Finite Group Databasebeta

← #13150979

Subgroups — #13150979

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721836C211yesmaximal, characteristic, complemented
4C9×#936C211yesmaximal, characteristic, complemented
5#11721269C214yesmaximal, complemented
6#11721253C214yesmaximal, complemented
7#11720943C211yesmaximal, characteristic, complemented
8#11720932C211yesmaximal, characteristic, complemented
9#11720930C211yesmaximal, characteristic, complemented
10#10595147—31nomaximal
11#10589777—31nomaximal
12C2×C3×D4×C9C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C3×#611C2212yes—
15C22×C3×Dic9C2212yes—
16C3×#3699—24no—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×Dic3×D9C2212yes—
21C2×#6189C2212yes—
22D9×Dic6—24no—
23#88551C2214yes—
24C4×C3⋊Dic9C2211yescharacteristic
25C9×#217C2214yes—
26C4×C9×Dic3C2211yescharacteristic
27(C2×D9)⋊Dic3C2212yes—
28Dic3⋊Dic9C2214yes—
29#88442C2214yes—
30Dic9⋊Dic3C2212yes—
31C2×C3×#237—31no—
32C2×D3×Dic6—31no—
33#64145—31no—
34C3×#936D311yescharacteristic
35#63969—34no—
36#63964—34no—
37#63873—31no—
38#63866—31no—
39#63855—31no—
40C2×Q8×D9—31no—
41C2×#3699D311yescharacteristic
42C9×#124—31no—
43#63506—31no—
44#63498—31no—
45#63498—61no—
46#63445—34no—
47#63440—34no—
48#63434—31no—
49#63431—31no—
50C23×C3×C9C2312yes—
51C22×C3×D9C2311yescharacteristic
52C3×D4×C9—22no—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2312yes—
55C2×C3⋊Dic9C2312yes—
56#6226—22no—
57C3×#611—24no—
58C2×C9×Dic3C2312yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9—24no—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2311yescharacteristic
63C3×C4×D9D414yes—
64C9×Dic6—22no—
65C3×Dic18—22no—
66Dic3×D9—24no—
67#6189—24no—
68#5825—91no—
69#5675—91no—
70C2×C32×D4C2×D311yescharacteristic
71C2×#634—31no—
72C2×C3×#67—32no—
73C22×C3×Dic3—32no—
74C3×#237—64no—
75C2×C3×C4×D3—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#627—32no—
79C2×D3×Dic3—32no—
80D3×Dic6—64no—
81#3757—34no—
82C4×C3⋊Dic3—31no—
83C3×#217C2×D314yes—
84C3×C4×Dic3C2×D311yescharacteristic
85#3724—34no—
86Dic3⋊Dic3—32no—
87Dic3⋊Dic3—34no—
88(C2×D3)⋊Dic3—32no—
89C2×Q8×C9—31no—
90C2×D4×C9—21no—
91C2×D4×C9C2×D311yescharacteristic
92C2×#611C2×D312yes—
93C22×Dic9C2×D312yes—
94Q8×D9—64no—
95#3699—24no—
96C2×C4×D9—32no—
97C2×C4×D9C2×D311yescharacteristic
98C2×Dic18—31no—
99C2×Dic18—32no—
100C2×Dic18—61no—
101C2×Dic18C2×D311yescharacteristic
102C9×C22⋊C4—34no—
103C42×C9—31no—
104#3676—34no—
105#3676—64no—
106(C2×D9)⋊C4—32no—
107C4⋊Dic9—34no—
108Dic9⋊C4—34no—
109Dic9⋊C4—32no—
110C4×Dic9—31no—
111C4×Dic9—61no—
112C22×C3×C9—22no—
113C22×C3×C9—24no—
114C22×C3×C9C2411yescharacteristic
115C2×C3×D9—22no—
116C2×C3×D9C2×D412yes—
117C3×C4×C9C2×D412yes—
118C3⋊Dic9—22no—
119C3⋊Dic9—22no—
120C9×Dic3—22no—
121C9×Dic3—22no—
122C3×Dic9—22no—
123C3×Dic9C2×D412yes—
124C2×C3×#41—91no—
125C2×D3×Q8—91no—
126C2×#237—31no—
127C22×Dic6—91no—
128C3×#124—31no—
129#944—91no—
130#944—91no—
131#944—91no—
132#936—91no—
133#936—91no—
