Finite Group Databasebeta

← #13139272

Subgroups — #13139272

Order 1728.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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
1#11721805C211yesmaximal, characteristic
2#11720829C211yesmaximal, characteristic, complemented
3C9×#829C211yesmaximal, characteristic, complemented
4C3×#63388C211yesmaximal, characteristic, complemented
5#11720243C211yesmaximal, characteristic, complemented
6#11720234C211yesmaximal, characteristic, complemented
7#11720231C211yesmaximal, characteristic, complemented
8#10591044—31nomaximal
9#10588598—31nomaximal
10#88746—21no—
11(C3⋊D9)⋊Q8—21no—
12C3×C9×Q16C2211yescharacteristic
13#88548—21no—
14#88525C2211yescharacteristic
15C8×C3⋊D9C2211yescharacteristic
16C9×Dic12C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88427—21no—
19C9×C3⋊C16C2211yescharacteristic
20C3×C9⋊C16C2211yescharacteristic
21#64128—31no—
22#63802—31no—
23C3×#829—31no—
24C3×#829D311yescharacteristic
25#63549—31no—
26#63549—31no—
27#63544—31no—
28#63481—31no—
29#63481—61no—
30Q16×D9—31no—
31C9×Q32—31no—
32#63388D311yescharacteristic, complemented
33#63387—31no—
34#63359—31no—
35#63357—31no—
36C3×Q8×C9—21no—
37C4⋊(C3⋊D9)—21no—
38C4⋊(C3⋊D9)—41no—
39C4×C3⋊D9—41no—
40C4×C3⋊D9D411yescharacteristic
41C9×Dic6—21no—
42C3×Dic18—21no—
43(C3⋊D9)⋊C4—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4918—91no—
48#4918—91no—
49#3832—61no—
50#3797—61no—
51C32×Q16C2×D311yescharacteristic
52#3755—61no—
53#3745—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×Dic12C2×D311yescharacteristic
57#3718—61no—
58#3701—61no—
59#3701—62no—
60Q8×D9—61no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—21no—
64C9×Q16—31no—
65C9×Q16C2×D311yescharacteristic
66#3675—61no—
67#3675—62no—
68#3674—61no—
69D72—31no—
70D72—61no—
71C8×D9—31no—
72C8×D9—61no—
73Dic36C2×D311yescharacteristic
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic
76C2×C3⋊D9—41no—
77C2×C3⋊D9D811yescharacteristic
78C3×C4×C9—41no—
79C3×C4×C9C2×D411yescharacteristic
80C3⋊Dic9D811yescharacteristic
81C9×Dic3—41no—
82C3×Dic9—41no—
83#919—91no—
84#919—91no—
85#919—181no—
86D3×Q16—91no—
87D3×Q16—91no—
88C3×Q32—31no—
89C3×Q32—91no—
90#829—31no—
91#829D911yescharacteristic, complemented
92#828—91no—
93#828—91no—
94#800—91no—
95#800—91no—
96#798—91no—
97#798—91no—
98C32×Q8—21no—
99C4⋊(C3⋊D3)—61no—
100C4⋊(C3⋊D3)—121no—
101C4×C3⋊D3—31no—
102C4×C3⋊D3—121no—
103C3×Dic6—21no—
104C3×Dic6—61no—
105#627—121no—
106#624—121no—
107C32×C8C22×D311yescharacteristic
108C3⋊(C3⋊C8)—31no—
109Q8×C9—22no—
110Q8×C9—21no—
111Q8×C9—61no—
112D36—62no—
113D36—61no—
114D36—121no—
115D36—122no—
116C4×D9—31no—
117C4×D9—61no—
118C4×D9—121no—
119C4×D9—121no—
120C4×D9—122no—
121Dic18—21no—
122Dic18—121no—
123C8×C9—21no—
124C8×C9C22×D311yescharacteristic
125C9⋊C8—31no—
126C9⋊C8—61no—
127#510—271noSylow
128C2×C3×C9C2×D811yescharacteristic
129C3⋊D9—21no—
130C3⋊D9—81no—
131#239—181no—
132#239—181no—
133#239—182no—
134D3×Q8—181no—
135D3×Q8—181no—
136C3×Q16—21no—
137C3×Q16—31no—
138C3×Q16—91no—
139C3×Q16C2×D911yescharacteristic
140C3×Q16D3211yescharacteristic
141#216—181no—
142#216—181no—
143#215—181no—
144#215—181no—
145#215—182no—
