Finite Group Databasebeta

← #13176802

Subgroups — #13176802

Order 1728.

1652 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
2C3×#64133C211yesmaximal, characteristic, complemented
3C32×C4×#67C211yesmaximal, characteristic
4#11723603C212yesmaximal
5#11723598C212yesmaximal, complemented
6#11723454C211yesmaximal, characteristic, complemented
7#11723430C212yesmaximal, complemented
8#11723425C211yesmaximal, characteristic, complemented
9#11723424C211yesmaximal, characteristic, complemented
10#11723409C212yesmaximal, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12#10596012—31nomaximal
13#10594853—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C2×C32×C4×D3C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C4⋊(C3⋊(C3⋊Dic3))C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C3×#3757C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3750C2212yes—
25C32×#217C2211yescharacteristic
26C32×(C2×D3)⋊C4C2211yescharacteristic
27C32×Dic3⋊C4C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64144—32no—
39#64135—31no—
40C2×C4⋊(C3⋊Dic3)—31no—
41C2×C4⋊(C3⋊Dic3)—34no—
42C2×C4⋊(C3⋊Dic3)—64no—
43#64133—31no—
44#64133D311yescharacteristic, complemented
45#64098—32no—
46#64088—32no—
47#64086—32no—
48#64079—31no—
49C4×#634—31no—
50C3×C4×#67D314yes—
51C3×#924—34no—
52#63962—38no—
53#63957—38no—
54#63869—34no—
55#63861—38no—
56#63856—34no—
57#63845—38no—
58Dic3×Dic6—34no—
59#63841—34no—
60C23×C33C2311yescharacteristic
61C22×C32×D3C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)—24no—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C3×#634—22no—
71C32×#67—22no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×C4×D3—22no—
76#5642—91no—
77#5541—94no—
78C2×C32×D4—31no—
79C22×C32×C4—24no—
80C22×C32×C4—31no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3—32no—
84C22×C3⋊Dic3—38no—
85C22×C3⋊Dic3—68no—
86C2×#634C2×D311yescharacteristic
87C2×C3×#67C2×D314yes—
88C2×C3×C4×D3C2×D314yes—
89C2×C3×Dic6—34no—
90C32×C4⋊C4—31no—
91C32×C22⋊C4—31no—
92C32×C22⋊C4—31no—
93C32×C42—31no—
94#3757—32no—
95#3757—32no—
96#3757C2×D312yes—
97C4⋊(C3⋊Dic3)—31no—
98C4⋊(C3⋊Dic3)—31no—
99C4⋊(C3⋊Dic3)—32no—
100C4⋊(C3⋊Dic3)—34no—
101C4⋊(C3⋊Dic3)—38no—
102C4⋊(C3⋊Dic3)—34no—
103C4⋊(C3⋊Dic3)—64no—
104C4⋊(C3⋊Dic3)—64no—
105C4⋊(C3⋊Dic3)—68no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—32no—
108#3750—32no—
109#3750—32no—
110#3750—32no—
111#3750C2×D312yes—
112C4×C3⋊Dic3—32no—
113C4×C3⋊Dic3—32no—
114C3×#217—38no—
115C3×#217C2×D314yes—
116C3×(C2×D3)⋊C4C2×D314yes—
117C3×C4⋊Dic3—34no—
118C3×Dic3⋊C4—38no—
119C3×Dic3⋊C4C2×D314yes—
120C3×C4×Dic3C2×D314yes—
121#3724—38no—
122#3724—38no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125(C2×D3)⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127Dic32—38no—
128Dic32—38no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33#4112yes—
132C22×C33C2411yescharacteristic
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C33×C4—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—22no—
139C3⋊(C3⋊Dic3)—22no—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—22no—
142C32×Dic3—21no—
143C32×Dic3—21no—
144C32×Dic3—21no—
145C3×C22⋊Q8—91no—
146C3×C4×D4—34no—
147#946—91no—
148Q8×Dic3—91no—
149#935—98no—
150C4×#67C3⋊D311yescharacteristic
151#926—94no—
152C2×C4⋊Dic3—94no—
153C2×C4⋊Dic3—91no—
154C2×C4⋊Dic3—184no—
155#924—34no—
156#924—94no—
157#898—92no—
158#892—91no—
159#890—92no—
160#890—91no—
