Finite Group Databasebeta

← #10592596⋊C3 #13142272

Subgroups — #10592596⋊C3 #13142272

Order 1728.

408 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#64117⋊C3C212yesmaximal, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#64110⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#108C211yesmaximal, characteristic, complemented
5#63650⋊C3C212yesmaximal, complemented
6#10592596C311yesmaximal, characteristic, complemented
7C3×#4897—31nomaximal
8#3828⋊C3C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10M16×H(3,𝔽3)C2212yes—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#3748⋊C3C2211yescharacteristic
13#3745⋊C3—22no—
14#3745⋊C3—21no—
15#3745⋊C3—21no—
16#3744⋊C3—22no—
17#3743⋊C3—21no—
18C8×(C3⋊D3)⋊C3—21no—
19#64117C2×C312yescomplemented
20C2×#3745C2×C311yescharacteristic, complemented
21#64110C2×C311yescharacteristic, complemented
22C3×#908—32no—
23C2×C3×D24—31no—
24C3×#901—31no—
25C32×#108—21no—
26C32×#108D311yescharacteristic, complemented
27C32×#108C2×C311yescharacteristic, complemented
28#63650C2×C312yescomplemented
29C3×#823—32no—
30C22×(C3⋊D3)⋊C3—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32#638⋊C3—21no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
36C4×(C3⋊D3)⋊C3—21no—
37#634⋊C3D411yescharacteristic
38C8×H(3,𝔽3)—22no—
39C8×H(3,𝔽3)D411yescharacteristic
40C8×H(3,𝔽3)D411yescharacteristic
41(C3⋊(C3⋊C8))⋊C3—21no—
42C3×#472—91no—
43#4897—31no—
44#4897—91no—
45#3828C22×C311yescharacteristic
46C2×C4⋊(C3⋊D3)C22×C312yes—
47C3×#235—31no—
48C2×C3×D12—32no—
49C32×M16—22no—
50C32×M16C2×D312yes—
51C32×M16C22×C312yes—
52C2×C32×C8—21no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55#3748C22×C311yescharacteristic
56#3745—22no—
57#3745—21no—
58#3745—21no—
59#3744—22no—
60#3743—21no—
61C8×C3⋊D3—21no—
62C3×#208—31no—
63C3×D24—62no—
64C3×D24—61no—
65C3×D24—61no—
66C3×#204—62no—
67C3×ΓL(1,𝔽25)—61no—
68C3×D3×C8—61no—
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×(C3⋊D3)⋊C3—21no—
71C2×(C3⋊D3)⋊C3—42no—
72C2×(C3⋊D3)⋊C3—82no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75(C3⋊Dic3)⋊C3—21no—
76C3×AGL(1,ℤ/8)—92no—
77C2×C3×D8—91no—
78C3×#131—91no—
79#908—32no—
80#908—92no—
81C2×D24—31no—
82C2×D24—91no—
83#901—31no—
84#901—91no—
85C3×#108—31no—
86C3×#108—31no—
87C3×#108—61no—
88C3×#108C3×D311yescharacteristic
89C3×#100—92no—
90#823—32no—
91#823—92no—
92C22×C3⋊D3—22no—
93C22×C3×D3—62no—
94C2×C32×C4—21no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C23×C311yescharacteristic
97#638—21no—
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—22no—
100C4⋊(C3⋊D3)C3×D411yescharacteristic
101C4×C3⋊D3—21no—
102#634C3×D411yescharacteristic
103C3×#67—61no—
104C3×D12—31no—
105C3×D12—62no—
106C3×D12—62no—
107C3×C4×D3—61no—
108C3×Dic6—31no—
109C32×C8—22no—
110C32×C8—22no—
111C32×C8—21no—
112C32×C8—21no—
113C32×C8—42no—
114C32×C8D1211yescharacteristic
115C32×C8D1211yescharacteristic
116C32×C8C3×D411yescharacteristic
117C32×C8C3×D411yescharacteristic
118C3⋊(C3⋊C8)—21no—
119C3×C3⋊C8—61no—
120#472—91noSylow
121C2×H(3,𝔽3)—21no—
122C2×H(3,𝔽3)#12111yescharacteristic
123(C3⋊D3)⋊C3—41no—
124(C3⋊D3)⋊C3—82no—
125C3×#41—91no—
126C2×C3×D4—92no—
127#235—31no—
128#235—91no—
129C2×D12—32no—
130C2×D12—92no—
131C3×SD16—182no—
132C3×D8—181no—
133C3×D8—181no—
134C3×D8—182no—
135C3×M16—32no—
136C3×M16—32no—
137C3×M16—62no—
138C3×M16—91no—
139C3×M16—181no—
140C3×M16C2×C3×D312yes—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C2×C3×C8—61no—
144C2×C3×C8—181no—
145C2×C3×C8C2×C3×D311yescharacteristic
146#208—31no—
147#208—91no—
148D24—62no—
149D24—61no—
150D24—61no—
151D24—181no—
152D24—181no—
153D24—182no—
154#204—62no—
155#204—182no—
156ΓL(1,𝔽25)—61no—
157ΓL(1,𝔽25)—181no—
158D3×C8—61no—
159D3×C8—181no—
