Finite Group Databasebeta

← #13177389

Subgroups — #13177389

Order 1728.

2308 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×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4C32×#938C211yesmaximal, characteristic, complemented
5#11723634C212yesmaximal, complemented
6#11723631C212yesmaximal, complemented
7#11723621C212yesmaximal, complemented
8C3⋊D3×#217C212yesmaximal, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10#11723501C211yesmaximal, characteristic, complemented
11#11723431C211yesmaximal, characteristic, complemented
12#10596083—31nomaximal
13#10595228—34nomaximal
14C23×C3×C3⋊D3C2212yes—
15C2×C33×D4C2211yescharacteristic
16C2×#6308C2211yescharacteristic, complemented
17C2×C3×#638C2212yes—
18C3×D4×C3⋊D3—24no—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C32×#67C2212yescomplemented
22C2×C32×C4×D3C2211yescharacteristic, complemented
23C2×#6290C2211yescharacteristic
24C2×#6290C2212yescomplemented
25C2×Dic3×C3⋊D3C2212yescomplemented
26C2×Dic3×C3⋊D3C2212yescomplemented
27#89021—24no—
28#88864C2212yescomplemented
29#88864C2212yescomplemented
30C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31C32×#217C2212yescomplemented
32C32×C4⋊Dic3C2211yescharacteristic, complemented
33#88806C2211yescharacteristic, complemented
34Dic3⋊(C3⋊Dic3)C2212yescomplemented
35#88803C2211yescharacteristic
36#88803C2212yes—
37#88803C2212yescomplemented
38(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic, complemented
39C2×D4×C3⋊D3D311yescharacteristic, complemented
40C2×#3828—31no—
41C2×C3×D3×D4—34no—
42C2×#3795—34no—
43C32×#121—31no—
44#64147—32no—
45#64146—32no—
46#64145—31no—
47#64145—34no—
48#64145—64no—
49C4×#638—31no—
50#64106—31no—
51#64102—31no—
52#64092—32no—
53C22⋊C4×C3⋊D3—32no—
54C3×#938D314yescomplemented
55#63976—38no—
56#63970—38no—
57#63969—38no—
58D3×#217—38no—
59C4×#625—34no—
60#63898—34no—
61#63857—34no—
62C23×C33C2312yes—
63C22×C3×C3⋊D3—24no—
64C22×C3×C3⋊D3—24no—
65C22×C3×C3⋊D3—42no—
66C22×C3×C3⋊D3C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C22×C32×D3C2311yescharacteristic
69C33×D4—22no—
70C2×C33×C4C2311yescharacteristic
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C2×C3⋊(C3⋊Dic3)C2312yes—
74#6308—22no—
75C3×#638—24no—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)D414yescomplemented
78C3×C4×C3⋊D3—22no—
79C32×#67—42no—
80C2×C32×Dic3C2311yescharacteristic
81C2×C32×Dic3C2312yes—
82C32×C4×D3—22no—
83#6290—24no—
84#6290—22no—
85Dic3×C3⋊D3—24no—
86Dic3×C3⋊D3—42no—
87#5806—91no—
88#5612—94no—
89C23×C3⋊D3C2×D312yes—
90C23×C3×D3—38no—
91C2×C32×D4—24no—
92C2×C32×D4—32no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C22×C32×C4—31no—
96C2×#638—32no—
97C2×#638—31no—
98C2×#638C2×D312yes—