161#889—98no—
162#886—92no—
163#879—98no—
164#877—98no—
165#877—92no—
166#870—94no—
167C4×Dic6—94no—
168C23×C32—24no—
169C23×C32—31no—
170C23×C32C22×D314yes—
171C23×C32C22×D311yescharacteristic
172C22×C3×D3C22×D314yes—
173C32×D4—62no—
174C2×C32×C4—216no—
175C2×C32×C4—28no—
176C2×C32×C4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—31no—
180C2×C32×C4—31no—
181C2×C32×C4—31no—
182C2×C32×C4—62no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3—64no—
194C2×C3⋊Dic3—616no—
195C2×C3⋊Dic3—1216no—
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D312yes—
198#634—22no—
199C3×#67—28no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3—38no—
202C2×C3×Dic3C22×D314yes—
203C2×C3×Dic3C22×D314yes—
204C2×C3×Dic3C22×D314yes—
205C3×C4×D3—28no—
206C3×Dic6—68no—
207#552—271noSylow
208C2×C33—22no—
209C2×C33C2×#4111yescharacteristic
210C2×C33C2×#4111yescharacteristic
211C2×C332−1+411yescharacteristic
212C32×D3—41no—
213C2×C3×Q8—91no—
214C2×C3×D4—34no—
215C22×C3×C4—24no—
216C22×C3×C4—34no—
217C22×C3×C4D3214yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C2×#67C2×C3⋊D311yescharacteristic
220C22×Dic3—98no—
221C22×Dic3—92no—
222C22×Dic3—188no—
223C2×C4×D3C2×C3⋊D311yescharacteristic
224C2×Dic6—34no—
225C3×C4⋊C4—34no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—92no—
228C3×C22⋊C4—34no—
229C3×C22⋊C4—34no—
230C3×C22⋊C4—92no—
231C3×C42—34no—
232#217—38no—
233#217—98no—
234#217—98no—
235#217C2×C3⋊D311yescharacteristic
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
239C4⋊Dic3—34no—
240C4⋊Dic3—94no—
241C4⋊Dic3—94no—
242C4⋊Dic3—98no—
243C4⋊Dic3—92no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3—184no—
249C4⋊Dic3—184no—
250C4⋊Dic3—188no—
251Dic3⋊C4—38no—
252Dic3⋊C4—98no—
253Dic3⋊C4—98no—
254Dic3⋊C4—98no—
255Dic3⋊C4—98no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×C3⋊D311yescharacteristic
259C4×Dic3—98no—
260C4×Dic3—98no—
261C4×Dic3—92no—
262C4×Dic3—92no—
263C4×Dic3C2×C3⋊D311yescharacteristic
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—24no—
267C22×C32—21no—
268C22×C32—24no—
269C22×C32—21no—
270C22×C32—24no—
271C22×C32—28no—
272C22×C32—61no—
273C22×C32—61no—
274C22×C32—61no—
275C22×C32#23518yes—
276C22×C32#23912yes—
277C22×C32C23×D314yes—
278C22×C32C23×D311yescharacteristic
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C2×C3×D3—24no—
282C32×C4—28no—
283C32×C4—24no—
284C32×C4—21no—
285C32×C4—28no—
286C32×C4—61no—
287C32×C4—61no—
288C32×C4—61no—
289C32×C4#23718yes—
290C32×C4#23712yes—
291C3⋊Dic3—22no—
292C3⋊Dic3—22no—
293C3⋊Dic3—62no—
294C3⋊Dic3—62no—
295C3⋊Dic3—68no—
296C3⋊Dic3—68no—
297C3⋊Dic3—128no—
298C3⋊Dic3—128no—
299C3×Dic3—24no—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—68no—
303C3×Dic3—68no—
304#126—272no—
305#125—271no—
306#125—272no—
307#123—272no—
308C22⋊Q8—91no—
309C22⋊Q8—271no—
310C4×Q8—271no—
311C4×D4—31no—
312#117—272no—
313#117—271no—
314C2×C4⋊C4—271no—
315C33#55211yescharacteristic, complemented, Sylow
316C23×C3—24no—
317C23×C3—34no—
318C23×C3C2×D3214yes—
319C23×C3C22×C3⋊D311yescharacteristic
320C22×D3C22×C3⋊D311yescharacteristic
321C3×Q8—182no—
322C3×D4—68no—
323C2×C3×C4—216no—
324C2×C3×C4—28no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—68no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C22×C3⋊D311yescharacteristic
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—22no—
339C2×Dic3—38no—
340C2×Dic3—38no—
341C2×Dic3—98no—
