Finite Group Databasebeta

← #13177254

Subgroups — #13177254

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊(C3⋊Dic3))C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic, complemented
3C32×#930C211yesmaximal, characteristic, complemented
4#11723637C212yesmaximal, complemented
5#67⋊(C3⋊Dic3)C212yesmaximal, complemented
6#11723522C211yesmaximal, characteristic, complemented
7D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#11723480C212yesmaximal, complemented
9#11723447C211yesmaximal, characteristic, complemented
10#11723430C212yesmaximal, complemented
11#11723404C211yesmaximal, characteristic, complemented
12#10596089—31nomaximal
13#10594995—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×#67C2212yes—
19C2×C32×D12C2211yescharacteristic
20C2×#6290C2212yes—
21C2×D3×C3⋊Dic3C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2212yes—
25C3×#3757C2211yescharacteristic
26C3×#3752C2211yescharacteristic
27C3×#3750C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4⋊Dic3C2211yescharacteristic
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#121—31no—
38#64148—32no—
39#64147—31no—
40D4×C3⋊Dic3—32no—
41D4×C3⋊Dic3—31no—
42C4×#638D311yescharacteristic, complemented
43C2×C4⋊(C3⋊Dic3)—31no—
44C2×C4⋊(C3⋊Dic3)—34no—
45C2×C4⋊(C3⋊Dic3)—64no—
46#64101—31no—
47#64099—31no—
48#64095—32no—
49#64088—32no—
50C3×#930D314yescomplemented
51C3×C4×#67—34no—
52#63980—38no—
53#67⋊Dic3—38no—
54#63908—34no—
55Dic3×D12—34no—
56#63891—38no—
57#63869—34no—
58#63861—38no—
59#63841—34no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C32×D3C2312yes—
63C2×C33×C4—22no—
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)—42no—
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C3×#638—22no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4×C3⋊D3—22no—
74C32×#67—24no—
75C2×C32×Dic3C2312yes—
76C32×D12—22no—
77#6290—42no—
78D3×C3⋊Dic3—42no—
79#5618—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—
89C22×C3⋊Dic3—38no—
90C22×C3⋊Dic3—32no—
91C22×C3⋊Dic3—68no—
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×#67—34no—
94C2×C3×#67C2×D318yes—
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3—34no—
97C2×#625—38no—
98C2×D3×Dic3—38no—
99C32×C4⋊C4—31no—
100C32×C22⋊C4—32no—
101#3757—32no—
102#3757—32no—
103#3757C2×D311yescharacteristic
104#3752—32no—
105#3752C2×D311yescharacteristic
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—38no—
111C4⋊(C3⋊Dic3)—34no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)—68no—
114C4⋊(C3⋊Dic3)—64no—
115C4⋊(C3⋊Dic3)—64no—
116#3750—32no—
117#3750C2×D311yescharacteristic
118C4×C3⋊Dic3—32no—
119C4×C3⋊Dic3C2×D311yescharacteristic
120C3×#217—34no—
121C3×(C2×D3)⋊C4—34no—
122C3×(C2×D3)⋊C4C2×D318yes—
123C3×C4⋊Dic3C2×D314yes—
124C3×Dic3⋊C4—34no—
125C3×C4×Dic3—34no—
126#3724—38no—
127Dic3⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33#4112yes—
135C22×C33C2411yescharacteristic
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
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—98no—
153#939—92no—
154#938—94no—
155D4×Dic3—94no—
156D4×Dic3—98no—
157#930—91no—
158#930C3⋊D311yescharacteristic, complemented
159C4×#67—34no—
160#926—91no—
161C2×C4⋊Dic3—94no—
162C2×C4⋊Dic3—91no—
163C2×C4⋊Dic3—184no—
164#898—92no—
165#892—94no—
166#890—94no—
167#886—98no—
168#883—92no—
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—64no—
181C32×D4—62no—
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—122no—
