Finite Group Databasebeta

← C2×#11722045 #13160371

Subgroups — C2×#11722045 #13160371

Order 1728.

1452 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×#3828C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4#11722045C214yesmaximal, direct product factor
5C2×#63652—31nomaximal
6C2×#63568—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3⋊(C3⋊Dic3)—21no—
9C2×C3×#638—21no—
10C3×#3828—21no—
11C3×#3828—21no—
12C3×#3828C412yescomplemented
13C3×#3828C2212yes—
14C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×#634C411yescharacteristic, complemented
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2212yescomplemented
19C32×#208—21no—
20C32×#208C2212yescomplemented
21C2×C32×C3⋊C8—21no—
22C2×#3828D311yescharacteristic
23C2×C3×#235—34no—
24C2×C32×M16—31no—
25C2×C4×C3⋊Dic3—31no—
26C2×C4×C3⋊Dic3—34no—
27C2×C4×C3⋊Dic3—64no—
28C2×C3×#208D314yescomplemented
29#63652—34no—
30#63568—316no—
31C23×C33D411yescharacteristic
32C22×C3×C3⋊D3—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—24no—
40C2×C3⋊(C3⋊Dic3)—22no—
41C3×#638—22no—
42C3×#638—22no—
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)C2×C412yescomplemented
46C3×C4×C3⋊D3—22no—
47C3×C4×C3⋊D3—22no—
48C3×#634—21no—
49C3×#634C2×C412yescomplemented
50C32×C3⋊C8—22no—
51C2×#837—91no—
52C2×#825—94no—
53C22×C32×C4—24no—
54C22×C32×C4C2×D314yes—
55C22×C32×C4C2×D311yescharacteristic
56C2×#638—21no—
57C22×C3⋊Dic3—68no—
58C22×C3⋊Dic3—64no—
59C22×C3⋊Dic3—61no—
60#3828—21no—
61#3828—21no—
62#3828Dic312yescomplemented
63#3828C2×D312yes—
64C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
65C2×C4×C3⋊D3—21no—
66C2×#634Dic311yescharacteristic, complemented
67C2×C3×#67—64no—
68C3×#235—38no—
69C3×#235—38no—
70C3×#235—64no—
71C3×#235—64no—
72C2×C3×D12—34no—
73C2×C3×C4×D3—64no—
74C2×C3×Dic6—34no—
75C32×M16—32no—
76C32×M16—61no—
77C2×C32×C8—61no—
78C4×C3⋊Dic3—32no—
79C4×C3⋊Dic3—38no—
80C4×C3⋊Dic3—68no—
81C4×C3⋊Dic3—68no—
82C4×C3⋊Dic3—64no—
83C4×C3⋊Dic3—61no—
84C3×#208—24no—
85C3×#208C2×D318yescomplemented
86C2×C3×C3⋊C8—24no—
87C22×C33—22no—
88C22×C33C22⋊C411yescharacteristic
89C22×C33C22⋊C411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3⋊(C3⋊Dic3)—24no—
98C3×C3⋊Dic3—22no—
99C2×C3×#41—91no—
100C2×#235—34no—
101C2×C3×M16—34no—
102C2×C4×Dic3—94no—
103C2×C4×Dic3—91no—
104C2×C4×Dic3—184no—
105C2×#208C3⋊D311yescharacteristic, complemented
106#837—94no—
107#825—916no—
108C23×C32—24no—
109C23×C32D1214yes—
110C23×C32#6711yescharacteristic
111C22×C3⋊D3—21no—
112C22×C3×D3—64no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4C4×D314yes—
120C2×C32×C4C4×D314yes—
121C2×C32×C4D1214yes—
122C2×C32×C4C2×Dic311yescharacteristic
123C2×C32×C4C2×Dic311yescharacteristic
124C2×C32×C4#6714yes—
125C2×C32×C4#6714yes—
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—22no—
132#638—22no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—616no—
135C2×C3⋊Dic3—632no—
136C2×C3⋊Dic3—616no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—68no—
139C2×C3⋊Dic3—62no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C2×Dic312yescomplemented
142C4×C3⋊D3—22no—
