Finite Group Databasebeta

← #13175321

Subgroups — #13175321

Order 1728.

1652 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723911C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723522C211yesmaximal, characteristic, complemented
5#11723522C211yesmaximal, characteristic, complemented
6#11723519C212yesmaximal, complemented
7#11723513C212yesmaximal, complemented
8#11723506C212yesmaximal, complemented
9C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#11723427C212yesmaximal, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596029—31nomaximal
13#10593923—34nomaximal
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6290C2212yes—
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C33×C42C2211yescharacteristic
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25#88857C2212yes—
26#88857C2212yes—
27C3×#3752C2211yescharacteristic
28C3×#3752C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88807C2212yes—
33#88803C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64147—31no—
39#64147—31no—
40#64138—32no—
41C4×#638—32no—
42#64101—31no—
43C4⋊C4×C3⋊D3—31no—
44#64093—32no—
45#64088—32no—
46C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
47#64079—31no—
48#64079—34no—
49#64079—64no—
50C3×C4×D12—34no—
51C3×C4×D12D314yescomplemented
52#63908—34no—
53#63908—34no—
54#63905—38no—
55D3×C4⋊Dic3—34no—
56#63885—38no—
57#63879—38no—
58#63858—38no—
59#63841—34no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4⋊(C3⋊D3)—22no—
71C3×C4×C3⋊D3—22no—
72C3×C4×C3⋊D3—22no—
73C2×C32×Dic3C2312yes—
74C32×D12—22no—
75C32×C4×D3—42no—
76#6290—24no—
77#6290—42no—
78Dic3×C3⋊D3—42no—
79#5565—91no—
80#5558—94no—
81C2×C32×D4—31no—
82C22×C32×C4—32no—
83C2×#638—32no—
84C2×#638—32no—
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3—32no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×C4×C3⋊D3C2×D311yescharacteristic
89C2×C3×D12—34no—
90C2×C3×D12C2×D314yes—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—34no—
93C2×C3×C4×D3C2×D318yes—
94C2×#625—38no—
95C2×#625—38no—
96C2×D3×Dic3—38no—
97C32×C4⋊C4—31no—
98C32×C22⋊C4—32no—
99C32×C42—24no—
100C32×C42C2×D314yes—
101C32×C42C2×D311yescharacteristic
102#3757—32no—
103#3757—32no—
104#3752—32no—
105#3752C2×D311yescharacteristic
106#3752C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—34no—
111C4⋊(C3⋊Dic3)—64no—
112C4⋊(C3⋊Dic3)—64no—
113C4⋊(C3⋊Dic3)C2×D311yescharacteristic
114#3750—32no—
115#3750—32no—
116#3750—38no—
117#3750—38no—
118#3750—32no—
119#3750—68no—
120#3750—68no—
121C4×C3⋊Dic3—32no—
122C3×(C2×D3)⋊C4—34no—
123C3×(C2×D3)⋊C4—34no—
124C3×(C2×D3)⋊C4C2×D318yes—
125C3×C4⋊Dic3—34no—
126C3×C4⋊Dic3C2×D314yes—
127Dic3⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3#4112yes—
139C2×C32×D3—22no—
140C2×C32×D3—22no—
141C2×C32×D3—22no—
142C33×C4—21no—
143C33×C4—21no—
144C33×C4#4112yes—
145C3⋊(C3⋊Dic3)—22no—
146C3⋊(C3⋊Dic3)—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—22no—
150C3×C4×D4—34no—
151C3×C4×D4—91no—
152#939—92no—
153#938—94no—
154#938—94no—
155#930—91no—
156#930—91no—
157#929—98no—
158C4×#67—98no—
159#926—91no—
160C2×C4⋊Dic3—91no—
161#898—92no—
162#892—94no—
163D3×C4⋊C4—94no—
164#884—98no—
165#883—92no—
166#67⋊C4—92no—
167#879—98no—
168C4×D12—34no—
169C4×D12C3⋊D311yescharacteristic, complemented
170#870—94no—
171#870—91no—
172#870—184no—
173C23×C32—32no—
174C22×C3⋊D3C22×D311yescharacteristic
175C22×C3⋊D3C22×D311yescharacteristic
176C22×C3×D3—34no—
177C22×C3×D3—34no—
178C22×C3×D3C22×D318yes—
179C32×D4—62no—
180C2×C32×C4—24no—
181C2×C32×C4—24no—
182C2×C32×C4—24no—
183C2×C32×C4—32no—
