Finite Group Databasebeta

← C3×#82231 #11802844

Subgroups — C3×#82231 #11802844

Order 1152.

1133 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic
2C3×#5212C212yesmaximal
3C3×#4980C214yesmaximal, complemented
4C3×#4951C214yesmaximal
5C3×#4936C211yesmaximal, characteristic, complemented
6C3×#4935C211yesmaximal, characteristic, complemented
7C3×C4×C4⋊Dic3C212yesmaximal, complemented
8C3×#2314—31nomaximal
9#82231C311yesmaximal, characteristic, direct product factor
10C23×C32×C4C2211yescharacteristic
11C32×#117C418yescomplemented
12C2×C32×C4⋊C4C2212yes—
13C2×C32×C22⋊C4C2212yes—
14C2×C32×C42C2211yescharacteristic
15C2×C32×C42C2211yescharacteristic
16C2×C3×#217C2214yes—
17C2×C3×C4⋊Dic3C2212yes—
18C2×C3×C4⋊Dic3C2212yes—
19C2×C3×C4⋊Dic3C2214yes—
20C2×C3×C4×Dic3C2214yes—
21C3×C3⋊#95C2214yes—
22C3×C3⋊#95C2214yes—
23C3×C3⋊#95C2214yes—
24C2×C3×#117—21no—
25C2×C3×#117D311yescharacteristic
26C2×C3×#117C2×C311yescharacteristic
27C3×#388—34no—
28C3×#387—34no—
29C3×#384—31no—
30C3×#382—31no—
31C3×#380—32no—
32C3×C4×C4⋊C4—32no—
33#5212C2×C312yes—
34#4980C2×C314yescomplemented
35#4951C2×C314yes—
36#4936C2×C311yescharacteristic, complemented
37#4935C2×C311yescharacteristic, complemented
38C4×C4⋊Dic3C2×C312yescomplemented
39C24×C32C2311yescharacteristic
40C22×C32×C4—22no—
41C22×C32×C4—22no—
42C22×C32×C4C2×C414yes—
43C22×C32×C4C2312yes—
44C22×C32×C4C2312yes—
45C22×C32×C4C2311yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3×Dic3C2314yes—
48C22×C3×Dic3C2314yes—
49C32×C4⋊C4C2×C418yes—
50C32×C22⋊C4C2×C418yes—
51C32×C42C2×C414yes—
52C32×C42C2×C414yes—
53C3×#217—44no—
54C3×C4⋊Dic3—24no—
55C3×C4⋊Dic3—24no—
56C3×C4⋊Dic3—44no—
57C3×C4×Dic3—44no—
58#2314—31noSylow
59C23×C3×C4—21no—
60C23×C3×C4C2×D311yescharacteristic
61C23×C3×C4C22×C311yescharacteristic
62C3×#117—28no—
63C3×#117Dic318yescomplemented
64C3×#117C3×C418yescomplemented
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—32no—
67C2×C3×C4⋊C4—32no—
68C2×C3×C4⋊C4—34no—
69C2×C3×C4⋊C4C2×D312yes—
70C2×C3×C4⋊C4C22×C312yes—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—34no—
73C2×C3×C22⋊C4C2×D312yes—
74C2×C3×C22⋊C4C22×C312yes—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×C3×C42—34no—
78C2×C3×C42C2×D311yescharacteristic
79C2×C3×C42C2×D311yescharacteristic
80C2×C3×C42C22×C311yescharacteristic
81C2×C3×C42C22×C311yescharacteristic
82C2×#217C22×C314yes—
83C2×C4⋊Dic3C22×C312yes—
84C2×C4⋊Dic3C22×C312yes—
85C2×C4⋊Dic3C22×C314yes—
86C2×C4×Dic3C22×C314yes—
87C3×#95—34no—
88C3×#95—34no—
89C3×#95—34no—
90C3⋊#95C22×C314yes—
91C3⋊#95C22×C314yes—
92C3⋊#95C22×C314yes—
93C23×C32—21no—
94C23×C32—21no—
95C23×C32—22no—
96C23×C32—22no—
97C23×C32C22×C412yes—
98C23×C32C2411yescharacteristic
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—28no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4C22×C414yes—
107C2×C32×C4C22×C414yes—
108C2×C32×C4C22×C412yes—
109C2×C32×C4C22×C412yes—
110C2×C32×C4#4112yes—
111C2×C32×C4#4112yes—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—24no—
118C2×#117C3×D311yescharacteristic
119#388—34no—
120#387—34no—
121#384—31no—
122#382—31no—
123#380—32no—
124C4×C4⋊C4—32no—
