Finite Group Databasebeta

← #74316⋊C3 #11921155

Subgroups — #74316⋊C3 #11921155

Order 1152.

299 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 31 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
1C2×C3⋊(#126⋊C3)C211yesmaximal, characteristic, complemented
2C2×C3⋊(#127⋊C3)C211yesmaximal, characteristic, complemented
3C3×#383⋊C3C211yesmaximal, characteristic, complemented
4#74316C311yesmaximal, characteristic, complemented
5#2818⋊C3—31nomaximal
6A4×D12—41nomaximal
7C2×C3×C42⋊C3C2211yescharacteristic
8C3⋊(#126⋊C3)—21no—
9C3⋊(#127⋊C3)—21no—
10C2×#876C2×C311yescharacteristic, complemented
11C2×#874C2×C311yescharacteristic, complemented
12C2×#127⋊C3—31no—
13C2×#126⋊C3—31no—
14C3×#383C2×C311yescharacteristic, complemented
15#4672—31no—
16#4672—31no—
17#4671—31no—
18#4671—31no—
19#383⋊C3—21no—
20#383⋊C3D311yescharacteristic, complemented
21C2×D3×A4—41no—
22C2×D3×A4—41no—
23C3×C4×A4—41no—
24C3×C42⋊C3D411yescharacteristic, complemented
25#2818—31noSylow
26C22×D12—31no—
27C22×D12—31no—
28C22×D12—31no—
29C22×D12A411yescharacteristic
30D4×A4—121no—
31C2×C3×C42C22×C311yescharacteristic
32C2×(C2×D3)⋊C4—31no—
33C2×(C2×D3)⋊C4—31no—
34C2×(C2×D3)⋊C4—31no—
35C2×(C2×D3)⋊C4—31no—
36C2×(C2×D3)⋊C4—31no—
37C2×Dic3⋊C4—31no—
38#876—21no—
39#876—61no—
40#874—21no—
41#874—61no—
42#127⋊C3—61no—
43#126⋊C3—61no—
44C2×C42⋊C3—21no—
45C2×C42⋊C3C2×D311yescharacteristic
46C3×#95—31no—
47C3×#95—31no—
48C2×C3×A4—41no—
49D3×A4—81no—
50D3×A4—81no—
51C3×D12—161no—
52C2×#127—31no—
53C2×#126—31no—
54#394—91no—
55#394—91no—
56#392—91no—
57#392—91no—
58#383C3×D311yescharacteristic, complemented
59C23×D3—31no—
60C23×D3C2×A411yescharacteristic
61C23×D3C2×A411yescharacteristic
62C22×A4—121no—
63C22×A4—121no—
64C22×C3×C4—31no—
65C22×C3×C4—31no—
66C22×C3×C4C2×A411yescharacteristic
67C22×Dic3—31no—
68C2×D12—61no—
69C2×D12—61no—
70C2×D12—61no—
71C2×D12—61no—
72C2×D12—121no—
73C2×D12—121no—
74C2×D12—121no—
75C2×D12—121no—
76C2×D12—121no—
77C2×D12—121no—
78C2×D12—121no—
79C2×D12—121no—
80C4×A4—41no—
81C4×A4—81no—
82C3×C42—31no—
83C3×C42C3×D411yescharacteristic, complemented
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—61no—
86(C2×D3)⋊C4—121no—
87(C2×D3)⋊C4—121no—
88(C2×D3)⋊C4—121no—
89(C2×D3)⋊C4—121no—
90Dic3⋊C4—61no—
91Dic3⋊C4—61no—
92C42⋊C3—21no—
93C42⋊C3D1211yescharacteristic, complemented
94C2×C3×D3—161no—
95C2×C3×D3—161no—
96C3×A4—41no—
97C32×C4—161no—
98C22×D4—31no—
99C22×D4—91no—
100C22×D4—91no—
101C22×D4—91no—
102#127—61no—
103#127—181no—
104#126—61no—
105#126—181no—
106C2×C4⋊C4—91no—
107C2×C22⋊C4—91no—
108C2×C22⋊C4—91no—
109C2×C22⋊C4—91no—
110C2×C22⋊C4—91no—
111C2×C22⋊C4—91no—
112C2×C42C2×C3×D311yescharacteristic
113#95—31no—
114#95—31no—
115C23×C3C22×A411yescharacteristic
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—61no—
119C22×D3—61no—
120C22×D3—61no—
121C22×D3—61no—
122C22×D3—61no—
123C22×D3—61no—
124C22×D3—61no—
125C22×D3—61no—
126C22×D3—61no—
127C22×D3—61no—
128C22×D3—61no—
129C2×A4—41no—
130C2×A4—81no—
131C2×A4—241no—
132C2×A4—241no—
