—
49#64092—32no—
50#64091—32no—
51C22⋊C4×C3⋊D3—32no—
52C4×C4⋊(C3⋊D3)—31no—
53C4×C4⋊(C3⋊D3)—31no—
54C3×#936D314yescomplemented
55#63977—38no—
56#63971—38no—
57#63969—38no—
58D3×#217—38no—
59C4×#626—34no—
60Dic3×D12—34no—
61#63856—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×D4—22no—
70C2×C33×C4C2311yescharacteristic
71#6312—42no—
72C2×C3⋊(C3⋊Dic3)C2312yes—
73C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
74C4×C3⋊(C3⋊D3)—22no—
75C3×#638—24no—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)D414yescomplemented
78C3×C4×C3⋊D3—22no—
79C2×C32×Dic3C2311yescharacteristic
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2311yescharacteristic
82C32×Dic6—22no—
83#6292—24no—
84#6292—22no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—42no—
87#5806—91no—
88#5676—94no—
89C23×C3⋊D3C2×D312yes—
90C23×C3×D3—38no—
91C2×C32×Q8—31no—
92C2×C32×D4—24no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C2×#638—32no—
96C2×#638—38no—
97C2×#638—68no—
98C2×#638C2×D312yes—
99#3832—64no—
100D4×C3⋊D3—24no—
101C2×C4⋊(C3⋊D3)—31no—
102C2×C4⋊(C3⋊D3)—32no—
103C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
104C2×C4×C3⋊D3—32no—
105C2×C4×C3⋊D3—31no—
106C2×C4×C3⋊D3—34no—
107C2×C4×C3⋊D3—32no—
108C2×C4×C3⋊D3—64no—
109C2×C4×C3⋊D3C2×D311yescharacteristic
110C2×C3×#67—38no—
111C3×D3×D4—616no—
112C2×C3×D12—34no—
113C2×C3×C4×D3—34no—
114C2×C3×Dic6C2×D314yescomplemented
115C2×#626—38no—
116C2×#626—34no—
117C2×D3×Dic3—38no—
118C2×D3×Dic3—38no—
119#3796—616no—
120C32×C22⋊C4—32no—
121C32×C22⋊C4—32no—
122C32×C42—31no—
123#3757—32no—
124#3757—38no—
125#3757—68no—
126#3752—31no—
127#3752—32no—
128#3752—32no—
129#3752—31no—
130C4⋊(C3⋊Dic3)—31no—
131C4⋊(C3⋊Dic3)—34no—
132C4⋊(C3⋊Dic3)—31no—
133C4⋊(C3⋊Dic3)—64no—
134#3750—32no—
135C3×#217C2×D318yescomplemented
136C3×#217C2×D318yescomplemented
137C3×C4×Dic3C2×D314yescomplemented
138#3724—38no—
139Dic3⋊Dic3—34no—
140#3722—34no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144C22×C33—22no—
145C22×C33—22no—
146C22×C33—22no—
147C22×C33C2411yescharacteristic
148C2×C3⋊(C3⋊D3)—21no—
149C2×C3⋊(C3⋊D3)—21no—
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—48no—
156C2×C3×C3⋊D3C2×D414yes—
157C2×C3×C3⋊D3#4112yescomplemented
158C33×C4C2×D412yes—
159C3⋊(C3⋊Dic3)—21no—
160C3⋊(C3⋊Dic3)—22no—
161C3×C3⋊Dic3—21no—
162C32×Dic3—22no—
163C32×Dic3—21no—
164C32×Dic3—21no—
165C22×C3×D4—91no—
166C2×#239—94no—
167C2×#237—91no—
168C2×D3×D4—34no—
169C3×#124—34no—
170#953—92no—
171C2×#217—92no—
172#946—94no—
173#939—92no—
174#938—94no—
175#938—91no—
176#938—184no—
177#936C3⋊D311yescharacteristic, complemented
178#935—92no—
179D4×Dic3—91no—
180C4×#67—91no—
181#924—91no—
182#884—98no—
183#883—98no—
184#882—98no—
185D3×C22⋊C4—98no—
186C4×D12—94no—
187C4×D12—94no—
188C23×C32—28no—
189C23×C32C22×D318yes—
190C23×C32C22×D312yes—
191C22×C3⋊D3—24no—
192C22×C3⋊D3—24no—
193C22×C3⋊D3—31no—
194C22×C3⋊D3—34no—
195C22×C3⋊D3—42no—
196C22×C3⋊D3—64no—
197C22×C3⋊D3C22×D311yescharacteristic
198C22×C3⋊D3C22×D312yes—
199C22×C3×D3—34no—
200C22×C3×D3—38no—
201C22×C3×D3—616no—
