Finite Group Databasebeta

← #82081⋊C3 #11921251

Subgroups — #82081⋊C3 #11921251

Order 1152.

403 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 33 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×C3⋊(#127⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#384⋊C3)C211yesmaximal, characteristic, complemented
4#82081C311yesmaximal, characteristic, complemented
5#2856⋊C3—31nomaximal
6A4×#67—41nomaximal
7C2×C3×C42⋊C3—21no—
8C2×C3×C42⋊C3C2211yescharacteristic
9C3⋊(#127⋊C3)—21no—
10C22×C3×C42C2×C311yescharacteristic, complemented
11C2×#930—31no—
12C2×#930—31no—
13C2×#874C2×C311yescharacteristic, complemented
14C2×#127⋊C3—31no—
15C22×C42⋊C3—21no—
16C22×C42⋊C3D311yescharacteristic, complemented
17#4939—31no—
18#4939—31no—
19#4935C2×C311yescharacteristic, complemented
20#384⋊C3—31no—
21C22×C3×A4—41no—
22C2×D3×A4—41no—
23Dic3×A4—41no—
24C3×C42⋊C3D411yescharacteristic, complemented
25#2856—31noSylow
26C23×C3×C4—31no—
27C22×#67—31no—
28C22×#67A411yescharacteristic
29C22×D12—31no—
30C22×D12—31no—
31D4×A4—121no—
32C2×C3×C42—21no—
33C2×C3×C42—31no—
34C2×C3×C42—61no—
35C2×C3×C42C22×C311yescharacteristic
36#930—62no—
37#930—62no—
38#930—62no—
39#930—62no—
40C2×(C2×D3)⋊C4—32no—
41C2×(C2×D3)⋊C4—31no—
42C2×(C2×D3)⋊C4—31no—
43C2×C4⋊Dic3—31no—
44C2×C4⋊Dic3—31no—
45#874—21no—
46#874—61no—
47#127⋊C3—61no—
48C2×C42⋊C3—22no—
49C2×C42⋊C3—21no—
50C2×C42⋊C3—21no—
51C2×C42⋊C3C2×D311yescharacteristic
52C2×C3×A4—41no—
53C2×C3×A4—81no—
54D3×A4—81no—
55C3×#67—161no—
56C2×#127—31no—
57C2×#121—91no—
58C2×#121—91no—
59C22×C42C3×D311yescharacteristic, complemented
60#390—91no—
61#390—91no—
62#384—31no—
63C24×C3C2×A411yescharacteristic
64C23×D3—31no—
65C23×D3C2×A411yescharacteristic
66C22×A4—41no—
67C22×A4—81no—
68C22×A4—121no—
69C22×C3×C4—31no—
70C22×C3×C4—32no—
71C22×C3×C4—32no—
72C22×C3×C4—31no—
73C22×C3×C4—61no—
74C22×C3×C4—61no—
75C22×C3×C4—62no—
76C2×#67—61no—
77C2×#67—62no—
78C2×#67—61no—
79C2×#67—61no—
80C2×#67—62no—
81C2×#67—61no—
82C22×Dic3—31no—
83C22×Dic3C2×A411yescharacteristic
84C2×D12—61no—
85C2×D12—61no—
86C2×D12—61no—
87C2×D12—61no—
88C2×D12—62no—
89C2×D12—62no—
90C2×D12—62no—
91C2×D12—62no—
92C4×A4—121no—
93C3×C42—31no—
94C3×C42—61no—
95C3×C42—61no—
96C3×C42C3×D411yescharacteristic, complemented
97(C2×D3)⋊C4—62no—
98(C2×D3)⋊C4—62no—
99(C2×D3)⋊C4—64no—
100C4⋊Dic3—62no—
101C4⋊Dic3—62no—
102C42⋊C3—21no—
103C42⋊C3#6711yescharacteristic, complemented
104C22×C32—161no—
105C2×C3×D3—161no—
106C3×A4—41no—
107C3×Dic3—161no—
108C22×D4—31no—
109C22×D4—91no—
110C22×D4—91no—
111C22×D4—91no—
112C23×C4—31no—
113#127—61no—
114#127—181no—
115#121—182no—
116#121—182no—
117#121—182no—
118#121—182no—
119C2×C4⋊C4—91no—
120C2×C4⋊C4—91no—
121C2×C22⋊C4—91no—
122C2×C22⋊C4—91no—
123C2×C22⋊C4—92no—
124C2×C42—21no—
125C2×C42—31no—
126C2×C42—61no—
127C2×C42C2×C3×D311yescharacteristic
128C23×C3—21no—
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C23×C3C22×A411yescharacteristic
133C22×D3—21no—
134C22×D3—61no—
135C22×D3—61no—
136C22×D3—62no—
137C22×D3—62no—
138C22×D3—61no—
139C22×D3—61no—
140C22×D3—61no—
141C2×A4—41no—
142C2×A4—81no—
143C2×A4—82no—
144C2×A4—81no—
145C2×A4—241no—
146C3×D4—481no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—32no—
152C2×C3×C4—62no—
153C2×C3×C4—62no—
