Finite Group Databasebeta

← #13177199

Subgroups — #13177199

Order 1728.

1732 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723967C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4#11723631C212yesmaximal, complemented
5#11723614C212yesmaximal, complemented
6#11723522C211yesmaximal, characteristic, complemented
7#11723508C212yesmaximal, complemented
8C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
9D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
10#11723447C211yesmaximal, characteristic, complemented
11#11723432C212yesmaximal, complemented
12#10596085—31nomaximal
13#10594945—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×D12C2211yescharacteristic
19C2×#6290C2212yes—
20C2×Dic3×C3⋊D3C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22#88864C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2212yes—
26C3×#3757C2211yescharacteristic
27C3×#3752C2211yescharacteristic
28C3×#3750C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C32×(C2×D3)⋊C4C2212yes—
31C32×C4⋊Dic3C2211yescharacteristic
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#121—31no—
38#64147—32no—
39#64147—31no—
40#64144—32no—
41D4×C3⋊Dic3—31no—
42C4×#638D311yescharacteristic, complemented
43#64135—31no—
44#64135—34no—
45#64135—64no—
46C4⋊C4×C3⋊D3—31no—
47#64099—31no—
48#64090—32no—
49#64088—32no—
50C3×#930D314yescomplemented
51C3×C4×#67—34no—
52#63970—38no—
53#63969—38no—
54#63908—34no—
55#63902—38no—
56Dic3×D12—34no—
57D3×C4⋊Dic3—34no—
58#63869—34no—
59#63862—38no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C3×#638—22no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4×C3⋊D3—22no—
73C32×#67—42no—
74C2×C32×Dic3C2312yes—
75C32×D12—22no—
76#6290—24no—
77Dic3×C3⋊D3—42no—
78D3×C3⋊Dic3—42no—
79#5614—94no—
80#5565—91no—
81C2×C32×D4—32no—
82C2×C32×D4—31no—
83C22×C32×C4—24no—
84C22×C32×C4C2×D314yes—
85C22×C32×C4C2×D311yescharacteristic
86C2×#638—32no—
87C2×#638C2×D311yescharacteristic
88C22×C3⋊Dic3—32no—
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3C2×D311yescharacteristic
91C2×C3×#67—34no—
92C2×C3×#67C2×D318yes—
93C2×C3×D12C2×D314yes—
94C2×C3×C4×D3—34no—
95C2×#625—38no—
96C2×D3×Dic3—38no—
97C2×D3×Dic3—38no—
98C32×C4⋊C4—31no—
99C32×C22⋊C4—32no—
100#3757—32no—
101#3757—38no—
102#3757—32no—
103#3757—32no—
104#3757—68no—
105#3757C2×D311yescharacteristic
106#3752C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—34no—
111C4⋊(C3⋊Dic3)—32no—
112C4⋊(C3⋊Dic3)—64no—
113C4⋊(C3⋊Dic3)—64no—
114#3750—32no—
115#3750—32no—
116#3750C2×D311yescharacteristic
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3—38no—
119C4×C3⋊Dic3—68no—
120C4×C3⋊Dic3C2×D311yescharacteristic
121C3×#217—34no—
122C3×(C2×D3)⋊C4—34no—
123C3×(C2×D3)⋊C4C2×D318yes—
124C3×C4⋊Dic3C2×D314yes—
125C3×Dic3⋊C4—34no—
126C3×C4×Dic3—34no—
127#3724—38no—
128Dic3⋊Dic3—38no—
129Dic3⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131(C2×D3)⋊Dic3—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33—21no—
135C22×C33C2411yescharacteristic
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#4112yes—
144C3⋊(C3⋊Dic3)—22no—
145C3⋊(C3⋊Dic3)—22no—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C32×Dic3—22no—
150C3×#121—34no—
151C3×C4×D4—91no—
152#939—92no—
153#938—94no—
154#938—98no—
155#935—98no—
156D4×Dic3—94no—
157#930—91no—
158#930C3⋊D311yescharacteristic, complemented
159C4×#67—34no—
160#926—94no—
161#926—91no—
162#926—184no—
163C2×C4⋊Dic3—91no—
164#898—92no—
165D3×C4⋊C4—94no—
166#890—94no—
167#883—92no—
168#67⋊C4—98no—
169#67⋊C4—92no—
170#879—98no—
171C4×D12—91no—
172#870—91no—
173C23×C32—24no—
174C23×C32—32no—
175C23×C32C22×D314yes—
