Finite Group Databasebeta

← C2×#11720291 #13139416

Subgroups — C2×#11720291 #13139416

Order 1728.

618 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4#11720291C214yesmaximal, direct product factor
5C2×#63568—31nomaximal
6C2×#63384—31nomaximal
7C22×C3×C4×C9C2211yescharacteristic
8C22×C3⋊Dic9—21no—
9C2×C3×#611—21no—
10C3×#3697—21no—
11C3×#3697—21no—
12C3×#3697C412yescomplemented
13C3×#3697C2212yes—
14C2×C3×D36C411yescharacteristic, complemented
15C2×C3×C4×D9—21no—
16C2×C3×Dic18C411yescharacteristic, complemented
17C4×C3⋊Dic9—21no—
18C4×C3⋊Dic9C2212yescomplemented
19C9×#208—21no—
20C9×#208C2212yescomplemented
21C2×C9×C3⋊C8—21no—
22C2×C3×#235—31no—
23C2×C4×C3⋊Dic3—31no—
24C2×C3×#208D311yescharacteristic
25C2×#3697D311yescharacteristic
26#63568—34no—
27C2×C9×M16—31no—
28C2×C4×Dic9—31no—
29C2×C4×Dic9—61no—
30#63384—34no—
31C23×C3×C9D411yescharacteristic
32C22×C3×D9—21no—
33C2×C3×C4×C9C2×C411yescharacteristic
34C2×C3×C4×C9C2×C411yescharacteristic
35C2×C3×C4×C9D411yescharacteristic
36C2×C3×C4×C9D411yescharacteristic
37C2×C3×C4×C9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—24no—
40C2×C3⋊Dic9—22no—
41C3×#611—22no—
42C3×#611—22no—
43C2×C3×Dic9—21no—
44C3×D36—21no—
45C3×D36C2×C412yescomplemented
46C3×C4×D9—22no—
47C3×C4×D9—22no—
48C3×Dic18—21no—
49C3×Dic18C2×C412yescomplemented
50C9×C3⋊C8—22no—
51C2×#837—91no—
52C2×#825—91no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C3⋊Dic3—61no—
55C2×C3×#67—61no—
56C3×#235—32no—
57C3×#235—32no—
58C3×#235—61no—
59C3×#235—61no—
60C2×C3×D12—31no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—31no—
63C4×C3⋊Dic3—32no—
64C4×C3⋊Dic3—61no—
65C3×#208—21no—
66C3×#208C2×D312yes—
67C2×C3×C3⋊C8—21no—
68C22×C4×C9—21no—
69C22×C4×C9C2×D311yescharacteristic
70C2×#611—21no—
71C22×Dic9—62no—
72C22×Dic9—61no—
73#3697—21no—
74#3697—21no—
75#3697Dic312yescomplemented
76#3697C2×D312yes—
77C2×D36Dic311yescharacteristic, complemented
78C2×C4×D9—21no—
79C2×Dic18Dic311yescharacteristic, complemented
80C9×M16—32no—
81C9×M16—61no—
82C2×C8×C9—61no—
83C4×Dic9—32no—
84C4×Dic9—62no—
85C4×Dic9—62no—
86C4×Dic9—61no—
87C22×C3×C9—22no—
88C22×C3×C9C22⋊C411yescharacteristic
89C22×C3×C9C22⋊C411yescharacteristic
90C22×C3×C9C2×D411yescharacteristic
91C2×C3×D9—22no—
92C2×C3×D9—42no—
93C3×C4×C9C22⋊C411yescharacteristic
94C3×C4×C9C22⋊C411yescharacteristic
95C3×C4×C9C22×C411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9—24no—
98C3×Dic9—22no—
99C2×C3×#41—91no—
100C2×#235—31no—
101C2×C3×M16—31no—
102C2×C4×Dic3—91no—
103C2×C4×Dic3—91no—
104C2×C4×Dic3—181no—
105C2×#208D911yescharacteristic, complemented
106#837—94no—
107#825—94no—
108C23×C32D1211yescharacteristic
109C22×C3×D3—61no—
110C2×C32×C4C4×D311yescharacteristic
111C2×C32×C4C4×D311yescharacteristic
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—62no—
118C3×#67—62no—
119C3×#67—62no—
120C2×C3×Dic3—61no—
121C3×D12—32no—
122C3×D12—61no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×Dic6—32no—
126C3×Dic6—61no—
127C23×C9—21no—
128C23×C9#6711yescharacteristic
129C22×D9—21no—
130C3×C3⋊C8—22no—
131C2×C4×C9—21no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9C2×Dic311yescharacteristic
138C2×C4×C9C2×Dic311yescharacteristic
139C2×C4×C9#6711yescharacteristic
140C2×C4×C9#6711yescharacteristic
141C2×C4×C9#6711yescharacteristic