202C22×C3×D3—616no—
203C22×C3×D3—128no—
204C32×Q8—62no—
205C32×D4—22no—
206C32×D4—28no—
207C32×D4—48no—
208C2×C32×C4—24no—
209C2×C32×C4—32no—
210C2×C32×C4—31no—
211C2×C32×C4—31no—
212C2×C32×C4C22×D314yes—
213C2×C32×C4C22×D311yescharacteristic
214#638—24no—
215#638—122no—
216#638—128no—
217#638—1216no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—31no—
220C2×C3⋊Dic3—38no—
221C2×C3⋊Dic3—34no—
222C2×C3⋊Dic3—68no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C4⋊(C3⋊D3)—62no—
226C4⋊(C3⋊D3)—64no—
227C4⋊(C3⋊D3)#6714yescomplemented
228C4×C3⋊D3—22no—
229C4×C3⋊D3—64no—
230C4×C3⋊D3—62no—
231C4×C3⋊D3—68no—
232C4×C3⋊D3—128no—
233C4×C3⋊D3—122no—
234C3×#67—616no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3C22×D314yes—
237C2×C3×Dic3C22×D314yes—
238C2×C3×Dic3C22×D318yes—
239C3×D12—316no—
240C3×C4×D3—68no—
241C3×Dic6—28no—
242#626—616no—
243#626—68no—
244D3×Dic3—616no—
245D3×Dic3—128no—
246#546—271noSylow
247C2×C33—42no—
248C2×C33C22×D411yescharacteristic
249C2×C33C2×#4111yescharacteristic
250C2×C33H(5,𝔽2)11yescharacteristic
251C3⋊(C3⋊D3)—41no—
252C3×C3⋊D3—22no—
253C3×C3⋊D3—24no—
254C24×C3—92no—
255C23×D3—38no—
256C2×C3×Q8—34no—
257C2×C3×D4—24no—
258C2×C3×D4—92no—
259C2×C3×D4—91no—
260C2×C3×D4—184no—
261C2×C3×D4D3214yes—
262C2×C3×D4C2×C3⋊D311yescharacteristic
263C22×C3×C4—91no—
264C2×#67—38no—
265C2×#67—98no—
266C2×#67—92no—
267C2×#67—91no—
268C2×#67—92no—
269C2×#67—188no—
270C22×Dic3—92no—
271C22×Dic3—92no—
272#239—1816no—
273#237—184no—
274D3×D4—616no—
275C2×D12—34no—
276C2×D12—94no—
277C2×D12—98no—
278C2×C4×D3—34no—
279C2×C4×D3—98no—
280C2×C4×D3—94no—
281C2×C4×D3—91no—
282C2×C4×D3—98no—
283C2×C4×D3—184no—
284C2×Dic6C2×C3⋊D311yescharacteristic, complemented
285C3×C22⋊C4—38no—
286C3×C22⋊C4—38no—
287C3×C42—34no—
288#217—98no—
289#217—92no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—188no—
294#217C2×C3⋊D312yescomplemented
295#217C2×C3⋊D312yescomplemented
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—94no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—98no—
300(C2×D3)⋊C4—91no—
301C4⋊Dic3—94no—
302C4⋊Dic3—94no—
303C4⋊Dic3—91no—
304C4⋊Dic3—184no—
305Dic3⋊C4—98no—
306Dic3⋊C4—91no—
307Dic3⋊C4—92no—
308C4×Dic3C2×C3⋊D311yescharacteristic, complemented
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—22no—
312C22×C32—24no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—28no—
316C22×C32—48no—
317C22×C32—48no—
318C22×C32—48no—
319C22×C32C23×D314yes—
320C22×C32C23×D311yescharacteristic
321C2×C3⋊D3—22no—
322C2×C3⋊D3—22no—
323C2×C3⋊D3—21no—
324C2×C3⋊D3—21no—
325C2×C3⋊D3—48no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—61no—
328C2×C3⋊D3—61no—
329C2×C3⋊D3—64no—
330C2×C3⋊D3—61no—
331C2×C3⋊D3—64no—
332C2×C3⋊D3—64no—
333C2×C3⋊D3—124no—
334C2×C3⋊D3—124no—
335C2×C3⋊D3#23712yescomplemented
336C2×C3⋊D3C2×#6714yes—
337C2×C3×D3—316no—
338C2×C3×D3—38no—
339C2×C3×D3—64no—
340C2×C3×D3—68no—
341C2×C3×D3—68no—
342C2×C3×D3—64no—
343C2×C3×D3—1232no—
344C32×C4—28no—
345C32×C4—61no—
346C32×C4—61no—
347C32×C4—62no—
348C32×C4D3×D418yes—
349C32×C4C2×#6712yes—
350C3⋊Dic3—21no—