134#936—181no—
135#936D911yescharacteristic, complemented
136#935—94no—
137#926—91no—
138#924—94no—
139#883—94no—
140#878—94no—
141#872—91no—
142#869—91no—
143C23×C32C22×D312yes—
144C22×C3×D3—31no—
145C32×D4—22no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—32no—
149#634—62no—
150C3×#67—64no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3C22×D312yes—
155C2×C3×Dic3C22×D312yes—
156C3×C4×D3—34no—
157C3×Dic6—22no—
158C3×Dic6—62no—
159#627—64no—
160D3×Dic3—64no—
161C23×C9—22no—
162C23×C9C22×D312yes—
163C22×D9C22×D311yescharacteristic
164Q8×C9—62no—
165D4×C9—22no—
166D4×C9—42no—
167C2×C4×C9—21no—
168C2×C4×C9—32no—
169C2×C4×C9—32no—
170C2×C4×C9C22×D311yescharacteristic
171#611—24no—
172C2×Dic9—24no—
173C2×Dic9—32no—
174C2×Dic9—32no—
175C2×Dic9—62no—
176C2×Dic9—62no—
177C2×Dic9C22×D312yes—
178C2×Dic9C22×D311yescharacteristic
179C4×D9—64no—
180C4×D9#6714yes—
181Dic18—22no—
182Dic18—62no—
183Dic18—64no—
184Dic18—122no—
185#536—271noSylow
186C2×C3×C9—42no—
187C2×C3×C9C22×D411yescharacteristic
188C2×C3×C92−1+412yes—
189C3×D9—22no—
190C3×#41—184no—
191C2×C3×Q8—31no—
192C2×C3×Q8—91no—
193C2×C3×D4—21no—
194C2×C3×D4—92no—
195C2×C3×D4C2×D911yescharacteristic
196C2×C3×D4D3211yescharacteristic
197C22×C3×C4—91no—
198C22×C3×C4—92no—
199C2×#67—32no—
200C22×Dic3—32no—
201C22×Dic3—92no—
202D3×Q8—184no—
203#237—64no—
204C2×C4×D3—31no—
205C2×C4×D3—92no—
206C2×Dic6—31no—
207C2×Dic6—92no—
208C2×Dic6—91no—
209C2×Dic6—91no—
210C2×Dic6—92no—
211C2×Dic6—181no—
212C2×Dic6—184no—
213C2×Dic6C2×D911yescharacteristic
214C3×C22⋊C4—34no—
215C3×C42—31no—
216#217—94no—
217#217—94no—
218#217—92no—
219#217—184no—
220#217C2×D914yes—
221(C2×D3)⋊C4—92no—
222C4⋊Dic3—94no—
223C4⋊Dic3—92no—
224Dic3⋊C4—92no—
225Dic3⋊C4—94no—
226Dic3⋊C4—94no—
227Dic3⋊C4—94no—
228C4×Dic3—91no—
229C4×Dic3—91no—
230C4×Dic3—181no—
231C4×Dic3C2×D911yescharacteristic
232C22×C32—22no—
233C22×C32—24no—
234C22×C32C23×D311yescharacteristic
235C2×C3×D3—32no—
236C2×C3×D3—62no—
237C32×C4D3×D412yes—
238C3⋊Dic3—62no—
239C3⋊Dic3—62no—
240C3×Dic3—22no—
241C3×Dic3—22no—
242C3×Dic3—32no—
243C3×Dic3—62no—
244C22×C9—24no—
245C22×C9—22no—
246C22×C9—21no—
247C22×C9—42no—
248C22×C9—44no—
249C22×C9C23×D311yescharacteristic
250C2×D9—22no—
251C2×D9C2×#6712yes—
252C4×C9—22no—
253C4×C9—62no—
254C4×C9—62no—
255C4×C9C2×#6712yes—
256Dic9—22no—
257Dic9—62no—
258Dic9—62no—
259Dic9—122no—
260Dic9—122no—
261Dic9C2×#6712yes—
262C2×#41—91no—
263C22×Q8—271no—
264C4⋊Q8—271no—
265C4⋊Q8—271no—
266#124—31no—
267#124—271no—
268#123—274no—
269C22⋊Q8—274no—
270#117—271no—
271C3×C9#53611yescharacteristic, complemented, Sylow
272C23×C3—22no—
273C23×C3—91no—
274C23×C3C22×D912yes—
275C23×C3C2×D3212yes—
276C22×D3—31no—
277C3×Q8—62no—
278C3×Q8—182no—
279C3×D4—22no—
280C3×D4—22no—
281C3×D4—42no—
282C3×D4—184no—
283C2×C3×C4—21no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—91no—
287C2×C3×C4—94no—
288C2×C3×C4—92no—
289C2×C3×C4—184no—
290C2×C3×C4C22×D911yescharacteristic
291C2×C3×C4C2×D3211yescharacteristic
