Finite Group Databasebeta

← #13177585

Subgroups — #13177585

Order 1728.

2408 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic, complemented
2C2×C32×#237C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723635C212yesmaximal, complemented
5#11723612C212yesmaximal, complemented
6C2×#88807C212yesmaximal, complemented
7#11723603C212yesmaximal, complemented
8C4×#6291C211yesmaximal, characteristic, complemented
9#11723456C211yesmaximal, characteristic, complemented
10#11723455C211yesmaximal, characteristic, complemented
11#11723426C211yesmaximal, characteristic, complemented
12#10596295—31nomaximal
13#10595248—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C22×C3⋊(C3⋊Dic3)C2212yes—
17#89078—24no—
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C22×C3×C3⋊Dic3C2212yes—
21C2×C32×#67C2212yes—
22C22×C32×Dic3C2212yes—
23C32×#237—24no—
24C2×C32×C4×D3C2211yescharacteristic
25C2×C32×Dic6C2211yescharacteristic
26C2×#6291C2211yescharacteristic
27C3×#3757C2212yes—
28C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30#88807C2212yes—
31#88807C2212yes—
32#88807C2211yescharacteristic
33#88807C2212yes—
34#88807C2212yes—
35#88804C2211yescharacteristic
36#88804C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×C32×#41—31no—
40C2×#3830—31no—
41C2×#3830—34no—
42C2×#3830—64no—
43C2×C3×#237D314yescomplemented
44#64155—31no—
45#64153—31no—
46#64148—32no—
47#64144—32no—
48D4×C3⋊Dic3D311yescharacteristic, complemented
49#64138—32no—
50C4×#638—31no—
51#64133—31no—
52C2×#3750—32no—
53C3×D4×Dic3—34no—
54#63978—38no—
55#63966—38no—
56C2×Dic3⋊Dic3—38no—
57#63962—38no—
58C4×#626—34no—
59#63871—34no—
60#63866—34no—
61#63857—34no—
62C23×C33C2312yes—
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C33×D4D414yescomplemented
66C2×C33×C4C2311yescharacteristic
67#6312—24no—
68C2×C3⋊(C3⋊Dic3)—24no—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)—22no—
72#6308—22no—
73C2×C3×C3⋊Dic3—24no—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C32×#67—24no—
78C2×C32×Dic3—24no—
79C2×C32×Dic3C2312yes—
80C2×C32×Dic3C2311yescharacteristic
81C32×C4×D3—22no—
82C32×Dic6—22no—
83#6291—22no—
84#5824—94no—
85#5555—91no—
86C32×#41—64no—
87C2×C32×Q8—31no—
88C2×C32×D4—24no—
89C2×C32×D4—32no—
90C2×C32×D4C2×D311yescharacteristic
91C2×C32×D4C2×D314yes—
92C22×C32×C4—32no—
93C22×C32×C4—31no—
94C2×#638—32no—
95C2×#638—38no—
96C2×#638—31no—
97C2×#638—68no—
98C22×C3⋊Dic3—32no—
99C22×C3⋊Dic3—38no—
100C22×C3⋊Dic3—68no—
101C22×C3⋊Dic3C2×D312yes—
102#3830—632no—
103#3830—64no—
104#3830—616no—
105C2×C4×C3⋊D3—31no—
106C2×C4×C3⋊D3—34no—
107C2×C4×C3⋊D3—64no—
108C2×#634—31no—
109C2×#634—34no—
110C2×#634—64no—
111C2×C3×#67C2×D318yes—
112C22×C3×Dic3—38no—
113C22×C3×Dic3C2×D318yes—
114C3×#237—216no—
115C2×C3×C4×D3C2×D314yes—
116C2×C3×Dic6C2×D314yes—
117C2×#626—34no—
118#3757—32no—
119#3757—31no—
120#3757C2×D312yes—
121#3752—31no—
122#3752—32no—
123C4⋊(C3⋊Dic3)C2×D311yescharacteristic
124#3750—31no—
125#3750—32no—
126#3750—32no—
127#3750—32no—
128#3750—32no—
129C4×C3⋊Dic3C2×D311yescharacteristic
130C3×#217—38no—
131C3×C4⋊Dic3—34no—
132C3×C4×Dic3—34no—
133Dic3⋊Dic3—38no—
