Finite Group Databasebeta

← #84089⋊C3 #11921370

Subgroups — #84089⋊C3 #11921370

Order 1152.

517 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 83 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×#41⋊C3C211yesmaximal, characteristic
2C2×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
3#84089C311yesmaximal, characteristic, complemented
4#2341⋊C3—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C412yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8Dic3×SL(2,𝔽3)—22no—
9C3×C3⋊#95—41nomaximal
10C22×C3×#41C2×C311yescharacteristic
11C22×#41⋊C3—21no—
12C22×#41⋊C3D311yescharacteristic
13C2×Q8×Dic3C2×C312yes—
14C2×C4×SL(2,𝔽3)—32no—
15#5221—32no—
16#5212—31no—
17C4×#217—31no—
18C22×C32×C4—41no—
19C22×C3×Dic3—42no—
20C3×#41⋊C3—22no—
21C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
22C2×C3×SL(2,𝔽3)D412yes—
23#2341—31noSylow
24C2×C3×#41—22no—
25C2×C3×#41—32no—
26C2×C3×#41—62no—
27C2×C3×#41C3×C412yescomplemented
28C22×C3×Q8C22×C311yescharacteristic
29C22×C3×D4—31no—
30C23×C3×C4—31no—
31C2×#41⋊C3—24no—
32C2×#41⋊C3—22no—
33C2×#41⋊C3—22no—
34C2×#41⋊C3Dic312yescomplemented
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)C2×D311yescharacteristic
37C2×#217—32no—
38C2×#217—32no—
39Q8×Dic3—22no—
40Q8×Dic3—62no—
41C2×C4⋊Dic3—32no—
42C2×C4×Dic3—32no—
43C4×SL(2,𝔽3)—62no—
44C3×#95—121no—
45C3⋊#95—31no—
46C3⋊#95A411yescharacteristic
47C23×C32—41no—
48C2×C32×C4—42no—
49C2×C32×C4—82no—
50C2×C3×Dic3—44no—
51C2×C3×Dic3—82no—
52C2×C3×Dic3—82no—
53C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
54C22×#41C3×D311yescharacteristic
55C2×C4×Q8—32no—
56#390—92no—
57#380—91no—
58C4×C22⋊C4—91no—
59C24×C3—31no—
60C3×#41—22no—
61C3×#41—64no—
62C3×#41—64no—
63C3×#41—62no—
64C2×C3×Q8—32no—
65C2×C3×Q8—31no—
66C2×C3×Q8C2×C3×C411yescharacteristic
67C2×C3×Q8C3×D412yes—
68C2×C3×D4—32no—
69C2×C3×D4—32no—
70C2×C3×D4—62no—
71C2×C3×D4—62no—
72C22×C3×C4—31no—
73C22×C3×C4—32no—
74C22×C3×C4—32no—
75C22×C3×C4—41no—
76C22×C3×C4—62no—
77C22×C3×C4—62no—
78C22×C3×C4—81no—
79C22×C3×C4—122no—
80C22×C3×C4C2×A411yescharacteristic
81C22×Dic3—32no—
82C22×Dic3C2×A412yes—
83#41⋊C3—24no—
84#41⋊C3—22no—
85C2×SL(2,𝔽3)—21no—
86C2×SL(2,𝔽3)—22no—
87C2×SL(2,𝔽3)C2×Dic311yescharacteristic
88C2×SL(2,𝔽3)#6712yes—
89#217—64no—
90#217—122no—
91C4⋊Dic3—62no—
92C4⋊Dic3—62no—
93C4×Dic3—62no—
94C4×Dic3—62no—
95C22×C32—41no—
96C22×C32—42no—
97C22×C32—42no—
98C22×C32—41no—
99C22×C32—41no—
100C32×C4—82no—
101C3×Dic3—84no—
102C2×#41—22no—
103C2×#41—32no—
104C2×#41—62no—
105C2×#41C3×Dic312yescomplemented
106C22×Q8C2×C3×D311yescharacteristic
107C22×D4—31no—
108C23×C4—31no—
109C4×Q8—62no—
110C4×Q8—182no—
111C2×C4⋊C4—92no—
112C2×C22⋊C4—92no—
113C2×C22⋊C4—92no—
114C2×C42—92no—
115#95—31no—
116#95—91no—
117C23×C3—32no—
118C23×C3—41no—
119C23×C3—62no—
120C23×C3—62no—
121C23×C3—61no—
122C23×C3—61no—
123C23×C3—81no—
124C23×C3C22×A411yescharacteristic
125C3×Q8—31no—
126C3×Q8—32no—
127C3×Q8—32no—
128C3×Q8C3×C22⋊C411yescharacteristic
129C3×D4—62no—
130C3×D4—62no—
131C3×D4—64no—
132C2×C3×C4—22no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—32no—
136C2×C3×C4—32no—
137C2×C3×C4—42no—
