Finite Group Databasebeta

← #13139233

Subgroups — #13139233

Order 1728.

346 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 59 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
2#11720829C211yesmaximal, characteristic, complemented
3C9×#827C211yesmaximal, characteristic, complemented
4C3×#63386C211yesmaximal, characteristic, complemented
5#11720243C211yesmaximal, characteristic, complemented
6#11720234C211yesmaximal, characteristic, complemented
7(C3⋊D9)⋊C16C211yesmaximal, characteristic, complemented
8#10591029—31nomaximal
9D9×SD32—31nomaximal
10D4×C3⋊D9—21no—
11(C3⋊D9)⋊Q8—21no—
12C3×C9×D8C2211yescharacteristic
13#88544—21no—
14#88525C2211yescharacteristic
15C8×C3⋊D9C2211yescharacteristic
16C9×Dic12C2211yescharacteristic, complemented
17C3×Dic36C2211yescharacteristic, complemented
18#88427—21no—
19C9×C3⋊C16C2211yescharacteristic, complemented
20C3×C9⋊C16C2211yescharacteristic, complemented
21D8×C3⋊D3—31no—
22#63802—31no—
23C3×#827—31no—
24C3×#827D311yescharacteristic
25#63549—31no—
26#63549—31no—
27#63542—31no—
28Q16×D9—31no—
29D8×D9—31no—
30D8×D9—61no—
31C9×SD32—31no—
32#63387—31no—
33#63386D311yescharacteristic, complemented
34#63359—31no—
35C16×D9—31no—
36C22×C3⋊D9—41no—
37C3×D4×C9—21no—
38#6233—41no—
39C4⋊(C3⋊D9)—21no—
40C4×C3⋊D9D411yescharacteristic
41C9×Dic6—21no—
42C3×Dic18—21no—
43(C3⋊D9)⋊C4—41no—
44#6189—41no—
45C3×C8×C9C2311yescharacteristic
46C3⋊(C9⋊C8)D411yescharacteristic
47D3×SD32—91no—
48D3×SD32—91no—
49D4×C3⋊D3—61no—
50#3797—61no—
51C32×D8C2×D311yescharacteristic
52#3753—61no—
53#3745—31no—
54C8×C3⋊D3—31no—
55C3×Dic12—31no—
56C3×Dic12C2×D311yescharacteristic
57#3718—61no—
58Q8×D9—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#3674—61no—
67#3673—61no—
68#3673—62no—
69D72—31no—
70D72—61no—
71C8×D9—31no—
72C8×D9—61no—
73Dic36C2×D311yescharacteristic, complemented
74C9×C16—31no—
75C9⋊C16C2×D311yescharacteristic, complemented
76C22×C3×C9—41no—
77C2×C3⋊D9—41no—
78C2×C3⋊D9—82no—
79C2×C3⋊D9D811yescharacteristic
80C3×C4×C9C2×D411yescharacteristic
81C3⋊Dic9D811yescharacteristic
82C9×Dic3—41no—
83C3×Dic9—41no—
84D3×Q16—91no—
85D3×Q16—91no—
86D3×D8—91no—
87D3×D8—91no—
88D3×D8—181no—
89C3×SD32—31no—
90C3×SD32—91no—
91#828—91no—
92#828—91no—
93#827—31no—
94#827D911yescharacteristic, complemented
95#800—91no—
96#800—91no—
97D3×C16—91no—
98D3×C16—91no—
99C22×C3⋊D3—121no—
100C32×D4—21no—
101#638—121no—
102C4⋊(C3⋊D3)—61no—
103C4×C3⋊D3—31no—
104C3×Dic6—21no—
105C3×Dic6—61no—
106#627—121no—
107#624—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—62no—
118D36—61no—
119C4×D9—31no—
120C4×D9—61no—
121C4×D9—121no—
122Dic18—21no—
123Dic18—121no—
124C8×C9—21no—
125C8×C9C22×D311yescharacteristic
126C9⋊C8—31no—
127C9⋊C8—61no—
128C2×SD32—271noSylow
129C2×C3×C9—81no—
130C2×C3×C9C2×D811yescharacteristic
131C3⋊D9—81no—
132C3⋊D9SD3212yescomplemented
133D3×Q8—181no—
134D3×Q8—181no—
135D3×D4—181no—
136D3×D4—181no—
137D3×D4—182no—
138C3×Q16—31no—
139C3×Q16—91no—
140C3×D8—21no—
141C3×D8C2×D911yescharacteristic
142C3×D8D3211yescharacteristic
143#216—181no—
144#216—181no—
145#213—181no—
146#213—181no—
147#213—182no—
148Dic12—31no—
149Dic12C2×D911yescharacteristic, complemented
150D24—91no—
151D24—91no—
152D24—181no—
153D3×C8—91no—
154D3×C8—91no—
155D3×C8—181no—
156C3×C16—31no—
157C3×C16—91no—
158C3⋊C16—31no—
159C3⋊C16C2×D911yescharacteristic, complemented
160C22×C32—41no—
161C2×C3⋊D3—31no—
162C2×C3⋊D3—121no—
