Finite Group Databasebeta

← #13175508

Subgroups — #13175508

Order 1728.

2284 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89029C211yesmaximal, characteristic, complemented
2C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723918C211yesmaximal, characteristic, complemented
4C3×#64095C211yesmaximal, characteristic, complemented
5C32×#882C211yesmaximal, characteristic, complemented
6#11723629C211yesmaximal, characteristic, complemented
7#11723602C211yesmaximal, characteristic, complemented
8#11723508C212yesmaximal, complemented
9#11723500C212yesmaximal, complemented
10#11723499C212yesmaximal, complemented
11#11723474C212yesmaximal, complemented
12#10596057—31nomaximal
13#10594082—34nomaximal
14C23×C32×D3C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×#638C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C2×C32×#67C2211yescharacteristic
19C2×C32×D12C2212yes—
20C2×#6293C2211yescharacteristic, complemented
21C2×#6290C2212yes—
22C2×#6290C2211yescharacteristic, complemented
23C2×Dic3×C3⋊D3C2211yescharacteristic, complemented
24#89029—24no—
25C2×D3×C3⋊Dic3—24no—
26C2×D3×C3⋊Dic3C2212yes—
27C2×D3×C3⋊Dic3C2212yes—
28C2×D3×C3⋊Dic3C2211yescharacteristic, complemented
29C33×C22⋊C4C2211yescharacteristic
30#88864C2211yescharacteristic
31#88857C2212yescomplemented
32C4×C3⋊(C3⋊Dic3)C2212yescomplemented
33C3×#3752C2212yescomplemented
34C3×C4⋊(C3⋊Dic3)C2212yescomplemented
35C32×(C2×D3)⋊C4C2212yes—
36#88806C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
39C23×C3⋊Dic3—31no—
40C2×#3830—31no—
41C2×#3805—34no—
42C22×D3×Dic3—34no—
43C32×#120—31no—
44#64162—31no—
45#64148—32no—
46#64144—31no—
47D4×C3⋊Dic3—32no—
48#64095D311yescharacteristic, complemented
49#64090—32no—
50#64087—32no—
51#64086—31no—
52#64086—34no—
53#64086—64no—
54C3×#886—34no—
55C3×#882D314yescomplemented
56#63976—34no—
57#63961—34no—
58#63902—38no—
59#63899—38no—
60#63898—38no—
61#63885—38no—
62C23×C33C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3—22no—
65C22×C32×D3—24no—
66C22×C32×D3C2311yescharacteristic
67C22×C32×D3C2312yes—
68C2×C33×C4C2312yes—
69C2×C3⋊(C3⋊Dic3)D414yescomplemented
70C2×C3⋊(C3⋊Dic3)C2312yes—
71C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
72C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
73C3×#638—22no—
74C2×C3×C3⋊Dic3—22no—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C32×#67—22no—
78C2×C32×Dic3C2311yescharacteristic
79C32×D12—24no—
80#6293—22no—
81#6290—24no—
82#6290—22no—
83Dic3×C3⋊D3—22no—
84D3×C3⋊Dic3—24no—
85D3×C3⋊Dic3—24no—
86D3×C3⋊Dic3—22no—
87#5656—91no—
88#5586—94no—
89C24×C32—31no—
90C23×C3×D3C2×D314yes—
91C2×C32×D4—31no—
92C2×C32×D4—32no—
93C2×#638—32no—
94C2×#638—31no—
95C2×#638C2×D311yescharacteristic
96C22×C3⋊Dic3—31no—
97C22×C3⋊Dic3—34no—
98C22×C3⋊Dic3—32no—
99C22×C3⋊Dic3—32no—
100C22×C3⋊Dic3—31no—
101C22×C3⋊Dic3—64no—
102C22×C3⋊Dic3—64no—
103C22×C3⋊Dic3C2×D311yescharacteristic
104#3830—64no—
105C2×C4×C3⋊D3—31no—
106C2×#634—31no—
107C2×C3×#67—34no—
108C2×C3×#67C2×D314yes—
109C22×C3×Dic3—34no—
110C2×C3×D12C2×D318yes—
111C2×#627—34no—
112C2×#625—34no—
113C2×#625—38no—
114#3805—616no—
115C2×D3×Dic3—38no—
116C2×D3×Dic3—38no—
117C2×D3×Dic3—34no—
118C2×D3×Dic3—34no—
119C2×D3×Dic3—616no—
120C32×C22⋊C4—24no—