125C24×C3—21no—
126C24×C3C22×D311yescharacteristic
127C24×C3C23×C311yescharacteristic
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—34no—
140C22×C3×C4—34no—
141C22×C3×C4C2×Dic314yes—
142C22×C3×C4C2×C3×C414yes—
143C22×C3×C4C22×D311yescharacteristic
144C22×C3×C4C22×D312yes—
145C22×C3×C4C22×D312yes—
146C22×C3×C4C22×D311yescharacteristic
147C22×C3×C4C23×C311yescharacteristic
148C22×C3×C4C23×C312yes—
149C22×C3×C4C23×C312yes—
150C22×C3×C4C23×C311yescharacteristic
151C22×Dic3C23×C314yes—
152C22×Dic3C23×C314yes—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—64no—
155C3×C4⋊C4—64no—
156C3×C4⋊C4—124no—
157C3×C4⋊C4C2×Dic318yes—
158C3×C4⋊C4C2×C3×C418yes—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—124no—
161C3×C22⋊C4C2×Dic318yes—
162C3×C22⋊C4C2×C3×C418yes—
163C3×C42—24no—
164C3×C42—24no—
165C3×C42—124no—
166C3×C42C2×Dic314yes—
167C3×C42C2×Dic314yes—
168C3×C42C2×C3×C414yes—
169C3×C42C2×C3×C414yes—
170#217—44no—
171C4⋊Dic3—24no—
172C4⋊Dic3—24no—
173C4⋊Dic3—44no—
174C4×Dic3—44no—
175C22×C32—22no—
176C22×C32—24no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32C23×C411yescharacteristic
181C22×C32C2×#4111yescharacteristic
182C22×C32C2×#4111yescharacteristic
183C22×C32H(5,𝔽2)12yes—
184C22×C322−1+412yes—
185C32×C4—22no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4#11714yes—
189C3×Dic3—44no—
190C3×Dic3—44no—
191C23×C4C2×C3×D311yescharacteristic
192#117C3×Dic318yescomplemented
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—34no—
196C2×C4⋊C4C2×C3×D312yes—
197C2×C22⋊C4—34no—
198C2×C22⋊C4C2×C3×D312yes—
199C2×C42—34no—
200C2×C42C2×C3×D311yescharacteristic
201C2×C42C2×C3×D311yescharacteristic
202#95—34no—
203#95—34no—
204#95—34no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—21no—
216C23×C3—22no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3C22×Dic312yes—
220C23×C3C22×C3×C412yes—
221C23×C3C23×D311yescharacteristic
222C23×C3C24×C311yescharacteristic
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—216no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—28no—
239C2×C3×C4—28no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—64no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—64no—
255C2×C3×C4—64no—
256C2×C3×C4#23512yes—
257C2×C3×C4#23512yes—
258C2×C3×C4C22×Dic314yes—
259C2×C3×C4C22×Dic314yes—
260C2×C3×C4C22×Dic312yes—
261C2×C3×C4C22×Dic312yes—
262C2×C3×C4C22×C3×C414yes—
263C2×C3×C4C22×C3×C414yes—
264C2×C3×C4C22×C3×C412yes—
265C2×C3×C4C22×C3×C412yes—
266C2×C3×C4C3×#4112yes—
267C2×C3×C4C3×#4112yes—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—24no—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×C32—42no—
275C2×C32C2×#11711yescharacteristic
276C2×C32#52012yes—
277C2×C32#54011yescharacteristic
278C2×C32#54111yescharacteristic
279C2×C32#54212yes—
280C24C22×C3×D311yescharacteristic
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—34no—
284C22×C4—34no—
285C22×C4C2×C3×Dic314yes—
286C22×C4C22×C3×D311yescharacteristic
287C22×C4C22×C3×D312yes—
288C22×C4C22×C3×D312yes—
289C22×C4C22×C3×D311yescharacteristic
290C4⋊C4—64no—
291C4⋊C4—64no—
292C4⋊C4—124no—
293C4⋊C4C2×C3×Dic318yes—
294C22⋊C4—124no—
