Finite Group Databasebeta

← #13161783

Subgroups — #13161783

Order 1728.

490 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C3×Q16×C3⋊D3C211yesmaximal, characteristic, complemented
2#11723647C211yesmaximal, characteristic
3C3×#63655C212yesmaximal, complemented
4#11722068C212yesmaximal, complemented
5#11722065C211yesmaximal, characteristic, complemented
6#10591039—31nomaximal
7D3×#828—32nomaximal
8C3×Q8×C3⋊D3—21no—
9C4⋊#1113—21no—
10C33×Q16C2211yescharacteristic, complemented
11C3×#3756—21no—
12C3×#3746C2211yescharacteristic, complemented
13C3×#3745C2212yes—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88810—21no—
16C3×C3⋊(C3⋊C16)C2212yescomplemented
17Q16×C3⋊D3D311yescharacteristic, complemented
18C3×D3×Q16—32no—
19C3×D3×Q16—61no—
20#63797—31no—
21D3×D24—32no—
22#63655D312yescomplemented
23C3×#828—32no—
24C3×#828—32no—
25C3×#828—62no—
26#63549—32no—
27#63548—32no—
28#63547—32no—
29#63542—31no—
30D3×C3⋊C16—32no—
31C2×#1113—41no—
32C33×Q8—21no—
33C3×C4⋊(C3⋊D3)—22no—
34C3×C4×C3⋊D3—41no—
35C3×C4×C3⋊D3D411yescharacteristic
36C3×#634—21no—
37C3×#634—41no—
38#6295—41no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41C2×#828—91no—
42D3×SD32—92no—
43Q8×C3⋊D3—21no—
44C3×D3×Q8—62no—
45C3×D3×Q8—62no—
46#3802—61no—
47D3×D12—62no—
48C32×Q16—21no—
49C32×Q16—22no—
50C32×Q16—41no—
51C32×Q16C2×D312yescomplemented
52C32×Q16C2×D311yescharacteristic, complemented
53#3756—21no—
54#3746C2×D311yescharacteristic, complemented
55#3745C2×D312yes—
56C8×C3⋊D3C2×D311yescharacteristic
57C3×#216—62no—
58C3×#216—62no—
59C3×Dic12—32no—
60C3×Dic12—61no—
61C3×D24—32no—
62C3×D24—32no—
63C3×D24—62no—
64C3×D3×C8—32no—
65C3×D3×C8—61no—
66#3714—62no—
67#3713—61no—
68C3⋊(C3⋊C16)C2×D312yescomplemented
69C3×C3⋊C16—32no—
70C3×C3⋊C16—32no—
71C3×C3⋊C16—62no—
72C2×C3×C3⋊D3—42no—
73C2×C3×C3⋊D3D811yescharacteristic
74#1113—82no—
75C33×C4—41no—
76C33×C4C2×D411yescharacteristic
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3D811yescharacteristic
79C2×C3×Q16—91no—
80D3×Q16—32no—
81D3×Q16—61no—
82D3×D8—92no—
83C2×D24—91no—
84C3×SD32—92no—
85#828—32no—
86#828—32no—
87#828—62no—
88#828—92no—
89#827—92no—
90C2×C3⋊C16—91no—
91#800—92no—
92D3×C16—92no—
93C2×D32—122no—
94C2×D32—121no—
95C32×Q8—21no—
96C32×Q8—22no—
97C32×Q8—22no—
98C32×Q8—42no—
99C32×Q8—42no—
100C4⋊(C3⋊D3)—22no—
101C4×C3⋊D3—41no—
102C4×C3⋊D3#6711yescharacteristic
103#634—21no—
104#634—41no—
105C3×D12—62no—
106C3×D12—62no—
107C3×D12—122no—
108C3×C4×D3—32no—
109C3×C4×D3—61no—
110C3×C4×D3—122no—
111C3×C4×D3—122no—
112C3×Dic6—62no—
113C3×Dic6—62no—
114C3×Dic6—122no—
115C3×Dic6—122no—
116#626—122no—
117#625—121no—
118C32×C8—22no—
119C32×C8—21no—
120C32×C8—41no—
121C32×C8C22×D312yes—
122C32×C8C22×D311yescharacteristic
123C3⋊(C3⋊C8)#6711yescharacteristic
124C3×C3⋊C8—32no—
125C3×C3⋊C8—61no—
126C2×SD32—271noSylow
127C2×C33C2×D811yescharacteristic
128C3×C3⋊D3—82no—
129C3×C3⋊D3SD3212yescomplemented
130C2×C3×Q8—181no—
131D3×Q8—62no—
132D3×Q8—62no—
133D3×D4—182no—
134C2×D12—181no—
135C3×Q16—21no—
136C3×Q16—22no—
137C3×Q16—41no—
138C3×Q16—91no—
139C3×Q16—181no—
140C3×Q16D3211yescharacteristic, complemented
141C3×Q16D3212yescomplemented
142C3×D8—92no—
143C2×C3×C8—91no—
144#216—62no—
145#216—62no—
146#213—182no—
147Dic12—32no—
148Dic12—61no—
149D24—32no—
150D24—32no—
151D24—62no—
152D24—181no—
153D3×C8—32no—
154D3×C8—61no—
155C3×C16—92no—
156C3⋊C16—32no—
157C3⋊C16—32no—
158C3⋊C16—62no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3#21311yescharacteristic
161C2×C3×D3—32no—
162C2×C3×D3—61no—
163C2×C3×D3—122no—
164C2×C3×D3—122no—
