Finite Group Databasebeta

← C2×#11722043 #13160376

Subgroups — C2×#11722043 #13160376

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×C32×#235C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4#11722043C214yesmaximal, direct product factor
5C2×#63664—31nomaximal
6C2×#63568—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3⋊(C3⋊Dic3)—21no—
9C2×C32×#67—21no—
10C32×#235—21no—
11C32×#235—21no—
12C32×#235C412yescomplemented
13C32×#235C2212yes—
14C2×C32×D12C411yescharacteristic, complemented
15C2×C32×C4×D3—21no—
16C2×C32×Dic6C411yescharacteristic, complemented
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2212yescomplemented
19C3×#3748—21no—
20C3×#3748C2212yescomplemented
21C2×C3×C3⋊(C3⋊C8)—21no—
22C2×C32×#41—31no—
23C2×C3×#235D314yes—
24C2×C4×C3⋊Dic3—31no—
25C2×C4×C3⋊Dic3—34no—
26C2×C4×C3⋊Dic3—64no—
27C2×#3748D311yescharacteristic, complemented
28C2×C3×#208—34no—
29#63664—34no—
30#63568—316no—
31C23×C33D411yescharacteristic
32C22×C32×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—
41C32×#67—22no—
42C32×#67—22no—
43C2×C32×Dic3—21no—
44C32×D12—21no—
45C32×D12C2×C412yescomplemented
46C32×C4×D3—22no—
47C32×C4×D3—22no—
48C32×Dic6—21no—
49C32×Dic6C2×C412yescomplemented
50C3×C3⋊(C3⋊C8)—22no—
51C2×#837—94no—
52C2×#825—91no—
53C32×#41—32no—
54C32×#41—32no—
55C32×#41—61no—
56C32×#41—61no—
57C2×C32×Q8—31no—
58C2×C32×D4—31no—
59C2×C32×D4—61no—
60C22×C32×C4—24no—
61C22×C32×C4—61no—
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C2×D314yes—
64C22×C3⋊Dic3—68no—
65C22×C3⋊Dic3—61no—
66C22×C3⋊Dic3—64no—
67C2×C3×#67—24no—
68C3×#235—24no—
69C3×#235—24no—
70C3×#235Dic318yescomplemented
71C3×#235C2×D318yes—
72C2×C3×D12Dic314yescomplemented
73C2×C3×C4×D3—24no—
74C2×C3×Dic6Dic314yescomplemented
75C4×C3⋊Dic3—32no—
76C4×C3⋊Dic3—38no—
77C4×C3⋊Dic3—68no—
78C4×C3⋊Dic3—68no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—64no—
81#3748—21no—
82#3748C2×D312yescomplemented
83C2×C3⋊(C3⋊C8)—21no—
84C3×#208—38no—
85C3×#208—64no—
86C2×C3×C3⋊C8—64no—
87C22×C33—22no—
88C22×C33C22⋊C411yescharacteristic
89C22×C33C22⋊C411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C2×C32×D3—22no—
92C2×C32×D3—42no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3⋊(C3⋊Dic3)—24no—
98C32×Dic3—22no—
99C2×C3×#41—34no—
100C2×#235C3⋊D311yescharacteristic
101C2×C3×M16—91no—
102C2×C4×Dic3—94no—
103C2×C4×Dic3—91no—
104C2×C4×Dic3—184no—
105C2×#208—34no—
106#837—916no—
107#825—94no—
108C23×C32—24no—
109C23×C32—61no—
110C23×C32D1211yescharacteristic
111C23×C32#6714yes—
112C22×C3×D3—24no—
113C32×Q8—32no—
114C32×Q8—61no—
115C32×D4—32no—
116C32×D4—62no—
117C32×D4—62no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4—62no—
127C2×C32×C4—62no—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4C2×Dic314yes—
132C2×C32×C4C2×Dic314yes—
133C2×C32×C4#6714yes—
134C2×C32×C4#6714yes—
135C2×C32×C4#6714yes—
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—616no—
141C2×C3⋊Dic3—632no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—616no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—68no—
146C3×#67—28no—
147C3×#67—28no—
148C2×C3×Dic3—24no—
149C3×D12—24no—
150C3×D12C2×Dic318yescomplemented
151C3×C4×D3—28no—
152C3×C4×D3—28no—
153C3×Dic6—24no—
154C3×Dic6C2×Dic318yescomplemented
