Finite Group Databasebeta

← #13147457

Subgroups — #13147457

Order 1728.

646 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C9×#872C211yesmaximal, characteristic
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11720966C212yesmaximal
5#11720963C212yesmaximal
6#11720961C211yesmaximal, characteristic
7#11720960C211yesmaximal, characteristic
8#11720933C212yesmaximal, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#11720928C212yesmaximal, complemented
11#11720927C211yesmaximal, characteristic, complemented
12#10593880—31nomaximal
13#10589620—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C3×C42×C9C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19#88534C2212yes—
20C4×C3⋊Dic9C2212yes—
21C9×(C2×D3)⋊C4C2211yescharacteristic
22C9×(C2×D3)⋊C4C2211yescharacteristic
23C9×Dic3⋊C4C2211yescharacteristic
24C9×Dic3⋊C4C2211yescharacteristic
25C4×C9×Dic3C2211yescharacteristic
26C3×C4⋊Dic9C2211yescharacteristic
27C3×Dic9⋊C4C2212yes—
28C3×C4×Dic9C2212yes—
29(C2×D3)⋊Dic9C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×#872D311yescharacteristic
39C3×C4×Dic6—31no—
40#63870—32no—
41#63869—31no—
42#63867—32no—
43#63866—31no—
44#63862—32no—
45#63859—32no—
46#63857—31no—
47#63856—31no—
48C9×#117—31no—
49#63486—31no—
50#63486—31no—
51#63451—32no—
52#63451—32no—
53#63441—32no—
54#63441—32no—
55#63432—31no—
56#63431—31no—
57C4×Dic18—31no—
58C4×Dic18—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×D3×C9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C4×D3×C9—22no—
73C3×Dic18—22no—
74#5583—91no—
75#5518—91no—
76C2×#634—31no—
77C2×C3×C4×D3C2×D311yescharacteristic
78C2×C3×Dic6—31no—
79C32×C42C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)—31no—
81#3750—32no—
82C4×C3⋊Dic3—32no—
83C3×(C2×D3)⋊C4C2×D311yescharacteristic
84C3×(C2×D3)⋊C4C2×D311yescharacteristic
85C3×C4⋊Dic3—31no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4C2×D311yescharacteristic
88C3×Dic3⋊C4C2×D311yescharacteristic
89C3×C4×Dic3—32no—
90C3×C4×Dic3C2×D311yescharacteristic
91#3724—32no—
92#3724—32no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99C22×C4×C9—31no—
100C2×Dic18—31no—
101C2×Dic18—61no—
102C2×Dic18C2×D311yescharacteristic
103C9×C4⋊C4—31no—
104C9×C4⋊C4—31no—
105C9×C22⋊C4—31no—
106C9×C22⋊C4—31no—
107C42×C9—21no—
108C42×C9—31no—
109C42×C9C2×D311yescharacteristic
110#3676—32no—
111#3676—32no—
112C4⋊Dic9—31no—
113C4⋊Dic9—32no—
114C4⋊Dic9—32no—
115C4⋊Dic9—61no—
116C4⋊Dic9C2×D311yescharacteristic
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123Dic9⋊C4C2×D312yes—
124C4×Dic9—32no—
125C4×Dic9—62no—
126C4×Dic9C2×D312yes—
127C22×C3×C9C2411yescharacteristic
128C2×D3×C9—21no—
129C2×D3×C9—21no—
130C2×D3×C9—21no—
131C3×C4×C9—21no—
132C3×C4×C9—21no—
133C3×C4×C9#4112yes—
134C3⋊Dic9—22no—
135C3⋊Dic9—22no—
136C9×Dic3—21no—
137C9×Dic3—21no—
138C9×Dic3—21no—
139C3×Dic9—22no—
140C3×Dic9—22no—
141C3×C4×Q8—91no—
142C3×#117—31no—
143#946—91no—
144#944—91no—
145#924—91no—
146#924—91no—
147#898—92no—
148#897—91no—
149#890—91no—
150#889—92no—
151#889—92no—
152#889—92no—
153#879—92no—
154#879—92no—
155#876—92no—
156#872D911yescharacteristic
157#870—91no—
158#870—92no—
159#869—91no—
160C4×Dic6—31no—
161C4×Dic6—91no—
162C4×Dic6—91no—
163C4×Dic6—181no—
164C22×C3×D3C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—32no—
170#634—62no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3C22×D311yescharacteristic
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D311yescharacteristic
176C3×C4×D3—22no—
177C3×Dic6—62no—
178C23×C9—31no—
179C2×C4×C9—21no—
180C2×C4×C9—21no—
181C2×C4×C9—21no—
182C2×C4×C9—31no—
183C2×C4×C9—31no—
184C2×C4×C9—31no—
185C2×C4×C9—62no—
186C2×C4×C9C22×D311yescharacteristic
187C2×C4×C9C22×D311yescharacteristic
188C2×C4×C9C22×D311yescharacteristic
189C2×Dic9—32no—
190C2×Dic9—32no—
191C2×Dic9—62no—
192C2×Dic9—62no—
193C2×Dic9C22×D312yes—
194C2×Dic9C22×D312yes—
195Dic18—22no—
196Dic18—64no—
197Dic18—62no—
198#541—271noSylow
199C2×C3×C9C2×#4111yescharacteristic
200C2×C3×C92−1+411yescharacteristic
201C2×C3×C92−1+411yescharacteristic
202D3×C9—41no—
203C2×C3×Q8—91no—
204C22×C3×C4—31no—
205C2×C4×D3C2×D911yescharacteristic
206C2×Dic6—31no—
207C2×Dic6—91no—
208C2×Dic6—91no—
209C2×Dic6—181no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—91no—
214C3×C22⋊C4—31no—
215C3×C22⋊C4—31no—
216C3×C42—21no—
217C3×C42—31no—
218C3×C42—92no—
219C3×C42C2×D911yescharacteristic
220C3×C42D3211yescharacteristic
221#217—92no—
222#217—92no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4C2×D911yescharacteristic
226(C2×D3)⋊C4C2×D911yescharacteristic
227C4⋊Dic3—31no—
