Finite Group Databasebeta

← #13177576

Subgroups — #13177576

Order 1728.

1804 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3C32×#938C211yesmaximal, characteristic, complemented
4#11723631C212yesmaximal, complemented
5#11723626C212yesmaximal, complemented
6#11723603C212yesmaximal, complemented
7#11723597C212yesmaximal, complemented
8C4×#6290C211yesmaximal, characteristic, complemented
9C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11723450C211yesmaximal, characteristic, complemented
11#11723424C211yesmaximal, characteristic, complemented
12#10596085—31nomaximal
13#10595245—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×#67C2212yes—
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6290C2211yescharacteristic
21C2×Dic3×C3⋊D3C2212yes—
22#88864C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3757C2212yes—
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29#88807C2212yes—
30#88807C2212yes—
31#88806C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2211yescharacteristic
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#121—31no—
38#64147—32no—
39#64147D311yescharacteristic, complemented
40#64144—32no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44C4×#638—31no—
45#64135—31no—
46C4⋊C4×C3⋊D3—31no—
47#64099—31no—
48#64090—32no—
49#64088—32no—
50C3×#938—34no—
51C3×#938D314yescomplemented
52#67⋊Dic3—38no—
53#63970—38no—
54#63962—38no—
55#63956—38no—
56C4×#625—34no—
57D3×C4⋊Dic3—34no—
58#63872—34no—
59#63841—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2311yescharacteristic
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—42no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C3×#638—24no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C32×#67—42no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2312yes—
76C32×C4×D3—22no—
77#6290—22no—
78Dic3×C3⋊D3—42no—
79#5618—91no—
80#5614—94no—
81C2×C32×D4—24no—
82C2×C32×D4—32no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C2×D311yescharacteristic
85C22×C32×C4—31no—
86C2×#638—31no—
87C2×#638C2×D312yes—
88C22×C3⋊Dic3—32no—
89C22×C3⋊Dic3—38no—
90C22×C3⋊Dic3—68no—
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×#67—38no—
94C2×C3×#67C2×D318yes—
95C2×C3×C4×D3—34no—
96C2×C3×C4×D3C2×D314yes—
97C2×#625—34no—
98C2×D3×Dic3—38no—
99C32×C4⋊C4—31no—
100C32×C22⋊C4—32no—
101#3757—32no—
102#3757—38no—
103#3757—32no—
104#3757—31no—
105#3757—68no—
106#3757C2×D312yes—
107#3752—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—32no—
111C4⋊(C3⋊Dic3)—64no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—31no—
114#3750—32no—
115#3750—32no—
116C4×C3⋊Dic3—31no—
117C4×C3⋊Dic3—34no—
118C4×C3⋊Dic3—32no—
119C4×C3⋊Dic3—64no—
120C3×#217—38no—
121C3×#217C2×D318yes—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124#3724—34no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—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#4112yes—
139C2×C32×D3—21no—
140C2×C32×D3—21no—
141C2×C32×D3—21no—
142C33×C4#4112yes—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—21no—
149C32×Dic3—22no—
150C3×#121—34no—
151C3×#121—91no—
152#939—92no—
153#938—34no—
154#938—98no—
155#938C3⋊D311yescharacteristic, complemented
156#935—98no—
157D4×Dic3—94no—
158D4×Dic3—91no—
159D4×Dic3—92no—
160D4×Dic3—184no—
161C4×#67—94no—
162C4×#67—91no—
163#926—94no—
164C2×C4⋊Dic3—91no—
165#892—91no—
166D3×C4⋊C4—94no—
167#890—94no—
168#890—91no—
169#886—92no—
170#67⋊C4—98no—
171#879—98no—
172#879—92no—
173C23×C32—28no—
174C23×C32—31no—
175C23×C32C22×D318yes—
176C23×C32C22×D312yes—
