Finite Group Databasebeta

← #13177429

Subgroups — #13177429

Order 1728.

2636 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic, complemented
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#64144C211yesmaximal, characteristic, complemented
4C32×#937C211yesmaximal, characteristic, complemented
5#11723630C211yesmaximal, characteristic, complemented
6#67⋊(C3⋊Dic3)C212yesmaximal, complemented
7#11723616C212yesmaximal, complemented
8#11723608C211yesmaximal, characteristic, complemented
9#11723598C211yesmaximal, characteristic, complemented
10#11723496C212yesmaximal, complemented
11#11723462C212yesmaximal, complemented
12#10596057—31nomaximal
13#10595160—34nomaximal
14C23×C32×D3C2211yescharacteristic
15C2×C33×D4C2211yescharacteristic
16C2×#6312C2211yescharacteristic
17C2×#6312C2211yescharacteristic, complemented
18C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
20#89078—24no—
21C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
22C2×#6308C2211yescharacteristic, complemented
23C22×C3×C3⋊Dic3C2211yescharacteristic
24C2×C32×#67C2212yes—
25C2×#6291C2212yes—
26C2×D3×C3⋊Dic3—24no—
27C2×D3×C3⋊Dic3C2212yes—
28C2×D3×C3⋊Dic3C2212yes—
29C3×#3757C2211yescharacteristic
30C3×#3757C2212yescomplemented
31C3×#3750C2212yescomplemented
32C32×#217C2211yescharacteristic
33C32×(C2×D3)⋊C4C2212yes—
34#88806C2212yescomplemented
35Dic3⋊(C3⋊Dic3)C2212yescomplemented
36#88804C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38Dic3×C3⋊Dic3C2212yescomplemented
39C23×C3⋊Dic3—31no—
40C2×#3830—31no—
41C2×#3830—34no—
42C2×#3830—64no—
43C22×D3×Dic3—34no—
44C32×#120—31no—
45#64162—31no—
46#64148—32no—
47#64144D311yescharacteristic, complemented
48D4×C3⋊Dic3—32no—
49#64095—31no—
50#64090—32no—
51#64087—32no—
52#64086—31no—
53C3×#937D314yescomplemented
54C3×#935—34no—
55#63977—34no—
56#67⋊Dic3—38no—
57#63971—38no—
58#63966—34no—
59#63957—34no—
60#63896—38no—
61#63881—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3⋊(C3⋊D3)C2311yescharacteristic
65C22×C32×D3—22no—
66C22×C32×D3—24no—
67C22×C32×D3C2312yes—
68C33×D4—22no—
69C2×C33×C4C2311yescharacteristic
70#6312—22no—
71#6312—22no—
72C2×C3⋊(C3⋊Dic3)—22no—
73C2×C3⋊(C3⋊Dic3)D414yescomplemented
74C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
76C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
77C4×C3⋊(C3⋊D3)—22no—
78#6308—22no—
79C2×C3×C3⋊Dic3—22no—
80C2×C3×C3⋊Dic3C2312yes—
81C2×C3×C3⋊Dic3C2312yes—
82C32×#67—24no—
83C2×C32×Dic3C2312yes—
84#6291—24no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—24no—
87#5656—91no—
88#5586—94no—
89C24×C32—31no—
90C23×C3×D3C2×D314yes—
91C2×C32×D4—24no—
92C2×C32×D4—32no—
93C2×C32×D4C2×D314yes—
