Finite Group Databasebeta

← #13177280

Subgroups — #13177280

Order 1728.

1648 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6312C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic, complemented
3C32×#926C211yesmaximal, characteristic, complemented
4#11723596C212yesmaximal, complemented
5#11723592C212yesmaximal
6#11723536C211yesmaximal, characteristic, complemented
7#11723453C212yesmaximal, complemented
8#11723449C211yesmaximal, characteristic, complemented
9#11723413C211yesmaximal, characteristic, complemented
10#11723410C211yesmaximal, characteristic, complemented
11#11723389C212yesmaximal, complemented
12#10596004—31nomaximal
13#10594809—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6312C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×#6293C2212yes—
19#88864C2211yescharacteristic
20#88859C2211yescharacteristic
21#88857C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4×Dic3C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34#88804C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#117—31no—
38C4×#638—31no—
39C4×#638—34no—
40C4×#638—64no—
41#64133D311yescharacteristic, complemented
42#64106—31no—
43#64101—31no—
44#64097—31no—
45#64096—31no—
46#64094—32no—
47#64088—32no—
48#64085—32no—
49C4×#634—32no—
50C3×#926D314yescomplemented
51C3×#924—34no—
52#63956—38no—
53#63953—38no—
54#63919—34no—
55#63871—34no—
56#63868—38no—
57#63849—34no—
58#63843—34no—
59#63837—38no—
60C23×C33C2311yescharacteristic
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65#6312—22no—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)—22no—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C3×#634—22no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75#6293—24no—
76#5641—91no—
77#5533—94no—
78C22×C32×C4—24no—
79C22×C32×C4C2×D314yes—
80C22×C32×C4C2×D311yescharacteristic
81C2×#638—31no—
82C2×#638—34no—
83C2×#638—64no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—64no—
87C2×#634—32no—
88C2×#634C2×D311yescharacteristic
89C2×C3×Dic6—34no—
90C2×#627—38no—
91C32×C4⋊C4—31no—
92C32×C4⋊C4—31no—
93C32×C22⋊C4—32no—
94C32×C42—32no—
95#3757—31no—
96#3757—34no—
97#3757—64no—
98#3757C2×D312yes—
99#3752—31no—
100#3752—34no—
101#3752—32no—
102#3752—32no—
103#3752—64no—
104C4⋊(C3⋊Dic3)—32no—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750—31no—
107#3750—34no—
108#3750—32no—
109#3750—32no—
110#3750—64no—
111#3750C2×D312yes—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3—64no—
117C3×#217—38no—
118C3×#217C2×D318yes—
119C3×C4⋊Dic3—34no—
120C3×C4⋊Dic3C2×D314yes—
121C3×C4⋊Dic3C2×D314yes—
122C3×Dic3⋊C4—38no—
123C3×C4×Dic3C2×D318yes—
124#3724—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128#3722—38no—
129Dic32—38no—
130Dic32—38no—
131C22×C33—21no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33C2411yescharacteristic
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—21no—
141C3⋊(C3⋊Dic3)—21no—
142C3⋊(C3⋊Dic3)#4112yes—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—22no—
145C32×Dic3—22no—
146C32×Dic3—22no—
147C3×C22⋊Q8—91no—
148C3×#117—34no—
149#946—91no—
150Q8×Dic3—91no—
151C4×#67—94no—
152C4×#67—91no—
153C4×#67—184no—
154#926C3⋊D311yescharacteristic, complemented
155#924—34no—
156#898—92no—
157#897—94no—
158#892—94no—
159#892—91no—
160#888—94no—
161#888—91no—
162#887—94no—
163#887—92no—
164#885—98no—
165#885—92no—
166#879—98no—
167#879—92no—
168#876—98no—
169C4×Dic6—98no—
170C23×C32—24no—
171C23×C32C22×D314yes—
172C23×C32C22×D311yescharacteristic
173C22×C3⋊D3—31no—
174C22×C3⋊D3—34no—
