Finite Group Databasebeta

← #84126⋊C3 #11921384

Subgroups — #84126⋊C3 #11921384

Order 1152.

362 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 27 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
1C3×#5950C211yesmaximal, characteristic, complemented
2C3⋊(B(4,𝔽2)⋊C3)C212yesmaximal, complemented
3#84126C311yesmaximal, characteristic, complemented
4#3027⋊C3—31nomaximal
5C3×#997C2211yescharacteristic
6A4×D12—41nomaximal
7C3×#585C2×C311yescharacteristic, complemented
8#5950—21no—
9#5950D311yescharacteristic, complemented
10#5830—31no—
11#5244—32no—
12#5244C2×C312yescomplemented
13#5240—31no—
14B(4,𝔽2)⋊C3—32no—
15C2×D3×A4—42no—
16C3×#41⋊C3—81no—
17C3×C4×A4—41no—
18C3×C4×A4—81no—
19#3027—31noSylow
20C3×2−1+4—61no—
21C3×H(5,𝔽2)—31no—
22C3×H(5,𝔽2)—61no—
23C3×H(5,𝔽2)C22×C311yescharacteristic
24C2×C3×#41—31no—
25C2×C3×#41—61no—
26C2×C3×#41—61no—
27C22×D12A411yescharacteristic, complemented
28#997—21no—
29#997C2×D311yescharacteristic
30D4×A4—121no—
31#947—61no—
32#940—61no—
33#937—32no—
34#937—32no—
35#930—61no—
36#930—61no—
37#926—31no—
38#834—32no—
39#834—32no—
40C2×C3×A4—41no—
41C2×C3×A4—81no—
42D3×A4—82no—
43C3×D12—161no—
44C3×SL(2,𝔽3)—81no—
45#585C3×D311yescharacteristic, complemented
46#535—91no—
47B(4,𝔽2)—32no—
48B(4,𝔽2)—92no—
49#410—91no—
50C23×D3C2×A412yes—
51C22×A4—122no—
52C3×#41—21no—
53C3×#41—62no—
54C3×#41—62no—
55C3×#41—61no—
56C3×#41—62no—
57C3×#41—61no—
58C3×#41—62no—
59C3×#41—62no—
60C3×#41—61no—
61C2×C3×Q8—31no—
62C2×C3×Q8—61no—
63C2×C3×Q8—61no—
64C2×C3×D4—31no—
65C2×C3×D4—31no—
66C2×C3×D4—31no—
67C2×C3×D4—61no—
68C2×C3×D4—62no—
69C2×C3×D4—61no—
70C2×C3×D4—61no—
71C2×C3×D4—61no—
72C22×C3×C4—21no—
73C22×C3×C4—31no—
74C22×C3×C4—31no—
75C22×C3×C4—61no—
76C22×C3×C4C2×A411yescharacteristic
77C2×#67—62no—
78C2×#67—62no—
79C2×D12—61no—
80C2×D12—61no—
81#41⋊C3—81no—
82#41⋊C3—82no—
83C4×A4—41no—
84C4×A4—81no—
85C4×A4—82no—
86C4×A4—81no—
87#217—32no—
88(C2×D3)⋊C4—62no—
89(C2×D3)⋊C4—62no—
90C4⋊Dic3—61no—
91C4×Dic3—61no—
92C2×C3×D3—162no—
93C3×A4—41no—
94C3×A4—81no—
95C32×C4—161no—
962−1+4—61no—
97H(5,𝔽2)—31no—
98H(5,𝔽2)—61no—
99H(5,𝔽2)C2×C3×D311yescharacteristic
100C2×#41—31no—
101C2×#41—61no—
102C2×#41—61no—
103C22×D4—31no—
104#127—181no—
105#124—181no—
106#121—181no—
107#121—181no—
108#120—92no—
109#120—92no—
110#117—91no—
111#99—92no—
112#99—92no—
113C23×C3—21no—
114C23×C3—31no—
115C23×C3—31no—
116C23×C3—61no—
117C23×C3C22×A411yescharacteristic
118C22×D3—62no—
119C22×D3—82no—
120C2×A4—41no—
121C2×A4—81no—
122C2×A4—82no—
123C2×A4—81no—
124C2×A4—242no—
125C3×Q8—21no—
126C3×Q8—62no—
127C3×Q8—61no—
128C3×D4—62no—
129C3×D4—62no—
130C3×D4—62no—
131C3×D4—61no—
132C3×D4—61no—
133C3×D4—62no—
