Finite Group Databasebeta

← #13147552

Subgroups — #13147552

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C3×#63434C211yesmaximal, characteristic
3C9×#869C211yesmaximal, characteristic, complemented
4#11720999C212yesmaximal
5D9×C4⋊Dic3C211yesmaximal, characteristic
6#11720984C212yesmaximal
7#11720969C211yesmaximal, characteristic, complemented
8#11720935C211yesmaximal, characteristic, complemented
9#11720917C212yesmaximal, complemented
10#11720907C211yesmaximal, characteristic
11#11720903C212yesmaximal, complemented
12#10593954—31nomaximal
13#10589831—31nomaximal
14C2×C3×C4×D9C2211yescharacteristic
15C2×C9×Dic6C2212yes—
16C2×Dic3×D9C2212yes—
17C2×#6189C2212yes—
18C3×C42×C9C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22#88534C2212yes—
23C3×(C2×D9)⋊C4C2211yescharacteristic
24C3×(C2×D9)⋊C4C2211yescharacteristic
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C9×C4⋊Dic3C2212yes—
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64079—31no—
38C3×#872—31no—
39C3×#869D311yescharacteristic
40#63918—31no—
41#63899—32no—
42D3×C4⋊Dic3—31no—
43#63887—32no—
44#63869—31no—
45#63846—32no—
46#63841—31no—
47#63839—32no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50#63506—32no—
51#63458—32no—
52#63454—31no—
53C4⋊C4×D9—31no—
54#63452—31no—
55#63451—32no—
56#63450—31no—
57#63434D311yescharacteristic
58#63432—31no—
59#63432—61no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2312yes—
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C3×C4×D9Q814yescomplemented
72C9×Dic6—42no—
73Dic3×D9—42no—
74#6189—42no—
75#5729—91no—
76#5519—91no—
77C2×C3×C4×D3—31no—
78C2×C3×Dic6C2×D312yes—
79C2×#627—32no—
80C2×D3×Dic3—32no—
81C32×C42C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—31no—
84#3750—32no—
85#3750—32no—
86C3×(C2×D3)⋊C4—31no—
87C3×(C2×D3)⋊C4—31no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—31no—
92C3×Dic3⋊C4—31no—
93C3×C4×Dic3—31no—
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99Dic32—32no—
100C2×Q8×C9—32no—
101C2×C4×D9—32no—
102C2×C4×D9C2×D311yescharacteristic
103C2×Dic18—32no—
104C9×C4⋊C4—31no—
105C9×C4⋊C4—32no—
106C9×C4⋊C4—31no—
107C42×C9—21no—
108C42×C9C2×D311yescharacteristic
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4C2×D311yescharacteristic
111(C2×D9)⋊C4C2×D311yescharacteristic
112C4⋊Dic9—31no—
113C4⋊Dic9—31no—
114C4⋊Dic9—32no—
115C4⋊Dic9—61no—
116C4⋊Dic9—61no—
117Dic9⋊C4—32no—
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123Dic9⋊C4—62no—
124Dic9⋊C4C2×D311yescharacteristic
125Dic9⋊C4C2×D311yescharacteristic
126C4×Dic9—32no—
127C4×Dic9C2×D311yescharacteristic
128C22×C3×C9C2411yescharacteristic
129C2×C3×D9—21no—
130C2×C3×D9—21no—
131C2×C3×D9C2×Q812yes—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—22no—
136C3⋊Dic9—22no—
137C9×Dic3—22no—
138C9×Dic3—22no—
139C3×Dic9—21no—
140C3×Dic9—21no—
141C3×Dic9C2×Q812yes—
142C3×C4⋊Q8—31no—
143C3×#117—91no—
144#946—92no—
145#944—92no—
146#926—91no—
147C2×C4⋊Dic3—91no—
148#896—92no—
149#892—91no—
150D3×C4⋊C4—91no—
151#890—91no—
152#890—92no—
153#889—92no—
154#888—91no—