295C22⋊C4C2×C3×Dic318yes—
296C42—124no—
297C42C2×C3×Dic314yes—
298C42C2×C3×Dic314yes—
299C22×C3—24no—
300C22×C3—22no—
301C22×C3—24no—
302C22×C3—22no—
303C22×C3—24no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—44no—
310C22×C3—42no—
311C22×C3—44no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—42no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3C2×#23511yescharacteristic
320C22×C3C2×#23511yescharacteristic
321C22×C3#100912yes—
322C22×C3#101012yes—
323C22×C3C23×Dic311yescharacteristic
324C22×C3C23×C3×C411yescharacteristic
325C22×C3C2×C3×#4111yescharacteristic
326C22×C3C2×C3×#4111yescharacteristic
327C22×C3C3×H(5,𝔽2)12yes—
328C22×C3C3×2−1+412yes—
329C3×C4—24no—
330C3×C4—22no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—22no—
334C3×C4—24no—
335C3×C4—24no—
336C3×C4—24no—
337C3×C4—44no—
338C3×C4—44no—
339C3×C4—124no—
340C3×C4—124no—
341C3×C4#92614yes—
342C3×C4C3×#11714yes—
343Dic3—44no—
344Dic3—44no—
345C32#231411yescharacteristic, complemented, Sylow
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23C22×C3×Dic312yes—
351C23C23×C3×D311yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—64no—
360C2×C4—64no—
361C2×C4—64no—
362C2×C4—64no—
363C2×C4—64no—
364C2×C4—64no—
365C2×C4C3×#23512yes—
366C2×C4C3×#23512yes—
367C2×C4C22×C3×Dic314yes—
368C2×C4C22×C3×Dic314yes—
369C2×C4C22×C3×Dic312yes—
370C2×C4C22×C3×Dic312yes—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—22no—
374C2×C3—21no—
375C2×C3—22no—
376C2×C3—44no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3#551811yescharacteristic
380C2×C3#553412yes—
381C2×C3#553511yescharacteristic
382C2×C3C2×#92611yescharacteristic
383C2×C3#582712yes—
384C2×C3C2×C3×#11711yescharacteristic
385C2×C3C3×#52012yes—
386C2×C3C3×#54011yescharacteristic
387C2×C3C3×#54111yescharacteristic
388C2×C3C3×#54212yes—
389C22—22no—
390C22—24no—
391C22—41no—
392C22—42no—
393C22—41no—
394C22C2×C3×#23511yescharacteristic
395C22C2×C3×#23511yescharacteristic
396C22C3×#100912yes—
397C22C3×#101012yes—
398C22C23×C3×Dic311yescharacteristic
399C4—22no—
400C4—24no—
401C4—24no—
402C4—124no—
403C4—124no—
404C4C3×#92614yes—
405C3—21no—
406C3#8223111yescharacteristic, direct product factor
407C3C3×#231411yescharacteristic, complemented
408C2—42no—
409C2C3×#551811yescharacteristic
410C2C3×#553412yes—
411C2C3×#553511yescharacteristic
412C2C2×C3×#92611yescharacteristic
413C2C3×#582712yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC2×C32×#117C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC2×C3×#117D3
hypocenterC3C3×#2314
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
3. C3×#4980C2semidirect
5. C3×#4936C2semidirect
6. C3×#4935C2semidirect
7. C3×C4×C4⋊Dic3C2semidirect
9. #82231C3direct (N × H)
11. C32×#117C4semidirect
34. #4980C2×C3semidirect
36. #4936C2×C3semidirect
37. #4935C2×C3semidirect
38. C4×C4⋊Dic3C2×C3semidirect
63. C3×#117Dic3semidirect
64. C3×#117C3×C4semidirect
192. #117C3×Dic3semidirect
345. C32#2314semidirect
406. C3#82231direct (N × H)
407. C3C3×#2314semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C3×#52122
C3×#49802
C3×#49512
C3×#49362
C3×#49352
C3×C4×C4⋊Dic32
C3×#23143
#822313