Finite Group Databasebeta

← C2×#11720996 #13150215

Subgroups — C2×#11720996 #13150215

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6226C211yesmaximal, characteristic, complemented
2C22×Dic3×D9C211yesmaximal, characteristic
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
6C2×#88442C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720996C218yesmaximal, direct product factor
9C2×#63896—31nomaximal
10C2×#63458—31nomaximal
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×#6226—24no—
16C2×#6226C2214yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×Dic3×D9—24no—
21C2×Dic3×D9C2214yes—
22C3×(C2×D9)⋊C4C2214yes—
23C9×Dic3⋊C4C2214yescomplemented
24(C2×D9)⋊Dic3C2214yes—
25#88442C2214yescomplemented
26Dic9⋊Dic3C2214yescomplemented
27C22×#634—31no—
28C22×D3×Dic3—31no—
29C2×C3×(C2×D3)⋊C4—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×#3724—31no—
32C2×Dic3⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34#63896—38no—
35C22×C4×D9—31no—
36C22×Dic18—31no—
37C22×Dic18—61no—
38C2×C9×C4⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4D311yescharacteristic
41C2×Dic9⋊C4—31no—
42C2×Dic9⋊C4—31no—
43#63458—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3×D9—22no—
46C22×C3×D9—22no—
47C22×C3×D9Q814yescomplemented
48C22×C3×D9C2312yes—
49C2×C3×C4×C9—22no—
50C2×C3×C4×C9C2312yes—
51C2×C3⋊Dic9—22no—
52C2×C3⋊Dic9D414yes—
53C2×C3⋊Dic9C2312yes—
54C2×C3⋊Dic9C2312yes—
55#6226—28no—
56C2×C9×Dic3—22no—
57C2×C9×Dic3—22no—
58C2×C9×Dic3C2312yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9—22no—
61C2×C3×Dic9C2312yes—
62Dic3×D9—28no—
63C2×#896—91no—
64C2×#878—91no—
65C23×C3×D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3—31no—
69C2×#634—34no—
70C2×#634—64no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3C2×D311yescharacteristic
73C22×C3×Dic3C2×D311yescharacteristic
74C2×D3×Dic3—34no—
75C2×D3×Dic3—64no—
76C23×D9C2×D311yescharacteristic
77C3×(C2×D3)⋊C4—34no—
78C3×Dic3⋊C4C2×D314yes—
79#3724—34no—
80Dic3⋊Dic3—34no—
81(C2×D3)⋊Dic3—34no—
82C22×C4×C9—21no—
83C22×C4×C9—31no—
84C22×C4×C9—31no—
85C22×C4×C9C2×D311yescharacteristic
86C22×Dic9—31no—
87C22×Dic9—31no—
88C22×Dic9—61no—
89C22×Dic9—61no—
90C22×Dic9C2×D311yescharacteristic
91C2×C4×D9—34no—
92C2×C4×D9—64no—
93C2×Dic18—34no—
94C2×Dic18—64no—
95C2×Dic18—64no—
96C2×Dic18—68no—
97C9×C4⋊C4—34no—
98(C2×D9)⋊C4—34no—
99(C2×D9)⋊C4C2×D314yes—
100Dic9⋊C4—34no—
101Dic9⋊C4—34no—
102C22×C3×C9C2×D412yes—
103C22×C3×C9C2×Q812yes—
104C22×C3×C9#4112yes—
105C22×C3×C9C2411yescharacteristic
106C2×C3×D9—28no—
107C2×C3×D9—22no—
108C2×C3×D9—22no—
109C2×C3×D9C2×Q814yescomplemented
110C3×C4×C9—22no—
111C3⋊Dic9—22no—
112C3⋊Dic9C2×D414yes—
113C9×Dic3—22no—
114C9×Dic3—22no—
115C3×Dic9—22no—
116C23×Dic3—91no—
117C22×C4×D3—91no—
118C22×Dic6—91no—
119C22×Dic6—91no—
120C22×Dic6—181no—
121C2×C3×C4⋊C4—31no—
122C2×C3×C22⋊C4—91no—
123C2×#217—91no—
124C2×(C2×D3)⋊C4—31no—
125C2×(C2×D3)⋊C4—91no—
126C2×C4⋊Dic3—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4D911yescharacteristic, complemented
131#896—98no—
132#878—98no—
133C23×C32C22×D311yescharacteristic
134C22×C3×D3—34no—
135C22×C3×D3—32no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C2×C32×C4—22no—
139C2×C32×C4C22×D312yes—
140C2×C3⋊Dic3—34no—
141C2×C3⋊Dic3—32no—
142C2×C3⋊Dic3—32no—
143C2×C3⋊Dic3—62no—