143C4×C3⋊D3—22no—
144#634—21no—
145#634C2×Dic312yescomplemented
146C3×#67—68no—
147C3×#67—68no—
148C2×C3×Dic3—64no—
149C3×D12—38no—
150C3×D12—64no—
151C3×C4×D3—68no—
152C3×C4×D3—68no—
153C3×Dic6—38no—
154C3×Dic6—64no—
155C32×C8—62no—
156C3×C3⋊C8—28no—
157C2×#104—271noSylow
158C2×C33—21no—
159C2×C33—21no—
160C2×C33#10412yescomplemented
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—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—24no—
171C22×C3×C4—181no—
172C22×C3×C4D3214yes—
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—64no—
175C22×Dic3—188no—
176C22×Dic3—184no—
177C22×Dic3—181no—
178#235—38no—
179#235—38no—
180#235—64no—
181#235—64no—
182C2×D12—34no—
183C2×C4×D3—64no—
184C2×Dic6—34no—
185C3×M16—38no—
186C3×M16—64no—
187C2×C3×C8—64no—
188C4×Dic3—98no—
189C4×Dic3—92no—
190C4×Dic3—188no—
191C4×Dic3—188no—
192C4×Dic3—184no—
193C4×Dic3—181no—
194#208—21no—
195#208C2×C3⋊D312yescomplemented
196C2×C3⋊C8—21no—
197C22×C32—216no—
198C22×C32—28no—
199C22×C32—22no—
200C22×C32—24no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32(C2×D3)⋊C414yes—
204C22×C32(C2×D3)⋊C414yes—
205C22×C32#21711yescharacteristic
206C22×C32#21711yescharacteristic
207C22×C32C2×D1214yes—
208C22×C32C2×#6711yescharacteristic
209C2×C3⋊D3—22no—
210C2×C3⋊D3—42no—
211C2×C3×D3—68no—
212C2×C3×D3—128no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4(C2×D3)⋊C414yes—
218C32×C4(C2×D3)⋊C414yes—
219C32×C4#21711yescharacteristic
220C32×C4#21711yescharacteristic
221C32×C4C2×C4×D314yes—
222C32×C4C22×Dic311yescharacteristic
223C32×C4C2×#6714yes—
224C32×C4C2×#6711yescharacteristic
225C3⋊Dic3—22no—
226C3⋊Dic3—632no—
227C3⋊Dic3—616no—
228C3⋊Dic3—64no—
229C3×Dic3—68no—
230C2×#41—91no—
231C2×M16—31no—
232C2×C42—271no—
233#104—274no—
234C33C2×#10411yescharacteristic, complemented, Sylow
235C23×C3—24no—
236C23×C3—181no—
237C23×C3#62614yes—
238C23×C3C4⋊(C3⋊D3)11yescharacteristic
239C22×D3—64no—
240C3×Q8—92no—
241C3×Q8—181no—
242C3×D4—92no—
243C3×D4—182no—
244C3×D4—182no—
245C3×D4—181no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—181no—
253C2×C3×C4—182no—
254C2×C3×C4—182no—
255C2×C3×C4D3×Dic314yes—
256C2×C3×C4D3×Dic314yes—
257C2×C3×C4#62514yes—
258C2×C3×C4#62514yes—
259C2×C3×C4#62614yes—
260C2×C3×C4C4×C3⋊D311yescharacteristic
261C2×C3×C4C4×C3⋊D311yescharacteristic
262C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
263C2×C3×C4#63811yescharacteristic
264C2×C3×C4#63811yescharacteristic
265C2×C3×C4C2×D3214yes—
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—68no—
268#67—68no—
269C2×Dic3—64no—
270C2×Dic3—1816no—
271C2×Dic3—1832no—
272C2×Dic3—1816no—
273C2×Dic3—184no—
274C2×Dic3—188no—
275C2×Dic3—182no—
276D12—38no—
277D12—64no—
278C4×D3—68no—
279C4×D3—68no—
280Dic6—38no—
281Dic6—64no—
282C3×C8—68no—
283C3⋊C8—22no—
284C2×C32—28no—
285C2×C32—28no—
286C2×C32—24no—
287C2×C32—24no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—28no—
291C2×C32—24no—
292C2×C32#82518yescomplemented
293C2×C32#83712yescomplemented
294C2×C32C2×(C2×D3)⋊C414yes—
295C2×C32C2×#21711yescharacteristic
296C3⋊D3—42no—
297C3×D3—128no—
