Finite Group Databasebeta

← #67×C42⋊C3 #11921250

Subgroups — #67×C42⋊C3 #11921250

Order 1152.

403 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C42⋊C3C211yesmaximal, characteristic, complemented
2C2×D3×C42⋊C3C211yesmaximal, characteristic, complemented
3Dic3×C42⋊C3C211yesmaximal, characteristic, complemented
4C42×#67C311yesmaximal, characteristic, complemented
5D4×C42⋊C3—31nomaximal
6A4×#67—41nomaximal
7C2×C3×C42⋊C3—21no—
8C2×C3×C42⋊C3C2211yescharacteristic
9D3×C42⋊C3—21no—
10C22×C3×C42C2×C311yescharacteristic, complemented
11C2×C4×#67—31no—
12C2×C42×D3C2×C311yescharacteristic, complemented
13C22×C42⋊C3—21no—
14C22×C42⋊C3—31no—
15C22×C42⋊C3D311yescharacteristic, complemented
16C4×#217—31no—
17C4×(C2×D3)⋊C4—31no—
18C4×Dic3⋊C4—31no—
19C42×Dic3C2×C311yescharacteristic, complemented
20C4×C42⋊C3—31no—
21C22×C3×A4—41no—
22C2×D3×A4—41no—
23Dic3×A4—41no—
24C3×C42⋊C3D411yescharacteristic, complemented
25C42×D4—31noSylow
26C23×C3×C4—31no—
27C22×#67A411yescharacteristic
28C22×C4×D3—31no—
29D4×A4—121no—
30C2×C3×C42—21no—
31C2×C3×C42—31no—
32C2×C3×C42—61no—
33C2×C3×C42C22×C311yescharacteristic
34C2×#217—31no—
35C4×#67—32no—
36C4×#67—32no—
37C4×#67—32no—
38C4×#67—32no—
39C2×(C2×D3)⋊C4—31no—
40C2×Dic3⋊C4—31no—
41C2×C4×Dic3—31no—
42C2×C4×Dic3—31no—
43C2×C4×Dic3—31no—
44C42×D3—21no—
45C42×D3—61no—
46C2×C42⋊C3—22no—
47C2×C42⋊C3—21no—
48C2×C42⋊C3—21no—
49C2×C42⋊C3—61no—
50C2×C42⋊C3C2×D311yescharacteristic
51C3⋊#95—32no—
52C2×C3×A4—41no—
53C2×C3×A4—81no—
54D3×A4—81no—
55C3×#67—161no—
56C2×C4×D4—91no—
57C22×C42—31no—
58C22×C42C3×D311yescharacteristic, complemented
59C4×C4⋊C4—91no—
60C4×C22⋊C4—91no—
61C4×C22⋊C4—91no—
62C43—31no—
63C24×C3C2×A411yescharacteristic
64C23×D3C2×A411yescharacteristic
65C22×A4—41no—
66C22×A4—81no—
67C22×A4—121no—
68C22×C3×C4—31no—
69C22×C3×C4—32no—
70C22×C3×C4—32no—
71C22×C3×C4—31no—
72C22×C3×C4—61no—
73C22×C3×C4—61no—
74C22×C3×C4—62no—
75C2×#67—31no—
76C2×#67—31no—
77C2×#67—31no—
78C2×#67—31no—
79C22×Dic3—31no—
80C22×Dic3—31no—
81C22×Dic3C2×A411yescharacteristic
82C2×C4×D3—32no—
83C2×C4×D3—32no—
84C2×C4×D3—61no—
85C2×C4×D3—61no—
86C2×C4×D3—62no—
87C4×A4—121no—
88C3×C42—31no—
89C3×C42—61no—
90C3×C42—61no—
91C3×C42C3×D411yescharacteristic, complemented
92#217—32no—
93#217—32no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98C4×Dic3—32no—
99C4×Dic3—32no—
100C4×Dic3—62no—
101C4×Dic3—61no—
102C4×Dic3—61no—
103C42⋊C3—21no—
104C42⋊C3#6711yescharacteristic, direct product factor
105C22×C32—161no—
106C2×C3×D3—161no—
107C3×A4—41no—
108C3×Dic3—161no—
109C22×D4—31no—
110C23×C4—31no—
111C23×C4—91no—
112C4×D4—92no—
113C4×D4—92no—
114C4×D4—92no—
115C4×D4—92no—
116C2×C4⋊C4—91no—
117C2×C22⋊C4—91no—
118C2×C22⋊C4—91no—
119C2×C42—21no—
120C2×C42—31no—
121C2×C42—61no—
122C2×C42—61no—
123C2×C42—91no—
124C2×C42—91no—
125C2×C42—91no—
126C2×C42—181no—
127C2×C42C2×C3×D311yescharacteristic
128#95—92no—
129C23×C3—21no—
130C23×C3—31no—
131C23×C3—31no—
132C23×C3—61no—
133C23×C3C22×A411yescharacteristic
134C22×D3—21no—
135C22×D3—31no—
136C22×D3—31no—
137C22×D3—61no—
138C2×A4—41no—
139C2×A4—81no—
140C2×A4—82no—
141C2×A4—81no—
142C2×A4—241no—
143C3×D4—481no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—32no—
149C2×C3×C4—62no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155#67—32no—
156#67—31no—
157#67—31no—
158#67—31no—
159#67C42⋊C311yescharacteristic, direct product factor
