Finite Group Databasebeta

← #13139287

Subgroups — #13139287

Order 1728.

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

normal subgroups only

This group has 311 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
1Q16×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11720851C211yesmaximal, characteristic
3#11720570C211yesmaximal, characteristic, complemented
4#63656⋊C3C211yesmaximal, characteristic, complemented
5#11720256C211yesmaximal, characteristic, complemented
6#11720250C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊C16C211yesmaximal, characteristic
8#10591043—31nomaximal
9D3×#829—31nomaximal
10Q8×(C3⋊D3)⋊C3—21no—
11((C3⋊D3)⋊C3)⋊Q8—21no—
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic
14#3756⋊C3—21no—
15C8×(C3⋊D3)⋊C3C2211yescharacteristic
16#3746⋊C3C2211yescharacteristic
17#3746⋊C3C2211yescharacteristic
18#88439—21no—
19#88386C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21Q16×C3⋊D3D311yescharacteristic
22C3×D3×Q16—31no—
23#63802—31no—
24D3×Dic12—31no—
25#63656D311yescharacteristic, complemented
26C3×#829—31no—
27C3×#829—31no—
28C3×#829—31no—
29C3×#829—61no—
30#63551—31no—
31#63551—31no—
32#63551—31no—
33#63550—31no—
34#63542—31no—
35D3×C3⋊C16—31no—
36Q8×H(3,𝔽3)—21no—
37#6229—21no—
38C4×(C3⋊D3)⋊C3—41no—
39C4×(C3⋊D3)⋊C3D411yescharacteristic
40#634⋊C3—21no—
41#634⋊C3—21no—
42#634⋊C3—41no—
43((C3⋊D3)⋊C3)⋊C4—41no—
44#6196—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
47C2×#829—91no—
48D3×Q32—91no—
49D3×Q32—91no—
50Q8×C3⋊D3—21no—
51C3×D3×Q8—61no—
52#3797—61no—
53D3×Dic6—61no—
54C32×Q16—21no—
55C32×Q16C2×D311yescharacteristic
56C32×Q16C2×D311yescharacteristic
57#3756—21no—
58#3746C2×D311yescharacteristic
59#3746C2×D311yescharacteristic
60C8×C3⋊D3C2×D311yescharacteristic
61C3×#216—61no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—31no—
65C3×Dic12—31no—
66C3×Dic12—61no—
67C3×D3×C8—31no—
68#3719—61no—
69#3718—61no—
70C3⋊(C3⋊C16)C2×D311yescharacteristic
71C3×C3⋊C16—31no—
72C3×C3⋊C16—31no—
73C3×C3⋊C16—31no—
74C3×C3⋊C16—61no—
75C2×(C3⋊D3)⋊C3D811yescharacteristic
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)C2×D411yescharacteristic
78#1084—41no—
79(C3⋊Dic3)⋊C3—41no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3D811yescharacteristic
82C2×C3×Q16—91no—
83D3×Q16—31no—
84D3×Q16—91no—
85D3×Q16—91no—
86D3×Q16—91no—
87C2×Dic12—91no—
88C3×Q32—91no—
89C3×Q32—91no—
90#829—31no—
91#829—31no—
92#829—31no—
93#829—61no—
94#829—91no—
95#829—91no—
96#829—91no—
97#829—91no—
98#829—91no—
99C2×C3⋊C16—91no—
100Dic24—91no—
101Dic24—91no—
102D3×C16—91no—
103D3×C16—91no—
104C32×Q8—21no—
105C32×Q8—21no—
106C32×Q8—41no—
107C4×C3⋊D3—41no—
108C4×C3⋊D3#6711yescharacteristic
109#634—21no—
110#634—21no—
111#634—41no—
112C3×C4×D3—31no—
113C3×C4×D3—121no—
114C3×Dic6—61no—
115C3×Dic6—61no—
116C3×Dic6—61no—
117C3×Dic6—61no—
118C3×Dic6—121no—
119C3×Dic6—121no—
120#627—121no—
121#627—121no—
122#624—121no—
123D3×Dic3—121no—
124C32×C8—21no—
125C32×C8C22×D311yescharacteristic
126C32×C8C22×D311yescharacteristic
127C3⋊(C3⋊C8)#6711yescharacteristic
128C3×C3⋊C8—31no—
129C2×Q32—271noSylow
130C2×H(3,𝔽3)C2×D811yescharacteristic
131(C3⋊D3)⋊C3Q3212yescomplemented
132C2×C3×Q8—181no—
133D3×Q8—61no—
134D3×Q8—181no—
135D3×Q8—181no—
136D3×Q8—181no—
137C2×Dic6—181no—
138C3×Q16—31no—
139C3×Q16—31no—
140C3×Q16—61no—
141C3×Q16—91no—
142C3×Q16—91no—
143C3×Q16—91no—
144C3×Q16—181no—
145C3×Q16D3211yescharacteristic
146C2×C3×C8—91no—
147#216—61no—
