Finite Group Databasebeta

← #13177688

Subgroups — #13177688

Order 1728.

1848 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723981C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723628C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6#11723612C212yesmaximal, complemented
7#11723592C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9#11723494C211yesmaximal, characteristic, complemented
10#11723441C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#10596030—31nomaximal
13#10595210—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C22×C32×Dic3C2212yes—
20C2×#6293C2211yescharacteristic
21C2×#6289C2212yes—
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3757C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2211yescharacteristic
30#88807C2212yes—
31#88806C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33#88804C2212yes—
34#88804C2212yes—
35#88804C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64149—34no—
40#64149—64no—
41#64145D311yescharacteristic, complemented
42#64138—32no—
43C4×#638—32no—
44#64102—31no—
45#64096—31no—
46#64094—32no—
47#64092—32no—
48#64084—31no—
49C4×#634—31no—
50C3×#936—34no—
51C3×D4×Dic3D314yescomplemented
52#63975—38no—
53#63966—38no—
54#63953—38no—
55#63952—38no—
56C4×#627—34no—
57#63894—34no—
58#63864—34no—
59#63843—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2312yes—
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64#6312—24no—
65#6312—42no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4⋊(C3⋊(C3⋊D3))—22no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C3×#634—22no—
73C2×C32×Dic3—42no—
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2312yes—
76C2×C32×Dic3C2311yescharacteristic
77#6293—22no—
78#6289—42no—
79#5681—91no—
80#5559—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—32no—
88C2×#638—38no—
89C2×#638—68no—
90C2×#638—68no—
91C2×C4⋊(C3⋊D3)—31no—
92C2×C4⋊(C3⋊D3)—34no—
93C2×C4⋊(C3⋊D3)—64no—
94C2×C4×C3⋊D3—32no—
95C2×#634—31no—
96C2×#634C2×D311yescharacteristic
97C22×C3×Dic3C2×D318yes—
98C2×C3×Dic6—34no—
99C2×#627—34no—
100C2×#624—38no—
101C32×C4⋊C4—31no—
102C32×C22⋊C4—32no—
103C32×C42—31no—
104#3757C2×D312yes—
105#3757C2×D312yes—
106#3752—32no—
107#3752—32no—
108#3752—32no—
109C4⋊(C3⋊Dic3)—31no—
110#3750—31no—
111#3750—31no—
112#3750—32no—
113C4×C3⋊Dic3—31no—
114C4×C3⋊Dic3—34no—
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3—64no—
117C4×C3⋊Dic3C2×D311yescharacteristic
118C3×#217—38no—
119C3×#217—38no—
120C3×#217C2×D318yes—
121C3×C4⋊Dic3C2×D314yes—
122C3×C4×Dic3—34no—
123C3×C4×Dic3C2×D314yes—
124#3724—34no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128#3722—38no—
129#3722—38no—
130#3722—38no—
131Dic32—38no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—22no—
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)—22no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—21no—
141C3⋊(C3⋊Dic3)#4112yes—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—21no—
145C32×Dic3—21no—
146C32×Dic3—22no—
147C32×Dic3—21no—
148C3×#124—91no—
149C3×C4×D4—34no—
150#947—91no—
151Q8×Dic3—91no—
152#940—94no—
153#940—91no—
154#940—184no—
155#936—34no—
156D4×Dic3C3⋊D311yescharacteristic, complemented
157#929—98no—
158C4×#67—98no—
159#893—94no—
160#887—94no—
161#886—92no—
162#885—98no—
163#885—92no—
164#883—98no—
165#883—92no—
166#67⋊C4—92no—
167#875—94no—
168C4×D12—91no—
169C4×Dic6—94no—
170C4×Dic6—91no—
171C23×C32—28no—
172C23×C32C22×D318yes—
