Finite Group Databasebeta

← #13177581

Subgroups — #13177581

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723979C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723624C212yesmaximal, complemented
5Dic3×#638C212yesmaximal, complemented
6#11723612C212yesmaximal, complemented
7#11723596C212yesmaximal, complemented
8C4×#6292C211yesmaximal, characteristic, complemented
9C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11723454C211yesmaximal, characteristic, complemented
11#11723413C211yesmaximal, characteristic, complemented
12#10596029—31nomaximal
13#10595182—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C22×C32×Dic3C2212yes—
20C2×#6292C2211yescharacteristic
21C2×Dic3×C3⋊D3C2212yes—
22#88864C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C3×#3757C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33#88804C2211yescharacteristic
34#88804C2212yes—
35#88803C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64147—31no—
39#64147—34no—
40#64147—64no—
41#64147D311yescharacteristic, complemented
42#64138—32no—
43C4×#638—32no—
44#64101—31no—
45C4⋊C4×C3⋊D3—31no—
46#64093—32no—
47#64088—32no—
48C4×C4⋊(C3⋊D3)—31no—
49#64079—31no—
50C3×#938—34no—
51C3×D4×Dic3D314yescomplemented
52Dic3×#67—38no—
53#63971—38no—
54#63966—38no—
55#63956—38no—
56C4×#626—34no—
57D3×C4⋊Dic3—34no—
58#63869—34no—
59#63849—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2311yescharacteristic
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65#6312—42no—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4×C3⋊(C3⋊D3)—22no—
69C3×#638—24no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C2×C32×Dic3—42no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77#6292—22no—
78Dic3×C3⋊D3—42no—
79#5618—91no—
80#5558—94no—
81C2×C32×D4—24no—
82C2×C32×D4C2×D314yes—
83C2×C32×D4C2×D311yescharacteristic
84C22×C32×C4—32no—
85C2×#638—32no—
86C2×#638—38no—
87C2×#638—68no—
88C2×#638C2×D312yes—
89C2×C4⋊(C3⋊D3)—31no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—34no—
92C2×C4×C3⋊D3—32no—
93C2×C4×C3⋊D3—64no—
94C2×C4×C3⋊D3C2×D311yescharacteristic
95C2×C3×#67—38no—
96C22×C3×Dic3C2×D318yes—
97C2×C3×C4×D3—34no—
98C2×#626—34no—
99C2×D3×Dic3—38no—
100C32×C4⋊C4—31no—
101C32×C22⋊C4—32no—
102C32×C42—31no—
103#3757—32no—
104#3757—38no—
105#3757—68no—
106#3757C2×D312yes—
107#3752—31no—
108#3752—31no—
109#3752—32no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—34no—
112C4⋊(C3⋊Dic3)—31no—
113C4⋊(C3⋊Dic3)—64no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750—32no—
116#3750—32no—
117#3750—32no—
118C4×C3⋊Dic3—32no—
119C3×#217—38no—
120C3×#217C2×D318yes—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3C2×D314yes—
123C3×C4×Dic3C2×D314yes—
124#3724—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—34no—
128#3722—38no—
129#3722—34no—
130(C2×D3)⋊Dic3—34no—
131Dic32—38no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3×C3⋊D3—21no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3#4112yes—
142C33×C4#4112yes—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—22no—
145C3×C3⋊Dic3—21no—
146C3×C3⋊Dic3—22no—
147C32×Dic3—21no—
148C32×Dic3—22no—
149C32×Dic3—21no—
150C3×#121—91no—
151C3×C4×D4—34no—
152#939—92no—
153#938—34no—
154#938—94no—
155#938—91no—
156#938—184no—
157D4×Dic3—92no—
158D4×Dic3C3⋊D311yescharacteristic, complemented
159#929—98no—
160C4×#67—98no—
161C4×#67—91no—
162C2×C4⋊Dic3—91no—
163#892—94no—
164#892—91no—
165D3×C4⋊C4—94no—
166#890—91no—
167#886—92no—
168#884—98no—
169#879—98no—
170#879—92no—
171C4×D12—94no—
172#870—94no—
173C23×C32—28no—
174C23×C32C22×D318yes—
175C23×C32C22×D312yes—
176C22×C3⋊D3—31no—
177C22×C3⋊D3—34no—
178C22×C3⋊D3—64no—
179C22×C3⋊D3C22×D311yescharacteristic
180C22×C3×D3—34no—