134Dic3⋊Dic3—38no—
135Dic3⋊Dic3—38no—
136Dic3⋊Dic3—38no—
137Dic3⋊Dic3—34no—
138#3722—38no—
139#3722—34no—
140(C2×D3)⋊Dic3—38no—
141(C2×D3)⋊Dic3—34no—
142C22×C33—22no—
143C22×C33—22no—
144C22×C33C2×D414yes—
145C22×C33C2411yescharacteristic
146C2×C3⋊(C3⋊D3)—21no—
147C2×C3⋊(C3⋊D3)—21no—
148C2×C3⋊(C3⋊D3)—21no—
149C2×C32×D3—21no—
150C2×C32×D3—21no—
151C2×C32×D3—21no—
152C33×C4C2×D412yes—
153C3⋊(C3⋊Dic3)—22no—
154C3⋊(C3⋊Dic3)—21no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3#4112yescomplemented
158C32×Dic3—22no—
159C32×Dic3—21no—
160C2×C3×#41—34no—
161C2×#237—94no—
162C2×#237—91no—
163C2×#237—184no—
164C2×#237C3⋊D311yescharacteristic, complemented
165C3×C4×D4—91no—
166#946—94no—
167#944—94no—
168#939—98no—
169#935—98no—
170D4×Dic3—34no—
171#930—92no—
172#929—98no—
173C4×#67—94no—
174C2×C4⋊Dic3—92no—
175#924—94no—
176C2×Dic3⋊C4—98no—
177#897—91no—
178#897—91no—
179#886—92no—
180#886—92no—
181C4×D12—91no—
182#869—91no—
183C23×C32—28no—
184C23×C32—31no—
185C23×C32C22×D318yes—
186C23×C32C22×D312yes—
187C22×C3⋊D3—31no—
188C22×C3⋊D3—34no—
189C22×C3⋊D3—64no—
190C22×C3×D3C22×D314yes—
191C32×Q8—62no—
192C32×D4—216no—
193C32×D4—64no—
194C32×D4D1214yescomplemented
195C32×D4#67116yescomplemented
196C2×C32×C4—24no—
197C2×C32×C4—31no—
198C2×C32×C4—32no—
199C2×C32×C4—64no—
200C2×C32×C4—62no—
201C2×C32×C4C22×D314yes—
202C2×C32×C4C22×D311yescharacteristic
203#638—632no—
204#638—64no—
205#638—616no—
206#638—62no—
207C2×C3⋊Dic3—24no—
208C2×C3⋊Dic3—32no—
209C2×C3⋊Dic3—31no—
210C2×C3⋊Dic3—38no—
211C2×C3⋊Dic3—34no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—64no—
214C2×C3⋊Dic3—632no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3—616no—
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D311yescharacteristic
220C4×C3⋊D3—616no—
221C4×C3⋊D3—62no—
222C4×C3⋊D3—68no—
223#634—616no—
224#634—62no—
225#634—68no—
226C3×#67—216no—
227C2×C3×Dic3—216no—
228C2×C3×Dic3—34no—
229C2×C3×Dic3—34no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—616no—
232C2×C3×Dic3C22×D318yes—
233C2×C3×Dic3C22×D314yes—
234C3×C4×D3—28no—
235C3×Dic6—28no—
236#626—68no—
237#547—271noSylow
238C2×C33—24no—
239C2×C33C22×D411yescharacteristic
240C2×C33C2×#4111yescharacteristic
241C2×C332−1+411yescharacteristic
242C3⋊(C3⋊D3)—41no—
243C32×D3—41no—
244C3×#41—616no—
245C2×C3×Q8—34no—
246C2×C3×D4—24no—
247C2×C3×D4—38no—
248C2×C3×D4D3214yes—
249C2×C3×D4C2×C3⋊D311yescharacteristic
250C22×C3×C4—38no—
251C22×C3×C4—34no—
252C22×C3×C4—92no—
253C2×#67—94no—
254C2×#67—98no—
255C2×#67—92no—
256C2×#67—188no—
257C2×#67C2×C3⋊D312yes—
258C22×Dic3—38no—
259C22×Dic3—98no—
260C22×Dic3—92no—
261C22×Dic3—188no—
262C22×Dic3C2×C3⋊D312yes—
263#237—24no—
264#237—1832no—
265#237—1816no—
266#237—184no—
267C2×D12—91no—
268C2×C4×D3—94no—
269C2×C4×D3—91no—
270C2×C4×D3—184no—
271C2×C4×D3C2×C3⋊D311yescharacteristic
272C2×Dic6—94no—
273C2×Dic6—91no—
274C2×Dic6—184no—
275C2×Dic6C2×C3⋊D311yescharacteristic
276C3×C4⋊C4—91no—
277C3×C22⋊C4—92no—
278C3×C42—91no—
279#217—38no—
280#217—98no—
281#217—94no—
282(C2×D3)⋊C4—94no—
283(C2×D3)⋊C4—98no—
284(C2×D3)⋊C4—91no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—91no—