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—68no—
157C2×#104—271noSylow
158C2×C33—21no—
159C2×C33—21no—
160C2×C33#10412yescomplemented
161C2×C33C2×C22⋊C411yescharacteristic
162C32×D3—42no—
163C3×#41—38no—
164C3×#41—38no—
165C3×#41—64no—
166C3×#41—64no—
167C2×C3×Q8—34no—
168C2×C3×D4—34no—
169C2×C3×D4—64no—
170C22×C3×C4—24no—
171C22×C3×C4—64no—
172C22×C3×C4D3214yes—
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—21no—
175C22×Dic3—188no—
176C22×Dic3—184no—
177C22×Dic3—181no—
178#235—21no—
179#235—21no—
180#235C3⋊Dic312yescomplemented
181#235C2×C3⋊D312yes—
182C2×D12C3⋊Dic311yescharacteristic, complemented
183C2×C4×D3—21no—
184C2×Dic6C3⋊Dic311yescharacteristic, complemented
185C3×M16—92no—
186C3×M16—181no—
187C2×C3×C8—181no—
188C4×Dic3—98no—
189C4×Dic3—92no—
190C4×Dic3—188no—
191C4×Dic3—188no—
192C4×Dic3—184no—
193C4×Dic3—181no—
194#208—38no—
195#208—64no—
196C2×C3⋊C8—64no—
197C22×C32—216no—
198C22×C32—22no—
199C22×C32—28no—
200C22×C32—24no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—62no—
204C22×C32—122no—
205C22×C32(C2×D3)⋊C411yescharacteristic
206C22×C32(C2×D3)⋊C411yescharacteristic
207C22×C32#21714yes—
208C22×C32#21714yes—
209C22×C32C2×D1211yescharacteristic
210C22×C32C2×#6714yes—
211C2×C3×D3—28no—
212C2×C3×D3—48no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—62no—
218C32×C4(C2×D3)⋊C411yescharacteristic
219C32×C4(C2×D3)⋊C411yescharacteristic
220C32×C4#21714yes—
221C32×C4#21714yes—
222C32×C4C2×C4×D311yescharacteristic
223C32×C4C22×Dic314yes—
224C32×C4C2×#6714yes—
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—632no—
227C3⋊Dic3—64no—
228C3⋊Dic3—616no—
229C3×Dic3—28no—
230C2×#41—31no—
231C2×M16—91no—
232C2×C42—271no—
233#104—274no—
234C33C2×#10411yescharacteristic, complemented, Sylow
235C23×C3—24no—
236C23×C3—64no—
237C23×C3#62614yes—
238C23×C3#63811yescharacteristic
239C22×D3—21no—
240C3×Q8—38no—
241C3×Q8—64no—
242C3×D4—38no—
243C3×D4—68no—
244C3×D4—68no—
245C3×D4—64no—
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—64no—
253C2×C3×C4—68no—
254C2×C3×C4—68no—
255C2×C3×C4D3×Dic314yes—
256C2×C3×C4D3×Dic314yes—
257C2×C3×C4#62514yes—
258C2×C3×C4#62514yes—
259C2×C3×C4#62614yes—
260C2×C3×C4C2×C3⋊Dic311yescharacteristic
261C2×C3×C4C2×C3⋊Dic311yescharacteristic
262C2×C3×C4#63811yescharacteristic
263C2×C3×C4#63811yescharacteristic
264C2×C3×C4#63811yescharacteristic
265C2×C3×C4C2×D3214yes—
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—22no—
268#67—22no—
269C2×Dic3—21no—
270C2×Dic3—1816no—
271C2×Dic3—1832no—
272C2×Dic3—1816no—
273C2×Dic3—184no—
274C2×Dic3—188no—
275C2×Dic3—182no—
276D12—21no—
277D12C2×C3⋊Dic312yescomplemented
278C4×D3—22no—
279C4×D3—22no—
280Dic6—21no—
281Dic6C2×C3⋊Dic312yescomplemented
282C3×C8—182no—
283C3⋊C8—68no—
284C2×C32—28no—
285C2×C32—28no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—28no—
291C2×C32—24no—
292C2×C32—122no—
293C2×C32#82512yescomplemented
294C2×C32#83718yescomplemented
295C2×C32C2×(C2×D3)⋊C411yescharacteristic
296C2×C32C2×#21714yes—
297C3×D3—48no—
298#41—32no—
299#41—32no—
300#41—61no—
301#41—61no—
302C2×Q8—31no—
303C2×D4—31no—
304C2×D4—61no—
305C22×C4—61no—
306C22×C4—541no—
307C22×C4D3×C3⋊D311yescharacteristic
308M16—92no—
309M16—181no—
310C2×C8—181no—
311C42—272no—
312C42—541no—
313C22×C3—216no—