138C2×C3×C4—62no—
139C2×C3×C4—64no—
140C2×C3×C4—62no—
141C2×C3×C4—82no—
142C2×C3×C4—84no—
143C2×C3×C4—82no—
144C2×C3×C4—124no—
145C2×C3×C4—242no—
146C2×C3×C4—242no—
147C2×C3×C4C4×A412yes—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3#41⋊C314yescomplemented
154SL(2,𝔽3)—21no—
155SL(2,𝔽3)#21711yescharacteristic, complemented
156C2×C32—41no—
157C2×C32—41no—
158C2×C32—41no—
159C2×C32—42no—
160C2×C32—42no—
161C24—31no—
162#41—22no—
163#41—64no—
164#41—64no—
165#41—62no—
166C2×Q8—32no—
167C2×Q8—31no—
168C2×Q8C2×C3×Dic311yescharacteristic
169C2×Q8C3×#6712yes—
170C2×D4—32no—
171C2×D4—32no—
172C2×D4—62no—
173C2×D4—62no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—92no—
181C22×C4D3×A411yescharacteristic
182C4⋊C4—182no—
183C4⋊C4—182no—
184C22⋊C4—184no—
185C22⋊C4—362no—
186C42—182no—
187C42—182no—
188C22×C3—41no—
189C22×C3—42no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—62no—
194C22×C3—62no—
195C22×C3—62no—
196C22×C3—81no—
197C22×C3—82no—
198C22×C3—81no—
199C22×C3—81no—
200C22×C3—82no—
201C22×C3—122no—
202C22×C3—122no—
203C22×C3C2×C4×A411yescharacteristic
204C22×C3D4×A412yes—
205C22×C3C2×#41⋊C312yes—
206C22×C3#99411yescharacteristic
207C22×C32−1+4⋊C311yescharacteristic
208C3×C4—22no—
209C3×C4—32no—
210C3×C4—31no—
211C3×C4—31no—
212C3×C4—82no—
213C3×C4—84no—
214C3×C4—244no—
215Dic3—24no—
216Dic3—122no—
217C32—41noSylow
218C23—32no—
219C23—61no—
220C23—62no—
221C23—61no—
222C23—62no—
223C23C2×D3×A411yescharacteristic
224Q8—31no—
225Q8—32no—
226Q8—32no—
227Q8C3×#21711yescharacteristic
228D4—62no—
229D4—62no—
230D4—64no—
231C2×C4—22no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—34no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—64no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—182no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4Dic3×A412yes—
246C2×C3—41no—
247C2×C3—42no—
248C2×C3—41no—
249C2×C3—42no—
250C2×C3—41no—
251C2×C3—81no—
252C2×C3—82no—
253C2×C3—81no—
254C2×C3—82no—
255C2×C3—81no—
256C2×C3—122no—
257C2×C3A4×C22⋊C411yescharacteristic
258C2×C3C4×#41⋊C311yescharacteristic
259C2×C3#520⋊C311yescharacteristic
260C2×C3#548⋊C312yes—
261C2×C3C4⋊(#41⋊C3)12yes—
262C22—62no—
263C22—62no—
264C22—62no—
265C22—122no—
266C22—122no—
267C22C3⋊#99411yescharacteristic
268C22D3×#41⋊C312yes—
269C22#1010⋊C311yescharacteristic
270C22C2×Dic3×A411yescharacteristic
271C22A4×#6712yes—
272C4—22no—
273C4—31no—
274C4—31no—
275C4—32no—
276C4—64no—
277C4—362no—
278C3—41no—
279C3—81no—
280C3#2341⋊C311yescharacteristic, complemented
281C2—122no—
282C2Dic3×#41⋊C311yescharacteristic
283C2#5834⋊C312yes—
284C2#5827⋊C311yescharacteristic
285C2#5831⋊C312yes—
286C2A4×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC22×C3×#41C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×SL(2,𝔽3)C2semidirect
3. #84089C3semidirect
6. C2×C3×#41⋊C3C4semidirect
27. C2×C3×#41C3×C4semidirect
34. C2×#41⋊C3Dic3semidirect
53. C3×SL(2,𝔽3)C22⋊C4semidirect
105. C2×#41C3×Dic3semidirect
153. C2×Dic3#41⋊C3semidirect
155. SL(2,𝔽3)#217semidirect
280. C3#2341⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×Dic3×SL(2,𝔽3)2
#840893
#2341⋊C33
C3×C3⋊#954