160C2×Dic3—31no—
161C2×Dic3—31no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—62no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C4×D3—62no—
168C4×D3—62no—
169C4×D3—62no—
170C4×D3—62no—
171C2×C32—161no—
172C2×C32—321no—
173C3×D3—321no—
174C24—31no—
175C24D3×A411yescharacteristic
176C2×D4—91no—
177C2×D4—91no—
178C2×D4—91no—
179C2×D4—91no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—32no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—62no—
188C22×C4—92no—
189C22×C4—92no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—181no—
193C22×C4—181no—
194C22×C4—182no—
195C4⋊C4—92no—
196C4⋊C4—92no—
197C22⋊C4—92no—
198C22⋊C4—92no—
199C22⋊C4—92no—
200C22⋊C4—92no—
201C42—31no—
202C42—61no—
203C42—61no—
204C42—92no—
205C42—92no—
206C42—181no—
207C42—181no—
208C42—182no—
209C42C3×#6711yescharacteristic, complemented
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3—161no—
218C22×C3—321no—
219C22×C3—481no—
220C22×C3C2×C42⋊C311yescharacteristic, complemented
221C22×C3D4×A411yescharacteristic
222C2×D3—31no—
223C2×D3—61no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3C2×C42⋊C311yescharacteristic, complemented
228A4—41no—
229A4—81no—
230C3×C4—32no—
231C3×C4—32no—
232C3×C4—62no—
233C3×C4—481no—
234Dic3—31no—
235Dic3—62no—
236Dic3—62no—
237Dic3C2×C42⋊C311yescharacteristic, complemented
238C32—161noSylow
239C23—21no—
240C23—31no—
241C23—31no—
242C23—61no—
243C23—61no—
244C23—91no—
245C23—91no—
246C23—181no—
247C23C2×D3×A411yescharacteristic
248D4—31no—
249D4—91no—
250D4—91no—
251D4—92no—
252D4—91no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—32no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—92no—
267C2×C4—92no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C3—21no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—61no—
279C2×C3—161no—
280C2×C3—321no—
281C2×C3—322no—
282C2×C3—321no—
283C2×C3—961no—
284C2×C3C22×C42⋊C311yescharacteristic
285D3—21no—
286D3—61no—
287C22—31no—
288C22—31no—
289C22—31no—
290C22—31no—
291C22—61no—
292C22—61no—
293C22—61no—
294C22—61no—
295C22—91no—
296C22—181no—
297C22—181no—
298C22—181no—
299C22—181no—
300C22D3×C42⋊C311yescharacteristic, complemented
301C22A4×#6711yescharacteristic
302C4—32no—
303C4—32no—
304C4—31no—
305C4—62no—
306C4—91no—
307C4—182no—
308C4—182no—
309C3—161no—
310C3—321no—
311C3D4×C42⋊C311yescharacteristic, complemented
312C2—21no—
313C2—31no—
314C2—31no—
315C2—61no—
316C2—61no—
317C2—181no—
318C2C2×D3×C42⋊C311yescharacteristic
subgroupG/H
centerC2C2×D3×C42⋊C3
commutator, cosocleC2×C3×C42C22×C3
FrattiniC23C2×D3×A4
FittingC22×C3×C42C2×C3
socleC23×C3C22×A4
hypercenterC22D3×C42⋊C3
hypocenterC3×C42C3×D4
LCSG ▷ C2×C3×C42 ▷ C3×C42
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×C42⋊C3C2semidirect
2. C2×D3×C42⋊C3C2semidirect
3. Dic3×C42⋊C3C2semidirect
4. C42×#67C3semidirect
10. C22×C3×C42C2×C3semidirect
12. C2×C42×D3C2×C3semidirect
15. C22×C42⋊C3D3semidirect
19. C42×Dic3C2×C3semidirect
24. C3×C42⋊C3D4semidirect
58. C22×C42C3×D3semidirect
91. C3×C42C3×D4semidirect
104. C42⋊C3#67direct (N × H)
159. #67C42⋊C3direct (N × H)
209. C42C3×#67semidirect
220. C22×C3C2×C42⋊C3semidirect
227. C2×D3C2×C42⋊C3semidirect
237. Dic3C2×C42⋊C3semidirect
300. C22D3×C42⋊C3semidirect
311. C3D4×C42⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C42⋊C32
C2×D3×C42⋊C32
Dic3×C42⋊C32
C42×#673
D4×C42⋊C33
A4×#674