Finite Group Databasebeta

← #13139234

Subgroups — #13139234

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#11721791C211yesmaximal, characteristic
2#11720829C211yesmaximal, characteristic, complemented
3C9×#827C211yesmaximal, characteristic, complemented
4C3×#63386C211yesmaximal, characteristic, complemented
5#11720245C211yesmaximal, characteristic, complemented
6#11720235C211yesmaximal, characteristic, complemented
7#11720231C211yesmaximal, characteristic, complemented
8#10591030—31nomaximal
9#10588595—31nomaximal
10#88742—21no—
11(C3⋊D9)⋊Q8—21no—
12C3×C9×D8C2211yescharacteristic
13#88543—21no—
14C8×C3⋊D9C2211yescharacteristic
15#88521C2211yescharacteristic
16C9×Dic12C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88427—21no—
19C9×C3⋊C16C2211yescharacteristic
20C3×C9⋊C16C2211yescharacteristic
21#64121—31no—
22#63802—31no—
23C3×#827—31no—
24C3×#827D311yescharacteristic
25#63551—31no—
26#63551—31no—
27#63544—31no—
28Q16×D9—31no—
29#63474—31no—
30#63474—61no—
31C9×SD32—31no—
32#63388—31no—
33#63386D311yescharacteristic, complemented
34Dic72—31no—
35#63357—31no—
36C3×D4×C9—21no—
37#6233—41no—
38C2×C3⋊Dic9—41no—
39C4×C3⋊D9D411yescharacteristic
40#6226—21no—
41C9×Dic6—21no—
42C3×Dic18—21no—
43(C3⋊D9)⋊C4—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4915—91no—
48#4915—91no—
49#3830—61no—
50#3797—61no—
51C32×D8C2×D311yescharacteristic
52#3754—61no—
53#3746—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×Dic12C2×D311yescharacteristic
57#3718—61no—
58Q8×D9—61no—
59#3699—61no—
60#3699—62no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—31no—
64C9×D8—21no—
65C9×D8C2×D311yescharacteristic
66#3674—61no—
67#3672—61no—
68#3672—62no—
69C8×D9—31no—
70C8×D9—61no—
71Dic36—31no—
72Dic36—61no—
73Dic36C2×D311yescharacteristic
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic
76C22×C3×C9—41no—
77C2×C3⋊D9D811yescharacteristic
78C3×C4×C9C2×D411yescharacteristic
79C3⋊Dic9—41no—
80C3⋊Dic9D811yescharacteristic
81C9×Dic3—41no—
82C3×Dic9—41no—
83D3×Q16—91no—
84D3×Q16—91no—
85#912—91no—
86#912—91no—
87#912—181no—
88C3×SD32—31no—
89C3×SD32—91no—
90#829—91no—
91#829—91no—
92#827—31no—
93#827D911yescharacteristic, complemented
94Dic24—91no—
95Dic24—91no—
96#798—91no—
97#798—91no—
98C32×D4—21no—
99#638—121no—
100C2×C3⋊Dic3—121no—
101C4×C3⋊D3—31no—
102#634—61no—
103C3×Dic6—21no—
104C3×Dic6—61no—
105#627—121no—
106#624—121no—
107C32×C8C22×D311yescharacteristic
108C3⋊(C3⋊C8)—31no—
109Q8×C9—61no—
110D4×C9—22no—
111D4×C9—21no—
112#611—121no—
113#611—122no—
114C2×Dic9—121no—
115C2×Dic9—122no—
116C4×D9—31no—
117C4×D9—61no—
118C4×D9—121no—
119Dic18—21no—
120Dic18—62no—
121Dic18—61no—
122Dic18—121no—
123C8×C9—21no—
124C8×C9C22×D311yescharacteristic
125C9⋊C8—31no—
126C9⋊C8—61no—
127#510—271noSylow
128C2×C3×C9—81no—
129C2×C3×C9C2×D811yescharacteristic
130C3⋊D9—21no—
131D3×Q8—181no—
132D3×Q8—181no—
133#237—181no—
134#237—181no—
135#237—182no—
136C3×Q16—31no—
137C3×Q16—91no—
138C3×D8—21no—
139C3×D8C2×D911yescharacteristic
140C3×D8D3211yescharacteristic
141#216—181no—
142#216—181no—
143#214—181no—
144#214—181no—
145#214—182no—
146Dic12—31no—
147Dic12—91no—
148Dic12—91no—