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—34no—
179C22×C3×D3C22×D314yes—
180C32×D4—216no—
181C32×D4—28no—
182C32×D4—22no—
183C32×D4—122no—
184C2×C32×C4—24no—
185C2×C32×C4—31no—
186C2×C32×C4—32no—
187C2×C32×C4—62no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D311yescharacteristic
190#638—24no—
191#638—62no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—38no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3—122no—
202C2×C3⋊Dic3—128no—
203C2×C3⋊Dic3—1216no—
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D311yescharacteristic
206C4×C3⋊D3—22no—
207C4×C3⋊D3—122no—
208C3×#67—48no—
209C3×#67—616no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3C22×D314yes—
213C2×C3×Dic3C22×D318yes—
214C3×C4×D3—28no—
215C3×C4×D3—68no—
216#625—68no—
217D3×Dic3—128no—
218#553—271noSylow
219C2×C33—42no—
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—22no—
224C32×D3—41no—
225C2×C3×D4—24no—
226C2×C3×D4—38no—
227C2×C3×D4—92no—
228C2×C3×D4D3214yes—
229C2×C3×D4C2×C3⋊D311yescharacteristic
230C22×C3×C4—34no—
231C22×C3×C4—91no—
232C2×#67—38no—
233C2×#67—94no—
234C2×#67—91no—
235C2×#67C2×C3⋊D312yes—
236C22×Dic3—98no—
237C22×Dic3—92no—
238C22×Dic3—92no—
239C22×Dic3—188no—
240C2×C4×D3—34no—
241C2×C4×D3—98no—
242C2×C4×D3C2×C3⋊D311yescharacteristic
243C3×C4⋊C4—34no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—38no—
246C3×C22⋊C4—92no—
247#217—38no—
248#217—98no—
249#217—94no—
250#217—98no—
251#217—92no—
252#217—91no—
253#217—188no—
254#217C2×C3⋊D312yes—
255(C2×D3)⋊C4—94no—
256(C2×D3)⋊C4—91no—
257(C2×D3)⋊C4—92no—
258C4⋊Dic3—34no—
259C4⋊Dic3—98no—
260C4⋊Dic3—94no—
261C4⋊Dic3—91no—
262C4⋊Dic3—92no—
263C4⋊Dic3—92no—
264C4⋊Dic3—184no—
265C4⋊Dic3C2×C3⋊D311yescharacteristic
266Dic3⋊C4—94no—
267Dic3⋊C4—98no—
268Dic3⋊C4—98no—
269Dic3⋊C4—91no—
270Dic3⋊C4—92no—
271C4×Dic3—98no—
272C4×Dic3—94no—
273C4×Dic3—91no—
274C4×Dic3—92no—
275C4×Dic3—184no—
276C22×C32—216no—
277C22×C32—22no—
278C22×C32—28no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—24no—
282C22×C32—28no—
283C22×C32—28no—
284C22×C32—48no—
285C22×C32—48no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32—61no—
289C22×C32C23×D314yes—
290C22×C32C23×D311yescharacteristic
291C2×C3⋊D3—21no—
292C2×C3⋊D3—21no—
293C2×C3⋊D3#23712yes—
294C2×C3×D3—24no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—38no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C32×C4—28no—
301C32×C4—61no—
302C32×C4—62no—
303C32×C4#23718yes—
304C32×C4#23712yes—
305C3⋊Dic3—21no—
306C3⋊Dic3—22no—
307C3⋊Dic3—68no—
308C3⋊Dic3—61no—
309C3⋊Dic3—64no—
310C3⋊Dic3—62no—
311C3⋊Dic3—61no—
312C3⋊Dic3—68no—
313C3⋊Dic3—64no—
314C3⋊Dic3—124no—
315C3⋊Dic3—128no—
316C3×Dic3—24no—
317C3×Dic3—28no—
318C3×Dic3—64no—
319C3×Dic3—68no—
320#126—272no—
321#125—271no—
322#123—272no—
323#121—31no—
324#121—91no—
325#121—272no—
326C4×D4—272no—
327C4×D4—271no—
328C4×D4—271no—
329#117—271no—
330C2×C4⋊C4—271no—
331C33#55311yescharacteristic, complemented, Sylow
332C23×C3—28no—
333C23×C3—34no—
334C23×C3—91no—
335C23×C3C2×D3218yes—
336C23×C3C22×C3⋊D312yes—
337C22×D3—34no—
338C22×D3C22×C3⋊D311yescharacteristic
339C3×D4—216no—
340C3×D4—28no—
341C3×D4—22no—
342C3×D4—128no—
343C3×D4—184no—
344C2×C3×C4—24no—
345C2×C3×C4—34no—
346C2×C3×C4—38no—
347C2×C3×C4—68no—
348C2×C3×C4—92no—
349C2×C3×C4—91no—
350C2×C3×C4—182no—
351C2×C3×C4C2×D3214yes—
352C2×C3×C4C22×C3⋊D311yescharacteristic
353#67—42no—
354#67—616no—
355#67—182no—
356#67—188no—
357C2×Dic3—38no—
358C2×Dic3—34no—
359C2×Dic3—94no—
360C2×Dic3—94no—
361C2×Dic3—98no—
362C2×Dic3—91no—
363C2×Dic3—92no—
364C2×Dic3—91no—
365C2×Dic3—184no—