173C23×C32C22×D312yes—
174C22×C3⋊D3—32no—
175C22×C3⋊D3—38no—
176C22×C3⋊D3—68no—
177C32×D4—22no—
178C32×D4—28no—
179C32×D4—48no—
180C2×C32×C4—24no—
181C2×C32×C4—32no—
182C2×C32×C4—31no—
183C2×C32×C4—31no—
184C2×C32×C4—122no—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D311yescharacteristic
187#638—64no—
188#638—616no—
189#638—1216no—
190#638—122no—
191#638—128no—
192#638—1216no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—34no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4⋊(C3⋊D3)—62no—
203C4⋊(C3⋊D3)—68no—
204C4⋊(C3⋊D3)—128no—
205C4×C3⋊D3—122no—
206#634—22no—
207#634—62no—
208C2×C3×Dic3—38no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—48no—
212C2×C3×Dic3C22×D318yes—
213C2×C3×Dic3C22×D314yes—
214C2×C3×Dic3C22×D314yes—
215C3×Dic6—68no—
216#627—68no—
217#624—128no—
218#559—271noSylow
219C2×C33—42no—
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3⋊(C3⋊D3)—42no—
224C2×C3×Q8—91no—
225C2×C3×D4—24no—
226C2×C3×D4D3214yes—
227C2×C3×D4C2×C3⋊D311yescharacteristic
228C22×C3×C4—38no—
229C2×#67—98no—
230C2×#67—92no—
231C2×#67—98no—
232C2×#67—92no—
233C2×#67—188no—
234C2×#67—188no—
235C22×Dic3C2×C3⋊D312yes—
236C2×D12—94no—
237C2×D12—91no—
238C2×D12—184no—
239C2×C4×D3—98no—
240C2×C4×D3—92no—
241C2×Dic6—34no—
242C2×Dic6—94no—
243C2×Dic6—91no—
244C3×C4⋊C4—34no—
245C3×C22⋊C4—38no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C42—34no—
249C3×C42—91no—
250#217—38no—
251#217—38no—
252#217C2×C3⋊D312yes—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259C4⋊Dic3—94no—
260C4⋊Dic3—91no—
261C4⋊Dic3—92no—
262C4⋊Dic3C2×C3⋊D311yescharacteristic
263Dic3⋊C4—94no—
264Dic3⋊C4—94no—
265Dic3⋊C4—98no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268C4×Dic3—34no—
269C4×Dic3—98no—
270C4×Dic3—94no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3—184no—
274C4×Dic3C2×C3⋊D311yescharacteristic
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—28no—
280C22×C32—28no—
281C22×C32—28no—
282C22×C32—48no—
283C22×C32—48no—
284C22×C32—48no—
285C22×C32C23×D314yes—
286C22×C32C23×D311yescharacteristic
287C2×C3⋊D3—62no—
288C2×C3⋊D3—62no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—68no—
293C2×C3⋊D3—128no—
294C2×C3⋊D3—128no—
295C2×C3⋊D3—128no—
296C32×C4—28no—
297C32×C4—61no—
298C32×C4—62no—
299C32×C4—61no—
300C32×C4#23518yes—
301C32×C4#23712yes—
302C3⋊Dic3—22no—
303C3⋊Dic3—21no—
304C3⋊Dic3—21no—
305C3⋊Dic3—32no—
306C3⋊Dic3—38no—
307C3⋊Dic3—68no—
308C3⋊Dic3—68no—
309C3⋊Dic3—64no—
310C3⋊Dic3—61no—
311C3×Dic3—28no—
312C3×Dic3—24no—
313C3×Dic3—24no—
314C3×Dic3—68no—
315C3×Dic3—64no—
316C3×Dic3—64no—
317#127—271no—
318#124—91no—
319#124—271no—
320#124—272no—
321#123—272no—
322#123—272no—
323C4×Q8—271no—
324C4×Q8—271no—
325C4×D4—31no—
326C4×D4—271no—
327C4×D4—272no—
328C33#55911yescharacteristic, complemented, Sylow
329C23×C3—28no—
330C23×C3C2×D3218yes—
331C23×C3C22×C3⋊D312yes—
332C22×D3—98no—
333C22×D3—92no—
334C22×D3—188no—
335C3×Q8—182no—
336C3×D4—28no—
337C3×D4—22no—
338C3×D4—48no—
339C2×C3×C4—24no—
340C2×C3×C4—38no—
341C2×C3×C4—34no—
342C2×C3×C4—34no—
343C2×C3×C4—92no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—128no—
347C2×C3×C4C2×D3214yes—
348C2×C3×C4C22×C3⋊D311yescharacteristic
349#67—184no—
350#67—1816no—
351#67—3616no—
352#67—368no—
353#67—362no—
354#67—3616no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—38no—
358C2×Dic3—42no—
359C2×Dic3—94no—
360C2×Dic3—91no—
361C2×Dic3—94no—
362C2×Dic3—91no—
363C2×Dic3—184no—