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×C3⋊D311yescharacteristic
308M16—32no—
309M16—61no—
310C2×C8—61no—
311C42—272no—
312C42—541no—
313C22×C3—216no—
314C22×C3—28no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—24no—
319C22×C3—182no—
320C22×C3—362no—
321C22×C3(C2×D3)⋊Dic314yes—
322C22×C3(C2×D3)⋊Dic314yes—
323C22×C3#375211yescharacteristic
324C22×C3#375211yescharacteristic
325C22×C3C2×#62614yes—
326C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
327C2×D3—68no—
328C2×D3—128no—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—182no—
334C3×C4(C2×D3)⋊Dic314yes—
335C3×C4(C2×D3)⋊Dic314yes—
336C3×C4#375211yescharacteristic
337C3×C4#375211yescharacteristic
338C3×C4C2×D3×Dic314yes—
339C3×C4C2×#62514yes—
340C3×C4C2×C4×C3⋊D311yescharacteristic
341C3×C4C2×#63811yescharacteristic
342Dic3—68no—
343Dic3—1832no—
344Dic3—184no—
345Dic3—1816no—
346C32—24no—
347C32C2×#82514yescomplemented
348C32C2×#83711yescharacteristic, complemented
349C23—181no—
350C23#629211yescharacteristic
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×C3⋊D311yescharacteristic
363C2×C4Dic3×C3⋊D311yescharacteristic
364C2×C4#629011yescharacteristic
365C2×C4#629011yescharacteristic
366C2×C4#629211yescharacteristic
367C2×C4C2×D3×C3⋊D311yescharacteristic
368C8—62no—
369C2×C3—28no—
370C2×C3—28no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—24no—
374C2×C3—24no—
375C2×C3—28no—
376C2×C3—24no—
377C2×C3—362no—
378C2×C3#6356818yescomplemented
379C2×C3#6365212yescomplemented
380C2×C3C2×(C2×D3)⋊Dic314yes—
381C2×C3C2×#375211yescharacteristic
382D3—128no—
383C22—22no—
384C22—182no—
385C22—362no—
386C22#8880311yescharacteristic
387C22#8880311yescharacteristic
388C22C2×#629211yescharacteristic
389C4—182no—
390C4—544no—
391C4#8880311yescharacteristic
392C4#8880311yescharacteristic
393C4C2×Dic3×C3⋊D311yescharacteristic
394C4C2×#629011yescharacteristic
395C3—24no—
396C3C2×#6356814yescomplemented
397C3C2×#6365211yescharacteristic, complemented
398C2—21no—
399C2—21no—
400C2—362no—
401C2#1172204512yesdirect product factor
402C2C2×#8880311yescharacteristic
subgroupG/H
centerC2×C4#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊(C3⋊Dic3)C2semidirect
3. C2×C32×#208C2semidirect
4. #11722045C2direct (N × H)
12. C3×#3828C4semidirect
14. C2×C3×C4⋊(C3⋊D3)C4semidirect
16. C2×C3×#634C4semidirect
18. C4×C3⋊(C3⋊Dic3)C22semidirect
20. C32×#208C22semidirect
28. C2×C3×#208D3semidirect
45. C3×C4⋊(C3⋊D3)C2×C4semidirect
49. C3×#634C2×C4semidirect
62. #3828Dic3semidirect
64. C2×C4⋊(C3⋊D3)Dic3semidirect
66. C2×#634Dic3semidirect
85. C3×#208C2×D3semidirect
105. C2×#208C3⋊D3semidirect
141. C4⋊(C3⋊D3)C2×Dic3semidirect
145. #634C2×Dic3semidirect
160. C2×C33#104semidirect
195. #208C2×C3⋊D3semidirect
234. C33C2×#104semidirect
292. C2×C32#825semidirect
293. C2×C32#837semidirect
347. C32C2×#825semidirect
348. C32C2×#837semidirect
378. C2×C3#63568semidirect
379. C2×C3#63652semidirect
396. C3C2×#63568semidirect
397. C3C2×#63652semidirect
401. C2#11722045direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C4×C3⋊(C3⋊Dic3)2
C2×C32×#2082
#117220452
C2×#636523
C2×#635683