Finite Group Databasebeta

← #13139238

Subgroups — #13139238

Order 1728.

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

normal subgroups only

This group has 346 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
1D8×C3⋊D9C211yesmaximal, characteristic, complemented
2#11720831C211yesmaximal, characteristic, complemented
3C9×#827C211yesmaximal, characteristic, complemented
4C3×#63385C211yesmaximal, characteristic, complemented
5#11720243C211yesmaximal, characteristic, complemented
6#11720232C211yesmaximal, characteristic, complemented
7#11720231C211yesmaximal, characteristic, complemented
8#10591036—31nomaximal
9#10588594—31nomaximal
10D4×C3⋊D9—21no—
11#88634—21no—
12C3×C9×D8C2211yescharacteristic
13#88544—21no—
14#88525C2211yescharacteristic
15C8×C3⋊D9C2211yescharacteristic
16C9×Dic12C2211yescharacteristic, complemented
17C3×D72C2211yescharacteristic, complemented
18#88425—21no—
19C9×C3⋊C16C2211yescharacteristic, complemented
20C3×C9⋊C16C2211yescharacteristic, complemented
21D8×C3⋊D3—31no—
22#63810—31no—
23C3×#827D311yescharacteristic
24C3×#826—31no—
25#63549—31no—
26#63546—31no—
27#63544—31no—
28#63481—31no—
29D8×D9—31no—
30D8×D9—61no—
31C9×SD32—31no—
32#63387—31no—
33#63385D311yescharacteristic, complemented
34D144—31no—
35#63357—31no—
36C22×C3⋊D9—41no—
37C3×D4×C9—21no—
38#6233—41no—
39C4⋊(C3⋊D9)—21no—
40C4×C3⋊D9D411yescharacteristic
41C3×D36—21no—
42C9×Dic6—21no—
43#6192—41no—
44Dic3×D9—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47#4914—91no—
48#4911—91no—
49D4×C3⋊D3—61no—
50#3796—61no—
51C32×D8C2×D311yescharacteristic
52#3753—61no—
53#3745—31no—
54C8×C3⋊D3—31no—
55C3×Dic12C2×D311yescharacteristic
56C3×D24—31no—
57#3715—61no—
58#3701—61no—
59D4×D9—61no—
60D4×D9—62no—
61C3×C3⋊C16—31no—
62C3×C3⋊C16C2×D311yescharacteristic
63C9×Q16—31no—
64C9×D8—21no—
65C9×D8C2×D311yescharacteristic
66#3675—61no—
67#3673—61no—
68#3673—62no—
69D72—31no—
70D72—61no—
71D72C2×D311yescharacteristic, complemented
72C8×D9—31no—
73C8×D9—61no—
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic, complemented
76C22×C3×C9—41no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—82no—
79C2×C3⋊D9D811yescharacteristic
80C2×C3×D9—41no—
81C3×C4×C9C2×D411yescharacteristic
82C3⋊Dic9D811yescharacteristic
83C9×Dic3—41no—
84#919—91no—
85#912—91no—
86D3×D8—91no—
87D3×D8—91no—
88D3×D8—181no—
89C3×SD32—31no—
90C3×D16—91no—
91#828—91no—
92#827D911yescharacteristic, complemented
93#826—31no—
94#826—91no—
95#800—91no—
96D48—91no—
97#798—91no—
98#798—91no—
99C22×C3⋊D3—121no—
100C32×D4—21no—
101#638—121no—
102C4⋊(C3⋊D3)—61no—
103C4×C3⋊D3—31no—
104C3×D12—61no—
105C3×Dic6—21no—
106#626—121no—
107D3×Dic3—121no—
108C22×D9—121no—
109C22×D9—122no—
110C32×C8C22×D311yescharacteristic
111C3⋊(C3⋊C8)—31no—
112Q8×C9—61no—
113D4×C9—22no—
114D4×C9—21no—
115#611—121no—
116#611—122no—
117D36—21no—
118D36—62no—
119D36—61no—
120D36—121no—
121C4×D9—31no—
122C4×D9—61no—
123C4×D9—121no—
124C8×C9—21no—
125C8×C9C22×D311yescharacteristic
126C9⋊C8—31no—
127C9⋊C8—61no—
128#509—271noSylow
129C2×C3×C9—81no—
130C2×C3×C9C2×D811yescharacteristic
131C3⋊D9—21no—
132C3⋊D9—81no—
133C3×D9—81no—
134#239—181no—
135#237—181no—
136D3×D4—181no—
137D3×D4—181no—
138D3×D4—182no—
139C3×Q16—31no—
140C3×D8—21no—
141C3×D8—91no—
142C3×D8C2×D911yescharacteristic
143C3×D8D3211yescharacteristic
144#215—181no—
145#214—181no—
146#213—181no—
147#213—181no—
148#213—182no—
149Dic12C2×D911yescharacteristic, complemented
150D24—31no—
151D24—91no—
152D24—91no—
153D24—181no—
154D3×C8—91no—
155D3×C8—91no—
156D3×C8—181no—
157C3×C16—31no—
158C3×C16—91no—
