Finite Group Databasebeta

← #13175356

Subgroups — #13175356

Order 1728.

1544 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723911C211yesmaximal, characteristic, complemented
2C3×#64084C211yesmaximal, characteristic, complemented
3C32×#875C211yesmaximal, characteristic, complemented
4#11723523C212yesmaximal, complemented
5#11723519C212yesmaximal, complemented
6#11723487C212yesmaximal, complemented
7#11723480C212yesmaximal, complemented
8#11723476C212yesmaximal, complemented
9#11723474C212yesmaximal, complemented
10#10596155—31nomaximal
11#10593985—34nomaximal
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×D12C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C2×#6290C2212yes—
17C2×#6290C2212yes—
18C2×Dic3×C3⋊D3C2212yes—
19C2×D3×C3⋊Dic3C2212yes—
20C33×C42C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23#88857C2212yes—
24#88857C2212yes—
25C3×#3752C2212yes—
26C3×#3752C2212yes—
27C32×(C2×D3)⋊C4C2212yes—
28C32×(C2×D3)⋊C4C2212yes—
29#88807C2212yes—
30Dic3⋊(C3⋊Dic3)C2212yes—
31#88803C2212yes—
32(C2×D3)⋊(C3⋊Dic3)C2212yes—
33C32×#124—31no—
34#64155—32no—
35#64148—32no—
36#64095—32no—
37#64093—32no—
38#64092—32no—
39#64087—32no—
40#64084D311yescharacteristic, complemented
41#64079—31no—
42#64079—34no—
43#64079—64no—
44C3×#875—34no—
45C3×#875D314yescomplemented
46#63891—38no—
47#63891—38no—
48#63887—38no—
49#63887—38no—
50#63885—38no—
51#63885—38no—
52C22×C3×C3⋊D3C2312yes—
53C22×C32×D3C2312yes—
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56C2×C33×C4C2311yescharacteristic
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3×C3⋊Dic3C2312yes—
60C3×C4⋊(C3⋊D3)—41no—
61C3×#634—41no—
62C2×C32×Dic3C2312yes—
63C32×D12—41no—
64C32×Dic6—41no—
65#6290—42no—
66#6290—42no—
67Dic3×C3⋊D3—42no—
68D3×C3⋊Dic3—42no—
69#5684—94no—
70#5684—91no—
71C2×C32×Q8—31no—
72C2×C32×D4—31no—
73C2×#638—32no—
74C2×#638—32no—
75C22×C3⋊Dic3—32no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3—32no—
78C2×#634C2×D311yescharacteristic
79C2×C3×D12—34no—
80C2×C3×D12C2×D314yes—
81C2×C3×Dic6—34no—
82C2×C3×Dic6C2×D314yes—
83C2×#625—38no—
84C2×#625—38no—
85C2×D3×Dic3—38no—
86C2×D3×Dic3—38no—
87C32×C22⋊C4—32no—
88C32×C22⋊C4—32no—
89C32×C42—24no—
90C32×C42C2×D314yes—
91C32×C42C2×D311yescharacteristic
92#3757—32no—
93#3752—32no—
94#3752C2×D312yes—
95#3752C2×D312yes—
96C4⋊(C3⋊Dic3)—31no—
97C4⋊(C3⋊Dic3)—31no—
98C4⋊(C3⋊Dic3)—34no—
99C4⋊(C3⋊Dic3)—34no—
100C4⋊(C3⋊Dic3)—32no—
101C4⋊(C3⋊Dic3)—64no—
102C4⋊(C3⋊Dic3)—64no—
103#3750—32no—
104#3750—32no—
105#3750—38no—
106#3750—38no—
107#3750—32no—
108#3750—68no—
109#3750—68no—
110C3×(C2×D3)⋊C4—38no—
111C3×(C2×D3)⋊C4—38no—
112C3×(C2×D3)⋊C4C2×D318yes—
113C3×(C2×D3)⋊C4C2×D318yes—
114Dic3⋊Dic3—38no—
115Dic3⋊Dic3—38no—
116(C2×D3)⋊Dic3—38no—
117(C2×D3)⋊Dic3—38no—
118C22×C33C2411yescharacteristic
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C32×D3—22no—
123C2×C32×D3—22no—
124C2×C32×D3—22no—
125C33×C4—21no—
126C33×C4—21no—
127C33×C4—21no—
128C3⋊(C3⋊Dic3)—22no—
129C3⋊(C3⋊Dic3)—22no—
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C3×#124—34no—
133C3×#124—91no—
134#946—98no—
135#946—92no—
136#939—98no—
137#939—92no—
138#886—98no—
139#886—92no—
140#884—98no—
141#884—92no—
142#883—98no—
143#883—92no—
144#878—98no—
145#878—92no—
146#875—34no—
