Finite Group Databasebeta

← #13161785

Subgroups — #13161785

Order 1728.

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

normal subgroups only

This group has 454 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
2#11723649C211yesmaximal, characteristic
3C3×#63656C212yesmaximal, complemented
4#11722069C212yesmaximal, complemented
5#11722065C211yesmaximal, characteristic
6#10591043—31nomaximal
7D3×#829—32nomaximal
8C3×Q8×C3⋊D3—21no—
9#89040—21no—
10C33×Q16C2211yescharacteristic
11C3×#3756—21no—
12C3×#3746C2211yescharacteristic
13C3×#3746C2212yes—
14C3×C8×C3⋊D3C2211yescharacteristic
15#88812—21no—
16C3×C3⋊(C3⋊C16)C2212yes—
17Q16×C3⋊D3D311yescharacteristic
18C3×D3×Q16—32no—
19C3×D3×Q16—61no—
20#63802—31no—
21D3×Dic12—32no—
22#63656D312yescomplemented
23C3×#829—32no—
24C3×#829—32no—
25C3×#829—62no—
26#63551—32no—
27#63551—32no—
28#63550—32no—
29#63542—31no—
30D3×C3⋊C16—32no—
31C33×Q8—21no—
32C3×C4×C3⋊D3—41no—
33C3×C4×C3⋊D3D411yescharacteristic
34C3×#634—21no—
35C3×#634—22no—
36C3×#634—41no—
37#6296—41no—
38#6294—41no—
39C33×C8C2311yescharacteristic
40C3×C3⋊(C3⋊C8)D411yescharacteristic
41C2×#829—91no—
42D3×Q32—92no—
43Q8×C3⋊D3—21no—
44C3×D3×Q8—62no—
45C3×D3×Q8—62no—
46#3797—61no—
47D3×Dic6—62no—
48C32×Q16—21no—
49C32×Q16—22no—
50C32×Q16—41no—
51C32×Q16C2×D312yes—
52C32×Q16C2×D311yescharacteristic
53#3756—21no—
54#3746C2×D312yes—
55#3746C2×D311yescharacteristic
56C8×C3⋊D3C2×D311yescharacteristic
57C3×#216—62no—
58C3×#216—62no—
59C3×Dic12—32no—
60C3×Dic12—32no—
61C3×Dic12—32no—
62C3×Dic12—61no—
63C3×Dic12—62no—
64C3×D3×C8—32no—
65C3×D3×C8—61no—
66#3719—62no—
67#3718—61no—
68C3⋊(C3⋊C16)C2×D312yes—
69C3×C3⋊C16—32no—
70C3×C3⋊C16—32no—
71C3×C3⋊C16—62no—
72C2×C3×C3⋊D3D811yescharacteristic
73C33×C4—41no—
74C33×C4C2×D411yescharacteristic
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—42no—
77C3×C3⋊Dic3D811yescharacteristic
78C2×C3×Q16—91no—
79D3×Q16—32no—
80D3×Q16—61no—
81D3×Q16—92no—
82C2×Dic12—91no—
83C3×Q32—92no—
84#829—32no—
85#829—32no—
86#829—62no—
87#829—92no—
88#829—92no—
89C2×C3⋊C16—91no—
90Dic24—92no—
91D3×C16—92no—
92C32×Q8—21no—
93C32×Q8—22no—
94C32×Q8—22no—
95C32×Q8—42no—
96C32×Q8—42no—
97C4×C3⋊D3—41no—
98C4×C3⋊D3#6711yescharacteristic
99#634—21no—
100#634—22no—
101#634—41no—
102C3×C4×D3—32no—
103C3×C4×D3—61no—
104C3×C4×D3—122no—
105C3×C4×D3—122no—
106C3×Dic6—62no—
107C3×Dic6—62no—
108C3×Dic6—62no—
109C3×Dic6—62no—
110C3×Dic6—122no—
111C3×Dic6—122no—
112C3×Dic6—122no—
113#627—122no—
114#627—121no—
115#624—121no—
116D3×Dic3—122no—
117C32×C8—22no—
118C32×C8—21no—
119C32×C8—41no—
120C32×C8C22×D312yes—
121C32×C8C22×D311yescharacteristic
122C3⋊(C3⋊C8)#6711yescharacteristic
123C3×C3⋊C8—32no—
124C3×C3⋊C8—61no—
125C2×Q32—271noSylow
126C2×C33C2×D811yescharacteristic
127C3×C3⋊D3Q3212yescomplemented
128C2×C3×Q8—181no—
129D3×Q8—62no—
130D3×Q8—62no—
131D3×Q8—182no—
132C2×Dic6—181no—
133C3×Q16—21no—
134C3×Q16—22no—
135C3×Q16—41no—
136C3×Q16—91no—
137C3×Q16—92no—
138C3×Q16—181no—
139C3×Q16D3211yescharacteristic
140C3×Q16D3212yes—
141C2×C3×C8—91no—
142#216—62no—
143#216—62no—
144#216—182no—
145Dic12—32no—
146Dic12—32no—
147Dic12—32no—
148Dic12—61no—
149Dic12—62no—
150Dic12—181no—
151D3×C8—32no—