292#67—64no—
293C2×Dic3—31no—
294C2×Dic3—32no—
295C2×Dic3—64no—
296C2×Dic3—92no—
297C2×Dic3—92no—
298C2×Dic3—92no—
299C2×Dic3—92no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—184no—
303C2×Dic3C22×D912yes—
304C2×Dic3C22×D912yes—
305C4×D3—34no—
306C4×D3—184no—
307Dic6—22no—
308Dic6—62no—
309Dic6—184no—
310Dic6—184no—
311Dic6—182no—
312Dic6—182no—
313Dic6—362no—
314C2×C32—42no—
315C2×C32C2×D3×D411yescharacteristic
316C2×C32#101012yes—
317C3×D3—62no—
318C2×C9—22no—
319C2×C9—21no—
320C2×C9—42no—
321C2×C9—82no—
322C2×C9#101012yes—
323C2×C9C22×#6711yescharacteristic
324D9—22no—
325#41—184no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—272no—
329C2×Q8—271no—
330C2×Q8—544no—
331C2×D4—92no—
332C2×D4D3×D911yescharacteristic
333C22×C4—91no—
334C22×C4—92no—
335C22×C4—272no—
336C4⋊C4—272no—
337C4⋊C4—274no—
338C4⋊C4—274no—
339C22⋊C4—34no—
340C22⋊C4—272no—
341C22⋊C4—274no—
342C42—31no—
343C42—271no—
344C22×C3—24no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—24no—
348C22×C3—21no—
349C22×C3—42no—
350C22×C3—44no—
351C22×C3—92no—
352C22×C3—182no—
353C22×C3C23×D911yescharacteristic
354C22×C3C22×D3211yescharacteristic
355C2×D3—32no—
356C2×D3—62no—
357C3×C4—22no—
358C3×C4—62no—
359C3×C4—62no—
360C3×C4—92no—
361C3×C4—182no—
362C3×C4D4×D912yes—
363C3×C4D3×#6712yes—
364Dic3—22no—
365Dic3—22no—
366Dic3—32no—
367Dic3—62no—
368Dic3—182no—
369Dic3—182no—
370Dic3—182no—
371Dic3—182no—
372Dic3—362no—
373Dic3—362no—
374C32#567511yescharacteristic
375C9—21no—
376C9#582511yescharacteristic, complemented
377C23—91no—
378C23C2×D3×D912yes—
379Q8—62no—
380Q8—182no—
381Q8—544no—
382Q8—542no—
383D4—22no—
384D4—184no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—91no—
388C2×C4—94no—
389C2×C4—92no—
390C2×C4—184no—
391C2×C4—272no—
392C2×C4—272no—
393C2×C4—544no—
394C2×C4C2×D3×D911yescharacteristic
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—42no—
398C2×C3—42no—
399C2×C3—82no—
400C2×C3—182no—
401C2×C3C2×D4×D911yescharacteristic
402C2×C3#6371612yes—
403C2×C3#6430912yes—
404C2×C3C2×D3×#6711yescharacteristic
405D3—62no—
406C22—22no—
407C22—24no—
408C22—92no—
409C22—182no—
410C22C22×D3×D911yescharacteristic
411C4—62no—
412C4—62no—
413C4—92no—
414C4—182no—
415C4—542no—
416C4—542no—
417C4D9×#6712yes—
418C3—21no—
419C3#1058977711yescharacteristic, complemented
420C3#1059514711yescharacteristic
421C2—42no—
422C2—182no—
423C2#1172249412yes—
424C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#3699C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×D9×Dic6C2semidirect
3. #11721836C2semidirect
4. C9×#936C2semidirect
5. #11721269C2semidirect
6. #11721253C2semidirect
7. #11720943C2semidirect
8. #11720932C2semidirect
9. #11720930C2semidirect
135. #936D9semidirect
271. C3×C9#536semidirect
376. C9#5825semidirect
419. C3#10589777semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×D9×Dic62
#117218362
C9×#9362
#117212692
#117212532
#117209432
#117209322
#117209302
#105951473
#105897773