228C4⋊Dic3—92no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—181no—
235Dic3⋊C4—32no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—182no—
247Dic3⋊C4C2×D911yescharacteristic
248Dic3⋊C4C2×D911yescharacteristic
249C4×Dic3—32no—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C4×Dic3—182no—
253C4×Dic3C2×D911yescharacteristic
254C22×C32C23×D311yescharacteristic
255C2×C3×D3—21no—
256C2×C3×D3—21no—
257C2×C3×D3—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4#23512yes—
261C3⋊Dic3—62no—
262C3⋊Dic3—62no—
263C3×Dic3—21no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C22×C9—21no—
269C22×C9—61no—
270C22×C9—61no—
271C22×C9—61no—
272C22×C9C23×D311yescharacteristic
273C4×C9—22no—
274C4×C9—21no—
275C4×C9—21no—
276C4×C9—22no—
277C4×C9—41no—
278C4×C9—61no—
279C4×C9—61no—
280C4×C9—61no—
281C4×C9#23512yes—
282Dic9—22no—
283Dic9—22no—
284Dic9—64no—
285Dic9—62no—
286Dic9—62no—
287Dic9—122no—
288C4⋊Q8—271no—
289#126—272no—
290#126—272no—
291#125—271no—
292#125—272no—
293#125—272no—
294C22⋊Q8—271no—
295C22⋊Q8—271no—
296C4×Q8—91no—
297C4×Q8—271no—
298#117—31no—
299C3×C9#54111yescharacteristic, complemented, Sylow
300C23×C3—31no—
301C22×D3C22×D911yescharacteristic
302C3×Q8—182no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—62no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4C22×D911yescharacteristic
313C2×C3×C4C22×D911yescharacteristic
314C2×C3×C4C22×D911yescharacteristic
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—92no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C2×Dic3C22×D911yescharacteristic
327C2×Dic3C22×D911yescharacteristic
328C2×Dic3C22×D911yescharacteristic
329C4×D3—22no—
330Dic6—62no—
331Dic6—184no—
332Dic6—182no—
333Dic6—182no—
334C2×C32C2×#23511yescharacteristic
335C2×C32#101011yescharacteristic
336C2×C32#101011yescharacteristic
337C3×D3—41no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—21no—
341C2×C9—121no—
342C2×C9C2×#23511yescharacteristic
343C2×C9#100711yescharacteristic
344C2×C9#101011yescharacteristic
345C2×Q8—91no—
346C2×Q8—271no—
347C22×C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—91no—
351C4⋊C4—92no—
352C4⋊C4—272no—
353C4⋊C4—272no—
354C4⋊C4—272no—
355C4⋊C4—272no—
356C4⋊C4—272no—
357C4⋊C4—272no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C22⋊C4—31no—
361C22⋊C4—31no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C42—31no—
365C42—92no—
366C42—272no—
367C42D3×D911yescharacteristic
368C22×C3—21no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3C23×D911yescharacteristic
373C22×C3C22×D3211yescharacteristic
374C2×D3—21no—
375C2×D3—21no—
376C2×D3—21no—
377C3×C4—22no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—22no—
383C3×C4—41no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—182no—
388C3×C4—182no—
389C3×C4#369712yes—
390C3×C4#379812yes—
391Dic3—21no—
392Dic3—21no—
393Dic3—21no—
394Dic3—62no—
395Dic3—62no—
396Dic3—184no—
397Dic3—182no—
398Dic3—182no—
399Dic3—182no—
400Dic3—182no—
401Dic3—362no—
402C32#551811yescharacteristic
403C9—21no—
404C9#558311yescharacteristic, complemented
405C23—31no—
406Q8—182no—
407Q8—542no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—62no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—272no—
415C2×C4—272no—
416C2×C4C2×D3×D911yescharacteristic
417C2×C4C2×D3×D911yescharacteristic
418C2×C4C2×D3×D911yescharacteristic
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—121no—
423C2×C3C2×#369711yescharacteristic
424C2×C3#6371611yescharacteristic
425C2×C3#6371611yescharacteristic
426C2×C3#6429711yescharacteristic
427C2×C3#6429811yescharacteristic
428C2×C3C2×#379811yescharacteristic
429D3—41no—
430C22—61no—
431C22—61no—
432C22—61no—
433C22C22×D3×D911yescharacteristic
434C4—21no—
435C4—21no—
436C4—61no—
437C4—61no—
438C4—61no—
439C4—182no—
440C4—182no—
441C4—542no—
442C4—542no—
443C4#8864012yes—
444C3—21no—
445C3#1058962011yescharacteristic, complemented
446C3#1059388011yescharacteristic
447C2—121no—
448C2#1172246411yescharacteristic
449C2#1172246511yescharacteristic
450C2C2×#8864011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3×D3C22×D3
hypercenterC42D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
3. C3×C4×Dic18C2semidirect
8. #11720933C2semidirect
9. #11720932C2semidirect
10. #11720928C2semidirect
11. #11720927C2semidirect
59. C4×Dic18D3semidirect
299. C3×C9#541semidirect
404. C9#5583semidirect
445. C3#10589620semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C9×#8722
C3×C4×Dic182
#117209662
#117209632
#117209612
#117209602
#117209332
#117209322
#117209282
#117209272
#105938803
#105896203