Finite Group Databasebeta

← #83562⋊C3 #11921299

Subgroups — #83562⋊C3 #11921299

Order 1152.

297 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 47 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
3C3⋊C8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#83562C311yesmaximal, characteristic, complemented
5#3358⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)—21no—
7C2×C3⋊(#41⋊C3)—21no—
8C3×C4×SL(2,𝔽3)C2211yescharacteristic
9C3×#832—41nomaximal
10C3×D4×Q8C2×C311yescharacteristic
11#5577C2×C311yescharacteristic, complemented
12C4⋊(#41⋊C3)—31no—
13D4×SL(2,𝔽3)—21no—
14D4×SL(2,𝔽3)D311yescharacteristic, complemented
15#5088—31no—
16#5087—31no—
17Q8×C3⋊C8C2×C311yescharacteristic, complemented
18#5015—31no—
19C4×#213—31no—
20C8×SL(2,𝔽3)—31no—
21C2×C32×D4—41no—
22C2×C3×SL(2,𝔽3)—41no—
23C2×C3×SL(2,𝔽3)D411yescharacteristic
24C3⋊(#41⋊C3)—41no—
25C3×C4⋊Dic3—41no—
26C2×C3×C3⋊C8—41no—
27#3358—31noSylow
28C22×C3×Q8—21no—
29C2×#239—21no—
30C2×#41⋊C3—61no—
31C22×SL(2,𝔽3)—22no—
32C22×SL(2,𝔽3)—21no—
33C3×C4⋊Q8—31no—
34C3×C22⋊Q8—61no—
35C3×C4×Q8C22×C311yescharacteristic
36C3×C4×D4—31no—
37C2×#213—31no—
38#895—61no—
39#874—31no—
40C4×D12—31no—
41C4×SL(2,𝔽3)—21no—
42C4×SL(2,𝔽3)C2×D311yescharacteristic
43C3×#102—121no—
44#832A411yescharacteristic
45#809—32no—
46#809—31no—
47#807—31no—
48C4⋊(C3⋊C8)—31no—
49C4×C3⋊C8—31no—
50C23×C32—81no—
51C32×D4—42no—
52C32×D4—81no—
53C2×C32×C4—41no—
54C2×C3×Dic3—81no—
55C3×SL(2,𝔽3)D811yescharacteristic, complemented
56C3×C3⋊C8—81no—
57#550—31no—
58D4×Q8C3×D311yescharacteristic
59#486—91no—
60#474—91no—
61#459—91no—
62C8×Q8—31no—
63C4×D8—91no—
64C2×C3×Q8—41no—
65C2×C3×Q8—61no—
66C2×C3×Q8—121no—
67C2×C3×Q8C3×D411yescharacteristic
68C2×C3×D4—41no—
69C2×C3×D4—81no—
70C2×C3×D4C2×A411yescharacteristic
71C22×C3×C4—61no—
72#239—41no—
73#239—121no—
74C2×D12—31no—
75C2×D12—61no—
76C2×D12—61no—
77C2×C4×D3—61no—
78#41⋊C3—121no—
79C2×SL(2,𝔽3)—21no—
80C2×SL(2,𝔽3)—42no—
81C2×SL(2,𝔽3)—41no—
82C2×SL(2,𝔽3)#6711yescharacteristic
83C2×C3×C8—121no—
84C3×C4⋊C4—31no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—61no—
87C3×C4⋊C4—121no—
88C3×C22⋊C4—61no—
89C3×C42—31no—
90#213—62no—
91(C2×D3)⋊C4—61no—
92C4⋊Dic3C2×A411yescharacteristic
93C2×C3⋊C8—31no—
94C2×C3⋊C8C2×A411yescharacteristic
95C22×C32—41no—
96C22×C32—82no—
97C22×C32—161no—
98C22×C32—161no—
99C32×C4—41no—
100C32×C4—41no—
101C3×Dic3—161no—
102C2×#41—61no—
103C22×Q8—21no—
104C2×D8—91no—
105C4⋊Q8—31no—
106#127—91no—
107C22⋊Q8—61no—
108#121—181no—
109C4×Q8C2×C3×D311yescharacteristic
110C4×D4—31no—
111C4×D4—91no—
112#107—91no—
113C4⋊C8—91no—
114#102—31no—
115#102—91no—
116#102—92no—
117C4×C8—91no—
118C23×C3—21no—
119C23×C3—81no—
120C23×C3—82no—
121C22×D3—61no—
122C3×Q8—31no—
123C3×Q8—121no—
124C3×Q8—121no—
125C3×Q8C3×D811yescharacteristic
126C3×D4—21no—
127C3×D4—42no—
128C3×D4—81no—
129C3×D4—82no—
130C3×D4—82no—
131C3×D4#41⋊C312yescomplemented
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—41no—
135C2×C3×C4—61no—
136C2×C3×C4—81no—
