Finite Group Databasebeta

← C3×#74224 #11802847

Subgroups — C3×#74224 #11802847

Order 1152.

1045 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic
2C32×#382C211yesmaximal, characteristic, complemented
3C3×#4987C212yesmaximal
4C3×Dic3×C4⋊C4C212yesmaximal, complemented
5C3×#4935C211yesmaximal, characteristic, complemented
6C3×#4670C214yesmaximal
7C3×#4655C214yesmaximal, complemented
8C3×#2314—31nomaximal
9#74224C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×D3C2211yescharacteristic
11C2×C32×C4⋊C4C2212yes—
12C2×C32×C42C2211yescharacteristic
13C2×C3×(C2×D3)⋊C4C2214yes—
14C2×C3×(C2×D3)⋊C4C2212yes—
15C2×C3×C4⋊Dic3C2212yes—
16C2×C3×C4⋊Dic3C2214yes—
17C2×C3×Dic3⋊C4C2212yes—
18C2×C3×C4×Dic3C2214yes—
19C2×C3×C4×Dic3C2211yescharacteristic
20C3×#872C418yescomplemented
21C32×#95C2214yes—
22C3×C3⋊#95C2214yes—
23C3×C3⋊#95C2214yes—
24C2×C3×#117—31no—
25C2×#872C2×C311yescharacteristic
26C3×#388—34no—
27C3×#387—34no—
28C3×#384—31no—
29C3×#382—21no—
30C3×#382D311yescharacteristic, complemented
31C3×#382C2×C311yescharacteristic, complemented
32C3×#380—32no—
33C3×C4×C4⋊C4—32no—
34#4987C2×C312yes—
35Dic3×C4⋊C4C2×C312yescomplemented
36#4935C2×C311yescharacteristic, complemented
37#4670C2×C314yes—
38#4655C2×C314yescomplemented
39C23×C3×D3C2311yescharacteristic
40C22×C32×C4C2314yes—
41C22×C32×C4C2312yes—
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3C2314yes—
44C22×C3×Dic3C2312yes—
45C22×C3×Dic3C2311yescharacteristic
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3—22no—
48C2×C3×C4×D3C2×C414yes—
49C32×C4⋊C4—24no—
50C32×C42C2×C414yes—
51C3×(C2×D3)⋊C4—44no—
52C3×(C2×D3)⋊C4C2×C418yes—
53C3×C4⋊Dic3—24no—
54C3×C4⋊Dic3—44no—
55C3×Dic3⋊C4C2×C418yes—
56C3×C4×Dic3—44no—
57C3×C4×Dic3C2×C414yes—
58#2314—31noSylow
59C23×C3×C4—31no—
60C22×C4×D3C22×C311yescharacteristic
61C3×#117—38no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—32no—
64C2×C3×C4⋊C4—32no—
65C2×C3×C4⋊C4—34no—
66C2×C3×C4⋊C4C2×D312yes—
67C2×C3×C4⋊C4C22×C312yes—
68C2×C3×C22⋊C4—32no—
69C2×C3×C22⋊C4—34no—
70C2×C3×C42—21no—
71C2×C3×C42—31no—
72C2×C3×C42—34no—
73C2×C3×C42C2×D311yescharacteristic
74C2×C3×C42C22×C311yescharacteristic
75C2×(C2×D3)⋊C4C22×C314yes—
76C2×(C2×D3)⋊C4C22×C312yes—
77C2×C4⋊Dic3C22×C312yes—
78C2×C4⋊Dic3C22×C314yes—
79C2×Dic3⋊C4C22×C312yes—
80C2×C4×Dic3C22×C314yes—
81C2×C4×Dic3C22×C311yescharacteristic
82#872C3×C418yescomplemented
83C3×#95—24no—
84C3×#95—34no—
85C3×#95—34no—
86C3×#95C2×D314yes—
87C3×#95C22×C314yes—
88C3⋊#95C22×C314yes—
89C3⋊#95C22×C314yes—
90C23×C32C2411yescharacteristic
91C22×C3×D3—21no—
92C22×C3×D3—21no—
93C22×C3×D3—22no—
94C22×C3×D3—22no—
95C22×C3×D3C22×C412yes—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—24no—
99C2×C32×C4—24no—
100C2×C32×C4—24no—
101C2×C32×C4C22×C414yes—
102C2×C32×C4C22×C412yes—
103C2×C32×C4#4112yes—
104C2×C32×C4#4112yes—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—24no—
110C2×C3×Dic3—24no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3C22×C414yes—
113C2×C3×Dic3C22×C412yes—
114C3×C4×D3—28no—
115C2×#117—31no—
116#388—34no—
