Finite Group Databasebeta

← #13139229

Subgroups — #13139229

Order 1728.

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

normal subgroups only

This group has 314 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, complemented
2#11720824C211yesmaximal, characteristic, complemented
3C9×#826C211yesmaximal, characteristic, complemented
4C3×#63386C211yesmaximal, characteristic, complemented
5#11720241C211yesmaximal, characteristic, complemented
6#11720235C211yesmaximal, characteristic, complemented
7(C3⋊D9)⋊C16C211yesmaximal, characteristic, complemented
8#10591035—31nomaximal
9#10588592—31nomaximal
10#88742—21no—
11#88639—21no—
12C3×C9×D8C2211yescharacteristic
13#88543—21no—
14C8×C3⋊D9C2211yescharacteristic
15#88521C2211yescharacteristic
16C9×D24C2211yescharacteristic
17C3×Dic36C2211yescharacteristic
18#88423—21no—
19C9×C3⋊C16C2211yescharacteristic
20C3×C9⋊C16C2211yescharacteristic
21#64121—31no—
22#63810—31no—
23C3×#827—31no—
24C3×#826D311yescharacteristic
25#63551—31no—
26#63548—31no—
27#63542—31no—
28#63474—31no—
29#63474—31no—
30#63474—61no—
31C9×D16—31no—
32#63386—31no—
33#63386D311yescharacteristic, complemented
34Dic72—31no—
35C16×D9—31no—
36C3×D4×C9—21no—
37#6233—41no—
38C2×C3⋊Dic9—41no—
39C4×C3⋊D9D411yescharacteristic
40#6226—21no—
41C9×D12—21no—
42C3×Dic18—21no—
43#6195—41no—
44D3×Dic9—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4916—91no—
48#4912—91no—
49#3830—61no—
50#3796—61no—
51C32×D8C2×D311yescharacteristic
52#3754—61no—
53#3746—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×D24C2×D311yescharacteristic
57#3715—61no—
58#3699—61no—
59#3699—61no—
60#3699—62no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×D8—21no—
64C9×D8—31no—
65C9×D8C2×D311yescharacteristic
66#3672—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
78C2×D3×C9—41no—
79C3×C4×C9C2×D411yescharacteristic
80C3⋊Dic9—41no—
81C3⋊Dic9D811yescharacteristic
82C3×Dic9—41no—
83#919—91no—
84#912—91no—
85#912—91no—
86#912—91no—
87#912—181no—
88C3×SD32—91no—
89C3×D16—31no—
90#829—91no—
91#827—31no—
92#827—91no—
93#826D911yescharacteristic, complemented
94Dic24—91no—
95#800—91no—
96D3×C16—91no—
97D3×C16—91no—
98C32×D4—21no—
99#638—121no—
100C2×C3⋊Dic3—121no—
101C4×C3⋊D3—31no—
102#634—61no—
103C3×D12—21no—
104C3×Dic6—61no—
105#626—121no—
106D3×Dic3—121no—
107C32×C8C22×D311yescharacteristic
108C3⋊(C3⋊C8)—31no—
109D4×C9—22no—
110D4×C9—21no—
111D4×C9—61no—
112#611—121no—
113#611—121no—
114#611—122no—
115C2×Dic9—121no—
116C2×Dic9—121no—
117C2×Dic9—122no—
118C4×D9—31no—
119C4×D9—61no—
120Dic18—21no—
121Dic18—62no—
122Dic18—61no—
123C8×C9—21no—
124C8×C9C22×D311yescharacteristic
125C9⋊C8—31no—
126C9⋊C8—61no—
127#508—271noSylow
128C2×C3×C9—81no—
129C2×C3×C9C2×D811yescharacteristic
130C3⋊D9—21no—
131D3×C9—81no—
132#239—181no—
133#237—181no—
134#237—181no—
135#237—181no—
136#237—182no—
137C3×Q16—91no—
138C3×D8—21no—
139C3×D8—31no—
140C3×D8C2×D911yescharacteristic
141C3×D8D3211yescharacteristic
142#215—181no—
143#214—181no—
144#214—181no—
145#214—181no—
146#214—182no—
147Dic12—31no—
148Dic12—91no—
149Dic12—91no—
150Dic12—181no—
151D24C2×D911yescharacteristic
152D3×C8—91no—
153D3×C8—91no—