137C2×C3×C4—121no—
138C2×C3×C4—121no—
139C2×C3×C4—241no—
140C2×C3×C4C22×A411yescharacteristic
141C2×Dic3—21no—
142D12—61no—
143D12—61no—
144D12—121no—
145D12—121no—
146D12—241no—
147C4×D3—121no—
148C4×D3—121no—
149SL(2,𝔽3)—21no—
150SL(2,𝔽3)#21311yescharacteristic, complemented
151C3×C8—241no—
152C3⋊C8—21no—
153C3⋊C8—61no—
154C2×C32—41no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—162no—
158#41—121no—
159#41—361no—
160C2×Q8—41no—
161C2×Q8—61no—
162C2×Q8—121no—
163C2×Q8C3×#6711yescharacteristic
164C2×D4—91no—
165C2×D4—181no—
166C2×D4—181no—
167C2×D4D3×A411yescharacteristic
168C22×C4—61no—
169C22×C4—181no—
170D8—182no—
171C2×C8—31no—
172C2×C8—91no—
173C4⋊C4—31no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—181no—
179C42—31no—
180C22×C3—22no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3—82no—
185C22×C3—84no—
186C22×C3—81no—
187C22×C3—161no—
188C22×C3—161no—
189C22×C3—162no—
190C22×C3—162no—
191C22×C3D4×A411yescharacteristic
192C2×D3—121no—
193C2×D3—121no—
194C2×D3—121no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4—61no—
200C3×C4—81no—
201C3×C4—81no—
202C3×C4—121no—
203C3×C4—481no—
204C3×C4C2×#41⋊C311yescharacteristic
205C3×C42−1+4⋊C311yescharacteristic
206Dic3—41no—
207C32—41noSylow
208C23—21no—
209C23—181no—
210Q8—31no—
211Q8—121no—
212Q8—121no—
213Q8C3×#21311yescharacteristic
214D4—21no—
215D4—181no—
216D4—181no—
217D4—361no—
218D4—361no—
219D4—721no—
220D4C3⋊(#41⋊C3)12yescomplemented
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—121no—
226C2×C4—121no—
227C2×C4—361no—
228C2×C4—361no—
229C2×C4C2×D3×A411yescharacteristic
230C8—61no—
231C8—181no—
232C2×C3—42no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—81no—
237C2×C3—81no—
238C2×C3—81no—
239C2×C3—162no—
240C2×C3—164no—
241C2×C3#548⋊C311yescharacteristic
242C2×C3A4×D811yescharacteristic
243C2×C3#577⋊C311yescharacteristic
244D3—241no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—361no—
249C22—361no—
250C22—361no—
251C22A4×#6711yescharacteristic
252C4—31no—
253C4—31no—
254C4—61no—
255C4—121no—
256C4—121no—
257C4C2×C3⋊(#41⋊C3)11yescharacteristic
258C4#1007⋊C311yescharacteristic
259C3—41no—
260C3—81no—
261C3#3358⋊C311yescharacteristic, complemented
262C2—42no—
263C2—721no—
264C2A4×#21311yescharacteristic
265C2#5817⋊C311yescharacteristic
266C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutator, cosocleC3×C4×Q8C22×C3
FrattiniC2×C4C2×D3×A4
FittingC3×D4×Q8C2×C3
socleC22×C3D4×A4
hypercenterC2×D4D3×A4
hypocenterC3×Q8C3×D8
LCSG ▷ C3×C4×Q8 ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×SL(2,𝔽3)C2semidirect
2. C4⋊(C3⋊(#41⋊C3))C2semidirect
3. C3⋊C8×SL(2,𝔽3)C2semidirect
4. #83562C3semidirect
11. #5577C2×C3semidirect
14. D4×SL(2,𝔽3)D3semidirect
17. Q8×C3⋊C8C2×C3semidirect
55. C3×SL(2,𝔽3)D8semidirect
131. C3×D4#41⋊C3semidirect
150. SL(2,𝔽3)#213semidirect
220. D4C3⋊(#41⋊C3)semidirect
261. C3#3358⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
C3⋊C8×SL(2,𝔽3)2
#835623
#3358⋊C33
C3×#8324