144#634—68no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3C22×D312yes—
150C2×C3×Dic3C22×D312yes—
151D3×Dic3—68no—
152C23×C9—21no—
153C23×C9C22×D311yescharacteristic
154C22×D9—22no—
155C22×D9—22no—
156C22×D9Dic614yescomplemented
157C22×D9C22×D312yes—
158C2×C4×C9—24no—
159C2×C4×C9—22no—
160C2×C4×C9—22no—
161C2×C4×C9—32no—
162C2×C4×C9—32no—
163C2×C4×C9—62no—
164C2×C4×C9—62no—
165C2×C4×C9C22×D312yes—
166C2×Dic9—22no—
167C2×Dic9—34no—
168C2×Dic9—32no—
169C2×Dic9—32no—
170C2×Dic9—64no—
171C2×Dic9—62no—
172C2×Dic9—62no—
173C2×Dic9—64no—
174C2×Dic9—62no—
175C2×Dic9C22×D312yes—
176C4×D9—68no—
177Dic18—68no—
178Dic18—616no—
179C2×C22⋊Q8—271noSylow
180C2×C3×C9C22⋊Q814yescomplemented
181C2×C3×C9C22×D411yescharacteristic
182C2×C3×C9C22×Q811yescharacteristic
183C2×C3×C9C2×#4111yescharacteristic
184C3×D9—24no—
185C24×C3—91no—
186C23×D3—31no—
187C22×C3×C4—21no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—91no—
191C22×C3×C4C2×D911yescharacteristic
192C22×C3×C4D3211yescharacteristic
193C22×Dic3—31no—
194C22×Dic3—91no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—94no—
199C22×Dic3—181no—
200C22×Dic3—181no—
201C22×Dic3—184no—
202C22×Dic3C2×D911yescharacteristic
203C22×Dic3C2×D911yescharacteristic
204C2×C4×D3—94no—
205C2×C4×D3—184no—
206C2×Dic6—94no—
207C2×Dic6—94no—
208C2×Dic6—184no—
209C2×Dic6—184no—
210C2×Dic6—184no—
211C2×Dic6—188no—
212C3×C4⋊C4—34no—
213C3×C22⋊C4—94no—
214#217—94no—
215(C2×D3)⋊C4—34no—
216(C2×D3)⋊C4—94no—
217C4⋊Dic3—94no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220Dic3⋊C4—94no—
221Dic3⋊C4C2×D914yescomplemented
222C22×C32#23512yes—
223C22×C32D3×D412yes—
224C22×C32D3×Q812yes—
225C22×C32C23×D311yescharacteristic
226C2×C3×D3—34no—
227C2×C3×D3—68no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C32×C4—22no—
231C3⋊Dic3—34no—
232C3⋊Dic3—62no—
233C3×Dic3—22no—
234C3×Dic3—22no—
235C3×Dic3—62no—
236C22×C9—21no—
237C22×C9—22no—
238C22×C9—22no—
239C22×C9—22no—
240C22×C9C2×Dic612yes—
241C22×C9D3×D412yes—
242C22×C9#23712yes—
243C22×C9C23×D311yescharacteristic
244C2×D9—28no—
245C2×D9—22no—
246C2×D9—22no—
247C2×D9C2×Dic614yescomplemented
248C4×C9—24no—
249C4×C9—22no—
250C4×C9—62no—
251C4×C9—62no—
252Dic9—22no—
253Dic9—34no—
254Dic9—64no—
255Dic9—64no—
256Dic9—62no—
257C22×Q8—271no—
258C23×C4—271no—
259C22⋊Q8—278no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—271no—
262C2×C4⋊C4—271no—
263C2×C22⋊C4—91no—
264C2×C22⋊C4—271no—
265C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
266C23×C3—21no—
267C23×C3—94no—
268C23×C3—92no—
269C23×C3—182no—
270C23×C3—182no—
271C23×C3C22×D911yescharacteristic
272C23×C3C2×D3211yescharacteristic
273C22×D3—32no—
274C22×D3—34no—
275C22×D3—62no—
276C22×D3—62no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—32no—
282C2×C3×C4—32no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—92no—
286C2×C3×C4—182no—
287C2×C3×C4C22×D912yes—
288C2×C3×C4C2×D3212yes—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—32no—
292C2×Dic3—62no—
293C2×Dic3—92no—
294C2×Dic3—94no—
295C2×Dic3—92no—
296C2×Dic3—94no—
297C2×Dic3—92no—
298C2×Dic3—92no—
299C2×Dic3—182no—
300C2×Dic3—184no—
301C2×Dic3—182no—
302C2×Dic3—188no—
303C2×Dic3—184no—
304C2×Dic3—182no—
305C2×Dic3—182no—
306C2×Dic3C22×D912yes—
307C2×Dic3C22×D912yes—
308C4×D3—188no—
309Dic6—188no—