99D4×C3⋊D3—24no—
100#3828—64no—
101C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
102C2×C4×C3⋊D3—32no—
103C2×C4×C3⋊D3—32no—
104C2×C4×C3⋊D3C2×D311yescharacteristic
105C2×#634—31no—
106C2×#634—34no—
107C2×#634—64no—
108C2×C3×#67—38no—
109C2×C3×#67C2×D318yescomplemented
110C3×D3×D4—616no—
111C2×C3×D12—34no—
112C2×C3×C4×D3—34no—
113C2×C3×C4×D3C2×D314yescomplemented
114C2×#625—38no—
115C2×#625—34no—
116C2×D3×Dic3—38no—
117C2×D3×Dic3—38no—
118#3795—616no—
119C32×C4⋊C4—31no—
120C32×C22⋊C4—32no—
121#3757—32no—
122#3757—32no—
123#3757—38no—
124#3757—38no—
125#3757—31no—
126#3757—68no—
127#3757—68no—
128#3752—31no—
129#3752—32no—
130#3752—32no—
131C4⋊(C3⋊Dic3)—32no—
132#3750—31no—
133C4×C3⋊Dic3—31no—
134C4×C3⋊Dic3—34no—
135C4×C3⋊Dic3—64no—
136C3×#217C2×D318yescomplemented
137C3×C4⋊Dic3C2×D314yescomplemented
138#3724—34no—
139Dic3⋊Dic3—38no—
140(C2×D3)⋊Dic3—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—22no—
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—21no—
152C2×C3×C3⋊D3—48no—
153C2×C3×C3⋊D3C2×D414yes—
154C2×C3×C3⋊D3#4112yescomplemented
155C2×C32×D3—21no—
156C2×C32×D3—21no—
157C2×C32×D3—21no—
158C33×C4C2×D412yes—
159C3⋊(C3⋊Dic3)—22no—
160C3⋊(C3⋊Dic3)—21no—
161C3⋊(C3⋊Dic3)—21no—
162C3×C3⋊Dic3—21no—
163C32×Dic3—21no—
164C32×Dic3—22no—
165C22×C3×D4—91no—
166C2×#237—91no—
167C2×D3×D4—34no—
168C2×#235—94no—
169C3×#121—34no—
170#953—92no—
171C2×#217—92no—
172#939—92no—
173#938—98no—
174#938C3⋊D311yescharacteristic, complemented
175#937—98no—
176#936—94no—
177#936—91no—
178#936—184no—
179#935—92no—
180D4×Dic3—91no—
181C4×#67—94no—
182C4×#67—91no—
183#924—91no—
184#897—94no—
185#893—94no—
186#883—98no—
187D3×C22⋊C4—98no—
188C23×C32—28no—
189C23×C32—31no—
190C23×C32C22×D318yes—
191C23×C32C22×D312yes—
192C22×C3⋊D3—24no—
193C22×C3⋊D3—24no—
194C22×C3⋊D3—42no—
195C22×C3⋊D3C22×D311yescharacteristic
196C22×C3⋊D3C22×D312yes—
197C22×C3×D3—34no—
198C22×C3×D3—38no—
199C22×C3×D3—616no—
200C22×C3×D3—616no—
201C22×C3×D3—128no—
202C22×C3×D3C22×D314yes—
203C32×D4—22no—
204C32×D4—28no—
205C32×D4—48no—
206C32×D4—122no—
207C2×C32×C4—24no—
208C2×C32×C4—31no—
209C2×C32×C4—32no—
210C2×C32×C4—62no—
211C2×C32×C4C22×D314yes—
212C2×C32×C4C22×D311yescharacteristic
213#638—24no—
214#638—62no—
215#638—64no—
216C2×C3⋊Dic3—31no—
217C2×C3⋊Dic3—31no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—34no—
220C2×C3⋊Dic3—34no—
221C2×C3⋊Dic3—38no—
222C2×C3⋊Dic3—64no—
223C2×C3⋊Dic3—64no—
224C2×C3⋊Dic3—68no—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C4⋊(C3⋊D3)#6714yescomplemented
227C4×C3⋊D3—22no—
228C4×C3⋊D3—64no—
229C4×C3⋊D3—122no—
230#634—62no—
231#634—68no—
232#634—128no—
233C3×#67—48no—
234C3×#67—616no—
235C2×C3×Dic3—34no—
236C2×C3×Dic3C22×D318yes—