155#888—92no—
156#878—92no—
157#878—92no—
158#872—31no—
159#870—91no—
160#870—91no—
161#870—91no—
162#870—181no—
163#869—91no—
164#869D911yescharacteristic, complemented
165C22×C3×D3—31no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D312yes—
176C3×C4×D3—34no—
177C3×Dic6—42no—
178#627—122no—
179D3×Dic3—122no—
180C22×D9C22×D311yescharacteristic
181Q8×C9—122no—
182C2×C4×C9—21no—
183C2×C4×C9—21no—
184C2×C4×C9—21no—
185C2×C4×C9—32no—
186C2×C4×C9—32no—
187C2×C4×C9C22×D311yescharacteristic
188C2×C4×C9C22×D311yescharacteristic
189C2×C4×C9C22×D311yescharacteristic
190C2×Dic9—32no—
191C2×Dic9—32no—
192C2×Dic9—62no—
193C2×Dic9—62no—
194C2×Dic9C22×D311yescharacteristic
195C2×Dic9C22×D311yescharacteristic
196C2×Dic9C22×D311yescharacteristic
197C4×D9—122no—
198C4×D9Dic614yescomplemented
199Dic18—122no—
200#544—271noSylow
201C2×C3×C9C22×Q811yescharacteristic
202C2×C3×C9H(5,𝔽2)11yescharacteristic
203C2×C3×C92−1+411yescharacteristic
204C3×D9—22no—
205C2×C3×Q8—32no—
206C22×C3×C4—91no—
207C22×Dic3—92no—
208C2×C4×D3—31no—
209C2×C4×D3—92no—
210C2×Dic6—92no—
211C2×Dic6—92no—
212C2×Dic6C2×D912yes—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—91no—
218C3×C22⋊C4—91no—
219C3×C22⋊C4—91no—
220C3×C42—21no—
221C3×C42—91no—
222C3×C42C2×D911yescharacteristic
223C3×C42D3211yescharacteristic
224#217—92no—
225(C2×D3)⋊C4—31no—
226(C2×D3)⋊C4—31no—
227(C2×D3)⋊C4—92no—
228C4⋊Dic3—92no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—92no—
234C4⋊Dic3—92no—
235C4⋊Dic3—181no—
236C4⋊Dic3—181no—
237C4⋊Dic3C2×D911yescharacteristic
238C4⋊Dic3C2×D911yescharacteristic
239C4⋊Dic3C2×D912yes—
240Dic3⋊C4—31no—
241Dic3⋊C4—31no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—182no—
252Dic3⋊C4—182no—
253C4×Dic3—31no—
254C4×Dic3—92no—
255C4×Dic3—92no—
256C22×C32C23×D311yescharacteristic
257C2×C3×D3—32no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4D3×Q812yes—
263C3⋊Dic3—62no—
264C3⋊Dic3—62no—
265C3×Dic3—22no—
266C3×Dic3—22no—
267C3×Dic3—32no—
268C3×Dic3—61no—
269C3×Dic3—61no—
270C22×C9—21no—
271C22×C9C23×D311yescharacteristic
272C2×D9—21no—
273C2×D9—21no—
274C2×D9C2×Dic612yes—
275C4×C9—21no—
276C4×C9—21no—
277C4×C9—22no—
278C4×C9—41no—
279C4×C9—41no—
280C4×C9—62no—
281C4×C9—62no—
282C4×C9C2×Dic612yes—
283Dic9—21no—
284Dic9—21no—
285Dic9—62no—
286Dic9—62no—
287Dic9—122no—
288Dic9—122no—
289Dic9C2×Dic612yes—
290C4⋊Q8—31no—
291C4⋊Q8—271no—
292C4⋊Q8—272no—
293#125—271no—
294#125—272no—
295#125—271no—
296C22⋊Q8—272no—
297C22⋊Q8—272no—
298#117—91no—
299#117—271no—
300C2×C4⋊C4—271no—
301C3×C9#54411yescharacteristic, complemented, Sylow
302C23×C3—91no—
303C22×D3—31no—
304C3×Q8—122no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—91no—
311C2×C3×C4—94no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4C22×D911yescharacteristic
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×C3×C4C2×D3211yescharacteristic
319C2×C3×C4C2×D3211yescharacteristic
320C2×Dic3—31no—
321C2×Dic3—31no—
322C2×Dic3—31no—
323C2×Dic3—92no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326C2×Dic3—92no—
327C2×Dic3—182no—
