Finite Group Databasebeta

← #13175327

Subgroups — #13175327

Order 1728.

1720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723916C211yesmaximal, characteristic, complemented
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic
5#11723521C212yesmaximal, complemented
6#11723510C212yesmaximal, complemented
7#11723494C211yesmaximal, characteristic, complemented
8#11723429C211yesmaximal, characteristic, complemented
9#11723422C212yesmaximal, complemented
10#11723414C212yesmaximal, complemented
11#11723406C211yesmaximal, characteristic, complemented
12#10596030—31nomaximal
13#10593978—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6291C2212yes—
20C2×#6291C2212yes—
21C2×#6289C2212yes—
22C33×C42C2211yescharacteristic
23#88859C2212yes—
24#88859C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×#3750C2211yescharacteristic
27C3×#3750C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88806C2212yes—
33#88804C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64149—31no—
39#64145—31no—
40#64138—32no—
41C4×#638—32no—
42#64102—31no—
43#64096—31no—
44#64094—32no—
45#64092—32no—
46#64084—31no—
47#64084—34no—
48#64084—64no—
49C4×#634D311yescharacteristic, complemented
50C3×C4×D12D314yescomplemented
51C3×C4×Dic6—34no—
52#63911—34no—
53#63907—38no—
54#63904—38no—
55#63894—34no—
56#63860—34no—
57#63854—38no—
58#63850—38no—
59#63844—34no—
60C22×C3⋊(C3⋊D3)C2312yes—
61C22×C32×D3C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C4⋊(C3⋊(C3⋊D3))—22no—
67#6308—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×#634—22no—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×C4×D3—42no—
76#6291—24no—
77#6291—42no—
78#6289—42no—
79#5585—91no—
80#5559—94no—
81C2×C32×D4—31no—
82C22×C32×C4—32no—
83C2×#638—32no—
84C2×#638—32no—
85C2×C4⋊(C3⋊D3)—31no—
86C2×C4⋊(C3⋊D3)—34no—
87C2×C4⋊(C3⋊D3)—64no—
88C2×C4×C3⋊D3—32no—
89C2×#634—31no—
90C2×#634—34no—
91C2×#634—64no—
92C2×#634C2×D311yescharacteristic
93C2×C3×D12C2×D314yes—
94C2×C3×C4×D3C2×D318yes—
95C2×C3×Dic6—34no—
96C2×#626—38no—
97C2×#626—38no—
98C2×#624—38no—
99C32×C4⋊C4—31no—
100C32×C22⋊C4—32no—
101C32×C42—24no—
102C32×C42C2×D314yes—
103C32×C42C2×D311yescharacteristic
104#3757—32no—
105#3757—32no—
106#3752—32no—
107#3752—32no—
108#3752—38no—
109#3752—38no—
110#3752—32no—
111#3752—68no—
112#3752—68no—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114#3750—32no—
115#3750C2×D311yescharacteristic
116#3750C2×D311yescharacteristic
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3C2×D311yescharacteristic
119C4×C3⋊Dic3C2×D311yescharacteristic
120C3×(C2×D3)⋊C4C2×D318yes—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3C2×D314yes—
123C3×Dic3⋊C4—34no—
124C3×Dic3⋊C4—34no—
125C3×C4×Dic3—34no—
126C3×C4×Dic3—34no—
127#3724—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—22no—
135C2×C3⋊(C3⋊D3)—22no—
136C2×C32×D3—22no—
137C2×C32×D3—22no—
138C2×C32×D3—22no—
139C33×C4—21no—
140C33×C4—21no—
141C33×C4#4112yes—
142C3⋊(C3⋊Dic3)—22no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3—21no—
146C3×C3⋊Dic3#4112yes—
147C32×Dic3—22no—
148C3×C4×Q8—91no—
149C3×C4×D4—34no—
150#947—92no—
151#940—94no—
152#936—94no—
153#929—98no—
154C4×#67—98no—
155#894—92no—
156#894—92no—
157#893—94no—
158#893—92no—
159#887—94no—
160#885—98no—
161#883—98no—
162#875—91no—
163#875—94no—
164#875—91no—
165#875—184no—
166#874—91no—