121C32×C22⋊C4—32no—
122C32×C22⋊C4C2×D314yes—
123C32×C22⋊C4C2×D311yescharacteristic
124#3757—31no—
125#3757—34no—
126#3757—32no—
127#3757—32no—
128#3757—64no—
129#3752C2×D312yescomplemented
130C4⋊(C3⋊Dic3)C2×D312yescomplemented
131#3750—32no—
132#3750—38no—
133#3750—32no—
134#3750—68no—
135C4×C3⋊Dic3—32no—
136C4×C3⋊Dic3—38no—
137C4×C3⋊Dic3—68no—
138C3×(C2×D3)⋊C4—38no—
139C3×(C2×D3)⋊C4C2×D318yes—
140C3×C4⋊Dic3—38no—
141#3724—38no—
142(C2×D3)⋊Dic3—38no—
143(C2×D3)⋊Dic3—38no—
144C22×C33—22no—
145C22×C33C2×D412yes—
146C22×C33C2411yescharacteristic
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3—22no—
149C2×C32×D3—28no—
150C2×C32×D3—22no—
151C2×C32×D3—22no—
152C2×C32×D3—21no—
153C2×C32×D3—22no—
154C2×C32×D3#4114yescomplemented
155C33×C4—22no—
156C3⋊(C3⋊Dic3)—22no—
157C3⋊(C3⋊Dic3)C2×D412yes—
158C3⋊(C3⋊Dic3)C2×D412yes—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3—21no—
161C32×Dic3—21no—
162C23×Dic3—94no—
163C2×#237—94no—
164C2×#237—91no—
165C22×C4×D3—91no—
166C3×#123—91no—
167C3×#120—34no—
168#953—94no—
169#939—98no—
170#937—91no—
171#935—94no—
172D4×Dic3—98no—
173#929—91no—
174#896—92no—
175#893—92no—
176#886—34no—
177#884—92no—
178#882C3⋊D311yescharacteristic, complemented
179#67⋊C4—98no—
180#67⋊C4—92no—
181#878—98no—
182#877—94no—
183#877—91no—
184#877—184no—
185C23×C32—24no—
186C23×C32—32no—
187C23×C32—31no—
188C23×C32—64no—
189C23×C32—62no—
190C23×C32C22×D314yes—
191C23×C32C22×D311yescharacteristic
192C22×C3⋊D3C22×D311yescharacteristic
193C22×C3×D3—28no—
194C22×C3×D3—216no—
195C22×C3×D3—34no—
196C22×C3×D3C22×D314yes—
197C22×C3×D3C22×D318yes—
198C32×D4—64no—
199C32×D4—62no—
200C2×C32×C4—28no—
201C2×C32×C4—31no—
202C2×C32×C4C22×D318yes—
203C2×C32×C4C22×D312yes—
204#638—22no—
205#638—62no—
206#638—64no—
207C2×C3⋊Dic3—22no—
208C2×C3⋊Dic3—34no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—31no—
211C2×C3⋊Dic3—32no—
212C2×C3⋊Dic3—316no—
213C2×C3⋊Dic3—34no—
214C2×C3⋊Dic3—34no—
215C2×C3⋊Dic3—38no—
216C2×C3⋊Dic3—64no—
217C2×C3⋊Dic3—64no—
218C2×C3⋊Dic3—616no—
219C2×C3⋊Dic3—68no—
220C2×C3⋊Dic3—62no—
221C2×C3⋊Dic3—64no—
222C2×C3⋊Dic3—64no—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C4×C3⋊D3—62no—
226#634—62no—
227C3×#67—28no—
228C3×#67—68no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—68no—
232C2×C3×Dic3C22×D314yes—
233C3×D12—216no—
234#627—68no—
235#625—68no—
236#625—616no—
237D3×Dic3—68no—
238D3×Dic3—616no—
239D3×Dic3—616no—
240D3×Dic3—68no—
241#525—271noSylow
242C2×C33—22no—
243C2×C33C22×D411yescharacteristic
244C2×C33C2×#4112yes—
245C3×C3⋊D3—41no—
246C32×D3—24no—
247C32×D3—41no—
248C24×C3—34no—
249C23×D3C2×C3⋊D311yescharacteristic
250C2×C3×D4—34no—
251C2×C3×D4—38no—
252C2×C3×D4—91no—
253C22×C3×C4—91no—
254C2×#67—34no—
255C2×#67—98no—
256C2×#67—94no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67C2×C3⋊D311yescharacteristic
260C22×Dic3—34no—
261C22×Dic3—98no—
262C22×Dic3—98no—
263C22×Dic3—94no—
264C22×Dic3—94no—
265C22×Dic3—91no—
266C22×Dic3—91no—
267C22×Dic3—184no—
268C22×Dic3—1816no—
269#237—184no—
270#237—1816no—
271C2×D12C2×C3⋊D312yes—
272C2×C4×D3—94no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×C4×D3—91no—
276C2×C4×D3—184no—
277C2×Dic6—94no—
278C2×Dic6—91no—
279C3×C4⋊C4—92no—
280C3×C22⋊C4—24no—