237C2×C3×Dic3C22×D314yes—
238C3×D12—316no—
239C3×C4×D3—28no—
240C3×C4×D3—68no—
241#625—616no—
242#625—68no—
243D3×Dic3—616no—
244D3×Dic3—128no—
245#546—271noSylow
246C2×C33—42no—
247C2×C33C22×D411yescharacteristic
248C2×C33C2×#4111yescharacteristic
249C2×C33H(5,𝔽2)11yescharacteristic
250C3×C3⋊D3—22no—
251C3×C3⋊D3—24no—
252C32×D3—41no—
253C24×C3—92no—
254C23×D3—38no—
255C2×C3×D4—24no—
256C2×C3×D4—38no—
257C2×C3×D4—92no—
258C2×C3×D4—91no—
259C2×C3×D4—184no—
260C2×C3×D4D3214yes—
261C2×C3×D4C2×C3⋊D311yescharacteristic
262C22×C3×C4—34no—
263C22×C3×C4—91no—
264C2×#67—38no—
265C2×#67—94no—
266C2×#67—98no—
267C2×#67—91no—
268C2×#67—92no—
269C2×#67C2×C3⋊D312yescomplemented
270C22×Dic3—92no—
271C22×Dic3—92no—
272#237—184no—
273D3×D4—616no—
274#235—1816no—
275C2×D12—34no—
276C2×C4×D3—34no—
277C2×C4×D3—98no—
278C2×C4×D3—98no—
279C2×C4×D3C2×C3⋊D311yescharacteristic, complemented
280C2×Dic6—94no—
281C2×Dic6—91no—
282C2×Dic6—184no—
283C3×C4⋊C4—34no—
284C3×C22⋊C4—38no—
285#217—94no—
286#217—98no—
287#217—98no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—91no—
292#217—92no—
293#217—188no—
294#217—188no—
295#217C2×C3⋊D312yescomplemented
296(C2×D3)⋊C4—94no—
297(C2×D3)⋊C4—98no—
298(C2×D3)⋊C4—98no—
299(C2×D3)⋊C4—91no—
300C4⋊Dic3—98no—
301C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
302Dic3⋊C4—94no—
303Dic3⋊C4—92no—
304Dic3⋊C4—91no—
305C4×Dic3—94no—
306C4×Dic3—91no—
307C4×Dic3—184no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32—22no—
311C22×C32—24no—
312C22×C32—28no—
313C22×C32—28no—
314C22×C32—28no—
315C22×C32—48no—
316C22×C32—48no—
317C22×C32—48no—
318C22×C32—61no—
319C22×C32—61no—
320C22×C32—61no—
321C22×C32C23×D314yes—
322C22×C32C23×D311yescharacteristic
323C2×C3⋊D3—22no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3—21no—
326C2×C3⋊D3—21no—
327C2×C3⋊D3—48no—
328C2×C3⋊D3#23712yescomplemented
329C2×C3⋊D3C2×#6714yes—
330C2×C3×D3—24no—
331C2×C3×D3—24no—
332C2×C3×D3—24no—
333C2×C3×D3—316no—
334C2×C3×D3—38no—
335C2×C3×D3—64no—
336C2×C3×D3—68no—
337C2×C3×D3—64no—
338C2×C3×D3—68no—
339C2×C3×D3—1232no—
340C32×C4—28no—
341C32×C4—62no—
342C32×C4—61no—
343C32×C4D3×D418yes—
344C32×C4C2×#6712yes—
345C3⋊Dic3—21no—
346C3⋊Dic3—61no—
347C3⋊Dic3—61no—
348C3⋊Dic3—62no—
349C3⋊Dic3—68no—
350C3⋊Dic3—64no—
351C3⋊Dic3—64no—
352C3⋊Dic3—128no—
353C3⋊Dic3—124no—
354C3⋊Dic3—124no—
355C3×Dic3—24no—
356C3×Dic3—28no—
357C3×Dic3—64no—
358C2×#41—271no—
359C22×D4—91no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—31no—
364#121—272no—
365#120—272no—
366C4×D4—271no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C33#54611yescharacteristic, complemented, Sylow
370C23×C3—28no—
371C23×C3—34no—
372C23×C3—91no—
373C23×C3—92no—
374C23×C3—184no—
375C23×C3—184no—
376C23×C3—362no—