364C2×Dic3—184no—
365C2×Dic3C22×C3⋊D312yes—
366C2×Dic3C22×C3⋊D311yescharacteristic
367C2×Dic3C22×C3⋊D311yescharacteristic
368D12—188no—
369D12—182no—
370D12—368no—
371C4×D3—362no—
372C4×D3—368no—
373Dic6—68no—
374Dic6—182no—
375Dic6—188no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—42no—
380C2×C32—48no—
381C2×C32—88no—
382C2×C32C2×#23514yes—
383C2×C32C2×#23711yescharacteristic
384C2×C32#100414yes—
385C2×C32D3×#4114yes—
386C2×C32D3×#4111yescharacteristic
387C2×C32#100911yescharacteristic
388C3⋊D3—122no—
389C3⋊D3—128no—
390C3⋊D3—248no—
391C2×Q8—91no—
392C2×Q8—271no—
393C2×D4—272no—
394C2×D4—272no—
395C2×D4—271no—
396C2×D4D3×C3⋊D311yescharacteristic
397C22×C4—32no—
398C22×C4—272no—
399C4⋊C4—31no—
400C4⋊C4—271no—
401C4⋊C4—271no—
402C4⋊C4—272no—
403C4⋊C4—271no—
404C22⋊C4—32no—
405C22⋊C4—92no—
406C22⋊C4—92no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C42—31no—
411C42—91no—
412C42—272no—
413C42—271no—
414C22×C3—28no—
415C22×C3—28no—
416C22×C3—28no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—48no—
422C22×C3—48no—
423C22×C3—48no—
424C22×C3C22×D3214yes—
425C22×C3C23×C3⋊D311yescharacteristic
426C2×D3—188no—
427C2×D3—188no—
428C2×D3—188no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—368no—
433C2×D3—368no—
434C2×D3—368no—
435C3×C4—28no—
436C3×C4—64no—
437C3×C4—68no—
438C3×C4—64no—
439C3×C4—182no—
440C3×C4—181no—
441C3×C4—181no—
442C3×C4#380418yes—
443C3×C4#382812yes—
444Dic3—22no—
445Dic3—21no—
446Dic3—21no—
447Dic3—64no—
448Dic3—64no—
449Dic3—68no—
450Dic3—98no—
451Dic3—92no—
452Dic3—188no—
453Dic3—188no—
454Dic3—184no—
455Dic3—181no—
456C32—24no—
457C32#555914yescomplemented
458C32#568111yescharacteristic, complemented
459C23—272no—
460C23C2×D3×C3⋊D312yes—
461Q8—182no—
462Q8—542no—
463D4—22no—
464D4—544no—
465D4—542no—
466D4—1082no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—32no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—122no—
474C2×C4—271no—
475C2×C4—271no—
476C2×C4—1082no—
477C2×C4C2×D3×C3⋊D311yescharacteristic
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—48no—
482C2×C3—42no—
483C2×C3—88no—
484C2×C3#6431214yes—
485C2×C3#6431414yes—
486C2×C3C2×#380414yes—
487C2×C3C2×#382811yescharacteristic
488C2×C3#6436111yescharacteristic
489C2×C3#41×C3⋊D311yescharacteristic
490D3—368no—
491D3—362no—
492D3—728no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—542no—
497C22—542no—
498C22—542no—
499C22C22×D3×C3⋊D311yescharacteristic
500C4—61no—
501C4—61no—
502C4—62no—
503C4—181no—
504C4—181no—
505C4—182no—
506C4—272no—
507C4—541no—
508C4#8903112yes—
509C3—24no—
510C3#1059521014yescomplemented
511C3#1059603011yescharacteristic, complemented
512C2—42no—
513C2—1082no—
514C2#1172458611yescharacteristic
515C2#1172458811yescharacteristic
516C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723628C2semidirect
5. #11723618C2semidirect
6. #11723612C2semidirect
7. #11723592C2semidirect
8. C4×#6293C2semidirect
9. #11723494C2semidirect
10. #11723441C2semidirect
11. #11723410C2semidirect
41. #64145D3semidirect
51. C3×D4×Dic3D3semidirect
156. D4×Dic3C3⋊D3semidirect
328. C33#559semidirect
457. C32#5559semidirect
458. C32#5681semidirect
510. C3#10595210semidirect
511. C3#10596030semidirect

Maximal subgroups

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

subgroupindex
#117239812
C3×#641452
C32×D4×Dic32
#117236282
#117236182
#117236122
#117235922
C4×#62932
#117234942
#117234412
#117234102
#105960303
#105952103