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⋊Dic3—122no—
202C2×C3⋊Dic3—128no—
203C2×C3⋊Dic3—1216no—
204C2×C3⋊Dic3C22×D311yescharacteristic
205C2×C3⋊Dic3C22×D311yescharacteristic
206C2×C3⋊Dic3C22×D311yescharacteristic
207C4×C3⋊D3—22no—
208C3×#67—216no—
209C3×#67—68no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3C22×D318yes—
214C3×D12—28no—
215C3×C4×D3—68no—
216#625—128no—
217D3×Dic3—128no—
218#553—271noSylow
219C2×C33—22no—
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—41no—
224C32×D3—42no—
225C2×C3×D4—38no—
226C2×C3×D4—34no—
227C2×C3×D4—91no—
228C22×C3×C4—24no—
229C22×C3×C4—91no—
230C22×C3×C4D3214yes—
231C22×C3×C4C2×C3⋊D311yescharacteristic
232C2×#67—34no—
233C2×#67—98no—
234C2×#67—92no—
235C2×#67C2×C3⋊D312yes—
236C22×Dic3—98no—
237C22×Dic3—98no—
238C22×Dic3—92no—
239C22×Dic3—188no—
240C2×D12C2×C3⋊D311yescharacteristic
241C2×C4×D3—34no—
242C2×C4×D3—92no—
243C3×C4⋊C4—34no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—38no—
246C3×C22⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C42—91no—
249#217—34no—
250#217—98no—
251#217—98no—
252#217—92no—
253(C2×D3)⋊C4—34no—
254(C2×D3)⋊C4—98no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4C2×C3⋊D312yes—
258C4⋊Dic3—98no—
259C4⋊Dic3—98no—
260C4⋊Dic3—94no—
261C4⋊Dic3—94no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—92no—
265C4⋊Dic3—184no—
266C4⋊Dic3—184no—
267C4⋊Dic3—188no—
268C4⋊Dic3C2×C3⋊D311yescharacteristic
269Dic3⋊C4—34no—
270Dic3⋊C4—98no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273C4×Dic3—34no—
274C4×Dic3—98no—
275C4×Dic3—92no—
276C22×C32—28no—
277C22×C32—28no—
278C22×C32—24no—
279C22×C32—21no—
280C22×C32—24no—
281C22×C32—21no—
282C22×C32—24no—
283C22×C32—28no—
284C22×C32—62no—
285C22×C32—62no—
286C22×C32—62no—
287C22×C32#23512yes—
288C22×C32#23718yes—
289C22×C32C23×D314yes—
290C22×C32C23×D311yescharacteristic
291C2×C3⋊D3—21no—
292C2×C3⋊D3—21no—
293C2×C3⋊D3—21no—
294C2×C3×D3—28no—
295C2×C3×D3—28no—
296C2×C3×D3—28no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—64no—
300C32×C4—28no—
301C32×C4—24no—
302C32×C4—21no—
303C32×C4—28no—
304C32×C4—62no—
305C32×C4#23718yes—
306C32×C4#23712yes—
307C3⋊Dic3—21no—
308C3⋊Dic3—21no—
309C3⋊Dic3—21no—
310C3⋊Dic3—62no—
311C3⋊Dic3—62no—
312C3⋊Dic3—68no—
313C3⋊Dic3—68no—
314C3⋊Dic3—128no—
315C3⋊Dic3—128no—
316C3×Dic3—28no—
317C3×Dic3—64no—
318C3×Dic3—64no—
319C3×Dic3—64no—
320#126—272no—
321#125—271no—
322#123—272no—
323#121—31no—
324#121—271no—
325#121—272no—
326C4×D4—91no—
327C4×D4—271no—
328C4×D4—272no—
329#117—271no—
330C2×C4⋊C4—271no—
331C33#55311yescharacteristic, complemented, Sylow
332C23×C3—24no—
333C23×C3—38no—
334C23×C3—91no—
335C23×C3C2×D3214yes—
336C23×C3C22×C3⋊D311yescharacteristic
337C22×D3—34no—
338C22×D3C22×C3⋊D312yes—
339C3×D4—616no—
340C3×D4—68no—
341C3×D4—182no—
342C2×C3×C4—216no—
343C2×C3×C4—28no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—38no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—182no—
352C2×C3×C4C2×D3214yes—
353C2×C3×C4C2×D3214yes—
354C2×C3×C4C22×C3⋊D311yescharacteristic
355C2×C3×C4C22×C3⋊D311yescharacteristic
356#67—24no—
357#67—68no—
358#67—362no—
359#67—368no—
360C2×Dic3—34no—
361C2×Dic3—34no—
362C2×Dic3—34no—
363C2×Dic3—98no—
364C2×Dic3—98no—
365C2×Dic3—92no—
366C2×Dic3—92no—
367C2×Dic3—188no—
368C2×Dic3—188no—
369C2×Dic3—368no—
370C2×Dic3—368no—
371C2×Dic3—362no—
372C2×Dic3—3616no—
373C2×Dic3C22×C3⋊D312yes—
374D12—22no—
375C4×D3—68no—