281C3×C22⋊C4—38no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4D3214yes—
284C3×C22⋊C4C2×C3⋊D311yescharacteristic
285#217—98no—
286#217—98no—
287#217—94no—
288#217—91no—
289#217—184no—
290(C2×D3)⋊C4—38no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4C2×C3⋊D312yes—
294C4⋊Dic3—38no—
295Dic3⋊C4—98no—
296Dic3⋊C4—98no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—188no—
300C4×Dic3—98no—
301C4×Dic3—92no—
302C4×Dic3—188no—
303C22×C32—22no—
304C22×C32—24no—
305C22×C32—28no—
306C22×C32—28no—
307C22×C32—34no—
308C22×C32—48no—
309C22×C32—68no—
310C22×C32—62no—
311C22×C32—62no—
312C22×C32—61no—
313C22×C32—62no—
314C22×C32D3×D418yes—
315C22×C32D3×D412yes—
316C22×C32C23×D314yes—
317C22×C32C23×D311yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3×D3—232no—
321C2×C3×D3—28no—
322C2×C3×D3—28no—
323C2×C3×D3—24no—
324C2×C3×D3—28no—
325C2×C3×D3—64no—
326C2×C3×D3—68no—
327C2×C3×D3#237116yescomplemented
328C32×C4—22no—
329C32×C4—28no—
330C32×C4—48no—
331C32×C4—61no—
332C3⋊Dic3—21no—
333C3⋊Dic3—22no—
334C3⋊Dic3—32no—
335C3⋊Dic3—32no—
336C3⋊Dic3—38no—
337C3⋊Dic3—38no—
338C3⋊Dic3—68no—
339C3⋊Dic3—68no—
340C3⋊Dic3—62no—
341C3⋊Dic3—68no—
342C3⋊Dic3—128no—
343C3×Dic3—24no—
344C3×Dic3—68no—
345C3×Dic3—64no—
346C2×#41—271no—
347C23×C4—271no—
348#123—91no—
349#123—271no—
350C22⋊Q8—272no—
351#121—272no—
352#120—31no—
353#120—271no—
354C4×D4—272no—
355C4×D4—272no—
356#117—271no—
357C33#52511yescharacteristic, complemented, Sylow
358C23×C3—24no—
359C23×C3—38no—
360C23×C3—34no—
361C23×C3—616no—
362C23×C3—68no—
363C23×C3—91no—
364C23×C3C2×D3214yes—
365C23×C3C22×C3⋊D311yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—34no—
369C22×D3C22×C3⋊D311yescharacteristic
370C22×D3C22×C3⋊D312yes—
371C3×D4—616no—
372C3×D4—68no—
373C3×D4—182no—
374C2×C3×C4—28no—
375C2×C3×C4—34no—
376C2×C3×C4—92no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379C2×C3×C4C2×D3218yes—
380C2×C3×C4C22×C3⋊D312yes—
381#67—22no—
382#67—68no—
383#67—182no—
384#67—188no—
385#67—1816no—
386#67—184no—
387C2×Dic3—38no—
388C2×Dic3—34no—
389C2×Dic3—68no—
390C2×Dic3—94no—
391C2×Dic3—94no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—916no—
395C2×Dic3—94no—
396C2×Dic3—98no—
397C2×Dic3—92no—
398C2×Dic3—184no—
399C2×Dic3—184no—
400C2×Dic3—1816no—
401C2×Dic3—188no—
402C2×Dic3—182no—
403C2×Dic3—188no—
404C2×Dic3—1816no—
405C2×Dic3—1816no—
406C2×Dic3C22×C3⋊D311yescharacteristic
407D12—24no—
408C4×D3—188no—
409C4×D3—184no—
410C4×D3—184no—
411C4×D3—182no—
412Dic6—182no—
413Dic6—188no—
414C2×C32—22no—
415C2×C32—24no—
416C2×C32—28no—
417C2×C32—28no—
418C2×C32—48no—
419C2×C32—64no—
420C2×C32—121no—
421C2×C32C2×D3×D414yes—
422C2×C32C2×D3×D411yescharacteristic
423C2×C32C2×#23718yes—
424C2×C32D3×#4112yes—
425C3⋊D3—41no—
426C3×D3—216no—
427C3×D3—44no—
428C3×D3—124no—
429C24—31no—
430#41—544no—
431C2×Q8—271no—
432C2×D4—31no—
433C2×D4—32no—
434C2×D4—91no—
435C2×D4—272no—
436C2×D4—271no—
437C22×C4—91no—
438C22×C4—271no—
439C22×C4—272no—
440C22×C4—271no—
441C22×C4—272no—
442C22×C4—271no—
443C22×C4—544no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C22⋊C4—32no—
448C22⋊C4—92no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C22⋊C4—271no—
452C22⋊C4D3×C3⋊D311yescharacteristic
453C42—272no—
454C22×C3—28no—
455C22×C3—22no—