287(C2×D3)⋊C4—92no—
288C4⋊Dic3—34no—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3—91no—
292C4⋊Dic3—92no—
293C4⋊Dic3—92no—
294Dic3⋊C4—94no—
295Dic3⋊C4—98no—
296Dic3⋊C4—98no—
297Dic3⋊C4—98no—
298Dic3⋊C4—98no—
299C4×Dic3—34no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—24no—
303C22×C32—216no—
304C22×C32—28no—
305C22×C32—28no—
306C22×C32—48no—
307C22×C32—48no—
308C22×C32—61no—
309C22×C32—61no—
310C22×C32—61no—
311C22×C32C2×D1214yes—
312C22×C32C2×#67116yes—
313C22×C32C23×D314yes—
314C22×C32C23×D311yescharacteristic
315C2×C3⋊D3—68no—
316C2×C3⋊D3—61no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—64no—
320C2×C3⋊D3—64no—
321C2×C3⋊D3—64no—
322C2×C3⋊D3—124no—
323C2×C3⋊D3—124no—
324C2×C3×D3—24no—
325C2×C3×D3—24no—
326C2×C3×D3—24no—
327C32×C4—28no—
328C32×C4—61no—
329C32×C4—62no—
330C32×C4C2×D1212yes—
331C32×C4C2×#6718yes—
332C3⋊Dic3—22no—
333C3⋊Dic3—21no—
334C3⋊Dic3—616no—
335C3⋊Dic3—68no—
336C3⋊Dic3—62no—
337C3⋊Dic3—61no—
338C3⋊Dic3—68no—
339C3⋊Dic3—64no—
340C3⋊Dic3#23712yescomplemented
341C3×Dic3—28no—
342C3×Dic3—24no—
343C3×Dic3—38no—
344C3×Dic3—68no—
345C3×Dic3—64no—
346C2×#41—31no—
347C2×#41—271no—
348C4⋊Q8—271no—
349#123—272no—
350#123—272no—
351C22⋊Q8—271no—
352C22⋊Q8—271no—
353#121—272no—
354C4×D4—91no—
355C4×D4—271no—
356C2×C4⋊C4—272no—
357C33#54711yescharacteristic, complemented, Sylow
358C23×C3—28no—
359C23×C3—34no—
360C23×C3C2×D3218yes—
361C23×C3C22×C3⋊D312yes—
362C22×D3—94no—
363C22×D3—91no—
364C22×D3—184no—
365C22×D3C22×C3⋊D311yescharacteristic
366C3×Q8—68no—
367C3×D4—216no—
368C3×D4—616no—
369C3×D4#626116yescomplemented
370C3×D4#63814yescomplemented
371C2×C3×C4—24no—
372C2×C3×C4—34no—
373C2×C3×C4—38no—
374C2×C3×C4—616no—
375C2×C3×C4—68no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—92no—
379C2×C3×C4—184no—
380C2×C3×C4C2×D3214yes—
381C2×C3×C4C22×C3⋊D311yescharacteristic
382#67—24no—
383#67—188no—
384#67—1832no—
385#67—1816no—
386#67—184no—
387C2×Dic3—24no—
388C2×Dic3—34no—
389C2×Dic3—34no—
390C2×Dic3—38no—
391C2×Dic3—616no—
392C2×Dic3—98no—
393C2×Dic3—94no—
394C2×Dic3—92no—
395C2×Dic3—91no—
396C2×Dic3—188no—
397C2×Dic3—184no—
398C2×Dic3—1832no—
399C2×Dic3—1816no—
400C2×Dic3—184no—
401C2×Dic3C22×C3⋊D312yes—
402C2×Dic3C22×C3⋊D311yescharacteristic
403D12—182no—
404C4×D3—22no—
405C4×D3—1816no—
406C4×D3—188no—
407C4×D3—182no—
408Dic6—22no—
409Dic6—1816no—
410Dic6—188no—
411Dic6—182no—
412C2×C32—24no—
413C2×C32—24no—
414C2×C32—24no—
415C2×C32—24no—
416C2×C32—216no—
417C2×C32—416no—
418C2×C32—121no—
419C2×C32C22×D1211yescharacteristic
420C2×C32C2×#23711yescharacteristic
421C2×C32D3×#4114yes—
422C2×C32#101014yes—
423C2×C32#101011yescharacteristic
424C2×C32C22×#6714yes—
425C3⋊D3—128no—
426C3⋊D3—121no—
427C3⋊D3—124no—
428C3×D3—44no—
429#41—64no—
430#41—544no—
431C2×Q8—31no—
432C2×Q8—271no—
433C2×D4—32no—
434C2×D4—271no—
435C2×D4—272no—
436C2×D4D3×C3⋊D311yescharacteristic
437C22×C4—32no—
438C22×C4—31no—
439C22×C4—92no—
440C22×C4—272no—
441C22×C4—271no—
442C4⋊C4—91no—
443C4⋊C4—271no—
444C4⋊C4—272no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C4⋊C4—272no—
448C22⋊C4—92no—
449C22⋊C4—271no—
450C22⋊C4—272no—