181C32×D4—22no—
182C32×D4—28no—
183C32×D4—48no—
184C2×C32×C4—24no—
185C2×C32×C4—32no—
186C2×C32×C4—31no—
187C2×C32×C4—31no—
188C2×C32×C4—122no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191#638—24no—
192#638—122no—
193#638—128no—
194#638—1216no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—32no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—38no—
199C2×C3⋊Dic3—68no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3C22×D312yes—
202C2×C3⋊Dic3C22×D311yescharacteristic
203C4⋊(C3⋊D3)—62no—
204C4×C3⋊D3—22no—
205C4×C3⋊D3—62no—
206C4×C3⋊D3—68no—
207C4×C3⋊D3—128no—
208C4×C3⋊D3—122no—
209C3×#67—616no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—48no—
213C2×C3×Dic3C22×D318yes—
214C2×C3×Dic3C22×D314yes—
215C2×C3×Dic3C22×D314yes—
216C3×C4×D3—68no—
217#626—68no—
218D3×Dic3—128no—
219#553—271noSylow
220C2×C33—42no—
221C2×C33C2×#4111yescharacteristic
222C2×C33C2×#4111yescharacteristic
223C2×C33H(5,𝔽2)11yescharacteristic
224C3⋊(C3⋊D3)—41no—
225C3×C3⋊D3—22no—
226C2×C3×D4—24no—
227C2×C3×D4—92no—
228C2×C3×D4D3214yes—
229C2×C3×D4C2×C3⋊D311yescharacteristic
230C22×C3×C4—38no—
231C22×C3×C4—91no—
232C2×#67—38no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67—91no—
236C2×#67—188no—
237C22×Dic3—92no—
238C22×Dic3C2×C3⋊D312yes—
239C2×D12—94no—
240C2×C4×D3—34no—
241C2×C4×D3—94no—
242C2×C4×D3—91no—
243C2×C4×D3—98no—
244C2×C4×D3—184no—
245C3×C4⋊C4—34no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—38no—
248C3×C22⋊C4—92no—
249C3×C42—34no—
250#217—38no—
251#217—98no—
252#217—92no—
253#217—91no—
254#217—188no—
255#217C2×C3⋊D312yes—
256(C2×D3)⋊C4—94no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—94no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—92no—
261C4⋊Dic3—34no—
262C4⋊Dic3—94no—
263C4⋊Dic3—94no—
264C4⋊Dic3—91no—
265C4⋊Dic3—92no—
266C4⋊Dic3—92no—
267C4⋊Dic3—184no—
268C4⋊Dic3C2×C3⋊D311yescharacteristic
269Dic3⋊C4—98no—
270Dic3⋊C4—98no—
271Dic3⋊C4—98no—
272Dic3⋊C4—91no—
273Dic3⋊C4—92no—
274C4×Dic3—98no—
275C4×Dic3—92no—
276C4×Dic3C2×C3⋊D311yescharacteristic
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—24no—
281C22×C32—28no—
282C22×C32—28no—
283C22×C32—28no—
284C22×C32—48no—
285C22×C32—48no—
286C22×C32—48no—
287C22×C32C23×D314yes—
288C22×C32C23×D311yescharacteristic
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—61no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—61no—
296C2×C3⋊D3—64no—
297C2×C3⋊D3—64no—
298C2×C3⋊D3—124no—
299C2×C3⋊D3—124no—
300C2×C3⋊D3#23712yes—
301C2×C3×D3—38no—
302C2×C3×D3—64no—
303C2×C3×D3—64no—
304C32×C4—28no—
305C32×C4—61no—
306C32×C4—62no—
307C32×C4—61no—
308C32×C4#23518yes—
309C32×C4#23712yes—
310C3⋊Dic3—21no—
311C3⋊Dic3—22no—
312C3⋊Dic3—61no—
313C3⋊Dic3—62no—
314C3⋊Dic3—64no—
315C3⋊Dic3—68no—
316C3⋊Dic3—124no—
317C3⋊Dic3—128no—
318C3×Dic3—28no—
319C3×Dic3—24no—
320C3×Dic3—24no—
321C3×Dic3—64no—
322C3×Dic3—68no—
323#126—272no—
324#125—271no—
325#123—272no—
326#121—91no—
327#121—272no—
328#121—271no—
329C4×D4—31no—
330C4×D4—271no—
331C4×D4—272no—
332#117—271no—
333C2×C4⋊C4—271no—
334C33#55311yescharacteristic, complemented, Sylow
335C23×C3—28no—
336C23×C3—91no—
337C23×C3C2×D3218yes—
338C23×C3C22×C3⋊D312yes—
339C22×D3—34no—
340C22×D3—94no—
341C22×D3—91no—
342C22×D3—184no—
343C3×D4—28no—
344C3×D4—22no—
345C3×D4—48no—
346C3×D4—184no—
347C2×C3×C4—24no—
348C2×C3×C4—38no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—92no—
352C2×C3×C4—91no—
353C2×C3×C4—128no—
354C2×C3×C4—182no—
355C2×C3×C4C2×D3214yes—
356C2×C3×C4C22×C3⋊D311yescharacteristic
357#67—616no—
358#67—182no—
359#67—368no—
360#67—362no—
361#67—3616no—
362C2×Dic3—38no—
363C2×Dic3—34no—
364C2×Dic3—42no—
365C2×Dic3—94no—
366C2×Dic3—98no—
367C2×Dic3—91no—
368C2×Dic3—92no—
369C2×Dic3—188no—