351C3⋊Dic3—62no—
352C3⋊Dic3—61no—
353C3⋊Dic3—64no—
354C3⋊Dic3—68no—
355C3⋊Dic3—124no—
356C3⋊Dic3—128no—
357C3×Dic3—28no—
358C3×Dic3—24no—
359C3×Dic3—24no—
360C3×Dic3—64no—
361C2×#41—271no—
362C22×D4—91no—
363#124—31no—
364#123—272no—
365C22⋊Q8—271no—
366#121—272no—
367#121—271no—
368#120—272no—
369C4×D4—271no—
370C4×D4—271no—
371C2×C22⋊C4—272no—
372C33#54611yescharacteristic, complemented, Sylow
373C23×C3—28no—
374C23×C3—91no—
375C23×C3—92no—
376C23×C3—184no—
377C23×C3—184no—
378C23×C3—362no—
379C23×C3C2×D3218yes—
380C23×C3C22×C3⋊D312yes—
381C22×D3—34no—
382C22×D3—38no—
383C22×D3—616no—
384C22×D3—616no—
385C22×D3—94no—
386C22×D3—91no—
387C22×D3—128no—
388C22×D3—184no—
389C3×Q8—68no—
390C3×D4—28no—
391C3×D4—22no—
392C3×D4—48no—
393C3×D4—94no—
394C3×D4—184no—
395C2×C3×C4—24no—
396C2×C3×C4—38no—
397C2×C3×C4—34no—
398C2×C3×C4—34no—
399C2×C3×C4—91no—
400C2×C3×C4—182no—
401C2×C3×C4C2×D3214yes—
402C2×C3×C4C22×C3⋊D311yescharacteristic
403#67—616no—
404#67—184no—
405#67—182no—
406#67—368no—
407#67—362no—
408#67—3616no—
409C2×Dic3—34no—
410C2×Dic3—98no—
411C2×Dic3—94no—
412C2×Dic3—92no—
413C2×Dic3—91no—
414C2×Dic3—188no—
415C2×Dic3—184no—
416C2×Dic3—184no—
417C2×Dic3—362no—
418C2×Dic3C22×C3⋊D311yescharacteristic
419C2×Dic3C22×C3⋊D311yescharacteristic
420C2×Dic3C22×C3⋊D312yes—
421D12—316no—
422D12—188no—
423D12—1816no—
424C4×D3—68no—
425C4×D3—1816no—
426C4×D3—188no—
427C4×D3—182no—
428C4×D3—368no—
429C4×D3—368no—
430Dic6—22no—
431C2×C32—24no—
432C2×C32—24no—
433C2×C32—24no—
434C2×C32—42no—
435C2×C32—48no—
436C2×C32—88no—
437C2×C32C2×D3×D414yes—
438C2×C32C2×#23711yescharacteristic
439C2×C32#100411yescharacteristic
440C2×C32D3×#4114yes—
441C2×C32#100914yes—
442C2×C32C22×#6711yescharacteristic
443C3⋊D3—22no—
444C3⋊D3—24no—
445C3⋊D3—128no—
446C3⋊D3—124no—
447C3⋊D3—121no—
448C3×D3—616no—
449C3×D3—68no—
450C24—92no—
451#41—544no—
452C2×Q8—31no—
453C2×D4—92no—
454C2×D4—91no—
455C2×D4—184no—
456C2×D4—271no—
457C2×D4—272no—
458C2×D4—272no—
459C2×D4D3×C3⋊D311yescharacteristic
460C22×C4—91no—
461C22×C4—272no—
462C22×C4—272no—
463C22×C4—271no—
464C4⋊C4—271no—
465C4⋊C4—272no—
466C4⋊C4—271no—
467C22⋊C4—32no—
468C22⋊C4—32no—
469C22⋊C4—271no—
470C22⋊C4—271no—
471C22⋊C4—272no—
472C22⋊C4—272no—
473C22⋊C4—272no—
474C42—31no—
475C22×C3—28no—
476C22×C3—28no—
477C22×C3—28no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—24no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—48no—
485C22×C3—94no—
486C22×C3—92no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3—182no—
490C22×C3—181no—
491C22×C3—368no—
492C22×C3C22×D3214yes—
493C22×C3C23×C3⋊D311yescharacteristic
494C2×D3—38no—
495C2×D3—316no—
496C2×D3—68no—
497C2×D3—68no—
498C2×D3—64no—
499C2×D3—64no—
500C2×D3—1232no—
501C2×D3—188no—
502C2×D3—184no—
503C2×D3—184no—
504C2×D3—184no—
505C2×D3—181no—
506C2×D3—181no—
507C2×D3—181no—
508C2×D3—364no—
509C2×D3—364no—
510C3×C4—28no—
511C3×C4—64no—
512C3×C4—64no—
513C3×C4—68no—
514C3×C4—181no—
515C3×C4D3×#6718yes—
516C3×C4D4×C3⋊D312yes—
517Dic3—22no—
518Dic3—21no—
519Dic3—21no—
520Dic3—64no—
521Dic3—188no—