148#216—181no—
149#216—181no—
150#216—181no—
151Dic12—31no—
152Dic12—31no—
153Dic12—31no—
154Dic12—31no—
155Dic12—61no—
156Dic12—91no—
157Dic12—91no—
158Dic12—181no—
159D3×C8—31no—
160D3×C8—91no—
161C3×C16—91no—
162C3×C16—91no—
163C3⋊C16—31no—
164C3⋊C16—31no—
165C3⋊C16—31no—
166C3⋊C16—61no—
167C3⋊C16—91no—
168C2×C3⋊D3#21311yescharacteristic
169C2×C3×D3—31no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—81no—
174C32×C4D3×D411yescharacteristic
175C32×C4C2×#6711yescharacteristic
176C3⋊Dic3—41no—
177C3⋊Dic3—41no—
178C3⋊Dic3#21311yescharacteristic
179C3×Dic3—31no—
180C3×Dic3—121no—
181C3×Dic3—121no—
182C3×Dic3—121no—
183C3×Dic3—121no—
184C3×Dic3—241no—
185C2×Q16—91no—
186C2×Q16—271no—
187Q32—91no—
188Q32—91no—
189Q32—271no—
190Q32—271no—
191C2×C16—271no—
192H(3,𝔽3)C2×Q3211yescharacteristic, complemented, Sylow
193C3×Q8—21no—
194C3×Q8—61no—
195C3×Q8—61no—
196C3×Q8—121no—
197C3×Q8—181no—
198C3×Q8—181no—
199C3×Q8—181no—
200C3×Q8—361no—
201C2×C3×C4—91no—
202C2×C3×C4—361no—
203C2×Dic3—361no—
204C4×D3—31no—
205C4×D3—91no—
206C4×D3—121no—
207C4×D3—361no—
208C4×D3—361no—
209C4×D3—361no—
210Dic6—61no—
211Dic6—61no—
212Dic6—61no—
213Dic6—61no—
214Dic6—121no—
215Dic6—121no—
216Dic6—181no—
217Dic6—181no—
218Dic6—361no—
219Dic6—361no—
220Dic6—361no—
221Dic6—361no—
222C3×C8—31no—
223C3×C8—31no—
224C3×C8—61no—
225C3×C8—91no—
226C3×C8C2×D3211yescharacteristic
227C3⋊C8—31no—
228C3⋊C8—91no—
229C2×C32—21no—
230C2×C32D3×D811yescharacteristic
231C2×C32C2×#21311yescharacteristic
232C3⋊D3#82912yescomplemented
233C3×D3—32no—
234C2×Q8—181no—
235C2×Q8—541no—
236Q16—91no—
237Q16—91no—
238Q16—91no—
239Q16—181no—
240Q16—541no—
241Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C2×C8—91no—
243C16—91no—
244C16—91no—
245C22×C3—91no—
246C2×D3—31no—
247C2×D3—91no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—41no—
251C3×C4—61no—
252C3×C4—91no—
253C3×C4—121no—
254C3×C4—121no—
255C3×C4—241no—
256C3×C4—361no—
257C3×C4—361no—
258C3×C4—361no—
259C3×C4D3×#6711yescharacteristic
260Dic3—31no—
261Dic3—91no—
262Dic3—121no—
263Dic3—121no—
264Dic3—121no—
265Dic3—121no—
266Dic3—241no—
267Dic3—361no—
268Dic3—361no—
269C32—21no—
270C32D3×Q3211yescharacteristic, complemented
271C32C2×#82911yescharacteristic, complemented
272Q8—21no—
273Q8—181no—
274Q8—181no—
275Q8—181no—
276Q8—361no—
277Q8—1081no—
278C2×C4—91no—
279C2×C4—361no—
280C2×C4—1081no—
281C8—91no—
282C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—61no—
286C2×C3—92no—
287C2×C3D3×#21311yescharacteristic
288D3—32no—
289D3—92no—
290C22—91no—
291C4—41no—
292C4—91no—
293C4—361no—
294C4—361no—
295C4—361no—
296C4#8867611yescharacteristic
297C3—31no—
298C3—31no—
299C3—61no—
300C3D3×#82911yescharacteristic
301C2—92no—
302C2#1172119411yescharacteristic
subgroupG/H
centerC2#11721194
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3D3×#213
prime radicalC2×(C3⋊D3)⋊C3D8
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×Q32
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
3. #11720570C2semidirect
4. #63656⋊C3C2semidirect
5. #11720256C2semidirect
6. #11720250C2semidirect
25. #63656D3semidirect
131. (C3⋊D3)⋊C3Q32semidirect
192. H(3,𝔽3)C2×Q32semidirect
232. C3⋊D3#829semidirect
270. C32D3×Q32semidirect
271. C32C2×#829semidirect

Maximal subgroups

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

subgroupindex
Q16×(C3⋊D3)⋊C32
#117208512
#117205702
#63656⋊C32
#117202562
#117202502
((C3⋊D3)⋊C3)⋊C162
#105910433
D3×#8293