Finite Group Databasebeta

← #13139263

Subgroups — #13139263

Order 1728.

330 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 59 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⋊D9C211yesmaximal, characteristic, complemented
2#11720818C211yesmaximal, characteristic
3C9×#828C211yesmaximal, characteristic, complemented
4C3×#63387C211yesmaximal, characteristic, complemented
5#11720241C211yesmaximal, characteristic, complemented
6#11720233C211yesmaximal, characteristic, complemented
7(C3⋊D9)⋊C16C211yesmaximal, characteristic, complemented
8#10591039—31nomaximal
9D9×SD32—31nomaximal
10Q8×C3⋊D9—21no—
11#88646—21no—
12C3×C9×Q16C2211yescharacteristic, complemented
13#88547—21no—
14C8×C3⋊D9C2211yescharacteristic
15#88521C2211yescharacteristic, complemented
16C9×D24C2211yescharacteristic
17C3×D72C2211yescharacteristic
18#88421—21no—
19C9×C3⋊C16C2211yescharacteristic, complemented
20C3×C9⋊C16C2211yescharacteristic, complemented
21Q16×C3⋊D3—31no—
22#63797—31no—
23C3×#828—31no—
24C3×#828D311yescharacteristic
25#63548—31no—
26#63548—31no—
27#63542—31no—
28Q16×D9—31no—
29Q16×D9—61no—
30D8×D9—31no—
31C9×SD32—31no—
32#63387D311yescharacteristic, complemented
33#63386—31no—
34#63359—31no—
35C16×D9—31no—
36C2×D3×D9—41no—
37C3×Q8×C9—21no—
38C4×C3⋊D9—41no—
39C4×C3⋊D9D411yescharacteristic
40#6226—21no—
41#6226—41no—
42C9×D12—21no—
43C3×D36—21no—
44#6194—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47D3×SD32—91no—
48D3×SD32—91no—
49Q8×C3⋊D3—61no—
50#3802—61no—
51C32×Q16C2×D311yescharacteristic
52#3756—61no—
53#3746—31no—
54C8×C3⋊D3—31no—
55C3×D24—31no—
56C3×D24C2×D311yescharacteristic
57#3713—61no—
58Q8×D9—61no—
59Q8×D9—62no—
60D4×D9—61no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—21no—
64C9×Q16C2×D311yescharacteristic, complemented
65C9×D8—31no—
66#3674—61no—
67#3674—62no—
68#3673—61no—
69D72C2×D311yescharacteristic
70C8×D9—31no—
71C8×D9—61no—
72Dic36—31no—
73Dic36—61no—
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic, complemented
76C2×C3⋊D9D811yescharacteristic
77C2×D3×C9—41no—
78C2×C3×D9—41no—
79D3×D9—82no—
80C3×C4×C9—41no—
81C3×C4×C9C2×D411yescharacteristic
82C3⋊Dic9—41no—
83C3⋊Dic9D811yescharacteristic
84D3×Q16—91no—
85D3×Q16—91no—
86D3×Q16—181no—
87D3×D8—91no—
88D3×D8—91no—
89C3×SD32—31no—
90C3×SD32—91no—
91#828—31no—
92#828D911yescharacteristic, complemented
93#827—91no—
94#827—91no—
95#800—91no—
96#800—91no—
97D3×C16—91no—
98D3×C16—91no—
99C2×D32—121no—
100C32×Q8—21no—
101C4×C3⋊D3—31no—
102C4×C3⋊D3—121no—
103#634—61no—
104#634—121no—
105C3×D12—21no—
106C3×D12—61no—
107#625—121no—
108C22×D9—121no—
109C32×C8C22×D311yescharacteristic
110C3⋊(C3⋊C8)—31no—
111Q8×C9—22no—
112Q8×C9—21no—
113D4×C9—61no—
114#611—121no—
115D36—21no—
116C4×D9—31no—
117C4×D9—61no—
118C4×D9—121no—
119C4×D9—122no—
120Dic18—62no—
121Dic18—61no—
122Dic18—121no—
123Dic18—122no—
124C8×C9—21no—
125C8×C9C22×D311yescharacteristic
126C9⋊C8—31no—
127C9⋊C8—61no—
128C2×SD32—271noSylow
129C2×C3×C9C2×D811yescharacteristic
130C3⋊D9SD3212yescomplemented
131D3×C9—81no—
132C3×D9—81no—
133D3×Q8—181no—
134D3×Q8—181no—
135D3×Q8—182no—
136D3×D4—181no—
137D3×D4—181no—
138C3×Q16—21no—
139C3×Q16C2×D911yescharacteristic, complemented
140C3×Q16D3211yescharacteristic
141C3×D8—31no—
142C3×D8—91no—
143#216—181no—
144#216—181no—
145#216—182no—
146#213—181no—
147#213—181no—
148Dic12—91no—
149Dic12—91no—
150Dic12—181no—
151D24—31no—
