Finite Group Databasebeta

← #13176544

Subgroups — #13176544

Order 1728.

2300 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×#89026C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723938C211yesmaximal, characteristic, complemented
4C3×#64102C211yesmaximal, characteristic, complemented
5C32×#895C211yesmaximal, characteristic, complemented
6#11723540C212yesmaximal, complemented
7C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
8#11723523C212yesmaximal, complemented
9#11723522C212yesmaximal, complemented
10C4×#6291C211yesmaximal, characteristic, complemented
11#11723429C211yesmaximal, characteristic, complemented
12#10596083—31nomaximal
13#10594630—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic, complemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2212yescomplemented
19C2×C32×D12C2211yescharacteristic
20C2×C32×D12C2212yescomplemented
21C2×C32×C4×D3C2211yescharacteristic, complemented
22C2×#6291C2211yescharacteristic
23C2×#6290C2212yes—
24C2×#6290C2212yescomplemented
25C2×Dic3×C3⋊D3C2212yescomplemented
26#89026—24no—
27#89021—24no—
28C33×C4⋊C4C2211yescharacteristic, complemented
29#88859C2212yes—
30C4⋊(C3⋊(C3⋊Dic3))C2212yescomplemented
31C3×#3752C2212yescomplemented
32C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
33C32×(C2×D3)⋊C4C2212yescomplemented
34#88807C2211yescharacteristic, complemented
35#88804C2211yescharacteristic
36(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic, complemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C2×D4×C3⋊D3—31no—
40C2×#3828—31no—
41C2×#3802—34no—
42C2×#3795—34no—
43C32×#121—31no—
44#64147—32no—
45#64146—32no—
46#64145—31no—
47C4×#638—31no—
48#64106—31no—
49#64106—34no—
50#64106—64no—
51#64102D311yescharacteristic, complemented
52#64092—32no—
53C22⋊C4×C3⋊D3—32no—
54C3×#895D314yescomplemented
55C3×#893—34no—
56#63922—38no—
57D3×(C2×D3)⋊C4—38no—
58#63908—38no—
59C4×#626—34no—
60#63891—38no—
61#63860—34no—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C22×C3×C3⋊D3C2312yes—
64C22×C32×D3C2312yes—
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)C2312yes—
73C4×C3⋊(C3⋊D3)—22no—
74#6308—22no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)D414yescomplemented
78C3×C4×C3⋊D3—42no—
79C2×C32×Dic3C2311yescharacteristic
80C32×D12—22no—
81C32×D12—42no—
82C32×C4×D3—22no—
83#6291—22no—
84#6290—24no—
85#6290—24no—
86Dic3×C3⋊D3—24no—
87#5612—94no—
88#5554—91no—
89C23×C3⋊D3—32no—
90C22×D32—38no—
91C2×C32×D4—31no—
92C2×C32×D4—32no—
93C22×C32×C4—31no—
94C2×#638—32no—
95C2×#638—32no—
96C2×#638—31no—
97D4×C3⋊D3—64no—
98#3828—64no—
99C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
100C2×C4×C3⋊D3—32no—
101C2×C4×C3⋊D3—31no—
102C2×C4×C3⋊D3—34no—
103C2×C4×C3⋊D3—64no—
104C2×C4×C3⋊D3C2×D312yescomplemented
105C2×#634—31no—
106C2×#634—34no—