175C22×C3⋊D3—64no—
176C2×C32×C4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—28no—
180C2×C32×C4—32no—
181C2×C32×C4—32no—
182C2×C32×C4—48no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187#638—62no—
188#638—68no—
189#638—616no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—34no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C4×C3⋊D3—62no—
202C4×C3⋊D3—68no—
203C4×C3⋊D3—128no—
204#634—22no—
205#634—64no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3C22×D318yes—
209C2×C3×Dic3C22×D318yes—
210C3×Dic6—68no—
211#627—616no—
212#556—271noSylow
213C2×C33—41no—
214C2×C33C2×#4111yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C332−1+411yescharacteristic
217C3⋊(C3⋊D3)—41no—
218C2×C3×Q8—91no—
219C22×C3×C4—24no—
220C22×C3×C4D3214yes—
221C22×C3×C4C2×C3⋊D311yescharacteristic
222C2×#67—94no—
223C2×#67—91no—
224C2×#67—184no—
225C2×C4×D3—94no—
226C2×C4×D3—91no—
227C2×C4×D3—184no—
228C2×Dic6—34no—
229C2×Dic6—98no—
230C2×Dic6—92no—
231C3×C4⋊C4—34no—
232C3×C4⋊C4—34no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—91no—
235C3×C22⋊C4—38no—
236C3×C22⋊C4—92no—
237C3×C42—38no—
238#217—38no—
239#217—94no—
240#217—91no—
241#217—184no—
242#217C2×C3⋊D312yes—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—91no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—184no—
250C4⋊Dic3—34no—
251C4⋊Dic3—98no—
252C4⋊Dic3—92no—
253C4⋊Dic3C2×C3⋊D311yescharacteristic
254C4⋊Dic3C2×C3⋊D311yescharacteristic
255Dic3⋊C4—38no—
256Dic3⋊C4—98no—
257Dic3⋊C4—98no—
258Dic3⋊C4—94no—
259Dic3⋊C4—91no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—184no—
263C4×Dic3—98no—
264C4×Dic3—98no—
265C4×Dic3—94no—
266C4×Dic3—91no—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3—184no—
270C4×Dic3C2×C3⋊D312yes—
271C22×C32—28no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—24no—
275C22×C32—21no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—24no—
279C22×C32—44no—
280C22×C32—44no—
281C22×C32C23×D314yes—
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—68no—
284C2×C3⋊D3—61no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—64no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—64no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—124no—
292C32×C4—21no—
293C32×C4—28no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—62no—
297C32×C4—62no—
298C32×C4#23518yes—
299C32×C4#23712yes—
300C3⋊Dic3—22no—
301C3⋊Dic3—22no—
302C3⋊Dic3—32no—
303C3⋊Dic3—38no—
304C3⋊Dic3—68no—
305C3⋊Dic3—61no—
306C3⋊Dic3—61no—
307C3⋊Dic3—64no—
308C3⋊Dic3—64no—
309C3⋊Dic3—124no—
310C3⋊Dic3—124no—
311C3×Dic3—28no—
312C3×Dic3—28no—
313C3×Dic3—68no—
314C3×Dic3—68no—
315C4⋊Q8—271no—
316#126—272no—
317#126—272no—
318#124—272no—
319C22⋊Q8—91no—
320C22⋊Q8—271no—
321C4×Q8—272no—
322C4×Q8—271no—
323C4×D4—271no—
324#117—31no—
325#117—271no—
326C33#55611yescharacteristic, complemented, Sylow
327C23×C3—24no—
328C23×C3C2×D3214yes—
329C23×C3C22×C3⋊D311yescharacteristic
330C22×D3—94no—
331C22×D3—91no—
332C22×D3—184no—
333C3×Q8—182no—
334C2×C3×C4—28no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—38no—
339C2×C3×C4—38no—
340C2×C3×C4—48no—
341C2×C3×C4—92no—
342C2×C3×C4—92no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C2×D3214yes—
345C2×C3×C4C22×C3⋊D311yescharacteristic
346C2×C3×C4C22×C3⋊D311yescharacteristic
347#67—188no—
348#67—182no—
349#67—1816no—
350C2×Dic3—38no—
351C2×Dic3—38no—
352C2×Dic3—94no—
353C2×Dic3—91no—
354C2×Dic3—94no—
355C2×Dic3—94no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—184no—
359C2×Dic3—184no—