342C2×Dic3—98no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—188no—
346C2×Dic3—188no—
347C2×Dic3—1816no—
348C2×Dic3—184no—
349C2×Dic3—3616no—
350C2×Dic3C22×C3⋊D311yescharacteristic
351C2×Dic3C22×C3⋊D311yescharacteristic
352C2×Dic3C22×C3⋊D311yescharacteristic
353C4×D3—22no—
354Dic6—68no—
355C2×C32—216no—
356C2×C32—22no—
357C2×C32—28no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—121no—
362C2×C32C2×#23514yes—
363C2×C32C2×#23714yes—
364C2×C32C2×#23711yescharacteristic
365C2×C32C2×#23911yescharacteristic
366C2×C32#101014yes—
367C2×C32#101011yescharacteristic
368C3×D3—44no—
369C2×Q8—91no—
370C2×D4—31no—
371C22×C4—31no—
372C22×C4—272no—
373C22×C4D3×C3⋊D311yescharacteristic
374C4⋊C4—31no—
375C4⋊C4—91no—
376C4⋊C4—92no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—271no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C22⋊C4—31no—
385C22⋊C4—31no—
386C22⋊C4—92no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—31no—
390C42—272no—
391C42—272no—
392C22×C3—28no—
393C22×C3—28no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—24no—
399C22×C3—28no—
400C22×C3—64no—
401C22×C3—64no—
402C22×C3—64no—
403C22×C3#379918yes—
404C22×C3#382812yes—
405C22×C3C22×D3214yes—
406C22×C3C23×C3⋊D311yescharacteristic
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C3×C4—28no—
411C3×C4—24no—
412C3×C4—21no—
413C3×C4—28no—
414C3×C4—64no—
415C3×C4—64no—
416C3×C4—64no—
417C3×C4—182no—
418C3×C4—182no—
419C3×C4#380518yes—
420C3×C4#383012yes—
421Dic3—21no—
422Dic3—21no—
423Dic3—21no—
424Dic3—68no—
425Dic3—68no—
426Dic3—188no—
427Dic3—188no—
428Dic3—182no—
429Dic3—182no—
430Dic3—368no—
431Dic3—368no—
432C32—24no—
433C32#554114yescomplemented
434C32#564211yescharacteristic, complemented
435C23—31no—
436C23C2×D3×C3⋊D311yescharacteristic
437Q8—182no—
438D4—62no—
439C2×C4—22no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—62no—
444C2×C4—92no—
445C2×C4—92no—
446C2×C4—272no—
447C2×C4—272no—
448C2×C4—544no—
449C2×C4C2×D3×C3⋊D311yescharacteristic
450C2×C4C2×D3×C3⋊D311yescharacteristic
451C2×C3—216no—
452C2×C3—28no—
453C2×C3—22no—
454C2×C3—24no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—124no—
458C2×C3#6429814yes—
459C2×C3C2×#379914yes—
460C2×C3C2×#380514yes—
461C2×C3C2×#382811yescharacteristic
462C2×C3C2×#383011yescharacteristic
463C2×C3#6436711yescharacteristic
464D3—41no—
465C22—21no—
466C22—21no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22#8902212yes—
471C22C22×D3×C3⋊D311yescharacteristic
472C4—21no—
473C4—61no—
474C4—61no—
475C4—61no—
476C4—182no—
477C4—182no—
478C4—542no—
479C4—542no—
480C4#8902912yes—
481C3—24no—
482C3#1059485314yescomplemented
483C3#1059601211yescharacteristic, complemented
484C2—22no—
485C2—121no—
486C2#1172456311yescharacteristic
487C2C2×#8902211yescharacteristic
488C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×C32×#67C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
2. C3×#64133C2semidirect
5. #11723598C2semidirect
6. #11723454C2semidirect
7. #11723430C2semidirect
8. #11723425C2semidirect
9. #11723424C2semidirect
10. #11723409C2semidirect
11. Dic3×#634C2semidirect
44. #64133D3semidirect
315. C33#552semidirect
433. C32#5541semidirect
434. C32#5642semidirect
482. C3#10594853semidirect
483. C3#10596012semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊(C3⋊Dic3))2
C3×#641332
C32×C4×#672
#117236032
#117235982
#117234542
#117234302
#117234252
#117234242
#117234092
Dic3×#6342
#105960123
#105948533