147#875C3⋊D311yescharacteristic, complemented
148#870—94no—
149#870—91no—
150#870—184no—
151C23×C32—32no—
152C22×C3⋊D3C22×D312yes—
153C22×C3×D3—38no—
154C22×C3×D3C22×D318yes—
155C32×Q8—121no—
156C32×D4—121no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—32no—
161C2×C32×C4C22×D314yes—
162C2×C32×C4C22×D314yes—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167#638—122no—
168#638—122no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—38no—
172C2×C3⋊Dic3—38no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—122no—
176C2×C3⋊Dic3C22×D312yes—
177C4⋊(C3⋊D3)—41no—
178C4×C3⋊D3—122no—
179#634—41no—
180C2×C3×Dic3—38no—
181C2×C3×Dic3C22×D318yes—
182C3×D12—44no—
183C3×D12—124no—
184C3×Dic6—44no—
185C3×Dic6—124no—
186#625—128no—
187#625—128no—
188D3×Dic3—128no—
189D3×Dic3—128no—
190#562—271noSylow
191C2×C33H(5,𝔽2)11yescharacteristic
192C2×C33H(5,𝔽2)11yescharacteristic
193C2×C332−1+411yescharacteristic
194C3×C3⋊D3—42no—
195C32×D3—42no—
196C2×C3×Q8—34no—
197C2×C3×Q8—91no—
198C2×C3×D4—34no—
199C2×C3×D4—91no—
200C2×#67—98no—
201C2×#67—98no—
202C2×#67—92no—
203C2×#67—92no—
204C22×Dic3—98no—
205C22×Dic3—92no—
206C2×D12—34no—
207C2×D12C2×C3⋊D311yescharacteristic
208C2×C4×D3—98no—
209C2×C4×D3—92no—
210C2×Dic6—34no—
211C2×Dic6C2×C3⋊D311yescharacteristic
212C3×C22⋊C4—38no—
213C3×C22⋊C4—38no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—92no—
216C3×C42—24no—
217C3×C42D3214yes—
218C3×C42C2×C3⋊D311yescharacteristic
219#217—98no—
220#217—92no—
221(C2×D3)⋊C4—38no—
222(C2×D3)⋊C4—38no—
223(C2×D3)⋊C4—98no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4C2×C3⋊D312yes—
226(C2×D3)⋊C4C2×C3⋊D312yes—
227C4⋊Dic3—98no—
228C4⋊Dic3—94no—
229C4⋊Dic3—94no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—184no—
234C4⋊Dic3—184no—
235Dic3⋊C4—98no—
236Dic3⋊C4—98no—
237Dic3⋊C4—98no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—188no—
242Dic3⋊C4—188no—
243C22×C32—24no—
244C22×C32—62no—
245C22×C32—62no—
246C22×C32—62no—
247C22×C32C23×D314yes—
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3—22no—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3×D3—28no—
253C2×C3×D3—28no—
254C2×C3×D3—28no—
255C2×C3×D3—68no—
256C2×C3×D3—68no—
257C2×C3×D3—68no—
258C32×C4—28no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—24no—
262C32×C4—21no—
263C32×C4—24no—
264C32×C4—24no—
265C32×C4—44no—
266C32×C4—44no—
267C32×C4—62no—
268C3⋊Dic3—22no—
269C3⋊Dic3—62no—
270C3⋊Dic3—62no—
271C3⋊Dic3—68no—
272C3⋊Dic3—68no—
273C3⋊Dic3—128no—
274C3⋊Dic3—128no—
275C3×Dic3—28no—
276C3×Dic3—68no—
277#125—271no—
278#124—31no—
279#124—91no—
280#123—272no—
281#123—272no—
282C22⋊Q8—272no—
283C22⋊Q8—272no—
284#121—272no—
285#121—272no—
286C33#56211yescharacteristic, complemented, Sylow
287C23×C3—38no—
288C23×C3—92no—
289C22×D3—38no—
290C22×D3C22×C3⋊D312yes—
291C3×Q8—124no—
292C3×Q8—361no—
293C3×D4—124no—
294C3×D4—361no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—38no—
299C2×C3×C4—92no—
300C2×C3×C4C2×D3214yes—
301C2×C3×C4C2×D3214yes—
302C2×C3×C4C2×D3214yes—
303C2×C3×C4C22×C3⋊D311yescharacteristic
304C2×C3×C4C22×C3⋊D311yescharacteristic
305C2×C3×C4C22×C3⋊D311yescharacteristic
306#67—362no—
307#67—362no—
308#67—368no—
309#67—368no—
310C2×Dic3—38no—
311C2×Dic3—98no—