377C23×C3C2×D3218yes—
378C23×C3C22×C3⋊D312yes—
379C22×D3—34no—
380C22×D3—38no—
381C22×D3—616no—
382C22×D3—616no—
383C22×D3—128no—
384C22×D3C22×C3⋊D311yescharacteristic
385C3×D4—28no—
386C3×D4—22no—
387C3×D4—48no—
388C3×D4—94no—
389C3×D4—128no—
390C3×D4—184no—
391C2×C3×C4—24no—
392C2×C3×C4—34no—
393C2×C3×C4—38no—
394C2×C3×C4—68no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4C2×D3214yes—
398C2×C3×C4C22×C3⋊D311yescharacteristic
399#67—42no—
400#67—616no—
401#67—184no—
402#67—182no—
403#67—1816no—
404#67—188no—
405C2×Dic3—34no—
406C2×Dic3—94no—
407C2×Dic3—94no—
408C2×Dic3—98no—
409C2×Dic3—91no—
410C2×Dic3—91no—
411C2×Dic3—92no—
412C2×Dic3—184no—
413C2×Dic3—184no—
414C2×Dic3—188no—
415C2×Dic3—184no—
416C2×Dic3—362no—
417C2×Dic3C22×C3⋊D312yes—
418C2×Dic3C22×C3⋊D311yescharacteristic
419D12—316no—
420C4×D3—22no—
421C4×D3—68no—
422C4×D3—1816no—
423C4×D3—368no—
424Dic6—188no—
425Dic6—182no—
426Dic6—368no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—42no—
431C2×C32—48no—
432C2×C32—88no—
433C2×C32—121no—
434C2×C32C2×D3×D414yes—
435C2×C32C2×#23711yescharacteristic
436C2×C32#100414yes—
437C2×C32#100411yescharacteristic
438C2×C32D3×#4114yes—
439C2×C32C22×#6711yescharacteristic
440C3⋊D3—22no—
441C3⋊D3—24no—
442C3×D3—44no—
443C3×D3—616no—
444C3×D3—68no—
445C24—92no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—32no—
449C2×D4—92no—
450C2×D4—91no—
451C2×D4—184no—
452C2×D4—271no—
453C2×D4—272no—
454C2×D4D3×C3⋊D311yescharacteristic
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—272no—
458C22×C4—272no—
459C4⋊C4—31no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C22⋊C4—32no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—271no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—271no—
470C22×C3—28no—
471C22×C3—28no—
472C22×C3—28no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—24no—
477C22×C3—48no—
478C22×C3—48no—
479C22×C3—48no—
480C22×C3—64no—
481C22×C3—64no—
482C22×C3—64no—
483C22×C3—94no—
484C22×C3—92no—
485C22×C3—181no—
486C22×C3—182no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3—368no—
490C22×C3C22×D3214yes—
491C22×C3C23×C3⋊D311yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—38no—
496C2×D3—316no—
497C2×D3—68no—
498C2×D3—68no—
499C2×D3—64no—
500C2×D3—64no—
501C2×D3—1232no—
502C3×C4—28no—
503C3×C4—68no—
504C3×C4—64no—
505C3×C4—181no—
506C3×C4D3×#6718yes—
507C3×C4D4×C3⋊D312yes—
508Dic3—21no—
509Dic3—22no—
510Dic3—64no—
511Dic3—184no—
512Dic3—184no—
513Dic3—188no—
514Dic3—182no—
515Dic3—181no—
516Dic3—181no—
517Dic3—368no—
518Dic3—364no—
519Dic3—364no—
520C32—24no—
521C32#561214yescomplemented
522C32#580611yescharacteristic, complemented
523C23—31no—
524C23—91no—
525C23—92no—
526C23—184no—
