Finite Group Databasebeta

← #13177605

Subgroups — #13177605

Order 1728.

1792 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#11723637C212yesmaximal, complemented
5#11723600C212yesmaximal, complemented
6#11723598C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723536C211yesmaximal, characteristic, complemented
9C4×#6290C211yesmaximal, characteristic, complemented
10#11723424C211yesmaximal, characteristic, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596087—31nomaximal
13#10595253—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×#6293C2212yes—
21C2×#6290C2211yescharacteristic
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#88806C2212yes—
30#88806C2211yescharacteristic
31#88803C2211yescharacteristic
32#88803C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#121—31no—
38#64148—32no—
39#64147D311yescharacteristic, complemented
40#64145—32no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44C4×#638—31no—
45#64135—31no—
46#64101—31no—
47#64097—31no—
48#64094—32no—
49#64086—32no—
50C3×#938—34no—
51C3×#938D314yescomplemented
52#63980—38no—
53#63959—38no—
54#63959—38no—
55#63957—38no—
56#63919—34no—
57C4×#625—34no—
58#63841—34no—
59#63841—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2311yescharacteristic
63C33×D4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—24no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C3×#638—42no—
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#6293—42no—
78#6290—22no—
79#5616—94no—
80#5616—91no—
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⋊D3C2×D311yescharacteristic
92C2×#634—32no—
93C2×C3×#67—38no—
94C2×C3×#67C2×D318yes—
95C2×C3×C4×D3—34no—
96C2×C3×C4×D3C2×D314yes—
97C2×#627—38no—
98C2×#625—34no—
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—
108#3752—32no—
109C4⋊(C3⋊Dic3)—31no—
110C4⋊(C3⋊Dic3)—34no—
111C4⋊(C3⋊Dic3)—64no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—31no—
114#3750—32no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3—34no—
117C4×C3⋊Dic3—32no—
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—
125#3724—38no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—34no—
130Dic32—38no—
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×C32×D3—21no—
140C2×C32×D3—21no—
141C2×C32×D3—21no—
142C33×C4#4112yes—
143C3⋊(C3⋊Dic3)—22no—
144C3⋊(C3⋊Dic3)—21no—
145C3⋊(C3⋊Dic3)#4112yes—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—22no—
148C32×Dic3—21no—
149C32×Dic3—22no—
150C3×#121—34no—
151C3×#121—91no—
152#939—98no—
153#939—92no—
154#938—34no—
155#938C3⋊D311yescharacteristic, complemented
156#936—98no—
157#936—92no—
158D4×Dic3—94no—
159D4×Dic3—91no—
160D4×Dic3—184no—
161C4×#67—94no—
162C4×#67—91no—
163#926—94no—
164#926—91no—
165#892—94no—
166#892—91no—
167#888—94no—
168#888—91no—
169#885—98no—
170#885—92no—
171#877—98no—
172#877—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—42no—
191#638—62no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—34no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3—616no—
203C2×C3⋊Dic3—1216no—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D312yes—
206C4×C3⋊D3—22no—
207#634—122no—
208C3×#67—48no—
209C3×#67—128no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—38no—
212C2×C3×Dic3C22×D314yes—
213C2×C3×Dic3C22×D318yes—
214C3×C4×D3—28no—
215C3×C4×D3—68no—
216#627—128no—
217#625—68no—
218#555—271noSylow
219C2×C33—42no—
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—41no—
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—188no—
239C2×C4×D3—34no—
240C2×C4×D3C2×C3⋊D311yescharacteristic
241C2×Dic6—98no—
242C2×Dic6—92no—
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—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—91no—
259(C2×D3)⋊C4—92no—
260C4⋊Dic3—34no—
261C4⋊Dic3—94no—
262C4⋊Dic3—91no—
263C4⋊Dic3—184no—
264C4⋊Dic3C2×C3⋊D311yescharacteristic
265Dic3⋊C4—94no—
266Dic3⋊C4—98no—
267Dic3⋊C4—91no—
268Dic3⋊C4—92no—
269C4×Dic3—98no—
270C4×Dic3—98no—
271C4×Dic3—94no—
272C4×Dic3—91no—
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3—184no—