160C22×C32—21no—
161C22×C32#23911yescharacteristic
162C22×C32C3×#4111yescharacteristic
163C2×C3⋊D3—21no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—82no—
166C2×C3×D3—61no—
167C2×C3×D3—122no—
168C2×C3×D3—242no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4C2×D1211yescharacteristic
172C32×C4D3×D411yescharacteristic
173C32×C4C2×C3×D411yescharacteristic
174C32×C4C2×C3×D411yescharacteristic
175C3⋊Dic3—21no—
176C3×Dic3—61no—
177AGL(1,ℤ/8)—92no—
178C2×D8—91no—
179#131—91no—
180#108(C3⋊D3)⋊C311yescharacteristic, complemented
181#100—92no—
182H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
183C23×C3—182no—
184C22×D3—62no—
185C22×D3—182no—
186C3×Q8—91no—
187C3×D4—91no—
188C3×D4—182no—
189C3×D4—182no—
190C3×D4—181no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×C3×C4C22×C3×D311yescharacteristic
196#67—61no—
197#67—181no—
198D12—31no—
199D12—62no—
200D12—62no—
201D12—91no—
202D12—182no—
203D12—182no—
204C4×D3—61no—
205C4×D3—181no—
206Dic6—31no—
207Dic6—91no—
208C3×C8—22no—
209C3×C8—31no—
210C3×C8—31no—
211C3×C8—31no—
212C3×C8—31no—
213C3×C8—62no—
214C3×C8—62no—
215C3×C8—61no—
216C3×C8—61no—
217C3×C8—122no—
218C3×C8—181no—
219C3×C8C3×D1211yescharacteristic
220C3×C8C3×D1211yescharacteristic
221C3⋊C8—61no—
222C3⋊C8—181no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32#89511yescharacteristic
228C2×C32C3×#12111yescharacteristic
229C3⋊D3—41no—
230C3⋊D3—82no—
231C3×D3—121no—
232C3×D3—242no—
233#41—91no—
234C2×D4—92no—
235SD16—182no—
236D8—182no—
237D8—181no—
238D8—181no—
239M16—91no—
240M16—181no—
241M16C2×(C3⋊D3)⋊C312yes—
242C2×C8—181no—
243C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—181no—
248C22×C3—362no—
249C22×C3—722no—
250C22×C3C3×#23911yescharacteristic
251C2×D3—61no—
252C2×D3—122no—
253C2×D3—181no—
254C2×D3—242no—
255C2×D3—362no—
256C2×D3—722no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—181no—
264C3×C4C2×C3×D1211yescharacteristic
265C3×C4C3×D3×D411yescharacteristic
266Dic3—61no—
267Dic3—181no—
268C32—21no—
269C32#489711yescharacteristic, complemented
270C32C3×#47211yescharacteristic, complemented
271C23—182no—
272Q8—91no—
273D4—91no—
274D4—181no—
275D4—182no—
276D4—182no—
277C2×C4—181no—
278C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
279C8—22no—
280C8—181no—
281C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
282C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
283C2×C3—21no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—121no—
290C2×C3—361no—
291C2×C3—722no—
292C2×C3C3×#89511yescharacteristic
293D3—121no—
294D3—242no—
295D3—361no—
296D3—722no—
297C22—181no—
298C22—362no—
299C22—722no—
300C22#8872211yescharacteristic
301C4—181no—
302C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
303C4D4×(C3⋊D3)⋊C311yescharacteristic
304C3—31no—
305C3—31no—
306C3—61no—
307C3C3×#489711yescharacteristic
308C2—21no—
309C2—361no—
310C2—722no—
311C2#64104⋊C311yescharacteristic
subgroupG/H
centerC2#64104⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#108C2
socleC2×C3C3×#895
hypercenter#108(C3⋊D3)⋊C3
hypocenterC32C3×#472
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
1. #64117⋊C3C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #64110⋊C3C2semidirect
4. H(3,𝔽3)×#108C2semidirect
5. #63650⋊C3C2semidirect
6. #10592596C3semidirect
19. #64117C2×C3semidirect
20. C2×#3745C2×C3semidirect
21. #64110C2×C3semidirect
26. C32×#108D3semidirect
27. C32×#108C2×C3semidirect
28. #63650C2×C3semidirect
180. #108(C3⋊D3)⋊C3semidirect
182. H(3,𝔽3)#472semidirect
269. C32#4897semidirect
270. C32C3×#472semidirect

Maximal subgroups

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

subgroupindex
#64117⋊C32
C2×#3745⋊C32
#64110⋊C32
H(3,𝔽3)×#1082
#63650⋊C32
#105925963
C3×#48973