94C2×C32×D4C2×D311yescharacteristic
95C2×#638—31no—
96C2×#638—31no—
97C2×#638—34no—
98C2×#638—34no—
99C2×#638—32no—
100C2×#638—64no—
101C2×#638—64no—
102C22×C3⋊Dic3—31no—
103C22×C3⋊Dic3—34no—
104C22×C3⋊Dic3—31no—
105C22×C3⋊Dic3—34no—
106C22×C3⋊Dic3—32no—
107C22×C3⋊Dic3—32no—
108C22×C3⋊Dic3—64no—
109C22×C3⋊Dic3—64no—
110C22×C3⋊Dic3—64no—
111C22×C3⋊Dic3C2×D311yescharacteristic
112#3830—64no—
113#3830—616no—
114#3830—632no—
115C2×C4×C3⋊D3—31no—
116C2×C4×C3⋊D3—34no—
117C2×C4×C3⋊D3—64no—
118C2×#634—31no—
119C2×#634—34no—
120C2×#634—64no—
121C2×C3×#67C2×D318yes—
122C22×C3×Dic3—34no—
123C2×#626—38no—
124C2×D3×Dic3—38no—
125C2×D3×Dic3—38no—
126C2×D3×Dic3—616no—
127C32×C22⋊C4—31no—
128C32×C22⋊C4—32no—
129#3757—32no—
130#3757C2×D311yescharacteristic
131#3757C2×D312yescomplemented
132#3752—32no—
133C4⋊(C3⋊Dic3)—32no—
134#3750—32no—
135#3750C2×D312yescomplemented
136C4×C3⋊Dic3—32no—
137C3×#217—34no—
138C3×#217—38no—
139C3×#217C2×D314yes—
140C3×(C2×D3)⋊C4C2×D318yes—
141C3×Dic3⋊C4—38no—
142#3724—38no—
143Dic3⋊Dic3—38no—
144#3722—38no—
145(C2×D3)⋊Dic3—38no—
146Dic32—38no—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33—22no—
150C22×C33C2×D412yes—
151C22×C33C2411yescharacteristic
152C2×C3⋊(C3⋊D3)—21no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C32×D3—28no—
155C2×C32×D3—22no—
156C2×C32×D3—22no—
157C2×C32×D3#4114yescomplemented
158C33×C4—21no—
159C3⋊(C3⋊Dic3)—21no—
160C3⋊(C3⋊Dic3)C2×D412yes—
161C3⋊(C3⋊Dic3)C2×D412yes—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—22no—
164C32×Dic3—22no—
165C23×Dic3—94no—
166C2×#237—94no—
167C2×#237—91no—
168C2×#237—184no—
169C22×C4×D3—91no—
170C3×#123—91no—
171C3×#120—34no—
172#953—94no—
173#939—98no—
174#937C3⋊D311yescharacteristic, complemented
175#935—34no—
176D4×Dic3—98no—
177#929—91no—
178#896—92no—
179#893—92no—
180#886—94no—
181#884—92no—
182#882—91no—
183#67⋊C4—98no—
184#67⋊C4—92no—
185#878—98no—
186#877—94no—
187#877—91no—
188C23×C32—24no—
189C23×C32—24no—
190C23×C32—32no—
191C23×C32—64no—
192C23×C32—62no—
193C23×C32C22×D314yes—
194C23×C32C22×D314yes—
195C23×C32C22×D311yescharacteristic
196C23×C32C22×D311yescharacteristic
197C22×C3⋊D3—31no—
198C22×C3⋊D3—34no—
199C22×C3⋊D3—64no—
200C22×C3×D3—28no—
201C22×C3×D3—216no—
202C22×C3×D3C22×D318yes—
203C32×D4—216no—
204C32×D4—28no—
205C32×D4—22no—
206C32×D4—64no—
207C2×C32×C4—24no—
208C2×C32×C4—32no—
209C2×C32×C4C22×D314yes—
210C2×C32×C4C22×D311yescharacteristic
211#638—64no—
212#638—62no—
213#638—68no—
214#638—616no—
215#638—616no—
216#638—68no—
217#638—62no—
218C2×C3⋊Dic3—22no—
219C2×C3⋊Dic3—34no—
220C2×C3⋊Dic3—31no—
221C2×C3⋊Dic3—31no—
222C2×C3⋊Dic3—31no—
223C2×C3⋊Dic3—316no—
224C2×C3⋊Dic3—34no—
225C2×C3⋊Dic3—34no—