276C22×C32—216no—
277C22×C32—28no—
278C22×C32—22no—
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—21no—
294C2×C3×D3—24no—
295C2×C3×D3—24no—
296C2×C3×D3—24no—
297C2×C3×D3—64no—
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—32no—
308C3⋊Dic3—38no—
309C3⋊Dic3—68no—
310C3⋊Dic3—61no—
311C3⋊Dic3—62no—
312C3⋊Dic3—64no—
313C3⋊Dic3—68no—
314C3⋊Dic3—124no—
315C3⋊Dic3—128no—
316C3×Dic3—24no—
317C3×Dic3—28no—
318C3×Dic3—64no—
319C3×Dic3—68no—
320C4⋊Q8—271no—
321#124—272no—
322#124—272no—
323#121—31no—
324#121—91no—
325#121—272no—
326C4×D4—271no—
327C4×D4—271no—
328#117—272no—
329#117—271no—
330#117—271no—
331C33#55511yescharacteristic, 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—362no—
344C2×C3×C4—24no—
345C2×C3×C4—34no—
346C2×C3×C4—38no—
347C2×C3×C4—68no—
348C2×C3×C4—91no—
349C2×C3×C4—92no—
350C2×C3×C4—182no—
351C2×C3×C4C2×D3214yes—
352C2×C3×C4C22×C3⋊D311yescharacteristic
353#67—42no—
354#67—128no—
355#67—182no—
356#67—188no—
357C2×Dic3—34no—
358C2×Dic3—38no—
359C2×Dic3—94no—
360C2×Dic3—91no—
361C2×Dic3—98no—
362C2×Dic3—94no—
363C2×Dic3—92no—
364C2×Dic3—91no—
365C2×Dic3—184no—
366C2×Dic3—184no—
367C2×Dic3—188no—
368C2×Dic3—184no—
369C2×Dic3—1816no—
370C2×Dic3—3616no—
371C2×Dic3C22×C3⋊D311yescharacteristic
372C2×Dic3C22×C3⋊D312yes—
373C4×D3—22no—
374C4×D3—68no—
375Dic6—362no—
376Dic6—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×C32#100414yes—
387C2×C32#100411yescharacteristic
388C2×C32D3×#4114yes—
389C2×C32D3×#4111yescharacteristic
390C3⋊D3—41no—
391C3×D3—44no—
392C3×D3—124no—
393C2×Q8—272no—
394C2×D4—32no—
395C2×D4—92no—
396C2×D4—271no—
397C2×D4D3×C3⋊D311yescharacteristic
398C22×C4—31no—
399C22×C4—91no—
400C22×C4—272no—
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—271no—
404C4⋊C4—272no—
405C4⋊C4—271no—
406C22⋊C4—32no—
407C22⋊C4—92no—
408C22⋊C4—272no—
409C22⋊C4—271no—
410C22⋊C4—272no—
411C22⋊C4—271no—
412C22⋊C4—272no—
413C42—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—181no—
430C22×C3—181no—
431C22×C3—181no—
432C22×C3C22×D3214yes—
433C22×C3C23×C3⋊D311yescharacteristic
434C2×D3—21no—
435C2×D3—21no—
436C2×D3—21no—
437C2×D3—64no—
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—98no—
452Dic3—92no—
453Dic3—188no—
454Dic3—188no—
455Dic3—184no—
456Dic3—181no—
457Dic3—182no—
458Dic3—368no—
459Dic3—364no—
460C32—24no—
461C32#561614yescomplemented
462C32#561611yescharacteristic, complemented
463C23—31no—
464C23—91no—
465C23C2×D3×C3⋊D312yes—
466Q8—1082no—
467D4—22no—
468D4—122no—
469D4—362no—
470D4—542no—
471C2×C4—31no—
472C2×C4—32no—
473C2×C4—62no—
474C2×C4—91no—
475C2×C4—92no—
476C2×C4—182no—
477C2×C4—271no—
478C2×C4—272no—
479C2×C4—271no—
480C2×C4—544no—
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—361no—
490C2×C3#6431214yes—
491C2×C3#6431314yes—
492C2×C3C2×#380514yes—
493C2×C3C2×#383011yescharacteristic
494C2×C3#6436111yescharacteristic
495C2×C3#41×C3⋊D311yescharacteristic
496D3—41no—
497D3—124no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—181no—
505C22—181no—
506C22—181no—
507C22C22×D3×C3⋊D311yescharacteristic
508C4—62no—
509C4—61no—
510C4—182no—
511C4—181no—
512C4—272no—
513C4—542no—
514C4—541no—
515C4#8902912yes—
516C3—24no—
517C3#1059525314yescomplemented
518C3#1059608711yescharacteristic, complemented
519C2—42no—
520C2—121no—
521C2—361no—
522C2#1172458611yescharacteristic
523C2#1172458711yescharacteristic
524C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#555
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. #11723637C2semidirect
5. #11723600C2semidirect
6. #11723598C2semidirect
7. #11723591C2semidirect
8. #11723536C2semidirect
9. C4×#6290C2semidirect
10. #11723424C2semidirect
11. #11723404C2semidirect
39. #64147D3semidirect
51. C3×#938D3semidirect
155. #938C3⋊D3semidirect
331. C33#555semidirect
461. C32#5616semidirect
462. C32#5616semidirect
517. C3#10595253semidirect
518. C3#10596087semidirect

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
#117236372
#117236002
#117235982
#117235912
#117235362
C4×#62902
#117234242
#117234042
#105960873
#105952533