451C22⋊C4—271no—
452C22⋊C4—272no—
453C42—91no—
454C22×C3—28no—
455C22×C3—28no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—24no—
459C22×C3—216no—
460C22×C3—48no—
461C22×C3—48no—
462C22×C3—64no—
463C22×C3—64no—
464C22×C3—64no—
465C22×C3C2×#626116yes—
466C22×C3C2×#63814yes—
467C22×C3C22×D3214yes—
468C22×C3C23×C3⋊D311yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3—21no—
472C2×D3—188no—
473C2×D3—184no—
474C2×D3—184no—
475C2×D3—184no—
476C2×D3—181no—
477C2×D3—181no—
478C2×D3—181no—
479C2×D3—364no—
480C2×D3—364no—
481C3×C4—28no—
482C3×C4—64no—
483C3×C4—68no—
484C3×C4—92no—
485C3×C4—182no—
486C3×C4—181no—
487C3×C4C2×#62618yes—
488C3×C4C2×#63812yes—
489Dic3—22no—
490Dic3—21no—
491Dic3—38no—
492Dic3—64no—
493Dic3—68no—
494Dic3—1816no—
495Dic3—188no—
496Dic3—188no—
497Dic3—184no—
498Dic3—182no—
499Dic3—181no—
500C32—24no—
501C32#555511yescharacteristic, complemented
502C32#582414yescomplemented
503C23—31no—
504C23—271no—
505C23C2×D3×C3⋊D312yes—
506Q8—62no—
507Q8—542no—
508D4—64no—
509D4—542no—
510D4—544no—
511D4#629114yescomplemented
512C2×C4—31no—
513C2×C4—32no—
514C2×C4—64no—
515C2×C4—62no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—92no—
519C2×C4—184no—
520C2×C4—271no—
521C2×C4—272no—
522C2×C4—544no—
523C2×C4—542no—
524C2×C4C2×D3×C3⋊D311yescharacteristic
525C2×C3—216no—
526C2×C3—24no—
527C2×C3—24no—
528C2×C3—24no—
529C2×C3—24no—
530C2×C3—416no—
531C2×C3—124no—
532C2×C3#6430914yes—
533C2×C3D3×#23714yes—
534C2×C3C22×#62614yes—
535C2×C3#41×C3⋊D311yescharacteristic
536C2×C3#6436711yescharacteristic
537C2×C3C22×#63811yescharacteristic
538D3—41no—
539D3—368no—
540D3—364no—
541D3—361no—
542C22—22no—
543C22—22no—
544C22—61no—
545C22—61no—
546C22—61no—
547C22—541no—
548C22—541no—
549C22—541no—
550C22C2×#629114yes—
551C22C22×D3×C3⋊D311yescharacteristic
552C4—61no—
553C4—62no—
554C4—92no—
555C4—181no—
556C4—182no—
557C4—541no—
558C4—542no—
559C4C2×#629112yes—
560C3—24no—
561C3#1059524814yescomplemented
562C3#1059629511yescharacteristic, complemented
563C2—24no—
564C2—121no—
565C2—1081no—
566C2#1172458211yescharacteristic
567C2C3⋊D3×#23711yescharacteristic
568C2C22×#629111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#547
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. C2×C32×#237C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. #11723635C2semidirect
5. #11723612C2semidirect
6. C2×#88807C2semidirect
7. #11723603C2semidirect
8. C4×#6291C2semidirect
9. #11723456C2semidirect
10. #11723455C2semidirect
11. #11723426C2semidirect
43. C2×C3×#237D3semidirect
48. D4×C3⋊Dic3D3semidirect
65. C33×D4D4semidirect
157. C3×C3⋊Dic3#41semidirect
164. C2×#237C3⋊D3semidirect
194. C32×D4D12semidirect
195. C32×D4#67semidirect
340. C3⋊Dic3#237semidirect
357. C33#547semidirect
369. C3×D4#626semidirect
370. C3×D4#638semidirect
501. C32#5555semidirect
502. C32#5824semidirect
511. D4#6291semidirect
561. C3#10595248semidirect
562. C3#10596295semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×C32×#2372
C3×D4×C3⋊Dic32
#117236352
#117236122
C2×#888072
#117236032
C4×#62912
#117234562
#117234552
#117234262
#105962953
#105952483