314C22×C3—28no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—24no—
319C22×C3—68no—
320C22×C3—128no—
321C22×C3(C2×D3)⋊Dic314yes—
322C22×C3(C2×D3)⋊Dic314yes—
323C22×C3#375711yescharacteristic
324C22×C3#375711yescharacteristic
325C22×C3C2×#62614yes—
326C22×C3C2×#63811yescharacteristic
327C2×D3—22no—
328C2×D3—42no—
329C3×C4—24no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—68no—
334C3×C4(C2×D3)⋊Dic314yes—
335C3×C4(C2×D3)⋊Dic314yes—
336C3×C4#375711yescharacteristic
337C3×C4#375711yescharacteristic
338C3×C4C2×D3×Dic314yes—
339C3×C4C2×#62514yes—
340C3×C4C22×C3⋊Dic311yescharacteristic
341C3×C4C2×#63811yescharacteristic
342Dic3—22no—
343Dic3—1832no—
344Dic3—1816no—
345Dic3—184no—
346C32—24no—
347C32C2×#82511yescharacteristic, complemented
348C32C2×#83714yescomplemented
349C23—61no—
350C23#629111yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—32no—
354D4—62no—
355D4—62no—
356D4—61no—
357C2×C4—61no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—544no—
361C2×C4—542no—
362C2×C4D3×C3⋊Dic311yescharacteristic
363C2×C4D3×C3⋊Dic311yescharacteristic
364C2×C4#629011yescharacteristic
365C2×C4#629011yescharacteristic
366C2×C4#629111yescharacteristic
367C2×C4C2×D3×C3⋊D311yescharacteristic
368C8—182no—
369C2×C3—28no—
370C2×C3—28no—
371C2×C3—24no—
372C2×C3—24no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—28no—
376C2×C3—24no—
377C2×C3—128no—
378C2×C3#6356818yescomplemented
379C2×C3#6366412yescomplemented
380C2×C3C2×(C2×D3)⋊Dic314yes—
381C2×C3C2×#375711yescharacteristic
382D3—42no—
383C22—22no—
384C22—62no—
385C22—122no—
386C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
387C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
388C22C2×#629111yescharacteristic
389C4—62no—
390C4—544no—
391C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
392C4(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
393C4C2×D3×C3⋊Dic311yescharacteristic
394C4C2×#629011yescharacteristic
395C3—24no—
396C3C2×#6356814yescomplemented
397C3C2×#6366411yescharacteristic, complemented
398C2—21no—
399C2—21no—
400C2—122no—
401C2#1172204312yesdirect product factor
402C2C2×(C2×D3)⋊(C3⋊Dic3)11yescharacteristic
subgroupG/H
centerC2×C4#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC2×C32×#235C2
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×C3×#3748C2semidirect
4. #11722043C2direct (N × H)
12. C32×#235C4semidirect
14. C2×C32×D12C4semidirect
16. C2×C32×Dic6C4semidirect
18. C4×C3⋊(C3⋊Dic3)C22semidirect
20. C3×#3748C22semidirect
27. C2×#3748D3semidirect
45. C32×D12C2×C4semidirect
49. C32×Dic6C2×C4semidirect
70. C3×#235Dic3semidirect
72. C2×C3×D12Dic3semidirect
74. C2×C3×Dic6Dic3semidirect
82. #3748C2×D3semidirect
150. C3×D12C2×Dic3semidirect
154. C3×Dic6C2×Dic3semidirect
160. C2×C33#104semidirect
180. #235C3⋊Dic3semidirect
182. C2×D12C3⋊Dic3semidirect
184. C2×Dic6C3⋊Dic3semidirect
234. C33C2×#104semidirect
277. D12C2×C3⋊Dic3semidirect
281. Dic6C2×C3⋊Dic3semidirect
293. C2×C32#825semidirect
294. C2×C32#837semidirect
347. C32C2×#825semidirect
348. C32C2×#837semidirect
378. C2×C3#63568semidirect
379. C2×C3#63664semidirect
396. C3C2×#63568semidirect
397. C3C2×#63664semidirect
401. C2#11722043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×#37482
#117220432
C2×#636643
C2×#635683