370C2×Dic3—184no—
371C2×Dic3—362no—
372C2×Dic3C22×C3⋊D312yes—
373C2×Dic3C22×C3⋊D311yescharacteristic
374C2×Dic3C22×C3⋊D311yescharacteristic
375D12—188no—
376C4×D3—68no—
377C4×D3—188no—
378C4×D3—182no—
379C4×D3—368no—
380C4×D3—368no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—42no—
385C2×C32—48no—
386C2×C32—88no—
387C2×C32C2×#23514yes—
388C2×C32C2×#23711yescharacteristic
389C2×C32C2×#23711yescharacteristic
390C2×C32#100414yes—
391C2×C32D3×#4114yes—
392C2×C32#100911yescharacteristic
393C3⋊D3—22no—
394C3⋊D3—128no—
395C3⋊D3—124no—
396C3⋊D3—121no—
397C3×D3—68no—
398C2×D4—92no—
399C2×D4—271no—
400C2×D4—272no—
401C2×D4D3×C3⋊D311yescharacteristic
402C22×C4—32no—
403C22×C4—91no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—31no—
407C4⋊C4—91no—
408C4⋊C4—272no—
409C4⋊C4—272no—
410C4⋊C4—271no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—92no—
415C22⋊C4—272no—
416C22⋊C4—271no—
417C22⋊C4—271no—
418C22⋊C4—272no—
419C42—31no—
420C42—272no—
421C22×C3—28no—
422C22×C3—28no—
423C22×C3—28no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—24no—
428C22×C3—48no—
429C22×C3—48no—
430C22×C3—48no—
431C22×C3—92no—
432C22×C3—181no—
433C22×C3—181no—
434C22×C3C22×D3214yes—
435C22×C3C23×C3⋊D311yescharacteristic
436C2×D3—38no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—188no—
440C2×D3—184no—
441C2×D3—184no—
442C2×D3—184no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—181no—
446C2×D3—364no—
447C2×D3—364no—
448C3×C4—28no—
449C3×C4—64no—
450C3×C4—68no—
451C3×C4—64no—
452C3×C4—181no—
453C3×C4—182no—
454C3×C4#380418yes—
455C3×C4#382812yes—
456Dic3—21no—
457Dic3—22no—
458Dic3—21no—
459Dic3—68no—
460Dic3—64no—
461Dic3—188no—
462Dic3—184no—
463Dic3—181no—
464Dic3—182no—
465Dic3—364no—
466Dic3—368no—
467C32—24no—
468C32#555814yescomplemented
469C32#561811yescharacteristic, complemented
470C23—91no—
471C23—271no—
472C23C2×D3×C3⋊D312yes—
473D4—22no—
474D4—184no—
475D4—542no—
476D4—1082no—
477C2×C4—31no—
478C2×C4—31no—
479C2×C4—32no—
480C2×C4—92no—
481C2×C4—91no—
482C2×C4—122no—
483C2×C4—182no—
484C2×C4—271no—
485C2×C4—272no—
486C2×C4—542no—
487C2×C4—1082no—
488C2×C4C2×D3×C3⋊D311yescharacteristic
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—48no—
493C2×C3—42no—
494C2×C3—88no—
495C2×C3—182no—
496C2×C3D3×#23714yes—
497C2×C3#6431414yes—
498C2×C3C2×#380414yes—
499C2×C3C2×#382811yescharacteristic
500C2×C3#6436111yescharacteristic
501C2×C3#41×C3⋊D311yescharacteristic
502D3—68no—
503D3—368no—
504D3—364no—
505D3—361no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22—541no—
513C22—541no—
514C22—541no—
515C22C22×D3×C3⋊D311yescharacteristic
516C4—61no—
517C4—61no—
518C4—62no—
519C4—182no—
520C4—181no—
521C4—542no—
522C4—541no—
523C4#8903112yes—
524C3—24no—
525C3#1059518214yescomplemented
526C3#1059602911yescharacteristic, complemented
527C2—42no—
528C2—182no—
529C2—1081no—
530C2C3⋊D3×#23711yescharacteristic
531C2#1172458811yescharacteristic
532C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723979C2semidirect
2. C3×#64147C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723624C2semidirect
5. Dic3×#638C2semidirect
6. #11723612C2semidirect
7. #11723596C2semidirect
8. C4×#6292C2semidirect
9. C3⋊D3×C4⋊Dic3C2semidirect
10. #11723454C2semidirect
11. #11723413C2semidirect
41. #64147D3semidirect
51. C3×D4×Dic3D3semidirect
158. D4×Dic3C3⋊D3semidirect
334. C33#553semidirect
468. C32#5558semidirect
469. C32#5618semidirect
525. C3#10595182semidirect
526. C3#10596029semidirect

Maximal subgroups

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

subgroupindex
#117239792
C3×#641472
C32×D4×Dic32
#117236242
Dic3×#6382
#117236122
#117235962
C4×#62922
C3⋊D3×C4⋊Dic32
#117234542
#117234132
#105960293
#105951823