107C2×#634—64no—
108C2×C3×D12—34no—
109C2×C3×D12C2×D314yes—
110C2×C3×D12C2×D318yescomplemented
111C2×C3×C4×D3—38no—
112C2×C3×C4×D3C2×D314yescomplemented
113C2×#626—34no—
114C2×#625—38no—
115C2×#625—38no—
116C2×D3×Dic3—38no—
117#3802—616no—
118#3795—616no—
119C32×C4⋊C4—24no—
120C32×C4⋊C4C2×D314yescomplemented
121C32×C4⋊C4C2×D311yescharacteristic, complemented
122C32×C22⋊C4—32no—
123#3757—32no—
124#3757—32no—
125#3757—31no—
126#3752—32no—
127#3752—38no—
128#3752—31no—
129#3752—68no—
130#3752C2×D312yescomplemented
131C4⋊(C3⋊Dic3)—32no—
132C4⋊(C3⋊Dic3)—38no—
133C4⋊(C3⋊Dic3)—68no—
134#3750—31no—
135C4×C3⋊Dic3C2×D311yescharacteristic, complemented
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4C2×D318yescomplemented
138C3×C4×Dic3—34no—
139Dic3⋊Dic3—34no—
140#3722—34no—
141(C2×D3)⋊Dic3—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—34no—
144C22×C33C2411yescharacteristic
145C2×C3⋊(C3⋊D3)—21no—
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)#4112yescomplemented
148C2×C3×C3⋊D3—22no—
149C2×C3×C3⋊D3—22no—
150C2×C3×C3⋊D3C2×D414yes—
151C2×C32×D3—21no—
152C2×C32×D3—22no—
153C2×C32×D3—21no—
154C2×C32×D3—22no—
155C2×C32×D3—21no—
156C2×C32×D3—22no—
157D3×C3⋊D3—48no—
158C33×C4—22no—
159C33×C4C2×D412yes—
160C3⋊(C3⋊Dic3)—22no—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—21no—
163C3×C3⋊Dic3—21no—
164C32×Dic3—21no—
165C2×#237—91no—
166C2×D3×D4—94no—
167C2×D3×D4—91no—
168C2×#235—94no—
169C3×#121—34no—
170C3×C4×D4—91no—
171#938—98no—
172#937—98no—
173#936—94no—
174#930—92no—
175C4×#67—94no—
176C2×(C2×D3)⋊C4—92no—
177#897—94no—
178#897—91no—
179#897—184no—
180#895C3⋊D311yescharacteristic, complemented
181#893—34no—
182#886—92no—
183#883—98no—
184#882—92no—
185D3×C22⋊C4—98no—
186#875—91no—
187C4×D12—91no—
188C23×C32—32no—
189C23×C32—31no—
190C22×C3⋊D3—31no—
191C22×C3⋊D3—34no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—64no—
195C22×C3⋊D3—122no—
196C22×C3⋊D3C22×D312yes—
197C22×C3×D3—38no—
198C22×C3×D3C22×D318yes—
199C22×C3×D3C22×D314yes—
200C2×D32—616no—
201C2×D32—616no—
202C2×D32—128no—
203C32×D4—62no—
204C32×D4—122no—
205C2×C32×C4—24no—
206C2×C32×C4—28no—
207C2×C32×C4—31no—
208C2×C32×C4—62no—
209C2×C32×C4C22×D318yes—
210C2×C32×C4C22×D314yes—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D311yescharacteristic
213#638—62no—
214#638—64no—
215#638—64no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—38no—
219C2×C3⋊Dic3—34no—
220C2×C3⋊Dic3—64no—
221C2×C3⋊Dic3—68no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D311yescharacteristic
224C4⋊(C3⋊D3)D1214yescomplemented
225C4×C3⋊D3—42no—
226C4×C3⋊D3—64no—
227C4×C3⋊D3—62no—
228C4×C3⋊D3—68no—
229C4×C3⋊D3—128no—
230#634—62no—
231#634—68no—
232#634—128no—
233C2×C3×Dic3—34no—
234C2×C3×Dic3—34no—
235C2×C3×Dic3C22×D314yes—
236C3×D12—28no—
237C3×D12—316no—