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—34no—
179C22×C3×D3C22×D318yes—
180C32×D4—62no—
181C32×D4—122no—
182C2×C32×C4—216no—
183C2×C32×C4—28no—
184C2×C32×C4—22no—
185C2×C32×C4—24no—
186C2×C32×C4—24no—
187C2×C32×C4—32no—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192#638—22no—
193#638—64no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—32no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—38no—
198C2×C3⋊Dic3—68no—
199C2×C3⋊Dic3—68no—
200C2×C3⋊Dic3—122no—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D311yescharacteristic
203C2×C3⋊Dic3C22×D311yescharacteristic
204C4×C3⋊D3—22no—
205C4×C3⋊D3—122no—
206C3×#67—48no—
207C3×#67—68no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3C22×D318yes—
212C3×D12—28no—
213C3×C4×D3—68no—
214#625—616no—
215D3×Dic3—128no—
216D3×Dic3—128no—
217#553—271noSylow
218C2×C33—41no—
219C2×C33C2×#4111yescharacteristic
220C2×C33C2×#4111yescharacteristic
221C2×C33H(5,𝔽2)11yescharacteristic
222C3×C3⋊D3—22no—
223C32×D3—42no—
224C2×C3×D4—38no—
225C2×C3×D4—34no—
226C2×C3×D4—91no—
227C22×C3×C4—24no—
228C22×C3×C4—91no—
229C22×C3×C4D3214yes—
230C22×C3×C4C2×C3⋊D311yescharacteristic
231C2×#67—34no—
232C2×#67—98no—
233C2×#67—92no—
234C2×#67C2×C3⋊D312yes—
235C22×Dic3—98no—
236C22×Dic3—92no—
237C2×D12C2×C3⋊D311yescharacteristic
238C2×C4×D3—34no—
239C2×C4×D3—98no—
240C2×C4×D3—92no—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—91no—
243C3×C22⋊C4—38no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—91no—
246C3×C42—91no—
247#217—34no—
248#217—98no—
249#217—98no—
250#217—98no—
251#217—92no—
252#217—188no—
253(C2×D3)⋊C4—34no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4C2×C3⋊D312yes—
257C4⋊Dic3—98no—
258C4⋊Dic3—94no—
259C4⋊Dic3—94no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—92no—
263C4⋊Dic3—184no—
264C4⋊Dic3—184no—
265C4⋊Dic3C2×C3⋊D311yescharacteristic
266Dic3⋊C4—34no—
267Dic3⋊C4—98no—
268Dic3⋊C4—98no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271C4×Dic3—34no—
272C4×Dic3—98no—
273C4×Dic3—92no—
274C4×Dic3—188no—
275C22×C32—28no—
276C22×C32—21no—
277C22×C32—24no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—24no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—44no—
284C22×C32—44no—
285C22×C32—62no—
286C22×C32—62no—
287C22×C32—62no—
288C22×C32C23×D314yes—
289C22×C32C23×D311yescharacteristic
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3#23512yes—
293C2×C3×D3—28no—
294C2×C3×D3—28no—
295C2×C3×D3—28no—
296C2×C3×D3—38no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C32×C4—28no—
300C32×C4—24no—
301C32×C4—21no—
302C32×C4—28no—
303C32×C4—62no—
304C32×C4#23718yes—
305C32×C4#23712yes—
306C3⋊Dic3—21no—
307C3⋊Dic3—21no—
308C3⋊Dic3—21no—
309C3⋊Dic3—62no—
310C3⋊Dic3—62no—
311C3⋊Dic3—68no—
312C3⋊Dic3—68no—
313C3⋊Dic3—128no—
314C3⋊Dic3—128no—
315C3×Dic3—28no—
316C3×Dic3—64no—
317C3×Dic3—64no—
318C3×Dic3—64no—
319#126—272no—
320#125—271no—
321#123—272no—
322#121—31no—
323#121—271no—
324#121—272no—
325C4×D4—91no—
326C4×D4—271no—
327C4×D4—272no—
328#117—271no—
329C2×C4⋊C4—271no—
330C33#55311yescharacteristic, complemented, Sylow
331C23×C3—24no—
332C23×C3—38no—
333C23×C3—91no—
334C23×C3C2×D3214yes—
335C23×C3C22×C3⋊D311yescharacteristic
336C22×D3—34no—
337C22×D3C22×C3⋊D312yes—
338C3×D4—68no—
339C3×D4—128no—
340C3×D4—182no—
341C2×C3×C4—216no—
342C2×C3×C4—28no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—38no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—182no—
351C2×C3×C4C2×D3214yes—
352C2×C3×C4C2×D3214yes—
353C2×C3×C4C22×C3⋊D311yescharacteristic
354C2×C3×C4C22×C3⋊D311yescharacteristic
355#67—42no—
356#67—68no—
357#67—184no—
358#67—1816no—
359C2×Dic3—34no—