167C4×D12—91no—
168C4×D12C3⋊D311yescharacteristic, complemented
169C4×Dic6—34no—
170C4×Dic6—91no—
171C23×C32—32no—
172C22×C3⋊D3—32no—
173C22×C3⋊D3—38no—
174C22×C3⋊D3—68no—
175C22×C3×D3C22×D318yes—
176C32×D4—62no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—32no—
181C2×C32×C4—122no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188#638—64no—
189#638—122no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—38no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C4⋊(C3⋊D3)—62no—
198C4⋊(C3⋊D3)—68no—
199C4⋊(C3⋊D3)—128no—
200C4×C3⋊D3—122no—
201#634—22no—
202#634—62no—
203#634—68no—
204#634—128no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3C22×D318yes—
210C3×D12—28no—
211C3×C4×D3—48no—
212C3×Dic6—68no—
213#626—616no—
214#626—128no—
215#624—128no—
216#559—271noSylow
217C2×C33C2×#4111yescharacteristic
218C2×C33C2×#4111yescharacteristic
219C2×C33H(5,𝔽2)11yescharacteristic
220C3⋊(C3⋊D3)—42no—
221C32×D3—42no—
222C2×C3×Q8—91no—
223C2×C3×D4—34no—
224C22×C3×C4—38no—
225C2×#67—98no—
226C2×#67—98no—
227C2×D12—92no—
228C2×D12—94no—
229C2×D12—91no—
230C2×D12—92no—
231C2×D12—184no—
232C2×D12C2×C3⋊D311yescharacteristic
233C2×C4×D3—98no—
234C2×C4×D3—92no—
235C2×C4×D3C2×C3⋊D312yes—
236C2×Dic6—34no—
237C2×Dic6—94no—
238C2×Dic6—91no—
239C2×Dic6—184no—
240C3×C4⋊C4—34no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—38no—
245C3×C42—24no—
246C3×C42—91no—
247C3×C42—91no—
248C3×C42D3214yes—
249C3×C42C2×C3⋊D311yescharacteristic
250#217—98no—
251#217—98no—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—98no—
256(C2×D3)⋊C4—98no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—188no—
261(C2×D3)⋊C4—188no—
262(C2×D3)⋊C4C2×C3⋊D312yes—
263C4⋊Dic3—34no—
264C4⋊Dic3C2×C3⋊D311yescharacteristic
265Dic3⋊C4—34no—
266Dic3⋊C4—34no—
267Dic3⋊C4—98no—
268Dic3⋊C4—92no—
269C4×Dic3—34no—
270C4×Dic3—34no—
271C4×Dic3—98no—
272C4×Dic3—92no—
273C22×C32—24no—
274C22×C32—62no—
275C22×C32—62no—
276C22×C32—62no—
277C22×C32C23×D314yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C2×C3⋊D3—128no—
288C2×C3×D3—28no—
289C2×C3×D3—28no—
290C2×C3×D3—28no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—28no—
294C32×C4—24no—
295C32×C4—24no—
296C32×C4—44no—
297C32×C4—44no—
298C32×C4—62no—
299C32×C4#23518yes—
300C32×C4#23912yes—
301C3⋊Dic3—21no—
302C3⋊Dic3—21no—
303C3⋊Dic3—21no—
304C3⋊Dic3—62no—
305C3⋊Dic3—68no—
306C3⋊Dic3—128no—
307C3⋊Dic3#23512yes—
308C3×Dic3—28no—
309C3×Dic3—38no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3—64no—
313#127—271no—
314#124—271no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—91no—
320C4×Q8—271no—
321C4×D4—31no—
322C4×D4—271no—
323C4×D4—272no—
324C33#55911yescharacteristic, complemented, Sylow
325C23×C3—38no—
326C22×D3—98no—
327C22×D3—92no—
328C22×D3—188no—
329C22×D3C22×C3⋊D312yes—
330C3×Q8—182no—
331C3×D4—68no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—38no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—91no—
340C2×C3×C4—128no—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C2×D3214yes—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346C2×C3×C4C22×C3⋊D311yescharacteristic
347#67—1816no—
348#67—368no—
349C2×Dic3—34no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—34no—
353C2×Dic3—98no—
354C2×Dic3—92no—
355C2×Dic3—188no—
356C2×Dic3C22×C3⋊D312yes—
357D12—22no—
358D12—184no—
359D12—188no—