328C2×Dic3—182no—
329C2×Dic3—362no—
330C2×Dic3C22×D912yes—
331C2×Dic3C22×D912yes—
332C4×D3—34no—
333C4×D3—362no—
334Dic6—42no—
335Dic6—362no—
336Dic6—362no—
337C2×C32#100411yescharacteristic
338C2×C32C2×D3×Q811yescharacteristic
339C2×C32#100711yescharacteristic
340C3×D3—62no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9C22×Dic611yescharacteristic
345C2×C9#100911yescharacteristic
346C2×C9#101011yescharacteristic
347D9—22no—
348C2×Q8—32no—
349C2×Q8—272no—
350C22×C4—91no—
351C22×C4—272no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—32no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—271no—
362C4⋊C4—271no—
363C4⋊C4—272no—
364C4⋊C4—272no—
365C22⋊C4—91no—
366C22⋊C4—91no—
367C22⋊C4—272no—
368C42—91no—
369C42—272no—
370C42D3×D911yescharacteristic
371C22×C3—21no—
372C22×C3—92no—
373C22×C3—181no—
374C22×C3—181no—
375C22×C3C23×D911yescharacteristic
376C22×C3C22×D3211yescharacteristic
377C2×D3—32no—
378C2×D3—61no—
379C2×D3—61no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—41no—
386C3×C4—41no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4—92no—
390C3×C4—181no—
391C3×C4—181no—
392C3×C4Q8×D912yes—
393C3×C4D3×Dic612yes—
394Dic3—22no—
395Dic3—22no—
396Dic3—32no—
397Dic3—61no—
398Dic3—61no—
399Dic3—182no—
400Dic3—182no—
401Dic3—182no—
402Dic3—182no—
403Dic3—362no—
404Dic3—362no—
405C32#572911yescharacteristic
406C9—21no—
407C9#551911yescharacteristic, complemented
408C23—91no—
409Q8—122no—
410Q8—1082no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—91no—
414C2×C4—94no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—272no—
418C2×C4—272no—
419C2×C4—1082no—
420C2×C4C2×D3×D911yescharacteristic
421C2×C4C2×D3×D911yescharacteristic
422C2×C4C2×D3×D911yescharacteristic
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—182no—
427C2×C3#6371011yescharacteristic
428C2×C3C2×Q8×D911yescharacteristic
429C2×C3#6371311yescharacteristic
430C2×C3C2×D3×Dic611yescharacteristic
431C2×C3#6429711yescharacteristic
432C2×C3#6430711yescharacteristic
433D3—62no—
434C22—92no—
435C22—181no—
436C22—181no—
437C22C22×D3×D911yescharacteristic
438C4—21no—
439C4—21no—
440C4—62no—
441C4—62no—
442C4—92no—
443C4—181no—
444C4—181no—
445C4—542no—
446C4—542no—
447C4D9×Dic612yes—
448C3—21no—
449C3#1058983111yescharacteristic, complemented
450C3#1059395411yescharacteristic
451C2—182no—
452C2C2×D9×Dic611yescharacteristic
453C2#1172245811yescharacteristic
454C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC42D3×D9
hypocenterC3×C9#544
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
3. C9×#869C2semidirect
7. #11720969C2semidirect
8. #11720935C2semidirect
9. #11720917C2semidirect
11. #11720903C2semidirect
71. C3×C4×D9Q8semidirect
164. #869D9semidirect
198. C4×D9Dic6semidirect
301. C3×C9#544semidirect
407. C9#5519semidirect
449. C3#10589831semidirect

Maximal subgroups

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

subgroupindex
#117217072
C3×#634342
C9×#8692
#117209992
D9×C4⋊Dic32
#117209842
#117209692
#117209352
#117209172
#117209072
#117209032
#105939543
#105898313