360C2×Dic3—34no—
361C2×Dic3—34no—
362C2×Dic3—98no—
363C2×Dic3—98no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—188no—
367C2×Dic3—188no—
368C2×Dic3—362no—
369C2×Dic3—368no—
370C2×Dic3C22×C3⋊D312yes—
371D12—22no—
372C4×D3—68no—
373C4×D3—368no—
374C4×D3—362no—
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—48no—
379C2×C32—41no—
380C2×C32—44no—
381C2×C32—122no—
382C2×C32C2×#23511yescharacteristic
383C2×C32C2×#23714yes—
384C2×C32C2×#23711yescharacteristic
385C2×C32#100414yes—
386C2×C32D3×#4114yes—
387C2×C32#100911yescharacteristic
388C3⋊D3—22no—
389C3×D3—48no—
390C3×D3—68no—
391C2×D4—32no—
392C2×D4—31no—
393C2×D4—91no—
394C2×D4—272no—
395C22×C4—91no—
396C22×C4—272no—
397C22×C4—272no—
398C22×C4D3×C3⋊D311yescharacteristic
399C4⋊C4—31no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C4⋊C4—271no—
405C4⋊C4—271no—
406C22⋊C4—32no—
407C22⋊C4—91no—
408C22⋊C4—91no—
409C22⋊C4—272no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C42—91no—
413C42—272no—
414C22×C3—28no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—24no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—68no—
425C22×C3—68no—
426C22×C3—68no—
427C22×C3—92no—
428C22×C3—181no—
429C22×C3—181no—
430C22×C3C22×D3214yes—
431C22×C3C23×C3⋊D311yescharacteristic
432C2×D3—22no—
433C2×D3—22no—
434C2×D3—22no—
435C2×D3—38no—
436C2×D3—64no—
437C2×D3—64no—
438C3×C4—28no—
439C3×C4—24no—
440C3×C4—21no—
441C3×C4—28no—
442C3×C4—68no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4—181no—
446C3×C4#380518yes—
447C3×C4#383012yes—
448Dic3—22no—
449Dic3—64no—
450Dic3—64no—
451Dic3—64no—
452Dic3—188no—
453Dic3—188no—
454Dic3—182no—
455Dic3—182no—
456Dic3—368no—
457Dic3—368no—
458C32—24no—
459C32#556511yescharacteristic, complemented
460C32#561414yescomplemented
461C23—32no—
462C23—91no—
463C23C2×D3×C3⋊D311yescharacteristic
464D4—62no—
465D4—122no—
466D4—182no—
467D4—544no—
468C2×C4—22no—
469C2×C4—32no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—182no—
474C2×C4—272no—
475C2×C4—272no—
476C2×C4—1082no—
477C2×C4—1082no—
478C2×C4C2×D3×C3⋊D311yescharacteristic
479C2×C4C2×D3×C3⋊D311yescharacteristic
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—48no—
484C2×C3—44no—
485C2×C3—41no—
486C2×C3—128no—
487C2×C3—182no—
488C2×C3D3×#23514yes—
489C2×C3#6430714yes—
490C2×C3C2×#380514yes—
491C2×C3C2×#383011yescharacteristic
492C2×C3#6436111yescharacteristic
493C2×C3#41×C3⋊D311yescharacteristic
494D3—42no—
495D3—68no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—62no—
500C22—62no—
501C22—62no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C22×D3×C3⋊D311yescharacteristic
506C4—21no—
507C4—62no—
508C4—181no—
509C4—181no—
510C4—181no—
511C4—542no—
512C4—542no—
513C4#8902912yes—
514C3—24no—
515C3#1059494514yescomplemented
516C3#1059608511yescharacteristic, complemented
517C2—41no—
518C2—122no—
519C2—182no—
520C2C3⋊D3×#23511yescharacteristic
521C2#1172457911yescharacteristic
522C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723967C2semidirect
2. C3×C4×#638C2semidirect
3. C32×#930C2semidirect
4. #11723631C2semidirect
5. #11723614C2semidirect
6. #11723522C2semidirect
7. #11723508C2semidirect
8. C3⋊D3×C4⋊Dic3C2semidirect
9. D12×C3⋊Dic3C2semidirect
10. #11723447C2semidirect
11. #11723432C2semidirect
42. C4×#638D3semidirect
50. C3×#930D3semidirect
158. #930C3⋊D3semidirect
330. C33#553semidirect
459. C32#5565semidirect
460. C32#5614semidirect
515. C3#10594945semidirect
516. C3#10596085semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×C4×#6382
C32×#9302
#117236312
#117236142
#117235222
#117235082
C3⋊D3×C4⋊Dic32
D12×C3⋊Dic32
#117234472
#117234322
#105960853
#105949453