Finite Group Databasebeta

← #84124⋊C3 #11921382

Subgroups — #84124⋊C3 #11921382

Order 1152.

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

normal subgroups only

This group has 330 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
2C3⋊(#458⋊C3)C212yesmaximal, complemented
3#84124C311yesmaximal, characteristic, complemented
4#3028⋊C3—31nomaximal
5C3×#997C2211yescharacteristic
6A4×Dic6—41nomaximal
7C3×#585C2×C311yescharacteristic
8#5950—21no—
9#5950D311yescharacteristic
10#5825—31no—
11#5243—32no—
12#5243C2×C312yes—
13#5240—31no—
14#458⋊C3—32no—
15C3×#41⋊C3—81no—
16C3×C4×A4—41no—
17C3×C4×A4—81no—
18Dic3×A4—42no—
19#3028—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×Dic6A411yescharacteristic, complemented
28#997—21no—
29#997C2×D311yescharacteristic
30Q8×A4—121no—
31#944—61no—
32#936—61no—
33#935—32no—
34#935—32no—
35#926—31no—
36#924—61no—
37#924—61no—
38#834—32no—
39#834—32no—
40C2×C3×A4—41no—
41C2×C3×A4—81no—
42C3×Dic6—161no—
43C3×SL(2,𝔽3)—81no—
44#585C3×D311yescharacteristic
45#536—91no—
46#458—32no—
47#458—92no—
48#410—91no—
49C3×#41—21no—
50C3×#41—62no—
51C3×#41—62no—
52C3×#41—61no—
53C3×#41—62no—
54C3×#41—61no—
55C3×#41—62no—
56C3×#41—62no—
57C3×#41—61no—
58C2×C3×Q8—31no—
59C2×C3×Q8—61no—
60C2×C3×Q8—61no—
61C2×C3×D4—31no—
62C2×C3×D4—31no—
63C2×C3×D4—31no—
64C2×C3×D4—61no—
65C2×C3×D4—62no—
66C2×C3×D4—61no—
67C2×C3×D4—61no—
68C2×C3×D4—61no—
69C22×C3×C4—21no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4—61no—
73C22×C3×C4C2×A411yescharacteristic
74C22×Dic3C2×A412yes—
75C2×Dic6—61no—
76C2×Dic6—61no—
77#41⋊C3—81no—
78#41⋊C3—82no—
79C4×A4—41no—
80C4×A4—81no—
81C4×A4—82no—
82C4×A4—81no—
83C4×A4—122no—
84#217—32no—
85#217—62no—
86#217—62no—
87C4⋊Dic3—61no—
88Dic3⋊C4—62no—
89Dic3⋊C4—62no—
90C4×Dic3—61no—
91C3×A4—41no—
92C3×A4—81no—
93C32×C4—161no—
94C3×Dic3—162no—
952−1+4—61no—
96H(5,𝔽2)—31no—
97H(5,𝔽2)—61no—
98H(5,𝔽2)C2×C3×D311yescharacteristic
99C2×#41—31no—
100C2×#41—61no—
101C2×#41—61no—
102C22×Q8—31no—
103C4⋊Q8—181no—
104#124—181no—
105#123—92no—
106#123—92no—
107C22⋊Q8—181no—
108C22⋊Q8—181no—
109#117—91no—
110#99—92no—
111#99—92no—
112C23×C3—21no—
113C23×C3—31no—
114C23×C3—31no—
115C23×C3—61no—
116C23×C3C22×A411yescharacteristic
117C2×A4—41no—
118C2×A4—81no—
119C2×A4—82no—
120C2×A4—81no—
121C3×Q8—21no—
122C3×Q8—62no—
123C3×Q8—61no—
124C3×Q8—481no—
125C3×D4—62no—
126C3×D4—62no—
127C3×D4—62no—
128C3×D4—61no—
129C3×D4—61no—
130C3×D4—62no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—61no—
136C2×C3×C4—61no—
137C2×C3×C4—62no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×Dic3—62no—
143C2×Dic3—62no—
144Dic6—41no—
145Dic6—121no—
146SL(2,𝔽3)—81no—
147SL(2,𝔽3)—82no—
148C2×C32—161no—
149#41—21no—
150#41—62no—
151#41—62no—
152#41—61no—
153#41—62no—
154#41—61no—
155#41—62no—
156#41—62no—
157#41—61no—
158C2×Q8—31no—
159C2×Q8—61no—
160C2×Q8—61no—
161C2×Q8—181no—
162C2×Q8—181no—
163C2×D4—31no—
164C2×D4—31no—
165C2×D4—31no—
166C2×D4—61no—
167C2×D4—62no—
168C2×D4—61no—
169C2×D4—61no—
170C2×D4—61no—
171C22×C4—21no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—61no—
176C22×C4D3×A411yescharacteristic
177C4⋊C4—181no—
178C4⋊C4—182no—
179C4⋊C4—182no—
180C22⋊C4—92no—
181C22⋊C4—182no—
182C22⋊C4—182no—
183C42—181no—
184C22×C3—31no—
185C22×C3—41no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—81no—
189C22×C3—121no—
190C22×C3—121no—
191C22×C3—241no—
192A4—41no—
193A4—81no—
194A4—81no—
195A4—82no—
196C3×C4—31no—
197C3×C4—61no—
198C3×C4—61no—
199C3×C4—161no—
200C3×C4—321no—
201C3×C4—482no—
202C3×C4#120⋊C311yescharacteristic
203Dic3—42no—
204Dic3—122no—
205C32—161noSylow
206C23—21no—
207C23—31no—
208C23—31no—
209C23—61no—
210C23C2×D3×A411yescharacteristic
211Q8—21no—
212Q8—62no—
213Q8—61no—
214Q8—121no—
215Q8—361no—
216D4—62no—
217D4—62no—
218D4—62no—
219D4—61no—
220D4—61no—
221D4—62no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C3—61no—
236C2×C3—121no—
237C2×C3—121no—
238C2×C3—161no—
239C2×C3—321no—
240C2×C3C2×#120⋊C311yescharacteristic
241C22—31no—
242C22—41no—
243C22—61no—
244C22—61no—
245C22—81no—
246C22—121no—
247C22—121no—
248C22—241no—
249C4—31no—
250C4—61no—
251C4—61no—
252C4—122no—
253C4—362no—
254C4C3⋊(#120⋊C3)11yescharacteristic
255C3—161no—
256C3—321no—
257C3#3028⋊C311yescharacteristic, complemented
258C2—61no—
259C2—121no—
260C2—121no—
261C2C2×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
prime radicalC3×#5950C2
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
2. C3⋊(#458⋊C3)C2semidirect
3. #84124C3semidirect
27. C22×Dic6A4semidirect
257. C3#3028⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#59502
C3⋊(#458⋊C3)2
#841243
#3028⋊C33
A4×Dic64