152D3×C8—61no—
153C3×C16—92no—
154C3⋊C16—32no—
155C3⋊C16—32no—
156C3⋊C16—62no—
157C2×C3⋊D3#21311yescharacteristic
158C2×C3×D3—32no—
159C2×C3×D3—61no—
160C32×C4—22no—
161C32×C4—21no—
162C32×C4—41no—
163C32×C4—41no—
164C32×C4—42no—
165C32×C4—42no—
166C32×C4—82no—
167C32×C4—82no—
168C32×C4D3×D412yes—
169C32×C4C2×#6711yescharacteristic
170C3⋊Dic3—41no—
171C3⋊Dic3—42no—
172C3⋊Dic3#21311yescharacteristic
173C3×Dic3—32no—
174C3×Dic3—61no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C3×Dic3—122no—
178C3×Dic3—122no—
179C3×Dic3—242no—
180C2×Q16—91no—
181C2×Q16—271no—
182Q32—92no—
183Q32—272no—
184C2×C16—271no—
185C33C2×Q3211yescharacteristic, complemented, Sylow
186C3×Q8—22no—
187C3×Q8—22no—
188C3×Q8—21no—
189C3×Q8—42no—
190C3×Q8—42no—
191C3×Q8—182no—
192C3×Q8—181no—
193C3×Q8—361no—
194C2×C3×C4—91no—
195C2×C3×C4—361no—
196C2×Dic3—361no—
197C4×D3—32no—
198C4×D3—61no—
199C4×D3—122no—
200C4×D3—122no—
201C4×D3—362no—
202Dic6—62no—
203Dic6—62no—
204Dic6—62no—
205Dic6—62no—
206Dic6—122no—
207Dic6—122no—
208Dic6—122no—
209Dic6—362no—
210Dic6—361no—
211C3×C8—22no—
212C3×C8—21no—
213C3×C8—41no—
214C3×C8—91no—
215C3×C8C2×D3212yes—
216C3×C8C2×D3211yescharacteristic
217C3⋊C8—32no—
218C3⋊C8—61no—
219C2×C32—22no—
220C2×C32—21no—
221C2×C32—41no—
222C2×C32D3×D812yes—
223C2×C32C2×#21311yescharacteristic
224C3⋊D3#82912yescomplemented
225C3×D3—34no—
226C3×D3—62no—
227C2×Q8—181no—
228C2×Q8—541no—
229Q16—92no—
230Q16—91no—
231Q16—181no—
232Q16—541no—
233Q16#111311yescharacteristic
234C2×C8—91no—
235C16—92no—
236C22×C3—91no—
237C2×D3—32no—
238C2×D3—61no—
239C3×C4—21no—
240C3×C4—22no—
241C3×C4—42no—
242C3×C4—42no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—82no—
246C3×C4—82no—
247C3×C4—91no—
248C3×C4—362no—
249C3×C4—361no—
250C3×C4D3×#6712yes—
251C3×C4#381111yescharacteristic
252Dic3—32no—
253Dic3—61no—
254Dic3—122no—
255Dic3—122no—
256Dic3—122no—
257Dic3—122no—
258Dic3—242no—
259C32—22no—
260C32—21no—
261C32—41no—
262C32D3×Q3212yescomplemented
263C32C2×#82911yescharacteristic, complemented
264Q8—21no—
265Q8—181no—
266Q8—182no—
267Q8—361no—
268Q8—1081no—
269C2×C4—91no—
270C2×C4—361no—
271C2×C4—1081no—
272C8—91no—
273C8C2×#111311yescharacteristic
274C2×C3—21no—
275C2×C3—22no—
276C2×C3—41no—
277C2×C3—92no—
278C2×C3D3×#21312yes—
279C2×C3#6392611yescharacteristic
280D3—34no—
281D3—62no—
282C22—91no—
283C4—41no—
284C4—91no—
285C4—361no—
286C4—362no—
287C4#8904911yescharacteristic
288C3—21no—
289C3—22no—
290C3—41no—
291C3D3×#82912yescomplemented
292C3#1059104311yescharacteristic, complemented
293C2—92no—
294C2#1172370211yescharacteristic
subgroupG/H
centerC2#11723702
commutator, cosocleC33×C8C23
FrattiniC8C2×#1113
FittingC33×Q16C22
socleC2×C33C2×D8
prime radicalC2×C3×C3⋊D3D8
hypercenterQ16#1113
hypocenterC33C2×Q32
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
3. C3×#63656C2semidirect
4. #11722069C2semidirect
22. #63656D3semidirect
127. C3×C3⋊D3Q32semidirect
185. C33C2×Q32semidirect
224. C3⋊D3#829semidirect
262. C32D3×Q32semidirect
263. C32C2×#829semidirect
291. C3D3×#829semidirect
292. C3#10591043semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
#117236492
C3×#636562
#117220692
#117220652
#105910433
D3×#8293