376C4×D3—362no—
377C2×C32—216no—
378C2×C32—22no—
379C2×C32—28no—
380C2×C32—24no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—122no—
384C2×C32C2×#23511yescharacteristic
385C2×C32C2×#23714yes—
386C2×C32C2×#23714yes—
387C2×C32C2×#23711yescharacteristic
388C2×C32#100914yes—
389C2×C32#100911yescharacteristic
390C3⋊D3—41no—
391C3×D3—48no—
392C3×D3—124no—
393C2×D4—32no—
394C2×D4—31no—
395C2×D4—91no—
396C2×D4—272no—
397C22×C4—91no—
398C22×C4—272no—
399C22×C4—272no—
400C22×C4D3×C3⋊D311yescharacteristic
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C4⋊C4—271no—
406C4⋊C4—271no—
407C4⋊C4—272no—
408C22⋊C4—32no—
409C22⋊C4—91no—
410C22⋊C4—91no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42—91no—
415C42—272no—
416C22×C3—28no—
417C22×C3—28no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—68no—
425C22×C3—68no—
426C22×C3—68no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3—181no—
430C22×C3#379518yes—
431C22×C3#383012yes—
432C22×C3C22×D3214yes—
433C22×C3C23×C3⋊D311yescharacteristic
434C2×D3—22no—
435C2×D3—22no—
436C2×D3—22no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—64no—
440C3×C4—28no—
441C3×C4—24no—
442C3×C4—21no—
443C3×C4—28no—
444C3×C4—68no—
445C3×C4—181no—
446C3×C4—181no—
447C3×C4—181no—
448C3×C4#380518yes—
449C3×C4#383012yes—
450Dic3—22no—
451Dic3—64no—
452Dic3—64no—
453Dic3—64no—
454Dic3—188no—
455Dic3—188no—
456Dic3—182no—
457Dic3—182no—
458Dic3—368no—
459Dic3—368no—
460C32—24no—
461C32#556511yescharacteristic, complemented
462C32#561814yescomplemented
463C23—32no—
464C23—91no—
465C23C2×D3×C3⋊D311yescharacteristic
466D4—64no—
467D4—62no—
468D4—182no—
469D4—1082no—
470C2×C4—22no—
471C2×C4—32no—
472C2×C4—91no—
473C2×C4—91no—
474C2×C4—91no—
475C2×C4—182no—
476C2×C4—272no—
477C2×C4—272no—
478C2×C4—1082no—
479C2×C4—1082no—
480C2×C4C2×D3×C3⋊D311yescharacteristic
481C2×C4C2×D3×C3⋊D311yescharacteristic
482C2×C3—216no—
483C2×C3—28no—
484C2×C3—22no—
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—128no—
489C2×C3—361no—
490C2×C3C2×#379514yes—
491C2×C3#6430814yes—
492C2×C3C2×#380514yes—
493C2×C3C2×#383011yescharacteristic
494C2×C3C2×#383011yescharacteristic
495C2×C3#6436611yescharacteristic
496D3—42no—
497D3—124no—
498C22—21no—
499C22—21no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—181no—
504C22—181no—
505C22—181no—
506C22#8901412yes—
507C22C22×D3×C3⋊D311yescharacteristic
508C4—21no—
509C4—62no—
510C4—181no—
511C4—181no—
512C4—181no—
513C4—542no—
514C4—542no—
515C4#8902912yes—
516C3—24no—
517C3#1059499514yescomplemented
518C3#1059608911yescharacteristic, complemented
519C2—22no—
520C2—122no—
521C2—361no—
522C2C2×#8901411yescharacteristic
523C2#1172458111yescharacteristic
524C2C2×#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. C2×C4⋊(C3⋊(C3⋊Dic3))C2semidirect
2. C3×C4×#638C2semidirect
3. C32×#930C2semidirect
4. #11723637C2semidirect
5. #67⋊(C3⋊Dic3)C2semidirect
6. #11723522C2semidirect
7. D12×C3⋊Dic3C2semidirect
8. #11723480C2semidirect
9. #11723447C2semidirect
10. #11723430C2semidirect
11. #11723404C2semidirect
42. C4×#638D3semidirect
50. C3×#930D3semidirect
158. #930C3⋊D3semidirect
331. C33#553semidirect
461. C32#5565semidirect
462. C32#5618semidirect
517. C3#10594995semidirect
518. C3#10596089semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊(C3⋊Dic3))2
C3×C4×#6382
C32×#9302
#117236372
#67⋊(C3⋊Dic3)2
#117235222
D12×C3⋊Dic32
#117234802
#117234472
#117234302
#117234042
#105960893
#105949953