226C2×C3⋊Dic3—34no—
227C2×C3⋊Dic3—616no—
228C2×C3⋊Dic3—64no—
229C2×C3⋊Dic3—64no—
230C2×C3⋊Dic3—64no—
231C2×C3⋊Dic3—64no—
232C2×C3⋊Dic3—64no—
233C2×C3⋊Dic3—62no—
234C2×C3⋊Dic3—68no—
235C2×C3⋊Dic3—616no—
236C2×C3⋊Dic3C22×D312yes—
237C2×C3⋊Dic3C22×D312yes—
238C4×C3⋊D3—62no—
239C4×C3⋊D3—68no—
240C4×C3⋊D3—616no—
241#634—62no—
242#634—68no—
243#634—616no—
244C3×#67—216no—
245C2×C3×Dic3—38no—
246C2×C3×Dic3—38no—
247C2×C3×Dic3—68no—
248C2×C3×Dic3C22×D318yes—
249#626—616no—
250D3×Dic3—616no—
251D3×Dic3—616no—
252#525—271noSylow
253C2×C33—22no—
254C2×C33—41no—
255C2×C33C22×D411yescharacteristic
256C2×C33C2×#4112yes—
257C3⋊(C3⋊D3)—41no—
258C32×D3—24no—
259C24×C3—34no—
260C23×D3C2×C3⋊D311yescharacteristic
261C2×C3×D4—24no—
262C2×C3×D4—38no—
263C2×C3×D4D3214yes—
264C2×C3×D4C2×C3⋊D311yescharacteristic
265C22×C3×C4—91no—
266C2×#67—98no—
267C2×#67—94no—
268C2×#67—94no—
269C2×#67—91no—
270C2×#67—91no—
271C2×#67—184no—
272C2×#67—184no—
273C2×#67C2×C3⋊D312yes—
274C22×Dic3—34no—
275C22×Dic3—98no—
276C22×Dic3—98no—
277C22×Dic3—94no—
278C22×Dic3—91no—
279C22×Dic3—94no—
280C22×Dic3—91no—
281C22×Dic3—184no—
282C22×Dic3—184no—
283C22×Dic3—1816no—
284#237—1816no—
285#237—184no—
286#237—1832no—
287C2×D12—92no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—94no—
291C2×C4×D3—91no—
292C2×C4×D3—184no—
293C2×C4×D3—184no—
294C2×Dic6—94no—
295C2×Dic6—91no—
296C2×Dic6—184no—
297C3×C4⋊C4—92no—
298C3×C22⋊C4—34no—
299C3×C22⋊C4—38no—
300C3×C22⋊C4—91no—
301C3×C22⋊C4—92no—
302#217—34no—
303#217—38no—
304#217—98no—
305#217C2×C3⋊D311yescharacteristic
306(C2×D3)⋊C4—98no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4C2×C3⋊D312yes—
310C4⋊Dic3—98no—
311Dic3⋊C4—38no—
312Dic3⋊C4—98no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315C4×Dic3—98no—
316C4×Dic3—92no—
317C22×C32—22no—
318C22×C32—28no—
319C22×C32—24no—
320C22×C32—21no—
321C22×C32—22no—
322C22×C32—24no—
323C22×C32—28no—
324C22×C32—28no—
325C22×C32—28no—
326C22×C32—34no—
327C22×C32—48no—
328C22×C32—48no—
329C22×C32—68no—
330C22×C32—62no—
331C22×C32—62no—
332C22×C32D3×D418yes—
333C22×C32D3×D412yes—
334C22×C32C23×D314yes—
335C22×C32C23×D311yescharacteristic
336C2×C3⋊D3—68no—
337C2×C3⋊D3—64no—
338C2×C3⋊D3—61no—
339C2×C3⋊D3—62no—
340C2×C3⋊D3—68no—
341C2×C3⋊D3—128no—
342C2×C3×D3—232no—
343C2×C3×D3—28no—
344C2×C3×D3—28no—
345C2×C3×D3#237116yescomplemented
346C32×C4—28no—
347C32×C4—24no—
348C32×C4—21no—
349C32×C4—62no—
350C3⋊Dic3—22no—
351C3⋊Dic3—22no—
352C3⋊Dic3—32no—
353C3⋊Dic3—32no—
354C3⋊Dic3—38no—
355C3⋊Dic3—38no—
356C3⋊Dic3—68no—
357C3⋊Dic3—68no—
358C3⋊Dic3—68no—
359C3⋊Dic3—64no—
360C3⋊Dic3—61no—
361C3×Dic3—28no—
362C3×Dic3—68no—
363C3×Dic3—68no—
364C2×#41—271no—
