Finite Group Databasebeta

← #13139268

Subgroups — #13139268

Order 1728.

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

normal subgroups only

This group has 306 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
1Q16×C3⋊D9C211yesmaximal, characteristic, complemented
2#11720824C211yesmaximal, characteristic
3C9×#828C211yesmaximal, characteristic, complemented
4C3×#63388C211yesmaximal, characteristic, complemented
5#11720241C211yesmaximal, characteristic, complemented
6#11720235C211yesmaximal, characteristic, complemented
7#11720231C211yesmaximal, characteristic, complemented
8#10591050—31nomaximal
9#10588595—31nomaximal
10Q8×C3⋊D9—21no—
11#88639—21no—
12C3×C9×Q16C2211yescharacteristic
13#88547—21no—
14C8×C3⋊D9C2211yescharacteristic
15#88521C2211yescharacteristic
16C9×D24C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88423—21no—
19C9×C3⋊C16C2211yescharacteristic
20C3×C9⋊C16C2211yescharacteristic
21Q16×C3⋊D3—31no—
22#63810—31no—
23C3×#829—31no—
24C3×#828D311yescharacteristic
25#63551—31no—
26#63548—31no—
27#63544—31no—
28Q16×D9—31no—
29Q16×D9—61no—
30#63474—31no—
31C9×SD32—31no—
32#63388D311yescharacteristic, complemented
33#63386—31no—
34Dic72—31no—
35#63357—31no—
36C3×Q8×C9—21no—
37C4×C3⋊D9—41no—
38C4×C3⋊D9D411yescharacteristic
39#6226—21no—
40#6226—41no—
41C9×D12—21no—
42C3×Dic18—21no—
43#6195—41no—
44D3×Dic9—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4918—91no—
48#4915—91no—
49Q8×C3⋊D3—61no—
50#3796—61no—
51C32×Q16C2×D311yescharacteristic
52#3756—61no—
53#3746—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×D24C2×D311yescharacteristic
57#3715—61no—
58Q8×D9—61no—
59Q8×D9—62no—
60#3699—61no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—21no—
64C9×Q16C2×D311yescharacteristic
65C9×D8—31no—
66#3674—61no—
67#3674—62no—
68#3672—61no—
69C8×D9—31no—
70C8×D9—61no—
71Dic36—31no—
72Dic36—61no—
73Dic36C2×D311yescharacteristic
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic
76C2×C3⋊D9D811yescharacteristic
77C2×D3×C9—41no—
78C3×C4×C9—41no—
79C3×C4×C9C2×D411yescharacteristic
80C3⋊Dic9—41no—
81C3⋊Dic9D811yescharacteristic
82C3×Dic9—41no—
83#919—91no—
84D3×Q16—91no—
85D3×Q16—91no—
86D3×Q16—181no—
87#912—91no—
88C3×Q32—91no—
89C3×SD32—31no—
90#829—31no—
91#829—91no—
92#828D911yescharacteristic, complemented
93#827—91no—
94Dic24—91no—
95#800—91no—
96#798—91no—
97#798—91no—
98C32×Q8—21no—
99C4×C3⋊D3—31no—
100C4×C3⋊D3—121no—
101#634—61no—
102#634—121no—
103C3×D12—21no—
104C3×Dic6—61no—
105#626—121no—
106D3×Dic3—121no—
107C32×C8C22×D311yescharacteristic
108C3⋊(C3⋊C8)—31no—
109Q8×C9—22no—
110Q8×C9—21no—
111D4×C9—61no—
112#611—121no—
113C2×Dic9—121no—
114C4×D9—31no—
115C4×D9—61no—
116C4×D9—121no—
117C4×D9—122no—
118Dic18—21no—
119Dic18—62no—
120Dic18—61no—
121Dic18—121no—
122Dic18—122no—
123C8×C9—21no—
124C8×C9C22×D311yescharacteristic
125C9⋊C8—31no—
126C9⋊C8—61no—
127#510—271noSylow
128C2×C3×C9C2×D811yescharacteristic
129C3⋊D9—21no—
130D3×C9—81no—
131#239—181no—
132D3×Q8—181no—
133D3×Q8—181no—
134D3×Q8—182no—
135#237—181no—
136C3×Q16—21no—