152D24C2×D911yescharacteristic
153D3×C8—91no—
154D3×C8—91no—
155D3×C8—181no—
156C3×C16—31no—
157C3×C16—91no—
158C3⋊C16—31no—
159C3⋊C16C2×D911yescharacteristic, complemented
160C2×C3⋊D3—31no—
161C2×C3×D3—41no—
162C2×C3×D3—121no—
163D32—242no—
164C32×C4—41no—
165C32×C4D3×D411yescharacteristic
166C3⋊Dic3—31no—
167C3⋊Dic3—121no—
168C22×C9—121no—
169C2×D9—31no—
170C2×D9—41no—
171C2×D9—61no—
172C2×D9—242no—
173C4×C9—21no—
174C4×C9—42no—
175C4×C9—41no—
176C4×C9D3×D411yescharacteristic
177Dic9—31no—
178Dic9—61no—
179Dic9—122no—
180Dic9—121no—
181C2×Q16—271no—
182C2×D8—271no—
183SD32—31no—
184SD32—91no—
185SD32—271no—
186SD32—271no—
187C2×C16—271no—
188C3×C9C2×SD3211yescharacteristic, complemented, Sylow
189C22×D3—361no—
190C22×D3—361no—
191C3×Q8—22no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×D4—61no—
195C3×D4—181no—
196#67—361no—
197#67—361no—
198D12—21no—
199D12—61no—
200C4×D3—91no—
201C4×D3—91no—
202C4×D3—181no—
203C4×D3—361no—
204C4×D3—361no—
205C4×D3—362no—
206Dic6—182no—
207Dic6—181no—
208Dic6—181no—
209Dic6—361no—
210Dic6—361no—
211Dic6—362no—
212C3×C8—21no—
213C3×C8C22×D911yescharacteristic
214C3×C8C2×D3211yescharacteristic
215C3⋊C8—91no—
216C3⋊C8—91no—
217C3⋊C8—181no—
218C2×C32D3×D811yescharacteristic
219C3⋊D3—32no—
220C3×D3—81no—
221C3×D3—241no—
222C2×C9—21no—
223C2×C9—241no—
224C2×C9D3×D811yescharacteristic
225D9—32no—
226D9—62no—
227D9—81no—
228C2×Q8—541no—
229C2×D4—541no—
230Q16—271no—
231Q16—541no—
232Q16D3×D911yescharacteristic, complemented
233D8—31no—
234D8—91no—
235D8—541no—
236C2×C8—271no—
237C16—31no—
238C16—91no—
239C22×C3—121no—
240C22×C3—361no—
241C2×D3—41no—
242C2×D3—91no—
243C2×D3—91no—
244C2×D3—121no—
245C2×D3—181no—
246C2×D3—722no—
247C2×D3—722no—
248C3×C4—21no—
249C3×C4—42no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4D4×D911yescharacteristic
253C3×C4#381111yescharacteristic
254Dic3—91no—
255Dic3—91no—
256Dic3—181no—
257Dic3—362no—
258Dic3—361no—
259Dic3—361no—
260C32D3×SD3211yescharacteristic
261C9—21no—
262C9D3×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×D911yescharacteristic
274C2×C3—21no—
275C2×C3—241no—
276C2×C3—721no—
277C2×C3D8×D911yescharacteristic
278C2×C3#6392611yescharacteristic
279D3—81no—
280D3—92no—
281D3—92no—
282D3—182no—
283D3—241no—
284C22—121no—
285C22—271no—
286C22—361no—
287C22—2162no—
288C4—41no—
289C4—271no—
290C4—1081no—
291C4#8866811yescharacteristic
292C3—21no—
293C3D9×SD3211yescharacteristic, complemented
294C3#1059103911yescharacteristic
295C2—241no—
296C2—272no—
297C2—721no—
298C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
prime radical#11720818C2
hypercenterQ16D3×D9
hypocenterC3×C9C2×SD32
LCSG ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. Q16×C3⋊D9C2semidirect
3. C9×#828C2semidirect
4. C3×#63387C2semidirect
5. #11720241C2semidirect
6. #11720233C2semidirect
7. (C3⋊D9)⋊C16C2semidirect
12. C3×C9×Q16C22semidirect
15. #88521C22semidirect
19. C9×C3⋊C16C22semidirect
20. C3×C9⋊C16C22semidirect
32. #63387D3semidirect
64. C9×Q16C2×D3semidirect
75. C9⋊C16C2×D3semidirect
92. #828D9semidirect
130. C3⋊D9SD32semidirect
139. C3×Q16C2×D9semidirect
159. C3⋊C16C2×D9semidirect
188. C3×C9C2×SD32semidirect
232. Q16D3×D9semidirect
262. C9D3×SD32semidirect
293. C3D9×SD32semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊D92
#117208182
C9×#8282
C3×#633872
#117202412
#117202332
(C3⋊D9)⋊C162
#105910393
D9×SD323