146Dic12—31no—
147Dic12C2×D911yescharacteristic
148D24—91no—
149D24—91no—
150D24—181no—
151D3×C8—91no—
152D3×C8—91no—
153D3×C8—181no—
154C3×C16—31no—
155C3×C16—91no—
156C3⋊C16—31no—
157C3⋊C16C2×D911yescharacteristic
158C2×C3⋊D3—31no—
159C2×C3⋊D3—121no—
160C32×C4—41no—
161C32×C4D3×D411yescharacteristic
162C3⋊Dic3—31no—
163C3×Dic3—41no—
164C3×Dic3—121no—
165C2×D9—31no—
166C2×D9—61no—
167C2×D9—122no—
168C2×D9—121no—
169C4×C9—21no—
170C4×C9—42no—
171C4×C9—41no—
172C4×C9—121no—
173C4×C9D3×D411yescharacteristic
174Dic9—31no—
175Dic9—41no—
176Dic9—61no—
177#135—271no—
178C2×Q16—271no—
179Q32—31no—
180Q32—91no—
181SD32—271no—
182SD32—271no—
183M32—271no—
184C3×C9#51011yescharacteristic, complemented, Sylow
185C3×Q8—22no—
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—61no—
189C3×Q8—181no—
190D12—182no—
191D12—181no—
192D12—181no—
193D12—361no—
194D12—361no—
195D12—362no—
196C4×D3—91no—
197C4×D3—91no—
198C4×D3—181no—
199C4×D3—361no—
200C4×D3—361no—
201C4×D3—361no—
202C4×D3—361no—
203C4×D3—362no—
204Dic6—21no—
205Dic6—61no—
206Dic6—361no—
207Dic6—361no—
208C3×C8—21no—
209C3×C8C22×D911yescharacteristic
210C3×C8C2×D3211yescharacteristic
211C3⋊C8—91no—
212C3⋊C8—91no—
213C3⋊C8—181no—
214C2×C32D3×D811yescharacteristic
215C3⋊D3—61no—
216C3⋊D3—241no—
217C2×C9—21no—
218C2×C9D3×D811yescharacteristic
219D9—61no—
220D9—121no—
221D9—242no—
222D9—241no—
223#41—541no—
224C2×Q8—541no—
225Q16—31no—
226Q16—91no—
227Q16—541no—
228Q16D3×D911yescharacteristic
229SD16—541no—
230D8—271no—
231C2×C8—271no—
232C16—31no—
233C16—91no—
234C2×D3—91no—
235C2×D3—91no—
236C2×D3—181no—
237C2×D3—362no—
238C2×D3—361no—
239C2×D3—361no—
240C3×C4—21no—
241C3×C4—42no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—121no—
245C3×C4—361no—
246C3×C4D4×D911yescharacteristic
247C3×C4#381111yescharacteristic
248Dic3—41no—
249Dic3—91no—
250Dic3—91no—
251Dic3—121no—
252Dic3—181no—
253C32#491811yescharacteristic
254C9—21no—
255C9#491811yescharacteristic, complemented
256Q8—21no—
257Q8—61no—
258Q8—181no—
259Q8—1081no—
260D4—541no—
261D4—1081no—
262C2×C4—271no—
263C2×C4—1081no—
264C2×C4—1081no—
265C8—271no—
266C8C2×D3×D911yescharacteristic
267C2×C3—21no—
268C2×C3D8×D911yescharacteristic
269C2×C3#6392611yescharacteristic
270D3—181no—
271D3—181no—
272D3—361no—
273D3—722no—
274D3—721no—
275D3—721no—
276C22—271no—
277C22—1081no—
278C4—41no—
279C4—121no—
280C4—271no—
281C4—361no—
282C4#8866811yescharacteristic
283C3—21no—
284C3#1058859811yescharacteristic, complemented
285C3#1059104411yescharacteristic
286C2—541no—
287C2—2161no—
288C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
prime radical#11721805C2
hypercenterQ16D3×D9
hypocenterC3×C9#510
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
2. #11720829C2semidirect
3. C9×#829C2semidirect
4. C3×#63388C2semidirect
5. #11720243C2semidirect
6. #11720234C2semidirect
7. #11720231C2semidirect
32. #63388D3semidirect
91. #829D9semidirect
184. C3×C9#510semidirect
255. C9#4918semidirect
284. C3#10588598semidirect

Maximal subgroups

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

subgroupindex
#117218052
#117208292
C9×#8292
C3×#633882
#117202432
#117202342
#117202312
#105910443
#105885983