Finite Group Databasebeta

← C2×#11721247 #13151960

Subgroups — C2×#11721247 #13151960

Order 1728.

870 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88551C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
5C2×#88442C211yesmaximal, characteristic, complemented
6C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7#11721247C218yesmaximal, direct product factor
8C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
9C2×#63956—31nomaximal
10C2×#63441—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18#88551C2214yes—
19C9×#217C2214yes—
20C3×#3676C2214yes—
21Dic3⋊Dic9C2214yescomplemented
22#88442C2214yescomplemented
23Dic9⋊Dic3C2214yescomplemented
24Dic3×Dic9C2214yescomplemented
25C2×#3757—31no—
26C2×C3×#217—31no—
27C2×C3×#217D311yescharacteristic
28C2×#3724—31no—
29C2×Dic3⋊Dic3—31no—
30C2×Dic3⋊Dic3—31no—
31#63956—38no—
32C2×Dic32—31no—
33C2×C9×C22⋊C4—31no—
34C2×#3676—31no—
35C2×#3676—61no—
36C2×#3676D311yescharacteristic
37C2×C4⋊Dic9—31no—
38C2×Dic9⋊C4—31no—
39C2×Dic9⋊C4—31no—
40C2×C4×Dic9—31no—
41#63441—38no—
42C23×C3×C9—22no—
43C23×C3×C9—22no—
44C23×C3×C9—22no—
45C23×C3×C9C2312yes—
46C23×C3×C9C2311yescharacteristic
47C2×C3⋊Dic9—22no—
48C2×C3⋊Dic9—22no—
49C2×C3⋊Dic9C2312yes—
50C2×C3⋊Dic9C2312yes—
51C2×C9×Dic3—22no—
52C2×C9×Dic3—22no—
53C2×C9×Dic3C2312yes—
54C2×C9×Dic3C2312yes—
55C2×C3×Dic9—22no—
56C2×C3×Dic9—22no—
57C2×C3×Dic9C2312yes—
58C2×C3×Dic9C2312yes—
59C2×#879—91no—
60C2×#879—91no—
61C24×C32C2×D311yescharacteristic
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—31no—
64C22×C3×Dic3—31no—
65C22×C3×Dic3—31no—
66C22×C3×Dic3C2×D311yescharacteristic
67C22×C3×Dic3C2×D311yescharacteristic
68C24×C9—21no—
69C24×C9C2×D311yescharacteristic
70#3757—34no—
71C3×#217—34no—
72C3×#217C2×D314yes—
73#3724—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—34no—
76Dic32—34no—
77C22×C4×C9—31no—
78C22×C4×C9—31no—
79C22×Dic9—31no—
80C22×Dic9—31no—
81C22×Dic9—61no—
82C22×Dic9—61no—
83C22×Dic9C2×D311yescharacteristic
84C22×Dic9C2×D311yescharacteristic
85C9×C22⋊C4—34no—
86#3676—34no—
87#3676—64no—
88#3676C2×D314yes—
89C4⋊Dic9—34no—
90Dic9⋊C4—34no—
91Dic9⋊C4—34no—
92C4×Dic9—34no—
93C22×C3×C9—22no—
94C22×C3×C9—22no—
95C22×C3×C9—22no—
96C22×C3×C9—44no—
97C22×C3×C9#4112yes—
98C22×C3×C9#4112yes—
99C22×C3×C9#4112yes—
100C22×C3×C9C2411yescharacteristic
101C3⋊Dic9—22no—
102C3⋊Dic9—22no—
103C9×Dic3—22no—
104C9×Dic3—22no—
105C3×Dic9—22no—
106C3×Dic9—22no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—91no—
109C2×#217—31no—
110C2×#217—91no—
111C2×#217—91no—
112C2×#217—181no—
113C2×#217D911yescharacteristic
114C2×C4⋊Dic3—91no—
115C2×C4⋊Dic3—91no—
116C2×Dic3⋊C4—91no—
117C2×Dic3⋊C4—91no—
118C2×Dic3⋊C4—91no—
119C2×Dic3⋊C4—91no—
120C2×C4×Dic3—91no—
121C2×C4×Dic3—91no—
122#879—98no—
123#879—98no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32C22×D312yes—
128C23×C32C22×D311yescharacteristic
129C2×C3⋊Dic3—32no—
130C2×C3⋊Dic3—32no—
131C2×C3⋊Dic3—62no—
132C2×C3⋊Dic3—62no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3C22×D312yes—