527C23—184no—
528C23—362no—
529C23C2×D3×C3⋊D312yes—
530Q8—542no—
531D4—22no—
532D4—94no—
533D4—122no—
534D4—184no—
535D4—544no—
536D4—542no—
537C2×C4—31no—
538C2×C4—32no—
539C2×C4—62no—
540C2×C4—91no—
541C2×C4—182no—
542C2×C4—271no—
543C2×C4—271no—
544C2×C4—272no—
545C2×C4—544no—
546C2×C4—1082no—
547C2×C4C2×D3×C3⋊D311yescharacteristic
548C2×C3—24no—
549C2×C3—24no—
550C2×C3—24no—
551C2×C3—48no—
552C2×C3—42no—
553C2×C3—88no—
554C2×C3—124no—
555C2×C3—182no—
556C2×C3—184no—
557C2×C3D3×#23714yes—
558C2×C3#6431314yes—
559C2×C3C2×D3×#6714yes—
560C2×C3C2×D4×C3⋊D311yescharacteristic
561C2×C3#6436111yescharacteristic
562C2×C3#41×C3⋊D311yescharacteristic
563D3—41no—
564D3—68no—
565D3—616no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—61no—
570C22—61no—
571C22—61no—
572C22—92no—
573C22—94no—
574C22—182no—
575C22—182no—
576C22—181no—
577C22—181no—
578C22—368no—
579C22C22×D3×C3⋊D311yescharacteristic
580C4—62no—
581C4—61no—
582C4—181no—
583C4—541no—
584C4—541no—
585C4—542no—
586C4C3⋊D3×#6712yes—
587C3—24no—
588C3#1059522814yescomplemented
589C3#1059608311yescharacteristic, complemented
590C2—42no—
591C2—121no—
592C2—182no—
593C2—184no—
594C2C3⋊D3×#23711yescharacteristic
595C2#1172458711yescharacteristic
596C2C2×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×#89021C2semidirect
3. #11723977C2semidirect
4. C32×#938C2semidirect
5. #11723634C2semidirect
6. #11723631C2semidirect
7. #11723621C2semidirect
8. C3⋊D3×#217C2semidirect
9. C4×#6290C2semidirect
10. #11723501C2semidirect
11. #11723431C2semidirect
16. C2×#6308C22semidirect
21. C2×C32×#67C22semidirect
22. C2×C32×C4×D3C22semidirect
24. C2×#6290C22semidirect
25. C2×Dic3×C3⋊D3C22semidirect
26. C2×Dic3×C3⋊D3C22semidirect
28. #88864C22semidirect
29. #88864C22semidirect
30. C4×C3⋊(C3⋊Dic3)C22semidirect
31. C32×#217C22semidirect
32. C32×C4⋊Dic3C22semidirect
33. #88806C22semidirect
34. Dic3⋊(C3⋊Dic3)C22semidirect
37. #88803C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
39. C2×D4×C3⋊D3D3semidirect
54. C3×#938D3semidirect
77. C3×C4⋊(C3⋊D3)D4semidirect
109. C2×C3×#67C2×D3semidirect
113. C2×C3×C4×D3C2×D3semidirect
136. C3×#217C2×D3semidirect
137. C3×C4⋊Dic3C2×D3semidirect
154. C2×C3×C3⋊D3#41semidirect
174. #938C3⋊D3semidirect
226. C4⋊(C3⋊D3)#67semidirect
269. C2×#67C2×C3⋊D3semidirect
279. C2×C4×D3C2×C3⋊D3semidirect
295. #217C2×C3⋊D3semidirect
301. C4⋊Dic3C2×C3⋊D3semidirect
328. C2×C3⋊D3#237semidirect
369. C33#546semidirect
521. C32#5612semidirect
522. C32#5806semidirect
588. C3#10595228semidirect
589. C3#10596083semidirect

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×#890212
#117239772
C32×#9382
#117236342
#117236312
#117236212
C3⋊D3×#2172
C4×#62902
#117235012
#117234312
#105960833
#105952283