Finite Group Databasebeta

← #13160229

Subgroups — #13160229

Order 1728.

780 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 98 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
1Q16×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2#11723322C211yesmaximal, characteristic
3C3×#63655C211yesmaximal, characteristic, complemented
4C32×#828C211yesmaximal, characteristic, complemented
5#11722011C211yesmaximal, characteristic, complemented
6#11722009C211yesmaximal, characteristic, complemented
7#11722000C211yesmaximal, characteristic, complemented
8C3⋊D3×SD32—31nomaximal
9#10591039—34nomaximal
10Q8×C3⋊(C3⋊D3)—21no—
11#89026—21no—
12C33×Q16C2211yescharacteristic, complemented
13#88863—21no—
14#88853C2211yescharacteristic, complemented
15C8×C3⋊(C3⋊D3)C2211yescharacteristic
16C3×#3745C2211yescharacteristic
17C32×D24C2211yescharacteristic
18#88789—21no—
19C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
20C32×C3⋊C16C2211yescharacteristic, complemented
21Q16×C3⋊D3—31no—
22Q16×C3⋊D3—34no—
23Q16×C3⋊D3—64no—
24D8×C3⋊D3—31no—
25#63797—34no—
26C32×SD32—31no—
27#63655D311yescharacteristic, complemented
28#63654—31no—
29#63627—31no—
30C16×C3⋊D3—31no—
31C3×#828—34no—
32C3×#828D314yescomplemented
33#63548—34no—
34#63548—34no—
35#63542—34no—
36C2×D3×C3⋊D3—41no—
37C33×Q8—21no—
38C4×C3⋊(C3⋊D3)—41no—
39C4×C3⋊(C3⋊D3)D411yescharacteristic
40#6308—21no—
41#6308—41no—
42C3×C4⋊(C3⋊D3)—21no—
43C32×D12—21no—
44#6290—41no—
45C33×C8C2311yescharacteristic
46C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
47D3×SD32—94no—
48D3×SD32—91no—
49Q8×C3⋊D3—61no—
50Q8×C3⋊D3—64no—
51Q8×C3⋊D3—68no—
52D4×C3⋊D3—61no—
53#3802—64no—
54C32×Q16—24no—
55C32×Q16C2×D314yescomplemented
56C32×Q16C2×D311yescharacteristic, complemented
57C32×D8—31no—
58#3756—61no—
59#3756—64no—
60#3756—68no—
61#3753—61no—
62#3746—31no—
63#3746—34no—
64#3746—64no—
65#3745C2×D311yescharacteristic
66C8×C3⋊D3—31no—
67C8×C3⋊D3—34no—
68C8×C3⋊D3—64no—
69C3×D24—34no—
70C3×D24C2×D314yes—
71#3713—64no—
72C32×C16—31no—
73C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
74C3×C3⋊C16—34no—
75C3×C3⋊C16C2×D314yescomplemented
76C2×C3⋊(C3⋊D3)D811yescharacteristic
77C2×C3×C3⋊D3—41no—
78C2×C32×D3—41no—
79D3×C3⋊D3—82no—
80C33×C4—41no—
81C33×C4C2×D411yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)D811yescharacteristic
84D3×Q16—94no—
85D3×Q16—91no—
86D3×Q16—184no—
87D3×D8—94no—
88D3×D8—91no—
89C3×SD32—34no—
90C3×SD32—91no—
91#828—34no—
92#828C3⋊D311yescharacteristic, complemented
93#827—94no—
94#827—91no—
95#800—94no—
96#800—91no—
97D3×C16—94no—
98D3×C16—91no—
99C22×C3⋊D3—121no—
100C2×D32—124no—
101C32×Q8—28no—
102C32×Q8—24no—
103C32×Q8—21no—
104C32×D4—61no—
105#638—121no—
106C4⋊(C3⋊D3)—21no—
107C4×C3⋊D3—31no—
108C4×C3⋊D3—34no—
109C4×C3⋊D3—64no—
110C4×C3⋊D3—121no—
111C4×C3⋊D3—124no—
112C4×C3⋊D3—128no—
113#634—68no—
114#634—64no—
115#634—61no—
116#634—121no—
117#634—124no—
118#634—128no—
119C3×D12—24no—
120C3×D12—64no—
121#625—124no—
122C32×C8—24no—
123C32×C8C22×D314yes—
124C32×C8C22×D311yescharacteristic
125C3⋊(C3⋊C8)—31no—
126C3⋊(C3⋊C8)—34no—
127C3⋊(C3⋊C8)—64no—
128C2×SD32—271noSylow
129C2×C33C2×D811yescharacteristic
130C3⋊(C3⋊D3)SD3212yescomplemented
131C3×C3⋊D3—81no—
132C32×D3—81no—
133D3×Q8—184no—
134D3×Q8—181no—
135D3×Q8—188no—
136D3×D4—181no—
137D3×D4—184no—
138C3×Q16—24no—
139C3×Q16D3214yescomplemented
140C3×Q16C2×C3⋊D311yescharacteristic, complemented
141C3×D8—34no—
142C3×D8—91no—
143#216—184no—
144#216—181no—
145#216—188no—
146#213—181no—
147#213—184no—
148Dic12—94no—
149Dic12—91no—
150Dic12—184no—
151D24—34no—
152D24C2×C3⋊D311yescharacteristic
153D3×C8—94no—
154D3×C8—91no—
155D3×C8—184no—
156C3×C16—34no—
157C3×C16—91no—