117#387—34no—
118#384—31no—
119#382C3×D311yescharacteristic, complemented
120#380—32no—
121C4×C4⋊C4—32no—
122C24×C3—31no—
123C23×D3C23×C311yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—31no—
128C22×C3×C4—32no—
129C22×C3×C4—34no—
130C22×C3×C4—34no—
131C22×C3×C4—62no—
132C22×C3×C4—62no—
133C22×C3×C4C22×D314yes—
134C22×C3×C4C22×D312yes—
135C22×C3×C4C22×D311yescharacteristic
136C22×C3×C4C23×C314yes—
137C22×C3×C4C23×C312yes—
138C22×C3×C4C23×C311yescharacteristic
139C22×Dic3C23×C314yes—
140C22×Dic3C23×C311yescharacteristic
141C22×Dic3C23×C312yes—
142C2×C4×D3—22no—
143C2×C4×D3—22no—
144C2×C4×D3C2×C3×C414yes—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—38no—
148C3×C4⋊C4—44no—
149C3×C4⋊C4—64no—
150C3×C4⋊C4—124no—
151C3×C22⋊C4—38no—
152C3×C22⋊C4—124no—
153C3×C42—24no—
154C3×C42—34no—
155C3×C42—124no—
156C3×C42C4×D314yes—
157C3×C42C2×C3×C414yes—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4C2×C3×C418yes—
160C4⋊Dic3—24no—
161C4⋊Dic3—44no—
162Dic3⋊C4C2×C3×C418yes—
163C4×Dic3—44no—
164C4×Dic3C2×C3×C414yes—
165C22×C32C23×C411yescharacteristic
166C22×C32C2×#4111yescharacteristic
167C22×C32C2×#4111yescharacteristic
168C22×C32H(5,𝔽2)12yes—
169C22×C322−1+412yes—
170C2×C3×D3—22no—
171C2×C3×D3—24no—
172C2×C3×D3—41no—
173C2×C3×D3—41no—
174C2×C3×D3—42no—
175C32×C4—24no—
176C32×C4—44no—
177C32×C4#11714yes—
178C3×Dic3—22no—
179C3×Dic3—24no—
180C3×Dic3—44no—
181C23×C4—31no—
182#117—38no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—34no—
186C2×C4⋊C4C2×C3×D312yes—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—34no—
189C2×C42—31no—
190C2×C42—34no—
191C2×C42C2×C3×D311yescharacteristic
192#95—34no—
193#95—34no—
194#95C2×C3×D314yes—
195C23×C3—21no—
196C23×C3—32no—
197C23×C3—61no—
198C23×C3—61no—
199C23×C3—62no—
200C23×C3—62no—
201C23×C3C23×D311yescharacteristic
202C23×C3C24×C311yescharacteristic
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3C22×C3×C412yes—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—34no—
223C2×C3×C4—32no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—44no—
227C2×C3×C4—44no—
228C2×C3×C4—44no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—68no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4—64no—
237C2×C3×C4C2×C4×D314yes—
238C2×C3×C4C2×C4×D312yes—
239C2×C3×C4#23712yes—
240C2×C3×C4#23912yes—
241C2×C3×C4C22×C3×C414yes—
242C2×C3×C4C22×C3×C412yes—
243C2×C3×C4C3×#4112yes—
244C2×C3×C4C3×#4112yes—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—22no—
248C2×Dic3—22no—
249C2×Dic3—24no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3C22×C3×C414yes—
253C2×Dic3C22×C3×C412yes—
254C4×D3—28no—
255C2×C32C2×#11711yescharacteristic
256C2×C32#52012yes—
257C2×C32#54011yescharacteristic
258C2×C32#54111yescharacteristic
259C2×C32#54212yes—
260C3×D3—42no—
261C24—31no—
262C22×C4—31no—
263C22×C4—32no—
264C22×C4—34no—
265C22×C4—34no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4C22×C3×D314yes—
269C22×C4C22×C3×D312yes—
270C22×C4C22×C3×D311yescharacteristic
271C4⋊C4—24no—
272C4⋊C4—38no—
273C4⋊C4—64no—
274C4⋊C4—124no—