238C3×D12—48no—
239C3×C4×D3—28no—
240C3×C4×D3—128no—
241#626—68no—
242#625—616no—
243#625—616no—
244D3×Dic3—616no—
245#546—271noSylow
246C2×C33C22×D411yescharacteristic
247C2×C33C2×#4111yescharacteristic
248C2×C33H(5,𝔽2)11yescharacteristic
249C3⋊(C3⋊D3)—22no—
250C3×C3⋊D3—24no—
251C32×D3—42no—
252C32×D3—41no—
253C23×D3—98no—
254C23×D3—92no—
255C2×C3×D4—34no—
256C2×C3×D4—38no—
257C2×C3×D4—91no—
258C22×C3×C4—34no—
259C22×C3×C4—92no—
260C2×#67—94no—
261C2×#67—98no—
262C2×#67—98no—
263C2×#67—92no—
264C2×#67—92no—
265C22×Dic3—92no—
266#237—184no—
267D3×D4—184no—
268D3×D4—1816no—
269#235—1816no—
270C2×D12—34no—
271C2×D12—91no—
272C2×D12C2×C3⋊D311yescharacteristic
273C2×D12C2×C3⋊D312yescomplemented
274C2×C4×D3—38no—
275C2×C4×D3—98no—
276C2×C4×D3—94no—
277C2×C4×D3—91no—
278C2×C4×D3—184no—
279C2×C4×D3C2×C3⋊D311yescharacteristic, complemented
280C2×Dic6—94no—
281C2×Dic6—91no—
282C2×Dic6—184no—
283C3×C4⋊C4—24no—
284C3×C4⋊C4D3214yescomplemented
285C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
286C3×C22⋊C4—38no—
287C3×C22⋊C4—92no—
288C3×C42—91no—
289#217—98no—
290#217—98no—
291#217—94no—
292(C2×D3)⋊C4—38no—
293(C2×D3)⋊C4—94no—
294(C2×D3)⋊C4—91no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—98no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—91no—
300(C2×D3)⋊C4—188no—
301(C2×D3)⋊C4C2×C3⋊D312yescomplemented
302C4⋊Dic3—98no—
303C4⋊Dic3—92no—
304C4⋊Dic3—91no—
305C4⋊Dic3—188no—
306Dic3⋊C4—94no—
307C4×Dic3—34no—
308C22×C32—24no—
309C22×C32—61no—
310C22×C32—62no—
311C22×C32—61no—
312C22×C32—62no—
313C22×C32—61no—
314C22×C32—62no—
315C22×C32C23×D314yes—
316C22×C32C23×D311yescharacteristic
317C2×C3⋊D3—22no—
318C2×C3⋊D3—22no—
319C2×C3⋊D3—32no—
320C2×C3⋊D3—38no—
321C2×C3⋊D3—68no—
322C2×C3⋊D3—61no—
323C2×C3⋊D3—61no—
324C2×C3⋊D3—64no—
325C2×C3⋊D3—64no—
326C2×C3⋊D3—124no—
327C2×C3⋊D3—124no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3C2×D1214yes—
330C2×C3×D3—24no—
331C2×C3×D3—28no—
332C2×C3×D3—24no—
333C2×C3×D3—28no—
334C2×C3×D3—24no—
335C2×C3×D3—28no—
336C2×C3×D3—316no—
337C2×C3×D3—68no—
338C2×C3×D3—68no—
339D32—1232no—
340C32×C4—22no—
341C32×C4—28no—
342C32×C4—28no—
343C32×C4—48no—
344C32×C4—61no—
345C32×C4C2×D1212yes—
346C32×C4D3×D418yes—
347C3⋊Dic3—21no—
348C3⋊Dic3—21no—
349C3⋊Dic3—62no—
350C3⋊Dic3—61no—
351C3⋊Dic3—68no—
352C3⋊Dic3—64no—
353C3⋊Dic3—128no—
354C3⋊Dic3—124no—
355C3×Dic3—24no—
356C3×Dic3—64no—
357C3×Dic3—64no—
358C2×#41—271no—
359C22×D4—271no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—31no—
364#121—272no—
365#120—272no—
366C4×D4—91no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C33#54611yescharacteristic, complemented, Sylow
370C23×C3—38no—
371C23×C3—34no—
372C23×C3—92no—
373C22×D3—38no—
374C22×D3—94no—
375C22×D3—91no—
376C22×D3—184no—
377C22×D3—184no—
378C22×D3—184no—
379C22×D3—1816no—