133C3×D4—481no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×Dic3—61no—
143C2×Dic3—61no—
144C2×Dic3—61no—
145D12—41no—
146D12—121no—
147D12—122no—
148D12—122no—
149D12—122no—
150D12—122no—
151D12—124no—
152C2×C32—161no—
153C3×D3—321no—
154C3×D3—321no—
155C24—31no—
156C24—31no—
157C24—91no—
158C2×D4—181no—
159C2×D4—181no—
160C2×D4—181no—
161C2×D4—181no—
162C2×D4—361no—
163C2×D4—361no—
164C2×D4—361no—
165C2×D4—361no—
166C2×D4—361no—
167C2×D4—361no—
168C2×D4—361no—
169C2×D4—361no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—91no—
173C22×C4D3×A411yescharacteristic
174C4⋊C4—181no—
175C4⋊C4—181no—
176C22⋊C4—181no—
177C22⋊C4—181no—
178C22⋊C4—361no—
179C22⋊C4—361no—
180C22⋊C4—361no—
181C22⋊C4—361no—
182C42—31no—
183C42C3×D1211yescharacteristic, complemented
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—481no—
187C22×C3—481no—
188C22×C3D4×A411yescharacteristic
189C2×D3—41no—
190C2×D3—41no—
191C2×D3—121no—
192C2×D3—121no—
193C2×D3—121no—
194C2×D3—121no—
195C2×D3—121no—
196C2×D3—121no—
197C2×D3—121no—
198C2×D3—121no—
199C2×D3—121no—
200C2×D3—121no—
201C2×D3—121no—
202A4—41no—
203A4—81no—
204C3×C4—41no—
205C3×C4—61no—
206C3×C4—61no—
207C3×C4—121no—
208C3×C4—161no—
209C3×C4—321no—
210Dic3—121no—
211C32—161noSylow
212C23—61no—
213C23—61no—
214C23—181no—
215C23—181no—
216C23—181no—
217C23—181no—
218C23—181no—
219C23—181no—
220C23—181no—
221C23—181no—
222C23—181no—
223C23—181no—
224C23—181no—
225C23C2×D3×A411yescharacteristic
226D4—121no—
227D4—361no—
228D4—362no—
229D4—362no—
230D4—362no—
231D4—362no—
232D4—364no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—181no—
242C2×C4—181no—
243C2×C4—181no—
244C2×C3—31no—
245C2×C3—31no—
246C2×C3—161no—
247C2×C3—321no—
248C2×C3—961no—
249C2×C3—961no—
250C2×C3#560⋊C311yescharacteristic
251D3—81no—
252D3—81no—
253D3—241no—
254C22—31no—
255C22—31no—
256C22—121no—
257C22—121no—
258C22—361no—
259C22—361no—
260C22—361no—
261C22—361no—
262C22—361no—
263C22—361no—
264C22—361no—
265C22—361no—
266C22—361no—
267C22—361no—
268C22—361no—
269C22A4×D1211yescharacteristic
270C4—41no—
271C4—61no—
272C4—61no—
273C4—121no—
274C4—361no—
275C3—161no—
276C3—321no—
277C3#2818⋊C311yescharacteristic, complemented
278C2—31no—
279C2—31no—
280C2—241no—
281C2—241no—
282C2—721no—
283C2(#126⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2(#126⋊C3)⋊D3
commutator, cosocleC2×C3×C42C22×C3
FrattiniC23C2×D3×A4
FittingC3×#383C2×C3
socleC23×C3C22×A4
hypocenterC3×C42C3×D4
LCSG ▷ C2×C3×C42 ▷ C3×C42
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(#126⋊C3)C2semidirect
2. C2×C3⋊(#127⋊C3)C2semidirect
3. C3×#383⋊C3C2semidirect
4. #74316C3semidirect
10. C2×#876C2×C3semidirect
11. C2×#874C2×C3semidirect
14. C3×#383C2×C3semidirect
20. #383⋊C3D3semidirect
24. C3×C42⋊C3D4semidirect
58. #383C3×D3semidirect
83. C3×C42C3×D4semidirect
93. C42⋊C3D12semidirect
183. C42C3×D12semidirect
277. C3#2818⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#126⋊C3)2
C2×C3⋊(#127⋊C3)2
C3×#383⋊C32
#743163
#2818⋊C33
A4×D124