275C22⋊C4—38no—
276C22⋊C4—124no—
277C42—34no—
278C42—124no—
279C42C3×C4×D314yes—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—62no—
286C22×C3—64no—
287C22×C3—122no—
288C22×C3—121no—
289C22×C3—121no—
290C22×C3C22×C4×D311yescharacteristic
291C22×C3C2×#23711yescharacteristic
292C22×C3C2×#23911yescharacteristic
293C22×C3#100912yes—
294C22×C3#101012yes—
295C22×C3C23×C3×C411yescharacteristic
296C22×C3C2×C3×#4111yescharacteristic
297C22×C3C2×C3×#4111yescharacteristic
298C22×C3C3×H(5,𝔽2)12yes—
299C22×C3C3×2−1+412yes—
300C2×D3—22no—
301C2×D3—24no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—42no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—44no—
311C3×C4—62no—
312C3×C4—64no—
313C3×C4—84no—
314C3×C4—124no—
315C3×C4#89214yes—
316C3×C4C3×#11714yes—
317Dic3—22no—
318Dic3—24no—
319Dic3—44no—
320C32#231411yescharacteristic, complemented, Sylow
321C23—32no—
322C23—61no—
323C23—61no—
324C23—62no—
325C23—62no—
326C23C23×C3×D311yescharacteristic
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—34no—
333C2×C4—32no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—68no—
339C2×C4—64no—
340C2×C4—64no—
341C2×C4—64no—
342C2×C4C2×C3×C4×D314yes—
343C2×C4C2×C3×C4×D312yes—
344C2×C4C3×#23712yes—
345C2×C4C3×#23912yes—
346C2×C3—21no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—122no—
352C2×C3C2×#89211yescharacteristic
353C2×C3#552312yes—
354C2×C3#568212yes—
355C2×C3#569711yescharacteristic
356C2×C3#569811yescharacteristic
357C2×C3C2×C3×#11711yescharacteristic
358C2×C3C3×#52012yes—
359C2×C3C3×#54011yescharacteristic
360C2×C3C3×#54111yescharacteristic
361C2×C3C3×#54212yes—
362D3—42no—
363C22—62no—
364C22—64no—
365C22—122no—
366C22—121no—
367C22—121no—
368C22C22×C3×C4×D311yescharacteristic
369C22C2×C3×#23711yescharacteristic
370C22C2×C3×#23911yescharacteristic
371C22C3×#100912yes—
372C22C3×#101012yes—
373C4—24no—
374C4—44no—
375C4—62no—
376C4—64no—
377C4—124no—
378C4C3×#89214yes—
379C3—21no—
380C3#7422411yescharacteristic, direct product factor
381C3C3×#231411yescharacteristic, complemented
382C2—122no—
383C2C2×C3×#89211yescharacteristic
384C2C3×#552312yes—
385C2C3×#568212yes—
386C2C3×#569711yescharacteristic
387C2C3×#569811yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#382C2
socleC23×C32C24
prime radicalC23×C3×D3C23
cosocleC23×C3C24×C3
hypercenterC3×#382D3
hypocenterC3C3×#2314
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#382

Complements

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

NH ≅ G/Ntype
2. C32×#382C2semidirect
4. C3×Dic3×C4⋊C4C2semidirect
5. C3×#4935C2semidirect
7. C3×#4655C2semidirect
9. #74224C3direct (N × H)
20. C3×#872C4semidirect
30. C3×#382D3semidirect
31. C3×#382C2×C3semidirect
35. Dic3×C4⋊C4C2×C3semidirect
36. #4935C2×C3semidirect
38. #4655C2×C3semidirect
82. #872C3×C4semidirect
119. #382C3×D3semidirect
320. C32#2314semidirect
380. C3#74224direct (N × H)
381. C3C3×#2314semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C32×#3822
C3×#49872
C3×Dic3×C4⋊C42
C3×#49352
C3×#46702
C3×#46552
C3×#23143
#742243