184C2×C32×C4—122no—
185C2×C32×C4C22×D314yes—
186C2×C32×C4C22×D314yes—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191#638—64no—
192#638—122no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—38no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—68no—
198C2×C3⋊Dic3—68no—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C4⋊(C3⋊D3)—22no—
202C4×C3⋊D3—22no—
203C4×C3⋊D3—22no—
204C4×C3⋊D3—122no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3C22×D318yes—
208C3×D12—28no—
209C3×D12—68no—
210C3×C4×D3—48no—
211C3×C4×D3—68no—
212C3×C4×D3—68no—
213#625—616no—
214#625—128no—
215D3×Dic3—128no—
216#553—271noSylow
217C2×C33C2×#4111yescharacteristic
218C2×C33C2×#4111yescharacteristic
219C2×C33H(5,𝔽2)11yescharacteristic
220C3×C3⋊D3—22no—
221C3×C3⋊D3—41no—
222C32×D3—42no—
223C2×C3×D4—34no—
224C2×C3×D4—91no—
225C22×C3×C4—38no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C2×#67—98no—
229C2×#67—92no—
230C2×#67—98no—
231C2×#67—92no—
232C22×Dic3—92no—
233C2×D12—34no—
234C2×D12C2×C3⋊D311yescharacteristic
235C2×C4×D3—34no—
236C2×C4×D3—34no—
237C2×C4×D3—98no—
238C2×C4×D3C2×C3⋊D312yes—
239C3×C4⋊C4—34no—
240C3×C4⋊C4—91no—
241C3×C22⋊C4—38no—
242C3×C22⋊C4—91no—
243C3×C22⋊C4—91no—
244C3×C42—24no—
245C3×C42D3214yes—
246C3×C42C2×C3⋊D311yescharacteristic
247#217—98no—
248#217—98no—
249#217—92no—
250(C2×D3)⋊C4—34no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4C2×C3⋊D312yes—
256C4⋊Dic3—34no—
257C4⋊Dic3—94no—
258C4⋊Dic3—94no—
259C4⋊Dic3—91no—
260C4⋊Dic3—91no—
261C4⋊Dic3—92no—
262C4⋊Dic3—184no—
263C4⋊Dic3—184no—
264C4⋊Dic3C2×C3⋊D311yescharacteristic
265Dic3⋊C4—98no—
266Dic3⋊C4—98no—
267Dic3⋊C4—98no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—188no—
271Dic3⋊C4—188no—
272C4×Dic3—98no—
273C4×Dic3—92no—
274C22×C32—24no—
275C22×C32—62no—
276C22×C32—62no—
277C22×C32—62no—
278C22×C32C23×D314yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—21no—
281C2×C3⋊D3—21no—
282C2×C3⋊D3—21no—
283C2×C3⋊D3—21no—
284C2×C3⋊D3—21no—
285C2×C3⋊D3#23512yes—
286C2×C3×D3—28no—
287C2×C3×D3—28no—
288C2×C3×D3—28no—
289C2×C3×D3—38no—
290C2×C3×D3—64no—
291C2×C3×D3—64no—
292C2×C3×D3—64no—
293C2×C3×D3—64no—
294C2×C3×D3—64no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—28no—
298C32×C4—24no—
299C32×C4—24no—
300C32×C4—44no—
301C32×C4—44no—
302C32×C4—62no—
303C32×C4#23518yes—
304C32×C4#23712yes—
305C3⋊Dic3—21no—
306C3⋊Dic3—21no—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3⋊Dic3—68no—
310C3⋊Dic3—68no—
311C3⋊Dic3—128no—
312C3⋊Dic3—128no—
313C3×Dic3—28no—
314C3×Dic3—64no—
315C3×Dic3—64no—
316#126—272no—
317#125—271no—
318#123—272no—
319#121—272no—
320#121—271no—
321#121—271no—
322C4×D4—31no—
323C4×D4—91no—
324C4×D4—272no—
325#117—271no—
326C2×C4⋊C4—271no—
327C33#55311yescharacteristic, complemented, Sylow
328C23×C3—38no—
329C23×C3—91no—
330C23×C3—91no—
331C22×D3—34no—
332C22×D3—34no—
333C22×D3C22×C3⋊D312yes—
334C3×D4—68no—
335C3×D4—182no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—38no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—128no—
343C2×C3×C4—182no—
344C2×C3×C4—182no—
345C2×C3×C4C2×D3214yes—
346C2×C3×C4C2×D3214yes—
347C2×C3×C4C2×D3214yes—
348C2×C3×C4C22×C3⋊D311yescharacteristic
349C2×C3×C4C22×C3⋊D311yescharacteristic
350C2×C3×C4C22×C3⋊D311yescharacteristic
351#67—184no—
352#67—1816no—
353#67—362no—
354#67—368no—
355C2×Dic3—34no—
356C2×Dic3—34no—
357C2×Dic3—98no—
358C2×Dic3—98no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—188no—
362C2×Dic3—188no—
363C2×Dic3—362no—
364C2×Dic3C22×C3⋊D312yes—
365D12—22no—
366D12—68no—
367C4×D3—42no—
368C4×D3—68no—
369C4×D3—68no—