142C2×C4×C9C22×D311yescharacteristic
143#611—22no—
144#611—22no—
145C2×Dic9—21no—
146C2×Dic9—64no—
147C2×Dic9—68no—
148C2×Dic9—64no—
149C2×Dic9—62no—
150D36—21no—
151D36C2×Dic312yescomplemented
152C4×D9—22no—
153C4×D9—22no—
154Dic18—21no—
155Dic18C2×Dic312yescomplemented
156C8×C9—62no—
157C2×#104—271noSylow
158C2×C3×C9—21no—
159C2×C3×C9—21no—
160C2×C3×C9#10412yescomplemented
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C3×D9—42no—
163C3×#41—92no—
164C3×#41—92no—
165C3×#41—181no—
166C3×#41—181no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—21no—
171C22×C3×C4—181no—
172C22×C3×C4C2×D911yescharacteristic
173C22×C3×C4D3211yescharacteristic
174C2×#67—61no—
175C22×Dic3—182no—
176C22×Dic3—181no—
177C22×Dic3—181no—
178#235—32no—
179#235—32no—
180#235—61no—
181#235—61no—
182C2×D12—31no—
183C2×C4×D3—61no—
184C2×Dic6—31no—
185C3×M16—32no—
186C3×M16—61no—
187C2×C3×C8—61no—
188C4×Dic3—92no—
189C4×Dic3—92no—
190C4×Dic3—182no—
191C4×Dic3—182no—
192C4×Dic3—181no—
193C4×Dic3—181no—
194#208—21no—
195#208C2×D912yescomplemented
196C2×C3⋊C8—21no—
197C22×C32—22no—
198C22×C32(C2×D3)⋊C411yescharacteristic
199C22×C32(C2×D3)⋊C411yescharacteristic
200C22×C32C2×D1211yescharacteristic
201C2×C3×D3—62no—
202C2×C3×D3—122no—
203C32×C4(C2×D3)⋊C411yescharacteristic
204C32×C4(C2×D3)⋊C411yescharacteristic
205C32×C4C2×C4×D311yescharacteristic
206C32×C4C2×#6711yescharacteristic
207C3⋊Dic3—64no—
208C3×Dic3—62no—
209C22×C9—24no—
210C22×C9—22no—
211C22×C9—21no—
212C22×C9—21no—
213C22×C9—21no—
214C22×C9#21711yescharacteristic
215C22×C9#21711yescharacteristic
216C22×C9C2×#6711yescharacteristic
217C2×D9—22no—
218C2×D9—42no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9#21711yescharacteristic
224C4×C9#21711yescharacteristic
225C4×C9C22×Dic311yescharacteristic
226C4×C9C2×#6711yescharacteristic
227Dic9—22no—
228Dic9—68no—
229Dic9—64no—
230C2×#41—91no—
231C2×M16—31no—
232C2×C42—271no—
233#104—274no—
234C3×C9C2×#10411yescharacteristic, complemented, Sylow
235C23×C3—21no—
236C23×C3—181no—
237C23×C3D3611yescharacteristic
238C23×C3#62611yescharacteristic
239C22×D3—61no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—92no—
243C3×D4—182no—
244C3×D4—182no—
245C3×D4—181no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—181no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4C4×D911yescharacteristic
256C2×C3×C4C4×D911yescharacteristic
257C2×C3×C4D3611yescharacteristic
258C2×C3×C4#61111yescharacteristic
259C2×C3×C4#61111yescharacteristic
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4D3×Dic311yescharacteristic
262C2×C3×C4D3×Dic311yescharacteristic
263C2×C3×C4#62511yescharacteristic
264C2×C3×C4#62511yescharacteristic
265C2×C3×C4#62611yescharacteristic
266C2×C3×C4C2×D3211yescharacteristic
267#67—62no—
268#67—62no—
269C2×Dic3—61no—
270C2×Dic3—184no—
271C2×Dic3—184no—
272C2×Dic3—188no—
273C2×Dic3—184no—
274C2×Dic3—182no—
275C2×Dic3—182no—
276D12—32no—
277D12—61no—
278C4×D3—62no—
279C4×D3—62no—
280Dic6—32no—
281Dic6—61no—
282C3×C8—62no—
283C3⋊C8—22no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#82512yes—
287C2×C32C2×(C2×D3)⋊C411yescharacteristic
288C3×D3—122no—
289C2×C9—22no—
290C2×C9—22no—
291C2×C9—21no—
292C2×C9—21no—
293C2×C9—22no—
294C2×C9—21no—
295C2×C9#83712yescomplemented
296C2×C9C2×#21711yescharacteristic