366C2×Dic3—184no—
367C2×Dic3—188no—
368C2×Dic3—362no—
369C2×Dic3—368no—
370C2×Dic3—362no—
371C2×Dic3—3616no—
372C2×Dic3C22×C3⋊D311yescharacteristic
373C2×Dic3C22×C3⋊D312yes—
374C4×D3—22no—
375C4×D3—68no—
376C4×D3—368no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—24no—
380C2×C32—416no—
381C2×C32—48no—
382C2×C32—42no—
383C2×C32—121no—
384C2×C32C2×#23714yes—
385C2×C32C2×#23711yescharacteristic
386C2×C32C2×#23711yescharacteristic
387C2×C32#100414yes—
388C2×C32D3×#4114yes—
389C2×C32#100911yescharacteristic
390C3⋊D3—22no—
391C3×D3—44no—
392C3×D3—68no—
393C2×D4—32no—
394C2×D4—92no—
395C2×D4—271no—
396C2×D4D3×C3⋊D311yescharacteristic
397C22×C4—31no—
398C22×C4—91no—
399C22×C4—272no—
400C22×C4—272no—
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C4⋊C4—271no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—32no—
409C22⋊C4—92no—
410C22⋊C4—271no—
411C22⋊C4—272no—
412C22⋊C4—271no—
413C22⋊C4—272no—
414C42—272no—
415C42—271no—
416C22×C3—216no—
417C22×C3—28no—
418C22×C3—28no—
419C22×C3—28no—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—24no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—92no—
430C22×C3—181no—
431C22×C3—181no—
432C22×C3C22×D3214yes—
433C22×C3C23×C3⋊D311yescharacteristic
434C2×D3—21no—
435C2×D3—21no—
436C2×D3—21no—
437C2×D3—38no—
438C2×D3—64no—
439C2×D3—64no—
440C3×C4—28no—
441C3×C4—68no—
442C3×C4—64no—
443C3×C4—181no—
444C3×C4—182no—
445C3×C4#380518yes—
446C3×C4#383012yes—
447Dic3—21no—
448Dic3—22no—
449Dic3—68no—
450Dic3—64no—
451Dic3—188no—
452Dic3—184no—
453Dic3—188no—
454Dic3—184no—
455Dic3—181no—
456Dic3—182no—
457Dic3—181no—
458Dic3—368no—
459Dic3—364no—
460C32—24no—
461C32#561414yescomplemented
462C32#561811yescharacteristic, complemented
463C23—31no—
464C23—91no—
465C23C2×D3×C3⋊D312yes—
466D4—22no—
467D4—122no—
468D4—184no—
469D4—542no—
470C2×C4—31no—
471C2×C4—32no—
472C2×C4—62no—
473C2×C4—92no—
474C2×C4—91no—
475C2×C4—182no—
476C2×C4—271no—
477C2×C4—271no—
478C2×C4—272no—
479C2×C4—1082no—
480C2×C4—1082no—
481C2×C4C2×D3×C3⋊D311yescharacteristic
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—416no—
486C2×C3—48no—
487C2×C3—42no—
488C2×C3—124no—
489C2×C3—182no—
490C2×C3D3×#23714yes—
491C2×C3#6431414yes—
492C2×C3C2×#380514yes—
493C2×C3C2×#383011yescharacteristic
494C2×C3#6436111yescharacteristic
495C2×C3#41×C3⋊D311yescharacteristic
496D3—41no—
497D3—68no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—92no—
505C22—181no—
506C22—181no—
507C22C22×D3×C3⋊D311yescharacteristic
508C4—62no—
509C4—61no—
510C4—182no—
511C4—181no—
512C4—541no—
513C4—541no—
514C4—542no—
515C4#8902912yes—
516C3—24no—
517C3#1059524514yescomplemented
518C3#1059608511yescharacteristic, complemented
519C2—42no—
520C2—121no—
521C2—182no—
522C2C3⋊D3×#23711yescharacteristic
523C2#1172458811yescharacteristic
524C2C2×#8902911yescharacteristic
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. D4×C3⋊(C3⋊Dic3)C2semidirect
2. C3×#64147C2semidirect
3. C32×#938C2semidirect
4. #11723631C2semidirect
5. #11723626C2semidirect
6. #11723603C2semidirect
7. #11723597C2semidirect
8. C4×#6290C2semidirect
9. C3⋊D3×C4⋊Dic3C2semidirect
10. #11723450C2semidirect
11. #11723424C2semidirect
39. #64147D3semidirect
51. C3×#938D3semidirect
155. #938C3⋊D3semidirect
331. C33#553semidirect
461. C32#5614semidirect
462. C32#5618semidirect
517. C3#10595245semidirect
518. C3#10596085semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C3×#641472
C32×#9382
#117236312
#117236262
#117236032
#117235972
C4×#62902
C3⋊D3×C4⋊Dic32
#117234502
#117234242
#105960853
#105952453