365C23×C4—271no—
366#123—91no—
367#123—271no—
368C22⋊Q8—272no—
369#121—272no—
370#120—31no—
371#120—271no—
372C4×D4—272no—
373C4×D4—272no—
374#117—271no—
375C33#52511yescharacteristic, complemented, Sylow
376C23×C3—24no—
377C23×C3—24no—
378C23×C3—38no—
379C23×C3—616no—
380C23×C3—68no—
381C23×C3C2×D3214yes—
382C23×C3C2×D3214yes—
383C23×C3C22×C3⋊D311yescharacteristic
384C23×C3C22×C3⋊D311yescharacteristic
385C22×D3—22no—
386C22×D3—24no—
387C22×D3—94no—
388C22×D3—91no—
389C22×D3—184no—
390C22×D3C22×C3⋊D312yes—
391C3×D4—216no—
392C3×D4—28no—
393C3×D4—22no—
394C3×D4—616no—
395C2×C3×C4—24no—
396C2×C3×C4—38no—
397C2×C3×C4—92no—
398C2×C3×C4—92no—
399C2×C3×C4—182no—
400C2×C3×C4C2×D3214yes—
401C2×C3×C4C22×C3⋊D311yescharacteristic
402#67—24no—
403#67—1816no—
404#67—188no—
405#67—182no—
406#67—1816no—
407#67—1816no—
408#67—188no—
409#67—182no—
410C2×Dic3—38no—
411C2×Dic3—38no—
412C2×Dic3—68no—
413C2×Dic3—94no—
414C2×Dic3—94no—
415C2×Dic3—91no—
416C2×Dic3—91no—
417C2×Dic3—916no—
418C2×Dic3—94no—
419C2×Dic3—94no—
420C2×Dic3—91no—
421C2×Dic3—184no—
422C2×Dic3—184no—
423C2×Dic3—184no—
424C2×Dic3—1816no—
425C2×Dic3—1816no—
426C2×Dic3—1816no—
427C2×Dic3—188no—
428C2×Dic3—182no—
429C2×Dic3—1816no—
430C2×Dic3C22×C3⋊D312yes—
431D12—184no—
432C4×D3—188no—
433C4×D3—182no—
434C4×D3—184no—
435C4×D3—184no—
436C4×D3—1816no—
437Dic6—188no—
438Dic6—182no—
439Dic6—1816no—
440C2×C32—22no—
441C2×C32—28no—
442C2×C32—24no—
443C2×C32—28no—
444C2×C32—48no—
445C2×C32—48no—
446C2×C32—41no—
447C2×C32—44no—
448C2×C32—64no—
449C2×C32C2×D3×D414yes—
450C2×C32C2×D3×D411yescharacteristic
451C2×C32C2×#23718yes—
452C2×C32D3×#4112yes—
453C3⋊D3—128no—
454C3⋊D3—121no—
455C3⋊D3—124no—
456C3×D3—216no—
457C24—31no—
458#41—544no—
459C2×Q8—271no—
460C2×D4—32no—
461C2×D4—272no—
462C2×D4—271no—
463C2×D4—271no—
464C2×D4D3×C3⋊D311yescharacteristic
465C22×C4—91no—
466C22×C4—272no—
467C22×C4—272no—
468C22×C4—271no—
469C22×C4—271no—
470C22×C4—271no—
471C22×C4—544no—
472C4⋊C4—92no—
473C4⋊C4—272no—
474C4⋊C4—272no—
475C22⋊C4—31no—
476C22⋊C4—32no—
477C22⋊C4—91no—
478C22⋊C4—92no—
479C22⋊C4—272no—
480C22⋊C4—272no—
481C42—272no—
482C22×C3—28no—
483C22×C3—22no—
484C22×C3—28no—
485C22×C3—28no—
486C22×C3—24no—
487C22×C3—21no—
488C22×C3—22no—
489C22×C3—24no—
490C22×C3—28no—
491C22×C3—316no—
492C22×C3—48no—
493C22×C3—48no—
494C22×C3—632no—
495C22×C3—68no—
496C22×C3—68no—
497C22×C3#381118yes—
498C22×C3D4×C3⋊D312yes—
499C22×C3C22×D3214yes—
500C22×C3C23×C3⋊D311yescharacteristic
501C2×D3—28no—
502C2×D3—22no—
503C2×D3—22no—
504C2×D3—188no—
505C2×D3—188no—
506C2×D3—184no—
507C2×D3—181no—
508C2×D3—182no—
509C2×D3—368no—
510C2×D3#383014yescomplemented
511C3×C4—28no—