149Dic12—181no—
150Dic12C2×D911yescharacteristic
151D3×C8—91no—
152D3×C8—91no—
153D3×C8—181no—
154C3×C16—31no—
155C3×C16—91no—
156C3⋊C16—31no—
157C3⋊C16C2×D911yescharacteristic
158C22×C32—41no—
159C2×C3⋊D3—31no—
160C32×C4D3×D411yescharacteristic
161C3⋊Dic3—31no—
162C3⋊Dic3—121no—
163C3×Dic3—41no—
164C3×Dic3—121no—
165C22×C9—42no—
166C22×C9—41no—
167C2×D9—31no—
168C2×D9—61no—
169C4×C9—21no—
170C4×C9—121no—
171C4×C9D3×D411yescharacteristic
172Dic9—31no—
173Dic9—41no—
174Dic9—61no—
175Dic9—122no—
176Dic9—121no—
177#135—271no—
178C2×Q16—271no—
179Q32—271no—
180Q32—271no—
181SD32—31no—
182SD32—91no—
183M32—271no—
184C3×C9#51011yescharacteristic, complemented, Sylow
185C3×Q8—61no—
186C3×Q8—181no—
187C3×D4—22no—
188C3×D4—21no—
189C3×D4—21no—
190#67—361no—
191#67—361no—
192#67—362no—
193C2×Dic3—361no—
194C2×Dic3—361no—
195C2×Dic3—362no—
196C4×D3—91no—
197C4×D3—91no—
198C4×D3—181no—
199C4×D3—361no—
200C4×D3—361no—
201Dic6—21no—
202Dic6—61no—
203Dic6—182no—
204Dic6—181no—
205Dic6—181no—
206Dic6—361no—
207Dic6—361no—
208C3×C8—21no—
209C3×C8C22×D911yescharacteristic
210C3×C8C2×D3211yescharacteristic
211C3⋊C8—91no—
212C3⋊C8—91no—
213C3⋊C8—181no—
214C2×C32—81no—
215C2×C32D3×D811yescharacteristic
216C3⋊D3—61no—
217C2×C9—21no—
218C2×C9—82no—
219C2×C9—81no—
220C2×C9D3×D811yescharacteristic
221D9—61no—
222D9—121no—
223#41—541no—
224C2×Q8—541no—
225Q16—31no—
226Q16—91no—
227Q16—271no—
228Q16—541no—
229SD16—541no—
230D8D3×D911yescharacteristic
231C2×C8—271no—
232C16—31no—
233C16—91no—
234C22×C3—42no—
235C22×C3—41no—
236C22×C3—41no—
237C2×D3—91no—
238C2×D3—91no—
239C2×D3—181no—
240C3×C4—21no—
241C3×C4—121no—
242C3×C4—361no—
243C3×C4D4×D911yescharacteristic
244C3×C4#381111yescharacteristic
245Dic3—41no—
246Dic3—91no—
247Dic3—91no—
248Dic3—121no—
249Dic3—181no—
250Dic3—362no—
251Dic3—361no—
252Dic3—361no—
253C32#491511yescharacteristic
254C9—21no—
255C9#491511yescharacteristic, complemented
256Q8—61no—
257Q8—181no—
258Q8—541no—
259Q8—1081no—
260D4—21no—
261D4—1081no—
262C2×C4—271no—
263C2×C4—1081no—
264C2×C4—1081no—
265C8—271no—
266C8C2×D3×D911yescharacteristic
267C2×C3—21no—
268C2×C3—82no—
269C2×C3—81no—
270C2×C3—81no—
271C2×C3D8×D911yescharacteristic
272C2×C3#6392611yescharacteristic
273D3—181no—
274D3—181no—
275D3—361no—
276C22—41no—
277C22—271no—
278C4—121no—
279C4—271no—
280C4—361no—
281C4—1081no—
282C4#8866811yescharacteristic
283C3—21no—
284C3#1058859511yescharacteristic, complemented
285C3#1059103011yescharacteristic
286C2—81no—
287C2—541no—
288C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×D8C22
socleC2×C32D3×D8
prime radical#11721791C2
hypercenterD8D3×D9
hypocenterC3×C9#510
LCSG ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
2. #11720829C2semidirect
3. C9×#827C2semidirect
4. C3×#63386C2semidirect
5. #11720245C2semidirect
6. #11720235C2semidirect
7. #11720231C2semidirect
33. #63386D3semidirect
93. #827D9semidirect
184. C3×C9#510semidirect
255. C9#4915semidirect
284. C3#10588595semidirect

Maximal subgroups

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

subgroupindex
#117217912
#117208292
C9×#8272
C3×#633862
#117202452
#117202352
#117202312
#105910303
#105885953