158C3⋊C16—34no—
159C3⋊C16C2×C3⋊D311yescharacteristic, complemented
160C22×C32—121no—
161C2×C3⋊D3—31no—
162C2×C3⋊D3—34no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—64no—
165C2×C3⋊D3—242no—
166C2×C3×D3—44no—
167C2×C3×D3—124no—
168D32—248no—
169C32×C4—24no—
170C32×C4—48no—
171C32×C4—44no—
172C32×C4—41no—
173C32×C4D3×D414yes—
174C32×C4D3×D411yescharacteristic
175C3⋊Dic3—31no—
176C3⋊Dic3—34no—
177C3⋊Dic3—64no—
178C3⋊Dic3—128no—
179C3⋊Dic3—124no—
180C3⋊Dic3—121no—
181C2×Q16—271no—
182C2×D8—271no—
183SD32—31no—
184SD32—91no—
185SD32—271no—
186SD32—271no—
187C2×C16—271no—
188C33C2×SD3211yescharacteristic, complemented, Sylow
189C22×D3—361no—
190C22×D3—364no—
191C3×Q8—28no—
192C3×Q8—24no—
193C3×Q8—21no—
194C3×D4—64no—
195C3×D4—181no—
196#67—361no—
197#67—364no—
198D12—21no—
199D12—64no—
200C4×D3—94no—
201C4×D3—91no—
202C4×D3—184no—
203C4×D3—361no—
204C4×D3—364no—
205C4×D3—368no—
206Dic6—188no—
207Dic6—184no—
208Dic6—181no—
209Dic6—361no—
210Dic6—364no—
211Dic6—368no—
212C3×C8—24no—
213C3×C8C2×D3214yes—
214C3×C8C22×C3⋊D311yescharacteristic
215C3⋊C8—94no—
216C3⋊C8—91no—
217C3⋊C8—184no—
218C2×C32—24no—
219C2×C32—241no—
220C2×C32D3×D814yes—
221C2×C32D3×D811yescharacteristic
222C3⋊D3—32no—
223C3⋊D3—38no—
224C3⋊D3—68no—
225C3⋊D3—81no—
226C3×D3—84no—
227C3×D3—244no—
228C2×Q8—541no—
229C2×D4—541no—
230Q16—271no—
231Q16—541no—
232Q16D3×C3⋊D311yescharacteristic, complemented
233D8—31no—
234D8—91no—
235D8—541no—
236C2×C8—271no—
237C16—31no—
238C16—91no—
239C22×C3—124no—
240C22×C3—361no—
241C2×D3—41no—
242C2×D3—91no—
243C2×D3—94no—
244C2×D3—124no—
245C2×D3—184no—
246C2×D3—722no—
247C2×D3—728no—
248C3×C4—24no—
249C3×C4—48no—
250C3×C4—41no—
251C3×C4—44no—
252C3×C4#381114yes—
253C3×C4D4×C3⋊D311yescharacteristic
254Dic3—91no—
255Dic3—94no—
256Dic3—184no—
257Dic3—368no—
258Dic3—361no—
259Dic3—364no—
260C32—24no—
261C32D3×SD3214yescomplemented
262C32D3×SD3211yescharacteristic, complemented
263C23—1081no—
264Q8—21no—
265Q8—541no—
266Q8—1081no—
267D4—61no—
268D4—181no—
269D4—1081no—
270C2×C4—271no—
271C2×C4—1081no—
272C8—271no—
273C8C2×D3×C3⋊D311yescharacteristic
274C2×C3—24no—
275C2×C3—244no—
276C2×C3—721no—
277C2×C3#6392614yes—
278C2×C3D8×C3⋊D311yescharacteristic
279D3—81no—
280D3—98no—
281D3—92no—
282D3—188no—
283D3—244no—
284C22—121no—
285C22—271no—
286C22—361no—
287C22—2162no—
288C4—41no—
289C4—271no—
290C4—1081no—
291C4#8903911yescharacteristic
292C3—24no—
293C3#1059103914yescomplemented
294C3C3⋊D3×SD3211yescharacteristic, complemented
295C2—241no—
296C2—272no—
297C2—721no—
298C2#1172354711yescharacteristic
subgroupG/H
centerC2#11723547
commutator, cosocleC33×C8C23
FrattiniC8C2×D3×C3⋊D3
FittingC33×Q16C22
socleC2×C33C2×D8
prime radical#11723322C2
hypercenterQ16D3×C3⋊D3
hypocenterC33C2×SD32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. Q16×C3⋊(C3⋊D3)C2semidirect
3. C3×#63655C2semidirect
4. C32×#828C2semidirect
5. #11722011C2semidirect
6. #11722009C2semidirect
7. #11722000C2semidirect
12. C33×Q16C22semidirect
14. #88853C22semidirect
19. C3×C3⋊(C3⋊C16)C22semidirect
20. C32×C3⋊C16C22semidirect
27. #63655D3semidirect
32. C3×#828D3semidirect
55. C32×Q16C2×D3semidirect
56. C32×Q16C2×D3semidirect
73. C3⋊(C3⋊C16)C2×D3semidirect
75. C3×C3⋊C16C2×D3semidirect
92. #828C3⋊D3semidirect
130. C3⋊(C3⋊D3)SD32semidirect
139. C3×Q16D32semidirect
140. C3×Q16C2×C3⋊D3semidirect
159. C3⋊C16C2×C3⋊D3semidirect
188. C33C2×SD32semidirect
232. Q16D3×C3⋊D3semidirect
261. C32D3×SD32semidirect
262. C32D3×SD32semidirect
293. C3#10591039semidirect
294. C3C3⋊D3×SD32semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊(C3⋊D3)2
#117233222
C3×#636552
C32×#8282
#117220112
#117220092
#117220002
C3⋊D3×SD323
#105910393