163C2×C3⋊D3—242no—
164C32×C4D3×D411yescharacteristic
165C3⋊Dic3—31no—
166C3×Dic3—41no—
167C3×Dic3—121no—
168C22×C9—42no—
169C22×C9—41no—
170C2×D9—31no—
171C2×D9—61no—
172C2×D9—122no—
173C2×D9—121no—
174C2×D9—242no—
175C2×D9—244no—
176C4×C9—21no—
177C4×C9—121no—
178C4×C9D3×D411yescharacteristic
179Dic9—31no—
180Dic9—41no—
181Dic9—61no—
182C2×Q16—271no—
183C2×D8—271no—
184SD32—31no—
185SD32—91no—
186SD32—271no—
187SD32—271no—
188C2×C16—271no—
189C3×C9C2×SD3211yescharacteristic, complemented, Sylow
190C22×D3—361no—
191C22×D3—361no—
192C22×D3—362no—
193C3×Q8—61no—
194C3×Q8—181no—
195C3×D4—22no—
196C3×D4—21no—
197C3×D4—21no—
198#67—361no—
199#67—361no—
200#67—362no—
201D12—182no—
202D12—181no—
203D12—181no—
204C4×D3—91no—
205C4×D3—91no—
206C4×D3—181no—
207C4×D3—361no—
208C4×D3—361no—
209Dic6—21no—
210Dic6—61no—
211Dic6—361no—
212Dic6—361no—
213C3×C8—21no—
214C3×C8C22×D911yescharacteristic
215C3×C8C2×D3211yescharacteristic
216C3⋊C8—91no—
217C3⋊C8—91no—
218C3⋊C8—181no—
219C2×C32—81no—
220C2×C32D3×D811yescharacteristic
221C3⋊D3—32no—
222C3⋊D3—241no—
223C2×C9—21no—
224C2×C9—82no—
225C2×C9—81no—
226C2×C9D3×D811yescharacteristic
227D9—32no—
228D9—62no—
229D9—242no—
230D9—241no—
231C2×Q8—541no—
232C2×D4—541no—
233Q16—31no—
234Q16—91no—
235Q16—541no—
236D8—271no—
237D8—541no—
238D8D3×D911yescharacteristic
239C2×C8—271no—
240C16—31no—
241C16—91no—
242C22×C3—42no—
243C22×C3—41no—
244C22×C3—41no—
245C2×D3—91no—
246C2×D3—91no—
247C2×D3—181no—
248C2×D3—362no—
249C2×D3—361no—
250C2×D3—361no—
251C2×D3—722no—
252C2×D3—722no—
253C2×D3—724no—
254C3×C4—21no—
255C3×C4—121no—
256C3×C4—361no—
257C3×C4D4×D911yescharacteristic
258C3×C4#381111yescharacteristic
259Dic3—41no—
260Dic3—91no—
261Dic3—91no—
262Dic3—121no—
263Dic3—181no—
264C32D3×SD3211yescharacteristic
265C9—21no—
266C9D3×SD3211yescharacteristic, complemented
267C23—1081no—
268Q8—61no—
269Q8—181no—
270Q8—1081no—
271D4—21no—
272D4—541no—
273D4—1081no—
274C2×C4—271no—
275C2×C4—1081no—
276C8—271no—
277C8C2×D3×D911yescharacteristic
278C2×C3—21no—
279C2×C3—82no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3D8×D911yescharacteristic
283C2×C3#6392611yescharacteristic
284D3—92no—
285D3—92no—
286D3—182no—
287D3—722no—
288D3—721no—
289D3—721no—
290C22—41no—
291C22—271no—
292C22—1081no—
293C22—2162no—
294C4—121no—
295C4—271no—
296C4—361no—
297C4#8866811yescharacteristic
298C3—21no—
299C3D9×SD3211yescharacteristic, complemented
300C3#1059102911yescharacteristic
301C2—81no—
302C2—272no—
303C2—2161no—
304C2#1172117411yescharacteristic
subgroupG/H
centerC2#11721174
commutator, cosocleC3×C8×C9C23
FrattiniC3×C8C2×D32
FittingC3×C9×D8C22
socleC2×C32D3×D8
prime radicalD8×C3⋊D9C2
hypercenterD8D3×D9
hypocenterC3×C9C2×SD32
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. #11720243C2semidirect
6. #11720234C2semidirect
7. (C3⋊D9)⋊C16C2semidirect
16. C9×Dic12C22semidirect
17. C3×Dic36C22semidirect
19. C9×C3⋊C16C22semidirect
20. C3×C9⋊C16C22semidirect
33. #63386D3semidirect
73. Dic36C2×D3semidirect
75. C9⋊C16C2×D3semidirect
94. #827D9semidirect
132. C3⋊D9SD32semidirect
149. Dic12C2×D9semidirect
159. C3⋊C16C2×D9semidirect
189. C3×C9C2×SD32semidirect
266. C9D3×SD32semidirect
299. C3D9×SD32semidirect

Maximal subgroups

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

subgroupindex
D8×C3⋊D92
#117208292
C9×#8272
C3×#633862
#117202432
#117202342
(C3⋊D9)⋊C162
#105910293
D9×SD323