522Dic3—184no—
523Dic3—181no—
524Dic3—182no—
525Dic3—364no—
526Dic3—368no—
527C32—24no—
528C32#567614yescomplemented
529C32#580611yescharacteristic, complemented
530C23—91no—
531C23—92no—
532C23—184no—
533C23—184no—
534C23—271no—
535C23—362no—
536C23C2×D3×C3⋊D312yes—
537Q8—62no—
538D4—22no—
539D4—94no—
540D4—184no—
541D4—544no—
542D4—542no—
543D4—1082no—
544C2×C4—31no—
545C2×C4—31no—
546C2×C4—32no—
547C2×C4—91no—
548C2×C4—182no—
549C2×C4—272no—
550C2×C4—271no—
551C2×C4—544no—
552C2×C4—542no—
553C2×C4—1082no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C2×C3—24no—
556C2×C3—24no—
557C2×C3—24no—
558C2×C3—42no—
559C2×C3—48no—
560C2×C3—88no—
561C2×C3—182no—
562C2×C3—184no—
563C2×C3D3×#23714yes—
564C2×C3#6431414yes—
565C2×C3C2×D3×#6714yes—
566C2×C3C2×D4×C3⋊D311yescharacteristic
567C2×C3#41×C3⋊D311yescharacteristic
568C2×C3#6436611yescharacteristic
569D3—68no—
570D3—616no—
571D3—368no—
572D3—361no—
573D3—364no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—92no—
578C22—94no—
579C22—182no—
580C22—182no—
581C22—181no—
582C22—181no—
583C22—368no—
584C22—541no—
585C22—541no—
586C22—541no—
587C22C22×D3×C3⋊D311yescharacteristic
588C4—61no—
589C4—61no—
590C4—62no—
591C4—181no—
592C4—542no—
593C4—541no—
594C4C3⋊D3×#6712yes—
595C3—24no—
596C3#1059522914yescomplemented
597C3#1059614711yescharacteristic, complemented
598C2—42no—
599C2—182no—
600C2—184no—
601C2—1081no—
602C2C3⋊D3×#23711yescharacteristic
603C2#1172458911yescharacteristic
604C2C2×C3⋊D3×#6711yescharacteristic
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×C3×D4×C3⋊D3C2semidirect
2. C2×#89017C2semidirect
3. #11723979C2semidirect
4. C32×#936C2semidirect
5. #11723636C2semidirect
6. #11723624C2semidirect
7. #11723621C2semidirect
8. C3⋊D3×#217C2semidirect
9. C4×#6292C2semidirect
10. Dic3×C4⋊(C3⋊D3)C2semidirect
11. #11723435C2semidirect
16. C2×#6312C22semidirect
17. C2×C4×C3⋊(C3⋊D3)C22semidirect
22. C2×C32×Dic6C22semidirect
24. C2×#6292C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
26. C2×Dic3×C3⋊D3C22semidirect
28. #88864C22semidirect
29. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
30. C32×#217C22semidirect
31. C32×#217C22semidirect
32. C32×C4×Dic3C22semidirect
33. #88806C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
35. #88804C22semidirect
38. #88803C22semidirect
40. C2×D4×C3⋊D3D3semidirect
54. C3×#936D3semidirect
77. C3×C4⋊(C3⋊D3)D4semidirect
114. C2×C3×Dic6C2×D3semidirect
135. C3×#217C2×D3semidirect
136. C3×#217C2×D3semidirect
137. C3×C4×Dic3C2×D3semidirect
157. C2×C3×C3⋊D3#41semidirect
177. #936C3⋊D3semidirect
227. C4⋊(C3⋊D3)#67semidirect
284. C2×Dic6C2×C3⋊D3semidirect
294. #217C2×C3⋊D3semidirect
295. #217C2×C3⋊D3semidirect
308. C4×Dic3C2×C3⋊D3semidirect
335. C2×C3⋊D3#237semidirect
372. C33#546semidirect
528. C32#5676semidirect
529. C32#5806semidirect
596. C3#10595229semidirect
597. C3#10596147semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890172
#117239792
C32×#9362
#117236362
#117236242
#117236212
C3⋊D3×#2172
C4×#62922
Dic3×C4⋊(C3⋊D3)2
#117234352
#105961473
#105952293