380C22×D3—1816no—
381C22×D3—362no—
382C22×D3—368no—
383C22×D3C22×C3⋊D312yes—
384C22×D3C22×C3⋊D311yescharacteristic
385C3×D4—68no—
386C3×D4—94no—
387C3×D4—128no—
388C2×C3×C4—24no—
389C2×C3×C4—28no—
390C2×C3×C4—34no—
391C2×C3×C4—68no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—362no—
395C2×C3×C4C2×D3218yes—
396C2×C3×C4C2×D3214yes—
397C2×C3×C4C22×C3⋊D312yes—
398C2×C3×C4C22×C3⋊D311yescharacteristic
399#67—188no—
400#67—184no—
401#67—184no—
402#67—1816no—
403#67—1816no—
404C2×Dic3—34no—
405C2×Dic3—34no—
406C2×Dic3—94no—
407C2×Dic3—98no—
408C2×Dic3—91no—
409C2×Dic3—92no—
410C2×Dic3—184no—
411C2×Dic3—188no—
412C2×Dic3—184no—
413C2×Dic3C22×C3⋊D311yescharacteristic
414D12—22no—
415D12—316no—
416D12—42no—
417D12—182no—
418C4×D3—22no—
419C4×D3—128no—
420C4×D3—1816no—
421C4×D3—188no—
422C4×D3—182no—
423C4×D3—368no—
424Dic6—188no—
425Dic6—182no—
426Dic6—368no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—122no—
431C2×C32—121no—
432C2×C32C22×D1211yescharacteristic
433C2×C32C2×D3×D414yes—
434C2×C32#100414yes—
435C2×C32#100411yescharacteristic
436C2×C32D3×#4114yes—
437C2×C32D3×#4111yescharacteristic
438C3⋊D3—24no—
439C3⋊D3—62no—
440C3⋊D3—68no—
441C3⋊D3—128no—
442C3×D3—48no—
443C3×D3—44no—
444C3×D3—616no—
445C24—272no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—31no—
449C2×D4—32no—
450C2×D4—91no—
451C2×D4—271no—
452C2×D4—272no—
453C2×D4—272no—
454C2×D4—544no—
455C22×C4—31no—
456C22×C4—92no—
457C22×C4—272no—
458C22×C4—271no—
459C4⋊C4—271no—
460C4⋊C4—272no—
461C4⋊C4D3×C3⋊D311yescharacteristic, complemented
462C22⋊C4—32no—
463C22⋊C4—92no—
464C22⋊C4—271no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—91no—
470C22×C3—24no—
471C22×C3—64no—
472C22×C3—68no—
473C22×C3—64no—
474C22×C3—68no—
475C22×C3—64no—
476C22×C3—68no—
477C22×C3—94no—
478C22×C3—182no—
479C22×C3—182no—
480C22×C3C22×D3214yes—
481C22×C3C23×C3⋊D311yescharacteristic
482C2×D3—21no—
483C2×D3—22no—
484C2×D3—21no—
485C2×D3—22no—
486C2×D3—21no—
487C2×D3—22no—
488C2×D3—316no—
489C2×D3—68no—
490C2×D3—68no—
491C2×D3—98no—
492C2×D3—92no—
493C2×D3—188no—
494C2×D3—184no—
495C2×D3—184no—
496C2×D3—181no—
497C2×D3—181no—
498C2×D3—364no—
499C2×D3—364no—
500C2×D3—368no—
501C2×D3—3632no—
502C3×C4—28no—
503C3×C4—22no—
504C3×C4—28no—
505C3×C4—48no—
506C3×C4—64no—
507C3×C4—181no—
508C3×C4—181no—
509C3×C4D3×D1218yes—
510C3×C4D4×C3⋊D312yes—
511Dic3—21no—
512Dic3—64no—
513Dic3—64no—
514Dic3—188no—
515Dic3—184no—
516Dic3—182no—
517Dic3—181no—
518Dic3—368no—
519Dic3—364no—
520C32—24no—
521C32#555411yescharacteristic, complemented
522C32#561214yescomplemented
523C23—32no—
524C23—31no—
525C23—92no—
526C23—271no—
527C23—544no—
528C23—544no—
529C23—1082no—
530Q8—542no—
531D4—62no—
532D4—94no—
533D4—122no—
534D4—542no—
535D4—544no—
536D4—544no—
537C2×C4—31no—
538C2×C4—62no—