297D9—42no—
298#41—92no—
299#41—92no—
300#41—181no—
301#41—181no—
302C2×Q8—91no—
303C2×D4—91no—
304C2×D4—181no—
305C22×C4—181no—
306C22×C4—541no—
307C22×C4D3×D911yescharacteristic
308M16—32no—
309M16—61no—
310C2×C8—61no—
311C42—272no—
312C42—541no—
313C22×C3—24no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—182no—
320C22×C3—362no—
321C22×C3(C2×D9)⋊C411yescharacteristic
322C22×C3(C2×D9)⋊C411yescharacteristic
323C22×C3C2×D3611yescharacteristic
324C22×C3(C2×D3)⋊Dic311yescharacteristic
325C22×C3(C2×D3)⋊Dic311yescharacteristic
326C22×C3C2×#62611yescharacteristic
327C2×D3—62no—
328C2×D3—122no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—182no—
334C3×C4(C2×D9)⋊C411yescharacteristic
335C3×C4(C2×D9)⋊C411yescharacteristic
336C3×C4C2×C4×D911yescharacteristic
337C3×C4C2×#61111yescharacteristic
338C3×C4(C2×D3)⋊Dic311yescharacteristic
339C3×C4(C2×D3)⋊Dic311yescharacteristic
340C3×C4C2×D3×Dic311yescharacteristic
341C3×C4C2×#62511yescharacteristic
342Dic3—62no—
343Dic3—184no—
344Dic3—188no—
345Dic3—184no—
346C32C2×#82511yescharacteristic
347C9—21no—
348C9C2×#83711yescharacteristic, complemented
349C23—181no—
350C23#619211yescharacteristic
351Q8—92no—
352Q8—181no—
353D4—92no—
354D4—182no—
355D4—182no—
356D4—181no—
357C2×C4—181no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—544no—
361C2×C4—542no—
362C2×C4Dic3×D911yescharacteristic
363C2×C4Dic3×D911yescharacteristic
364C2×C4#619211yescharacteristic
365C2×C4#619411yescharacteristic
366C2×C4#619411yescharacteristic
367C2×C4C2×D3×D911yescharacteristic
368C8—62no—
369C2×C3—22no—
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—362no—
378C2×C3#6338412yescomplemented
379C2×C3C2×(C2×D9)⋊C411yescharacteristic
380C2×C3#6356812yes—
381C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
382D3—122no—
383C22—22no—
384C22—182no—
385C22—362no—
386C22(C2×D9)⋊Dic311yescharacteristic
387C22(C2×D9)⋊Dic311yescharacteristic
388C22C2×#619211yescharacteristic
389C4—182no—
390C4—544no—
391C4(C2×D9)⋊Dic311yescharacteristic
392C4(C2×D9)⋊Dic311yescharacteristic
393C4C2×Dic3×D911yescharacteristic
394C4C2×#619411yescharacteristic
395C3—21no—
396C3C2×#6338411yescharacteristic, complemented
397C3C2×#6356811yescharacteristic
398C2—21no—
399C2—21no—
400C2—362no—
401C2#1172029112yesdirect product factor
402C2C2×(C2×D9)⋊Dic311yescharacteristic
subgroupG/H
centerC2×C4#6194
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32(C2×D3)⋊C4
prime radicalC2×C3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#104
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊Dic9C2semidirect
3. C2×C9×#208C2semidirect
4. #11720291C2direct (N × H)
12. C3×#3697C4semidirect
14. C2×C3×D36C4semidirect
16. C2×C3×Dic18C4semidirect
18. C4×C3⋊Dic9C22semidirect
20. C9×#208C22semidirect
45. C3×D36C2×C4semidirect
49. C3×Dic18C2×C4semidirect
75. #3697Dic3semidirect
77. C2×D36Dic3semidirect
79. C2×Dic18Dic3semidirect
105. C2×#208D9semidirect
151. D36C2×Dic3semidirect
155. Dic18C2×Dic3semidirect
160. C2×C3×C9#104semidirect
195. #208C2×D9semidirect
234. C3×C9C2×#104semidirect
295. C2×C9#837semidirect
348. C9C2×#837semidirect
378. C2×C3#63384semidirect
396. C3C2×#63384semidirect
401. C2#11720291direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×C4×C3⋊Dic92
C2×C9×#2082
#117202912
C2×#635683
C2×#633843