159C3⋊C16—31no—
160C3⋊C16C2×D911yescharacteristic, complemented
161C22×C32—41no—
162C2×C3⋊D3—31no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—242no—
165C2×C3×D3—121no—
166C32×C4D3×D411yescharacteristic
167C3⋊Dic3—31no—
168C3×Dic3—41no—
169C22×C9—42no—
170C22×C9—41no—
171C2×D9—31no—
172C2×D9—41no—
173C2×D9—61no—
174C2×D9—122no—
175C2×D9—121no—
176C2×D9—244no—
177C2×D9—242no—
178C4×C9—21no—
179C4×C9—121no—
180C4×C9D3×D411yescharacteristic
181Dic9—31no—
182Dic9—61no—
183#135—271no—
184C2×D8—271no—
185SD32—31no—
186SD32—271no—
187D16—91no—
188D16—271no—
189M32—271no—
190C3×C9#50911yescharacteristic, complemented, Sylow
191C22×D3—361no—
192C22×D3—361no—
193C22×D3—362no—
194C3×Q8—61no—
195C3×D4—22no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—181no—
199#67—361no—
200#67—361no—
201#67—361no—
202#67—362no—
203C2×Dic3—361no—
204D12—61no—
205D12—182no—
206D12—181no—
207D12—181no—
208D12—361no—
209C4×D3—91no—
210C4×D3—91no—
211C4×D3—181no—
212C4×D3—361no—
213Dic6—21no—
214C3×C8—21no—
215C3×C8C22×D911yescharacteristic
216C3×C8C2×D3211yescharacteristic
217C3⋊C8—91no—
218C3⋊C8—91no—
219C3⋊C8—181no—
220C2×C32—81no—
221C2×C32D3×D811yescharacteristic
222C3⋊D3—61no—
223C3⋊D3—241no—
224C3×D3—241no—
225C2×C9—21no—
226C2×C9—82no—
227C2×C9—81no—
228C2×C9D3×D811yescharacteristic
229D9—62no—
230D9—61no—
231D9—81no—
232D9—242no—
233D9—241no—
234#41—541no—
235C2×D4—541no—
236Q16—31no—
237SD16—541no—
238D8—91no—
239D8—271no—
240D8—541no—
241D8D3×D911yescharacteristic
242C2×C8—271no—
243C16—31no—
244C16—91no—
245C22×C3—42no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—361no—
249C2×D3—91no—
250C2×D3—91no—
251C2×D3—121no—
252C2×D3—181no—
253C2×D3—362no—
254C2×D3—361no—
255C2×D3—361no—
256C2×D3—722no—
257C2×D3—724no—
258C2×D3—722no—
259C3×C4—21no—
260C3×C4—121no—
261C3×C4D4×D911yescharacteristic
262C3×C4#381111yescharacteristic
263Dic3—41no—
264Dic3—91no—
265Dic3—91no—
266Dic3—181no—
267C32#491411yescharacteristic
268C9—21no—
269C9#491111yescharacteristic, complemented
270C23—1081no—
271Q8—61no—
272D4—21no—
273D4—181no—
274D4—541no—
275D4—1081no—
276D4—1081no—
277C2×C4—271no—
278C2×C4—1081no—
279C8—271no—
280C8C2×D3×D911yescharacteristic
281C2×C3—21no—
282C2×C3—82no—
283C2×C3—81no—
284C2×C3—81no—
285C2×C3—721no—
286C2×C3D8×D911yescharacteristic
287C2×C3#6392611yescharacteristic
288D3—182no—
289D3—181no—
290D3—181no—
291D3—241no—
292D3—722no—
293D3—721no—
294D3—721no—
295C22—41no—
296C22—271no—
297C22—361no—
298C22—1081no—
299C22—2162no—
300C4—121no—
301C4—271no—
302C4#8866811yescharacteristic
303C3—21no—
304C3#1058859411yescharacteristic, complemented
305C3#1059103611yescharacteristic
306C2—81no—
307C2—541no—
308C2—721no—
309C2—2161no—
310C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×D8C22
socleC2×C32D3×D8
hypercenterD8D3×D9
hypocenterC3×C9#509
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. D8×C3⋊D9C2semidirect
2. #11720831C2semidirect
3. C9×#827C2semidirect
4. C3×#63385C2semidirect
5. #11720243C2semidirect
6. #11720232C2semidirect
7. #11720231C2semidirect
16. C9×Dic12C22semidirect
17. C3×D72C22semidirect
19. C9×C3⋊C16C22semidirect
20. C3×C9⋊C16C22semidirect
33. #63385D3semidirect
71. D72C2×D3semidirect
75. C9⋊C16C2×D3semidirect
92. #827D9semidirect
149. Dic12C2×D9semidirect
160. C3⋊C16C2×D9semidirect
190. C3×C9#509semidirect
269. C9#4911semidirect
304. C3#10588594semidirect

Maximal subgroups

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

subgroupindex
D8×C3⋊D92
#117208312
C9×#8272
C3×#633852
#117202432
#117202322
#117202312
#105910363
#105885943