154C2×C3×C4—62no—
155C2×C3×C4—62no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158#67—41no—
159#67—121no—
160#67—121no—
161#67—122no—
162#67—121no—
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×Dic3—62no—
166D12—122no—
167D12—122no—
168D12—122no—
169D12—122no—
170C2×C32—161no—
171C2×C32—321no—
172C3×D3—321no—
173C24—31no—
174C24—91no—
175C24D3×A411yescharacteristic
176C2×D4—181no—
177C2×D4—181no—
178C2×D4—181no—
179C2×D4—181no—
180C2×D4—182no—
181C2×D4—182no—
182C2×D4—182no—
183C2×D4—182no—
184C2×D4—181no—
185C2×D4—182no—
186C2×D4—181no—
187C2×D4—181no—
188C2×D4—182no—
189C2×D4—181no—
190C22×C4—31no—
191C22×C4—32no—
192C22×C4—32no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—62no—
198C22×C4—91no—
199C4⋊C4—182no—
200C4⋊C4—182no—
201C22⋊C4—182no—
202C22⋊C4—182no—
203C22⋊C4—184no—
204C42—31no—
205C42—61no—
206C42—61no—
207C42C3×#6711yescharacteristic, complemented
208C22×C3—31no—
209C22×C3—31no—
210C22×C3—31no—
211C22×C3—61no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—161no—
216C22×C3—321no—
217C22×C3—481no—
218C22×C3#127⋊C311yescharacteristic, complemented
219C22×C3D4×A411yescharacteristic
220C2×D3—41no—
221C2×D3—121no—
222C2×D3—121no—
223C2×D3—121no—
224C2×D3—122no—
225C2×D3—121no—
226C2×D3—122no—
227C2×D3—121no—
228A4—41no—
229A4—81no—
230C3×C4—32no—
231C3×C4—32no—
232C3×C4—62no—
233C3×C4—481no—
234Dic3—41no—
235Dic3—121no—
236C32—161noSylow
237C23—21no—
238C23—31no—
239C23—31no—
240C23—61no—
241C23—61no—
242C23—181no—
243C23—181no—
244C23—181no—
245C23—182no—
246C23—181no—
247C23—182no—
248C23—181no—
249C23C2×D3×A411yescharacteristic
250D4—121no—
251D4—362no—
252D4—362no—
253D4—362no—
254D4—362no—
255D4—361no—
256D4—361no—
257D4—362no—
258D4—361no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—182no—
273C2×C3—21no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—61no—
277C2×C3—161no—
278C2×C3—321no—
279C2×C3—322no—
280C2×C3—321no—
281C2×C3—961no—
282C2×C3C2×#127⋊C311yescharacteristic
283D3—81no—
284D3—241no—
285C22—31no—
286C22—31no—
287C22—31no—
288C22—61no—
289C22—61no—
290C22—61no—
291C22—61no—
292C22—121no—
293C22—361no—
294C22—361no—
295C22—361no—
296C22—362no—
297C22—361no—
298C22—362no—
299C22—361no—
300C22C3⋊(#127⋊C3)11yescharacteristic, complemented
301C22A4×#6711yescharacteristic
302C4—32no—
303C4—32no—
304C4—62no—
305C4—121no—
306C4—361no—
307C3—161no—
308C3—321no—
309C3#2856⋊C311yescharacteristic, complemented
310C2—21no—
311C2—31no—
312C2—31no—
313C2—61no—
314C2—241no—
315C2—721no—
316C2C2×C3⋊(#127⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#127⋊C3)
commutator, cosocleC2×C3×C42C22×C3
FrattiniC23C2×D3×A4
FittingC22×C3×C42C2×C3
socleC23×C3C22×A4
hypercenterC22C3⋊(#127⋊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×C3⋊(#127⋊C3)C2semidirect
3. C3⋊(#384⋊C3)C2semidirect
4. #82081C3semidirect
10. C22×C3×C42C2×C3semidirect
13. C2×#874C2×C3semidirect
16. C22×C42⋊C3D3semidirect
19. #4935C2×C3semidirect
24. C3×C42⋊C3D4semidirect
59. C22×C42C3×D3semidirect
96. C3×C42C3×D4semidirect
103. C42⋊C3#67semidirect
207. C42C3×#67semidirect
218. C22×C3#127⋊C3semidirect
300. C22C3⋊(#127⋊C3)semidirect
309. C3#2856⋊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×C3⋊(#127⋊C3)2
C3⋊(#384⋊C3)2
#820813
#2856⋊C33
A4×#674