360C2×Dic3—184no—
361C2×Dic3C22×C3⋊D312yes—
362C2×Dic3C22×C3⋊D312yes—
363C4×D3—188no—
364C4×D3—182no—
365C4×D3—368no—
366Dic6—68no—
367Dic6—184no—
368Dic6—1816no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—48no—
373C2×C32—41no—
374C2×C32—44no—
375C2×C32C2×#23514yes—
376C2×C32C2×#23711yescharacteristic
377C2×C32#100711yescharacteristic
378C2×C32D3×#4114yes—
379C2×C32D3×#4111yescharacteristic
380C2×C32#101014yes—
381C3⋊D3—128no—
382C3⋊D3—121no—
383C3⋊D3—124no—
384C2×Q8—91no—
385C2×Q8—272no—
386C2×D4—271no—
387C22×C4—271no—
388C22×C4D3×C3⋊D311yescharacteristic
389C4⋊C4—31no—
390C4⋊C4—31no—
391C4⋊C4—91no—
392C4⋊C4—92no—
393C4⋊C4—272no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—92no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—271no—
402C22⋊C4—271no—
403C42—32no—
404C42—272no—
405C42—272no—
406C42—271no—
407C22×C3—28no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—24no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—24no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3C22×D3214yes—
418C22×C3C23×C3⋊D311yescharacteristic
419C2×D3—188no—
420C2×D3—184no—
421C2×D3—184no—
422C2×D3—184no—
423C2×D3—181no—
424C2×D3—181no—
425C2×D3—181no—
426C2×D3—364no—
427C2×D3—364no—
428C3×C4—24no—
429C3×C4—21no—
430C3×C4—28no—
431C3×C4—44no—
432C3×C4—68no—
433C3×C4—68no—
434C3×C4—182no—
435C3×C4—182no—
436C3×C4#380418yes—
437C3×C4#382812yes—
438Dic3—22no—
439Dic3—22no—
440Dic3—68no—
441Dic3—68no—
442Dic3—98no—
443Dic3—92no—
444Dic3—188no—
445Dic3—184no—
446Dic3—184no—
447Dic3—181no—
448Dic3—181no—
449Dic3—364no—
450Dic3—364no—
451C32—24no—
452C32#553314yescomplemented
453C32#564111yescharacteristic, complemented
454C23—271no—
455C23C2×D3×C3⋊D311yescharacteristic
456Q8—182no—
457Q8—544no—
458D4—542no—
459C2×C4—22no—
460C2×C4—32no—
461C2×C4—32no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—271no—
465C2×C4—271no—
466C2×C4—271no—
467C2×C4—542no—
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C4C2×D3×C3⋊D311yescharacteristic
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—24no—
473C2×C3—48no—
474C2×C3—41no—
475C2×C3—44no—
476C2×C3#6429714yes—
477C2×C3#6430614yes—
478C2×C3C2×#380414yes—
479C2×C3C2×#382811yescharacteristic
480C2×C3#41×C3⋊D311yescharacteristic
481C2×C3#6436711yescharacteristic
482D3—368no—
483D3—361no—
484D3—364no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—541no—
489C22—541no—
490C22—541no—
491C22C22×D3×C3⋊D311yescharacteristic
492C4—21no—
493C4—62no—
494C4—62no—
495C4—182no—
496C4—182no—
497C4—272no—
498C4—541no—
499C4—541no—
500C4#8903112yes—
501C3—24no—
502C3#1059480914yescomplemented
503C3#1059600411yescharacteristic, complemented
504C2—41no—
505C2—1081no—
506C2#1172456211yescharacteristic
507C2#1172457811yescharacteristic
508C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×#6312C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64133C2semidirect
3. C32×#926C2semidirect
4. #11723596C2semidirect
6. #11723536C2semidirect
7. #11723453C2semidirect
8. #11723449C2semidirect
9. #11723413C2semidirect
10. #11723410C2semidirect
11. #11723389C2semidirect
41. #64133D3semidirect
50. C3×#926D3semidirect
154. #926C3⋊D3semidirect
326. C33#556semidirect
452. C32#5533semidirect
453. C32#5641semidirect
502. C3#10594809semidirect
503. C3#10596004semidirect

Maximal subgroups

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

subgroupindex
C4×#63122
C3×#641332
C32×#9262
#117235962
#117235922
#117235362
#117234532
#117234492
#117234132
#117234102
#117233892
#105960043
#105948093