165C2×C3×D3—242no—
166D32—244no—
167D32—242no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4—42no—
173C32×C4—42no—
174C32×C4—82no—
175C32×C4—82no—
176C32×C4D3×D412yes—
177C32×C4C2×#6711yescharacteristic
178C3⋊Dic3—41no—
179C3⋊Dic3#21311yescharacteristic
180C3×Dic3—32no—
181C3×Dic3—61no—
182C3×Dic3—122no—
183C3×Dic3—122no—
184C2×Q16—91no—
185C2×D8—271no—
186SD32—92no—
187SD32—272no—
188C2×C16—271no—
189C33C2×SD3211yescharacteristic, complemented, Sylow
190C22×D3—362no—
191C22×D3—361no—
192C3×Q8—22no—
193C3×Q8—22no—
194C3×Q8—21no—
195C3×Q8—42no—
196C3×Q8—42no—
197C3×Q8—181no—
198C3×Q8—361no—
199C3×D4—182no—
200C2×C3×C4—91no—
201C2×C3×C4—361no—
202#67—362no—
203D12—62no—
204D12—62no—
205D12—122no—
206D12—361no—
207C4×D3—32no—
208C4×D3—61no—
209C4×D3—122no—
210C4×D3—122no—
211Dic6—62no—
212Dic6—62no—
213Dic6—122no—
214Dic6—122no—
215C3×C8—22no—
216C3×C8—21no—
217C3×C8—41no—
218C3×C8—91no—
219C3×C8C2×D3212yes—
220C3×C8C2×D3211yescharacteristic
221C3⋊C8—32no—
222C3⋊C8—61no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32D3×D812yes—
227C2×C32C2×#21311yescharacteristic
228C3⋊D3—82no—
229C3⋊D3#82812yescomplemented
230C3×D3—34no—
231C3×D3—62no—
232C3×D3—242no—
233C3×D3—242no—
234C3×D3—482no—
235C2×Q8—181no—
236C2×D4—541no—
237Q16—91no—
238Q16—181no—
239Q16#111311yescharacteristic, complemented
240D8—92no—
241D8—541no—
242C2×C8—91no—
243C16—92no—
244C22×C3—91no—
245C22×C3—362no—
246C2×D3—32no—
247C2×D3—61no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—242no—
251C2×D3—724no—
252C2×D3—722no—
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—42no—
256C3×C4—42no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—82no—
260C3×C4—82no—
261C3×C4—91no—
262C3×C4—361no—
263C3×C4D3×#6712yes—
264C3×C4#381111yescharacteristic
265Dic3—32no—
266Dic3—61no—
267Dic3—122no—
268Dic3—122no—
269C32—22no—
270C32—21no—
271C32—41no—
272C32D3×SD3212yescomplemented
273C32C2×#82811yescharacteristic, complemented
274C23—1081no—
275Q8—21no—
276Q8—181no—
277Q8—361no—
278D4—182no—
279D4—1081no—
280C2×C4—91no—
281C2×C4—361no—
282C8—91no—
283C8C2×#111311yescharacteristic
284C2×C3—21no—
285C2×C3—22no—
286C2×C3—41no—
287C2×C3—92no—
288C2×C3—722no—
289C2×C3D3×#21312yes—
290C2×C3#6392611yescharacteristic
291D3—34no—
292D3—62no—
293D3—242no—
294D3—242no—
295D3—482no—
296C22—91no—
297C22—362no—
298C22—2162no—
299C4—41no—
300C4—91no—
301C4—361no—
302C4#8904911yescharacteristic
303C3—21no—
304C3—22no—
305C3—41no—
306C3D3×#82812yescomplemented
307C3#1059103911yescharacteristic, complemented
308C2—92no—
309C2—722no—
310C2#1172370211yescharacteristic
subgroupG/H
centerC2#11723702
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×Q16C22
socleC2×C33C2×D8
prime radical#11723647C2
hypercenterQ16#1113
hypocenterC33C2×SD32
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. C3×Q16×C3⋊D3C2semidirect
3. C3×#63655C2semidirect
4. #11722068C2semidirect
5. #11722065C2semidirect
10. C33×Q16C22semidirect
12. C3×#3746C22semidirect
16. C3×C3⋊(C3⋊C16)C22semidirect
17. Q16×C3⋊D3D3semidirect
22. #63655D3semidirect
51. C32×Q16C2×D3semidirect
52. C32×Q16C2×D3semidirect
54. #3746C2×D3semidirect
68. C3⋊(C3⋊C16)C2×D3semidirect
129. C3×C3⋊D3SD32semidirect
140. C3×Q16D32semidirect
141. C3×Q16D32semidirect
189. C33C2×SD32semidirect
229. C3⋊D3#828semidirect
239. Q16#1113semidirect
272. C32D3×SD32semidirect
273. C32C2×#828semidirect
306. C3D3×#828semidirect
307. C3#10591039semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
#117236472
C3×#636552
#117220682
#117220652
#105910393
D3×#8283