154D3×C8—181no—
155C3×C16—31no—
156C3×C16—91no—
157C3⋊C16—31no—
158C3⋊C16C2×D911yescharacteristic
159C22×C32—41no—
160C2×C3⋊D3—31no—
161C2×C3×D3—41no—
162C32×C4D3×D411yescharacteristic
163C3⋊Dic3—31no—
164C3⋊Dic3—121no—
165C3×Dic3—121no—
166C22×C9—42no—
167C22×C9—41no—
168C22×C9—121no—
169C2×D9—31no—
170C2×D9—61no—
171C4×C9—21no—
172C4×C9D3×D411yescharacteristic
173Dic9—31no—
174Dic9—41no—
175Dic9—61no—
176Dic9—122no—
177Dic9—121no—
178#135—271no—
179#135—271no—
180Q32—271no—
181SD32—91no—
182SD32—271no—
183D16—31no—
184C2×C16—271no—
185C3×C9#50811yescharacteristic, complemented, Sylow
186C3×Q8—181no—
187C3×D4—22no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—61no—
191#67—361no—
192#67—361no—
193#67—361no—
194#67—362no—
195C2×Dic3—361no—
196C2×Dic3—361no—
197C2×Dic3—361no—
198C2×Dic3—362no—
199D12—21no—
200D12—361no—
201C4×D3—91no—
202C4×D3—91no—
203C4×D3—181no—
204C4×D3—361no—
205Dic6—61no—
206Dic6—182no—
207Dic6—181no—
208Dic6—181no—
209C3×C8—21no—
210C3×C8C22×D911yescharacteristic
211C3×C8C2×D3211yescharacteristic
212C3⋊C8—91no—
213C3⋊C8—91no—
214C3⋊C8—181no—
215C2×C32—81no—
216C2×C32D3×D811yescharacteristic
217C3⋊D3—61no—
218C3×D3—81no—
219C2×C9—21no—
220C2×C9—82no—
221C2×C9—81no—
222C2×C9—241no—
223C2×C9D3×D811yescharacteristic
224D9—61no—
225D9—121no—
226#41—541no—
227#41—541no—
228Q16—91no—
229Q16—271no—
230SD16—541no—
231SD16—541no—
232D8—31no—
233D8D3×D911yescharacteristic
234C2×C8—271no—
235C16—31no—
236C16—91no—
237C22×C3—42no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—121no—
241C2×D3—41no—
242C2×D3—91no—
243C2×D3—91no—
244C2×D3—181no—
245C3×C4—21no—
246C3×C4—361no—
247C3×C4D4×D911yescharacteristic
248C3×C4#381111yescharacteristic
249Dic3—91no—
250Dic3—91no—
251Dic3—121no—
252Dic3—181no—
253Dic3—362no—
254Dic3—361no—
255Dic3—361no—
256C32#491211yescharacteristic
257C9—21no—
258C9#491611yescharacteristic, complemented
259Q8—181no—
260Q8—541no—
261D4—21no—
262D4—61no—
263D4—1081no—
264D4—1081no—
265C2×C4—271no—
266C2×C4—1081no—
267C2×C4—1081no—
268C8—271no—
269C8C2×D3×D911yescharacteristic
270C2×C3—21no—
271C2×C3—82no—
272C2×C3—81no—
273C2×C3—81no—
274C2×C3—241no—
275C2×C3D8×D911yescharacteristic
276C2×C3#6392611yescharacteristic
277D3—81no—
278D3—181no—
279D3—181no—
280D3—361no—
281C22—41no—
282C22—121no—
283C22—271no—
284C4—271no—
285C4—361no—
286C4—1081no—
287C4#8866811yescharacteristic
288C3—21no—
289C3#1058859211yescharacteristic, complemented
290C3#1059103511yescharacteristic
291C2—81no—
292C2—241no—
293C2—541no—
294C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×D8C22
socleC2×C32D3×D8
hypercenterD8D3×D9
hypocenterC3×C9#508
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
1. #11721791C2semidirect
2. #11720824C2semidirect
3. C9×#826C2semidirect
4. C3×#63386C2semidirect
5. #11720241C2semidirect
6. #11720235C2semidirect
7. (C3⋊D9)⋊C16C2semidirect
33. #63386D3semidirect
93. #826D9semidirect
185. C3×C9#508semidirect
258. C9#4916semidirect
289. C3#10588592semidirect

Maximal subgroups

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

subgroupindex
#117217912
#117208242
C9×#8262
C3×#633862
#117202412
#117202352
(C3⋊D9)⋊C162
#105910353
#105885923