512C3×C4—24no—
513C3×C4—21no—
514C3×C4—68no—
515C3×C4—182no—
516C3×C4—182no—
517Dic3—22no—
518Dic3—68no—
519Dic3—68no—
520Dic3—98no—
521Dic3—98no—
522Dic3—92no—
523Dic3—92no—
524Dic3—188no—
525Dic3—188no—
526Dic3—188no—
527Dic3—184no—
528Dic3—181no—
529C32—24no—
530C32#558614yescomplemented
531C32#565611yescharacteristic, complemented
532C23—32no—
533C23—64no—
534C23—62no—
535C23—271no—
536C23C2×D3×C3⋊D311yescharacteristic
537C23C2×D3×C3⋊D311yescharacteristic
538Q8—542no—
539D4—22no—
540D4—64no—
541D4—544no—
542D4—542no—
543D4—542no—
544C2×C4—32no—
545C2×C4—92no—
546C2×C4—92no—
547C2×C4—182no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4—274no—
551C2×C4—271no—
552C2×C4—542no—
553C2×C4—544no—
554C2×C4—544no—
555C2×C4—542no—
556C2×C4C2×D3×C3⋊D311yescharacteristic
557C2×C3—28no—
558C2×C3—22no—
559C2×C3—28no—
560C2×C3—24no—
561C2×C3—48no—
562C2×C3—48no—
563C2×C3—44no—
564C2×C3—41no—
565C2×C3—616no—
566C2×C3D3×#23718yes—
567C2×C3C2×#381114yes—
568C2×C3C2×D4×C3⋊D311yescharacteristic
569C2×C3C2×#383012yes—
570D3—24no—
571D3—368no—
572D3—364no—
573D3—361no—
574C22—22no—
575C22—22no—
576C22—21no—
577C22—34no—
578C22—68no—
579C22—62no—
580C22—62no—
581C22—542no—
582C22—541no—
583C22#8903912yes—
584C22C22×D3×C3⋊D311yescharacteristic
585C4—21no—
586C4—62no—
587C4—182no—
588C4—182no—
589C4—272no—
590C4—272no—
591C4—541no—
592C3—24no—
593C3#1059516014yescomplemented
594C3#1059605711yescharacteristic, complemented
595C2—22no—
596C2—41no—
597C2—64no—
598C2—1081no—
599C2D3×#383012yes—
600C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C22×D3×C3⋊Dic3C2semidirect
3. C3×#64144C2semidirect
4. C32×#937C2semidirect
5. #11723630C2semidirect
6. #67⋊(C3⋊Dic3)C2semidirect
7. #11723616C2semidirect
8. #11723608C2semidirect
9. #11723598C2semidirect
10. #11723496C2semidirect
11. #11723462C2semidirect
17. C2×#6312C22semidirect
19. C22×C3⋊(C3⋊Dic3)C22semidirect
21. C2×C4×C3⋊(C3⋊D3)C22semidirect
22. C2×#6308C22semidirect
30. C3×#3757C22semidirect
31. C3×#3750C22semidirect
34. #88806C22semidirect
35. Dic3⋊(C3⋊Dic3)C22semidirect
36. #88804C22semidirect
38. Dic3×C3⋊Dic3C22semidirect
47. #64144D3semidirect
53. C3×#937D3semidirect
73. C2×C3⋊(C3⋊Dic3)D4semidirect
131. #3757C2×D3semidirect
135. #3750C2×D3semidirect
157. C2×C32×D3#41semidirect
174. #937C3⋊D3semidirect
345. C2×C3×D3#237semidirect
375. C33#525semidirect
510. C2×D3#3830semidirect
530. C32#5586semidirect
531. C32#5656semidirect
593. C3#10595160semidirect
594. C3#10596057semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C22×D3×C3⋊Dic32
C3×#641442
C32×#9372
#117236302
#67⋊(C3⋊Dic3)2
#117236162
#117236082
#117235982
#117234962
#117234622
#105960573
#105951603