140C2×C3×Dic3C22×D312yes—
141C23×C9—24no—
142C23×C9—22no—
143C23×C9—22no—
144C23×C9—22no—
145C23×C9—21no—
146C23×C9—22no—
147C23×C9—42no—
148C23×C9—42no—
149C23×C9C22×D312yes—
150C23×C9C22×D311yescharacteristic
151C2×C4×C9—32no—
152C2×C4×C9—32no—
153C2×C4×C9—62no—
154C2×C4×C9—62no—
155C2×Dic9—22no—
156C2×Dic9—22no—
157C2×Dic9—32no—
158C2×Dic9—32no—
159C2×Dic9—62no—
160C2×Dic9—62no—
161C2×Dic9—62no—
162C2×Dic9—62no—
163C2×Dic9—122no—
164C2×Dic9—122no—
165C2×Dic9C22×D312yes—
166C2×Dic9C22×D312yes—
167C2×#126—271noSylow
168C2×C3×C9—42no—
169C2×C3×C9#12614yescomplemented
170C2×C3×C9C2×#4111yescharacteristic
171C2×C3×C9C2×#4111yescharacteristic
172C2×C3×C9C2×#4111yescharacteristic
173C24×C3—21no—
174C24×C3C2×D911yescharacteristic
175C24×C3D3211yescharacteristic
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—91no—
179C22×C3×C4—91no—
180C22×Dic3—31no—
181C22×Dic3—31no—
182C22×Dic3—91no—
183C22×Dic3—91no—
184C22×Dic3—91no—
185C22×Dic3—91no—
186C22×Dic3—181no—
187C22×Dic3—181no—
188C22×Dic3C2×D911yescharacteristic
189C22×Dic3C2×D911yescharacteristic
190C3×C22⋊C4—34no—
191C3×C22⋊C4—94no—
192#217—34no—
193#217—94no—
194#217—94no—
195#217—184no—
196#217C2×D914yes—
197C4⋊Dic3—94no—
198C4⋊Dic3—94no—
199Dic3⋊C4—94no—
200Dic3⋊C4—94no—
201Dic3⋊C4—94no—
202Dic3⋊C4—94no—
203C4×Dic3—94no—
204C4×Dic3—94no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—44no—
209C22×C32#23512yes—
210C22×C32#23712yes—
211C22×C32#23712yes—
212C22×C32C23×D311yescharacteristic
213C3⋊Dic3—62no—
214C3⋊Dic3—62no—
215C3×Dic3—22no—
216C3×Dic3—22no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C22×C9—24no—
220C22×C9—22no—
221C22×C9—22no—
222C22×C9—22no—
223C22×C9—21no—
224C22×C9—22no—
225C22×C9—22no—
226C22×C9—22no—
227C22×C9—42no—
228C22×C9—42no—
229C22×C9—48no—
230C22×C9—44no—
231C22×C9#23512yes—
232C22×C9#23712yes—
233C22×C9#23712yes—
234C22×C9C23×D311yescharacteristic
235C4×C9—62no—
236C4×C9—62no—
237Dic9—22no—
238Dic9—22no—
239Dic9—62no—
240Dic9—62no—
241Dic9—122no—
242Dic9—122no—
243#126—278no—
244C2×C4⋊C4—271no—
245C2×C4⋊C4—271no—
246C2×C4⋊C4—271no—
247C2×C22⋊C4—31no—
248C2×C22⋊C4—91no—
249C2×C22⋊C4—271no—
250C2×C42—271no—
251C3×C9C2×#12611yescharacteristic, complemented, Sylow
252C23×C3—24no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—21no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3C22×D912yes—
264C23×C3C22×D911yescharacteristic
265C23×C3C2×D3212yes—
266C23×C3C2×D3211yescharacteristic
267C2×C3×C4—32no—
268C2×C3×C4—32no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—32no—
278C2×Dic3—32no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—92no—
282C2×Dic3—92no—
283C2×Dic3—92no—
284C2×Dic3—92no—
285C2×Dic3—182no—
286C2×Dic3—182no—
287C2×Dic3—182no—
288C2×Dic3—182no—
289C2×Dic3—182no—
290C2×Dic3—182no—
291C2×Dic3—362no—
292C2×Dic3—362no—
293C2×Dic3C22×D912yes—
294C2×Dic3C22×D912yes—
295C2×C32—42no—
296C2×C32#87914yes—
297C2×C32C2×#23511yescharacteristic
298C2×C32C2×#23711yescharacteristic