134C3×D4—481no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—31no—
138C2×C3×C4—31no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—62no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146#67—242no—
147#67—242no—
148C2×Dic3—62no—
149D12—41no—
150D12—121no—
151SL(2,𝔽3)—81no—
152SL(2,𝔽3)—82no—
153C2×C32—161no—
154C3×D3—322no—
155C24—32no—
156#41—21no—
157#41—62no—
158#41—62no—
159#41—61no—
160#41—62no—
161#41—61no—
162#41—62no—
163#41—62no—
164#41—61no—
165C2×Q8—31no—
166C2×Q8—61no—
167C2×Q8—61no—
168C2×D4—31no—
169C2×D4—31no—
170C2×D4—31no—
171C2×D4—61no—
172C2×D4—62no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4—181no—
177C2×D4—181no—
178C2×D4—182no—
179C2×D4—182no—
180C22×C4—21no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—61no—
184C22×C4D3×A411yescharacteristic
185C4⋊C4—181no—
186C22⋊C4—92no—
187C22⋊C4—182no—
188C22⋊C4—182no—
189C42—181no—
190C22×C3—31no—
191C22×C3—41no—
192C22×C3—61no—
193C22×C3—61no—
194C22×C3—81no—
195C22×C3—121no—
196C22×C3—121no—
197C22×C3—241no—
198C22×C3—482no—
199C2×D3—42no—
200C2×D3—122no—
201C2×D3—122no—
202A4—41no—
203A4—81no—
204A4—81no—
205A4—82no—
206C3×C4—31no—
207C3×C4—61no—
208C3×C4—61no—
209C3×C4—161no—
210C3×C4—321no—
211C3×C4#120⋊C311yescharacteristic
212Dic3—122no—
213C32—161noSylow
214C23—21no—
215C23—31no—
216C23—31no—
217C23—61no—
218C23—182no—
219C23—242no—
220C23C2×D3×A411yescharacteristic
221Q8—21no—
222Q8—62no—
223Q8—61no—
224D4—62no—
225D4—62no—
226D4—62no—
227D4—61no—
228D4—61no—
229D4—62no—
230D4—121no—
231D4—361no—
232D4—722no—
233D4—722no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—182no—
246C2×C3—61no—
247C2×C3—121no—
248C2×C3—121no—
249C2×C3—161no—
250C2×C3—321no—
251C2×C3—962no—
252C2×C3C2×#120⋊C311yescharacteristic
253D3—82no—
254C22—31no—
255C22—41no—
256C22—61no—
257C22—61no—
258C22—81no—
259C22—121no—
260C22—121no—
261C22—122no—
262C22—241no—
263C22—362no—
264C22—362no—
265C4—31no—
266C4—61no—
267C4—61no—
268C4—362no—
269C4C3⋊(#120⋊C3)11yescharacteristic
270C3—161no—
271C3—321no—
272C3#3027⋊C311yescharacteristic, complemented
273C2—61no—
274C2—121no—
275C2—121no—
276C2—242no—
277C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
centerC2C2×C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC3×H(5,𝔽2)C22×C3
FrattiniC23C2×D3×A4
FittingC3×#585C2×C3
socleC2×C3C2×#120⋊C3
hypercenterC4C3⋊(#120⋊C3)
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3×#5950C2semidirect
2. C3⋊(B(4,𝔽2)⋊C3)C2semidirect
3. #84126C3semidirect
7. C3×#585C2×C3semidirect
9. #5950D3semidirect
12. #5244C2×C3semidirect
27. C22×D12A4semidirect
45. #585C3×D3semidirect
272. C3#3027⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#59502
C3⋊(B(4,𝔽2)⋊C3)2
#841263
#3027⋊C33
A4×D124