456C22×C3—24no—
457C22×C3—28no—
458C22×C3—316no—
459C22×C3—48no—
460C22×C3—632no—
461C22×C3—68no—
462C22×C3—68no—
463C22×C3—64no—
464C22×C3—68no—
465C22×C3—181no—
466C22×C3—182no—
467C22×C3#380218yes—
468C22×C3D4×C3⋊D312yes—
469C22×C3C22×D3214yes—
470C22×C3C23×C3⋊D311yescharacteristic
471C2×D3—28no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—21no—
475C2×D3—22no—
476C2×D3—64no—
477C2×D3—68no—
478C2×D3#383014yescomplemented
479C3×C4—28no—
480C3×C4—22no—
481C3×C4—48no—
482C3×C4—64no—
483C3×C4—181no—
484C3×C4—182no—
485Dic3—21no—
486Dic3—64no—
487Dic3—68no—
488Dic3—98no—
489Dic3—98no—
490Dic3—92no—
491Dic3—92no—
492Dic3—188no—
493Dic3—188no—
494Dic3—188no—
495Dic3—182no—
496Dic3—368no—
497C32—24no—
498C32#558614yescomplemented
499C32#565611yescharacteristic, complemented
500C23—32no—
501C23—31no—
502C23—64no—
503C23—62no—
504C23—91no—
505C23C2×D3×C3⋊D311yescharacteristic
506Q8—542no—
507D4—64no—
508D4—62no—
509D4—182no—
510D4—542no—
511D4—544no—
512C2×C4—31no—
513C2×C4—92no—
514C2×C4—91no—
515C2×C4—182no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—274no—
519C2×C4—272no—
520C2×C4—542no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4—542no—
524C2×C4C2×D3×C3⋊D312yes—
525C2×C3—28no—
526C2×C3—22no—
527C2×C3—24no—
528C2×C3—28no—
529C2×C3—48no—
530C2×C3—616no—
531C2×C3—124no—
532C2×C3—361no—
533C2×C3C2×#380214yes—
534C2×C3D3×#23718yes—
535C2×C3C2×D4×C3⋊D311yescharacteristic
536C2×C3C2×#383012yes—
537D3—24no—
538D3—41no—
539D3—124no—
540C22—22no—
541C22—34no—
542C22—68no—
543C22—62no—
544C22—62no—
545C22—61no—
546C22—62no—
547C22—181no—
548C22—182no—
549C22#8902612yes—
550C22C22×D3×C3⋊D311yescharacteristic
551C4—22no—
552C4—61no—
553C4—182no—
554C4—181no—
555C4—272no—
556C4—272no—
557C4—542no—
558C3—24no—
559C3#1059408214yescomplemented
560C3#1059605711yescharacteristic, complemented
561C2—22no—
562C2—64no—
563C2—121no—
564C2—361no—
565C2C2×#8902611yescharacteristic
566C2D3×#383012yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#525
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89029C2semidirect
2. C22×D3×C3⋊Dic3C2semidirect
3. #11723918C2semidirect
4. C3×#64095C2semidirect
5. C32×#882C2semidirect
6. #11723629C2semidirect
7. #11723602C2semidirect
8. #11723508C2semidirect
9. #11723500C2semidirect
10. #11723499C2semidirect
11. #11723474C2semidirect
20. C2×#6293C22semidirect
22. C2×#6290C22semidirect
23. C2×Dic3×C3⋊D3C22semidirect
28. C2×D3×C3⋊Dic3C22semidirect
31. #88857C22semidirect
32. C4×C3⋊(C3⋊Dic3)C22semidirect
33. C3×#3752C22semidirect
34. C3×C4⋊(C3⋊Dic3)C22semidirect
36. #88806C22semidirect
38. (C2×D3)⋊(C3⋊Dic3)C22semidirect
48. #64095D3semidirect
55. C3×#882D3semidirect
69. C2×C3⋊(C3⋊Dic3)D4semidirect
129. #3752C2×D3semidirect
130. C4⋊(C3⋊Dic3)C2×D3semidirect
154. C2×C32×D3#41semidirect
178. #882C3⋊D3semidirect
327. C2×C3×D3#237semidirect
357. C33#525semidirect
478. C2×D3#3830semidirect
498. C32#5586semidirect
499. C32#5656semidirect
559. C3#10594082semidirect
560. C3#10596057semidirect

Maximal subgroups

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

subgroupindex
C2×#890292
C22×D3×C3⋊Dic32
#117239182
C3×#640952
C32×#8822
#117236292
#117236022
#117235082
#117235002
#117234992
#117234742
#105960573
#105940823