299C2×C32C2×#23711yescharacteristic
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—21no—
303C2×C9—24no—
304C2×C9—44no—
305C2×C9—42no—
306C2×C9#87914yescomplemented
307C2×C9C2×#23511yescharacteristic
308C2×C9C2×#23711yescharacteristic
309C2×C9C2×#23711yescharacteristic
310C24D3×D911yescharacteristic
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—271no—
316C22×C4—271no—
317C4⋊C4—274no—
318C4⋊C4—274no—
319C4⋊C4—274no—
320C22⋊C4—34no—
321C22⋊C4—94no—
322C22⋊C4—274no—
323C42—274no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—48no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3#369712yes—
341C22×C3#369912yes—
342C22×C3#369912yes—
343C22×C3C23×D911yescharacteristic
344C22×C3#380412yes—
345C22×C3#380412yes—
346C22×C3#380512yes—
347C22×C3C22×D3211yescharacteristic
348C3×C4—62no—
349C3×C4—62no—
350C3×C4—182no—
351C3×C4—182no—
352Dic3—22no—
353Dic3—22no—
354Dic3—62no—
355Dic3—62no—
356Dic3—182no—
357Dic3—182no—
358Dic3—182no—
359Dic3—182no—
360Dic3—362no—
361Dic3—362no—
362C32C2×#87911yescharacteristic
363C9—21no—
364C9C2×#87911yescharacteristic, complemented
365C23—22no—
366C23—22no—
367C23—22no—
368C23C2×D3×D912yes—
369C23C2×D3×D911yescharacteristic
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—92no—
375C2×C4—92no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—272no—
379C2×C4—272no—
380C2×C4—542no—
381C2×C4—542no—
382C2×C3—21no—
383C2×C3—24no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—44no—
387C2×C3—42no—
388C2×C3—42no—
389C2×C3#6344114yescomplemented
390C2×C3C2×#369711yescharacteristic
391C2×C3C2×#369911yescharacteristic
392C2×C3C2×#369911yescharacteristic
393C2×C3#6395614yes—
394C2×C3C2×#380411yescharacteristic
395C2×C3C2×#380411yescharacteristic
396C2×C3C2×#380511yescharacteristic
397C22—22no—
398C22—22no—
399C22—22no—
400C22—44no—
401C22#8865812yes—
402C22#8866212yes—
403C22#8866312yes—
404C22C22×D3×D911yescharacteristic
405C4—62no—
406C4—62no—
407C4—182no—
408C4—182no—
409C4—542no—
410C4—542no—
411C3—21no—
412C3C2×#6344111yescharacteristic, complemented
413C3C2×#6395611yescharacteristic
414C2—42no—
415C2#1172124714yesdirect product factor
416C2C2×#8865811yescharacteristic
417C2C2×#8866211yescharacteristic
418C2C2×#8866311yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
hypercenterC24D3×D9
hypocenterC3×C9C2×#126
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic9C2semidirect
5. C2×#88442C2semidirect
6. C2×Dic9⋊Dic3C2semidirect
7. #11721247C2direct (N × H)
8. C2×Dic3×Dic9C2semidirect
21. Dic3⋊Dic9C22semidirect
22. #88442C22semidirect
23. Dic9⋊Dic3C22semidirect
24. Dic3×Dic9C22semidirect
169. C2×C3×C9#126semidirect
251. C3×C9C2×#126semidirect
306. C2×C9#879semidirect
364. C9C2×#879semidirect
389. C2×C3#63441semidirect
412. C3C2×#63441semidirect
415. C2#11721247direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#885512
C2×C9×#2172
C2×C3×#36762
C2×Dic3⋊Dic92
C2×#884422
C2×Dic9⋊Dic32
#117212472
C2×Dic3×Dic92
C2×#639563
C2×#634413