312C2×Dic3—98no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315C2×Dic3—188no—
316C2×Dic3—188no—
317C2×Dic3—362no—
318C2×Dic3—368no—
319C2×Dic3C22×C3⋊D312yes—
320D12—41no—
321D12—124no—
322C4×D3—368no—
323C4×D3—362no—
324Dic6—41no—
325Dic6—124no—
326C2×C32—24no—
327C2×C32—24no—
328C2×C32—24no—
329C2×C32—122no—
330C2×C32#100414yes—
331C2×C32#100411yescharacteristic
332C2×C32#100914yes—
333C2×C32#100911yescharacteristic
334C2×C32#101014yes—
335C2×C32#101011yescharacteristic
336C3⋊D3—42no—
337C3×D3—48no—
338C3×D3—128no—
339C2×Q8—31no—
340C2×Q8—91no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—272no—
344C2×D4—272no—
345C22×C4—272no—
346C22×C4—272no—
347C4⋊C4—272no—
348C4⋊C4—272no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C4⋊C4—271no—
352C4⋊C4—271no—
353C22⋊C4—32no—
354C22⋊C4—32no—
355C22⋊C4—92no—
356C22⋊C4—92no—
357C22⋊C4—272no—
358C22⋊C4—272no—
359C42D3×C3⋊D311yescharacteristic
360C22×C3—24no—
361C22×C3—68no—
362C22×C3—68no—
363C22×C3—68no—
364C22×C3—182no—
365C22×C3—182no—
366C22×C3—182no—
367C22×C3C22×D3214yes—
368C22×C3C23×C3⋊D311yescharacteristic
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—68no—
373C2×D3—68no—
374C2×D3—68no—
375C3×C4—28no—
376C3×C4—24no—
377C3×C4—24no—
378C3×C4—24no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—44no—
383C3×C4—44no—
384C3×C4—68no—
385C3×C4—182no—
386Dic3—22no—
387Dic3—68no—
388Dic3—188no—
389Dic3—188no—
390Dic3—182no—
391Dic3—182no—
392Dic3—368no—
393Dic3—368no—
394C32—24no—
395C32#568414yescomplemented
396C32#568411yescharacteristic, complemented
397C23—32no—
398C23—92no—
399Q8—121no—
400Q8—361no—
401D4—121no—
402D4—361no—
403D4—1082no—
404D4—1082no—
405C2×C4—32no—
406C2×C4—92no—
407C2×C4—272no—
408C2×C4—272no—
409C2×C4—1082no—
410C2×C4—1082no—
411C2×C4C2×D3×C3⋊D311yescharacteristic
412C2×C4C2×D3×C3⋊D311yescharacteristic
413C2×C4C2×D3×C3⋊D311yescharacteristic
414C2×C3—24no—
415C2×C3—24no—
416C2×C3—24no—
417C2×C3—128no—
418C2×C3—362no—
419C2×C3#6429814yes—
420C2×C3#6430714yes—
421C2×C3#6430714yes—
422C2×C3#6436111yescharacteristic
423C2×C3#6436611yescharacteristic
424C2×C3#6436711yescharacteristic
425D3—42no—
426D3—128no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—182no—
431C22—182no—
432C22—182no—
433C22C22×D3×C3⋊D311yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—62no—
438C4—182no—
439C4—542no—
440C4—542no—
441C3—24no—
442C3#1059398514yescomplemented
443C3#1059615511yescharacteristic, complemented
444C2—122no—
445C2—362no—
446C2#1172456311yescharacteristic
447C2#1172457911yescharacteristic
448C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#562
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723911C2semidirect
2. C3×#64084C2semidirect
3. C32×#875C2semidirect
4. #11723523C2semidirect
5. #11723519C2semidirect
6. #11723487C2semidirect
7. #11723480C2semidirect
8. #11723476C2semidirect
9. #11723474C2semidirect
40. #64084D3semidirect
45. C3×#875D3semidirect
147. #875C3⋊D3semidirect
286. C33#562semidirect
395. C32#5684semidirect
396. C32#5684semidirect
442. C3#10593985semidirect
443. C3#10596155semidirect

Maximal subgroups

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

subgroupindex
#117239112
C3×#640842
C32×#8752
#117235232
#117235192
#117234872
#117234802
#117234762
#117234742
#105961553
#105939853