370C4×D3—368no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—24no—
374C2×C32—122no—
375C2×C32C2×#23514yes—
376C2×C32C2×#23511yescharacteristic
377C2×C32C2×#23711yescharacteristic
378C2×C32#100414yes—
379C2×C32D3×#4114yes—
380C2×C32#100911yescharacteristic
381C3⋊D3—22no—
382C3⋊D3—41no—
383C3×D3—48no—
384C3×D3—68no—
385C3×D3—124no—
386C2×D4—31no—
387C2×D4—91no—
388C2×D4—272no—
389C2×D4—272no—
390C22×C4—32no—
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—272no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C4⋊C4—271no—
398C4⋊C4—271no—
399C4⋊C4—272no—
400C4⋊C4—272no—
401C22⋊C4—32no—
402C22⋊C4—91no—
403C22⋊C4—91no—
404C22⋊C4—272no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C42—272no—
408C42D3×C3⋊D311yescharacteristic
409C22×C3—24no—
410C22×C3—68no—
411C22×C3—68no—
412C22×C3—68no—
413C22×C3—92no—
414C22×C3—181no—
415C22×C3—181no—
416C22×C3—181no—
417C22×C3—181no—
418C22×C3—181no—
419C22×C3C22×D3214yes—
420C22×C3C23×C3⋊D311yescharacteristic
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—38no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—64no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—28no—
435C3×C4—44no—
436C3×C4—44no—
437C3×C4—68no—
438C3×C4—181no—
439C3×C4—181no—
440C3×C4#379518yes—
441C3×C4#382812yes—
442Dic3—22no—
443Dic3—64no—
444Dic3—64no—
445Dic3—188no—
446Dic3—188no—
447Dic3—182no—
448Dic3—182no—
449Dic3—368no—
450Dic3—368no—
451C32—24no—
452C32#555814yescomplemented
453C32#556511yescharacteristic, complemented
454C23—32no—
455C23—91no—
456C23—91no—
457D4—62no—
458D4—182no—
459D4—544no—
460D4—1082no—
461C2×C4—32no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—122no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4—272no—
468C2×C4—272no—
469C2×C4—1082no—
470C2×C4C2×D3×C3⋊D311yescharacteristic
471C2×C4C2×D3×C3⋊D311yescharacteristic
472C2×C4C2×D3×C3⋊D311yescharacteristic
473C2×C3—24no—
474C2×C3—24no—
475C2×C3—24no—
476C2×C3—128no—
477C2×C3—182no—
478C2×C3—361no—
479C2×C3C2×#379514yes—
480C2×C3D3×#23514yes—
481C2×C3#6430714yes—
482C2×C3C2×#382811yescharacteristic
483C2×C3#6436111yescharacteristic
484C2×C3#41×C3⋊D311yescharacteristic
485D3—42no—
486D3—68no—
487D3—124no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—92no—
492C22—181no—
493C22—181no—
494C22—181no—
495C22—181no—
496C22—181no—
497C22C22×D3×C3⋊D311yescharacteristic
498C4—21no—
499C4—21no—
500C4—62no—
501C4—181no—
502C4—181no—
503C4—542no—
504C4—542no—
505C4#8902112yes—
506C3—24no—
507C3#1059392314yescomplemented
508C3#1059602911yescharacteristic, complemented
509C2—122no—
510C2—182no—
511C2—361no—
512C2C2×#8902111yescharacteristic
513C2C3⋊D3×#23511yescharacteristic
514C2#1172457911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#553
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×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×D12C2semidirect
4. #11723522C2semidirect
5. #11723522C2semidirect
6. #11723519C2semidirect
7. #11723513C2semidirect
8. #11723506C2semidirect
9. C3⋊D3×C4⋊Dic3C2semidirect
10. #11723427C2semidirect
11. #11723404C2semidirect
46. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C4×D12D3semidirect
169. C4×D12C3⋊D3semidirect
327. C33#553semidirect
452. C32#5558semidirect
453. C32#5565semidirect
507. C3#10593923semidirect
508. C3#10596029semidirect

Maximal subgroups

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

subgroupindex
#117239112
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
#117235222
#117235192
#117235132
#117235062
C3⋊D3×C4⋊Dic32
#117234272
#117234042
#105960293
#105939233