360D12—182no—
361D12—368no—
362D12—362no—
363C4×D3—42no—
364C4×D3—362no—
365C4×D3—368no—
366Dic6—68no—
367Dic6—188no—
368Dic6—182no—
369Dic6—368no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—122no—
374C2×C32C2×#23514yes—
375C2×C32C2×#23511yescharacteristic
376C2×C32#100414yes—
377C2×C32C2×#23911yescharacteristic
378C2×C32D3×#4114yes—
379C2×C32#100911yescharacteristic
380C3⋊D3—122no—
381C3⋊D3—128no—
382C3⋊D3—248no—
383C3×D3—48no—
384C2×Q8—91no—
385C2×Q8—271no—
386C2×D4—31no—
387C2×D4—272no—
388C2×D4—272no—
389C2×D4—271no—
390C22×C4—32no—
391C22×C4—272no—
392C4⋊C4—31no—
393C4⋊C4—91no—
394C4⋊C4—91no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C22⋊C4—32no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—91no—
404C42—91no—
405C42—272no—
406C42D3×C3⋊D311yescharacteristic
407C22×C3—24no—
408C22×C3—68no—
409C22×C3—68no—
410C22×C3—68no—
411C22×C3C22×D3214yes—
412C22×C3C23×C3⋊D311yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—188no—
417C2×D3—188no—
418C2×D3—188no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—368no—
423C2×D3—368no—
424C2×D3—368no—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—28no—
430C3×C4—44no—
431C3×C4—44no—
432C3×C4—68no—
433C3×C4—92no—
434C3×C4—181no—
435C3×C4—181no—
436C3×C4—181no—
437C3×C4#379918yes—
438C3×C4#382812yes—
439Dic3—22no—
440Dic3—38no—
441Dic3—64no—
442Dic3—64no—
443Dic3—64no—
444Dic3—188no—
445Dic3—182no—
446Dic3—368no—
447C32—24no—
448C32#555914yescomplemented
449C32#558511yescharacteristic, complemented
450C23—32no—
451C23—272no—
452Q8—182no—
453Q8—542no—
454D4—62no—
455D4—544no—
456D4—542no—
457D4—1082no—
458C2×C4—32no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—122no—
464C2×C4—272no—
465C2×C4—1082no—
466C2×C4C2×D3×C3⋊D311yescharacteristic
467C2×C4C2×D3×C3⋊D311yescharacteristic
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—128no—
473C2×C3C2×#379914yes—
474C2×C3D3×#23514yes—
475C2×C3#6430714yes—
476C2×C3C2×#382811yescharacteristic
477C2×C3#6436111yescharacteristic
478C2×C3#41×C3⋊D311yescharacteristic
479D3—42no—
480D3—368no—
481D3—362no—
482D3—728no—
483C22—62no—
484C22—62no—
485C22—62no—
486C22—542no—
487C22—542no—
488C22—542no—
489C22C22×D3×C3⋊D311yescharacteristic
490C4—21no—
491C4—21no—
492C4—62no—
493C4—92no—
494C4—181no—
495C4—181no—
496C4—181no—
497C4—542no—
498C4#8902212yes—
499C3—24no—
500C3#1059397814yescomplemented
501C3#1059603011yescharacteristic, complemented
502C2—122no—
503C2—1082no—
504C2C2×#8902211yescharacteristic
505C2C3⋊D3×#23511yescharacteristic
506C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723524C2
hypercenterC42D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723916C2semidirect
2. C3×C4×#634C2semidirect
3. C32×C4×D12C2semidirect
5. #11723521C2semidirect
6. #11723510C2semidirect
7. #11723494C2semidirect
8. #11723429C2semidirect
9. #11723422C2semidirect
10. #11723414C2semidirect
11. #11723406C2semidirect
49. C4×#634D3semidirect
50. C3×C4×D12D3semidirect
168. C4×D12C3⋊D3semidirect
324. C33#559semidirect
448. C32#5559semidirect
449. C32#5585semidirect
500. C3#10593978semidirect
501. C3#10596030semidirect

Maximal subgroups

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

subgroupindex
#117239162
C3×C4×#6342
C32×C4×D122
#117235242
#117235212
#117235102
#117234942
#117234292
#117234222
#117234142
#117234062
#105960303
#105939783