310Dic6—188no—
311Dic6—1816no—
312C2×C32#89614yes—
313C2×C32C2×#23511yescharacteristic
314C2×C32C2×D3×D411yescharacteristic
315C2×C32C2×D3×Q811yescharacteristic
316C3×D3—64no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—24no—
321C2×C9#87814yescomplemented
322C2×C9C22×Dic611yescharacteristic
323C2×C9C2×D3×D411yescharacteristic
324C2×C9C2×#23711yescharacteristic
325D9—24no—
326C24—91no—
327C2×Q8—274no—
328C2×Q8—544no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—91no—
332C22×C4—274no—
333C22×C4—271no—
334C22×C4—271no—
335C22×C4—544no—
336C22×C4D3×D911yescharacteristic
337C4⋊C4—34no—
338C4⋊C4—274no—
339C4⋊C4—274no—
340C22⋊C4—94no—
341C22⋊C4—274no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—94no—
347C22×C3—182no—
348C22×C3—182no—
349C22×C3—188no—
350C22×C3#369712yes—
351C22×C3D4×D912yes—
352C22×C3Q8×D912yes—
353C22×C3C23×D911yescharacteristic
354C22×C3D3×Dic612yes—
355C22×C3#379512yes—
356C22×C3#381112yes—
357C22×C3C22×D3211yescharacteristic
358C2×D3—34no—
359C2×D3—62no—
360C2×D3—62no—
361C2×D3—68no—
362C3×C4—24no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—182no—
368Dic3—22no—
369Dic3—22no—
370Dic3—62no—
371Dic3—94no—
372Dic3—94no—
373Dic3—184no—
374Dic3—184no—
375Dic3—182no—
376Dic3—182no—
377C32C2×#89611yescharacteristic
378C9—21no—
379C9C2×#87811yescharacteristic, complemented
380C23—94no—
381C23—92no—
382C23—182no—
383C23—182no—
384C23C2×D3×D911yescharacteristic
385Q8—548no—
386C2×C4—22no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—92no—
392C2×C4—182no—
393C2×C4—272no—
394C2×C4—274no—
395C2×C4—272no—
396C2×C4—548no—
397C2×C4—542no—
398C2×C4C2×D3×D912yes—
399C2×C3—24no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—184no—
404C2×C3#6345814yescomplemented
405C2×C3C2×#369711yescharacteristic
406C2×C3C2×D4×D911yescharacteristic
407C2×C3C2×Q8×D911yescharacteristic
408C2×C3#6389614yes—
409C2×C3C2×D3×Dic611yescharacteristic
410C2×C3C2×#379511yescharacteristic
411C2×C3C2×#381111yescharacteristic
412D3—64no—
413C22—94no—
414C22—188no—
415C22—182no—
416C22—182no—
417C22D9×Dic612yes—
418C22#8864112yes—
419C22#8866812yes—
420C22C22×D3×D911yescharacteristic
421C4—22no—
422C4—62no—
423C4—62no—
424C4—182no—
425C4—274no—
426C4—542no—
427C3—21no—
428C3C2×#6345811yescharacteristic, complemented
429C3C2×#6389611yescharacteristic
430C2—184no—
431C2#1172099614yesdirect product factor
432C2C2×D9×Dic611yescharacteristic
433C2C2×#8864111yescharacteristic
434C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6226C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
6. C2×#88442C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720996C2direct (N × H)
16. C2×#6226C22semidirect
23. C9×Dic3⋊C4C22semidirect
25. #88442C22semidirect
26. Dic9⋊Dic3C22semidirect
47. C22×C3×D9Q8semidirect
109. C2×C3×D9C2×Q8semidirect
130. C2×Dic3⋊C4D9semidirect
156. C22×D9Dic6semidirect
180. C2×C3×C9C22⋊Q8semidirect
221. Dic3⋊C4C2×D9semidirect
247. C2×D9C2×Dic6semidirect
265. C3×C9C2×C22⋊Q8semidirect
321. C2×C9#878semidirect
379. C9C2×#878semidirect
404. C2×C3#63458semidirect
428. C3C2×#63458semidirect
431. C2#11720996direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62262
C22×Dic3×D92
C2×C3×(C2×D9)⋊C42
C2×C9×Dic3⋊C42
C2×(C2×D9)⋊Dic32
C2×#884422
C2×Dic9⋊Dic32
#117209962
C2×#638963
C2×#634583