137C3×Q16—91no—
138C3×Q16C2×D911yescharacteristic
139C3×Q16D3211yescharacteristic
140C3×D8—31no—
141#216—181no—
142#216—181no—
143#216—182no—
144#215—181no—
145#214—181no—
146Dic12—31no—
147Dic12—91no—
148Dic12—91no—
149Dic12—181no—
150D24C2×D911yescharacteristic
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—41no—
160C32×C4—41no—
161C32×C4D3×D411yescharacteristic
162C3⋊Dic3—31no—
163C3⋊Dic3—121no—
164C3×Dic3—121no—
165C22×C9—121no—
166C2×D9—31no—
167C2×D9—61no—
168C4×C9—21no—
169C4×C9—42no—
170C4×C9—41no—
171C4×C9D3×D411yescharacteristic
172Dic9—31no—
173Dic9—41no—
174Dic9—61no—
175Dic9—122no—
176Dic9—121no—
177#135—271no—
178C2×Q16—271no—
179Q32—91no—
180Q32—271no—
181SD32—31no—
182SD32—271no—
183M32—271no—
184C3×C9#51011yescharacteristic, complemented, Sylow
185C3×Q8—22no—
186C3×Q8—21no—
187C3×Q8—21no—
188C3×Q8—181no—
189C3×D4—61no—
190#67—361no—
191C2×Dic3—361no—
192D12—21no—
193D12—361no—
194C4×D3—91no—
195C4×D3—91no—
196C4×D3—181no—
197C4×D3—361no—
198C4×D3—361no—
199C4×D3—361no—
200C4×D3—362no—
201Dic6—61no—
202Dic6—182no—
203Dic6—181no—
204Dic6—181no—
205Dic6—361no—
206Dic6—361no—
207Dic6—362no—
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—81no—
217C2×C9—21no—
218C2×C9—241no—
219C2×C9D3×D811yescharacteristic
220D9—62no—
221D9—61no—
222#41—541no—
223C2×Q8—541no—
224Q16—91no—
225Q16—271no—
226Q16—541no—
227Q16D3×D911yescharacteristic
228SD16—541no—
229D8—31no—
230C2×C8—271no—
231C16—31no—
232C16—91no—
233C22×C3—121no—
234C2×D3—41no—
235C2×D3—91no—
236C2×D3—91no—
237C2×D3—181no—
238C3×C4—21no—
239C3×C4—42no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—361no—
243C3×C4D4×D911yescharacteristic
244C3×C4#381111yescharacteristic
245Dic3—91no—
246Dic3—91no—
247Dic3—121no—
248Dic3—181no—
249Dic3—362no—
250Dic3—361no—
251Dic3—361no—
252C32#491511yescharacteristic
253C9—21no—
254C9#491811yescharacteristic, complemented
255Q8—21no—
256Q8—181no—
257Q8—541no—
258Q8—1081no—
259D4—61no—
260D4—1081no—
261C2×C4—271no—
262C2×C4—1081no—
263C2×C4—1081no—
264C8—271no—
265C8C2×D3×D911yescharacteristic
266C2×C3—21no—
267C2×C3—241no—
268C2×C3D8×D911yescharacteristic
269C2×C3#6392611yescharacteristic
270D3—81no—
271D3—182no—
272D3—181no—
273D3—181no—
274C22—121no—
275C22—271no—
276C4—41no—
277C4—271no—
278C4—361no—
279C4—1081no—
280C4#8866811yescharacteristic
281C3—21no—
282C3#1058859511yescharacteristic, complemented
283C3#1059105011yescharacteristic
284C2—241no—
285C2—541no—
286C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×Q16C22
socleC2×C32D3×D8
prime radical#11720824C2
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
1. Q16×C3⋊D9C2semidirect
3. C9×#828C2semidirect
4. C3×#63388C2semidirect
5. #11720241C2semidirect
6. #11720235C2semidirect
7. #11720231C2semidirect
32. #63388D3semidirect
92. #828D9semidirect
184. C3×C9#510semidirect
254. C9#4918semidirect
282. C3#10588595semidirect

Maximal subgroups

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

subgroupindex
Q16×C3⋊D92
#117208242
C9×#8282
C3×#633882
#117202412
#117202352
#117202312
#105910503
#105885953