539C2×C4—91no—
540C2×C4—91no—
541C2×C4—272no—
542C2×C4—271no—
543C2×C4—362no—
544C2×C4—544no—
545C2×C4—542no—
546C2×C4C2×D3×C3⋊D312yes—
547C2×C4C2×D3×C3⋊D311yescharacteristic
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—128no—
552C2×C3—124no—
553C2×C3—184no—
554C2×C3C2×D3×D1214yes—
555C2×C3#6431314yes—
556C2×C3#6431914yes—
557C2×C3C2×D4×C3⋊D311yescharacteristic
558C2×C3#6436111yescharacteristic
559C2×C3#41×C3⋊D311yescharacteristic
560D3—42no—
561D3—41no—
562D3—616no—
563D3—188no—
564D3—182no—
565D3—368no—
566C22—61no—
567C22—62no—
568C22—61no—
569C22—62no—
570C22—61no—
571C22—62no—
572C22—94no—
573C22—182no—
574C22—182no—
575C22—272no—
576C22—541no—
577C22—541no—
578C22—1088no—
579C22C22×D3×C3⋊D311yescharacteristic
580C4—22no—
581C4—61no—
582C4—181no—
583C4—181no—
584C4—541no—
585C4—542no—
586C4C3⋊D3×D1212yes—
587C3—24no—
588C3#1059463014yescomplemented
589C3#1059608311yescharacteristic, complemented
590C2—122no—
591C2—121no—
592C2—184no—
593C2—542no—
594C2C2×C3⋊D3×D1211yescharacteristic
595C2#1172458711yescharacteristic
596C2#1172459611yescharacteristic
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×#89026C2semidirect
2. C2×#89021C2semidirect
3. #11723938C2semidirect
4. C3×#64102C2semidirect
5. C32×#895C2semidirect
6. #11723540C2semidirect
7. C3⋊D3×(C2×D3)⋊C4C2semidirect
8. #11723523C2semidirect
9. #11723522C2semidirect
10. C4×#6291C2semidirect
11. #11723429C2semidirect
16. C2×#6308C22semidirect
18. C2×C3×C4×C3⋊D3C22semidirect
20. C2×C32×D12C22semidirect
21. C2×C32×C4×D3C22semidirect
24. C2×#6290C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
28. C33×C4⋊C4C22semidirect
30. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
31. C3×#3752C22semidirect
32. C3×C4×C3⋊Dic3C22semidirect
33. C32×(C2×D3)⋊C4C22semidirect
34. #88807C22semidirect
36. (C2×D3)⋊(C3⋊Dic3)C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
51. #64102D3semidirect
54. C3×#895D3semidirect
77. C3×C4⋊(C3⋊D3)D4semidirect
104. C2×C4×C3⋊D3C2×D3semidirect
110. C2×C3×D12C2×D3semidirect
112. C2×C3×C4×D3C2×D3semidirect
120. C32×C4⋊C4C2×D3semidirect
121. C32×C4⋊C4C2×D3semidirect
130. #3752C2×D3semidirect
135. C4×C3⋊Dic3C2×D3semidirect
137. C3×(C2×D3)⋊C4C2×D3semidirect
147. C2×C3⋊(C3⋊D3)#41semidirect
180. #895C3⋊D3semidirect
224. C4⋊(C3⋊D3)D12semidirect
273. C2×D12C2×C3⋊D3semidirect
279. C2×C4×D3C2×C3⋊D3semidirect
284. C3×C4⋊C4D32semidirect
285. C3×C4⋊C4C2×C3⋊D3semidirect
301. (C2×D3)⋊C4C2×C3⋊D3semidirect
369. C33#546semidirect
461. C4⋊C4D3×C3⋊D3semidirect
521. C32#5554semidirect
522. C32#5612semidirect
588. C3#10594630semidirect
589. C3#10596083semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
C2×#890212
#117239382
C3×#641022
C32×#8952
#117235402
C3⋊D3×(C2×D3)⋊C42
#117235232
#117235222
C4×#62912
#117234292
#105960833
#105946303