Finite Group Databasebeta

← #10592571⋊C3 #13142231

Subgroups — #10592571⋊C3 #13142231

Order 1728.

505 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 505 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C2×#3745⋊C3C211yesmaximal, characteristic
2#64102⋊C3C212yesmaximal, complemented
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C212yesmaximal, complemented
6#10592571C311yesmaximal, characteristic, complemented
7C3×#4872—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×C4×(C3⋊D3)⋊C3—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)C2212yes—
12#3752⋊C3—22no—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15#3745⋊C3C414yescomplemented
16C2×#3745C2×C311yescharacteristic
17#64102C2×C312yescomplemented
18C2×C3×D24—31no—
19C3×#893—32no—
20C32×#107—21no—
21C32×#107D311yescharacteristic, complemented
22C32×#107C2×C311yescharacteristic, complemented
23C8×C3⋊Dic3C2×C311yescharacteristic, complemented
24#63636C2×C312yescomplemented
25C3×C8×Dic3—31no—
26C3×#809—32no—
27C22×(C3⋊D3)⋊C3—22no—
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3D411yescharacteristic
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
34C4×(C3⋊D3)⋊C3—44no—
35C8×H(3,𝔽3)C2×C412yescomplemented
36(C3⋊(C3⋊C8))⋊C3—21no—
37C3×C4×D8—91no—
38#4872—31no—
39#4872—91no—
40C2×C4⋊(C3⋊D3)C22×C312yes—
41C2×C4×C3⋊D3—22no—
42C2×C3×D12—32no—
43C2×C3×C4×D3—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D311yescharacteristic
46C2×C32×C8C22×C311yescharacteristic
47C32×C4⋊C4—22no—
48C32×C4⋊C4C2×D312yes—
49C32×C4⋊C4C22×C312yes—
50#3752—22no—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
53#3745C3×C414yescomplemented
54C3×(C2×D3)⋊C4—62no—
55C3×C4×Dic3—31no—
56C2×C3×C3⋊C8—31no—
57C3×D24—34no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C2×(C3⋊D3)⋊C3—24no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)—42no—
63C4×H(3,𝔽3)C22×C411yescharacteristic
64C4×H(3,𝔽3)#4111yescharacteristic
65(C3⋊Dic3)⋊C3—21no—
66(C3⋊Dic3)⋊C3D812yescomplemented
67C2×C3×D8—91no—
68C3×C4×D4—92no—
69C2×D24—31no—
70C2×D24—91no—
71#893—32no—
72#893—92no—
73C3×#107—31no—
74C3×#107—31no—
75C3×#107—61no—
76C3×#107C3×D311yescharacteristic
77C3×#102—92no—
78C3×C4×C8—91no—
79C8×Dic3—31no—
80C8×Dic3—91no—
81#809—32no—
82#809—92no—
83C22×C3⋊D3—22no—
84C22×C3×D3—62no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—42no—
89C2×C32×C4C22×D311yescharacteristic
90C2×C32×C4C23×C311yescharacteristic
91C2×C3⋊Dic3C3×D411yescharacteristic
92C2×C3⋊Dic3C3×D411yescharacteristic
93C4⋊(C3⋊D3)—22no—
94C4⋊(C3⋊D3)C2×C3×C414yescomplemented
95C4×C3⋊D3—44no—
96C2×C3×Dic3—31no—
97C2×C3×Dic3—31no—
98C3×D12—34no—
99C3×D12—62no—
100C3×C4×D3—124no—
101C32×C8—22no—
102C32×C8C4×D312yescomplemented
103C32×C8C2×C3×C412yescomplemented
104C3⋊(C3⋊C8)—21no—
105C3×C3⋊C8—61no—
106C4×D8—91noSylow
107C2×H(3,𝔽3)C4×D411yescharacteristic
108C2×H(3,𝔽3)C2×D811yescharacteristic
109C2×H(3,𝔽3)#13511yescharacteristic
110(C3⋊D3)⋊C3—44no—
111C2×C3×D4—92no—
112C22×C3×C4—182no—
113C2×D12—32no—
114C2×D12—92no—
115C2×C4×D3—62no—
116C2×C4×D3—182no—
117C3×D8—94no—
118C2×C3×C8—31no—
119C2×C3×C8—31no—
120C2×C3×C8—61no—
121C2×C3×C8—91no—
122C2×C3×C8C2×C3×D311yescharacteristic
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—62no—
126C3×C4⋊C4C2×C3×D312yes—
127C3×C22⋊C4—182no—
128C3×C42—91no—
129(C2×D3)⋊C4—62no—
130(C2×D3)⋊C4—182no—
131C4×Dic3—31no—
132C4×Dic3—91no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—91no—
135D24—34no—
136D24—94no—
137C22×C32—21no—
138C22×C32D3×D411yescharacteristic
139C22×C32C2×C3×D411yescharacteristic
140C2×C3⋊D3—24no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3×D3—64no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—42no—
149C32×C4—42no—
150C32×C4—82no—
151C32×C4C2×C4×D311yescharacteristic
152C32×C4#23911yescharacteristic
153C32×C4C22×C3×C411yescharacteristic
154C32×C4C3×#4111yescharacteristic
155C3⋊Dic3—21no—
156C3⋊Dic3C3×D812yescomplemented
157C3×Dic3—32no—
158C3×Dic3—61no—
159C2×D8—91no—
160C4×D4—92no—
161#107(C3⋊D3)⋊C311yescharacteristic, complemented
162#102—92no—
163C4×C8—91no—
164H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
165C23×C3—182no—
166C22×D3—62no—
167C22×D3—182no—
168C3×D4—94no—
169C3×D4—182no—
170C2×C3×C4—22no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—61no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—122no—
179C2×C3×C4—364no—
180C2×C3×C4C22×C3×D311yescharacteristic
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—91no—
184C2×Dic3—91no—
185D12—34no—
186D12—62no—
187D12—94no—
188D12—182no—
189C4×D3—124no—
190C4×D3—364no—
191C3×C8—32no—
192C3×C8—32no—
193C3×C8—62no—
194C3×C8—181no—
195C3×C8C3×C4×D312yes—
196C3⋊C8—61no—
197C3⋊C8—181no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32#89311yescharacteristic
202C2×C32D3×D811yescharacteristic
203C2×C32#91911yescharacteristic
204C2×C32C3×C4×D411yescharacteristic
205C2×C32C2×C3×D811yescharacteristic
206C2×C32C3×#13511yescharacteristic
207C3⋊D3—44no—
208C3×D3—124no—
209C2×D4—92no—
210C22×C4—182no—
211D8—94no—
212C2×C8—91no—
213C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
214C4⋊C4C2×(C3⋊D3)⋊C312yes—
215C22⋊C4—182no—
216C42—91no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—184no—
221C22×C3—362no—
222C22×C3—362no—
223C22×C3C3×D3×D411yescharacteristic
224C2×D3—64no—
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—184no—
228C2×D3—362no—
229C2×D3—362no—
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—42no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—92no—
238C3×C4—122no—
239C3×C4—122no—
240C3×C4—181no—
241C3×C4—242no—
242C3×C4C2×C3×C4×D311yescharacteristic
243C3×C4C3×#23911yescharacteristic
244Dic3—32no—
245Dic3—61no—
246Dic3—92no—
247Dic3—181no—
248C32—21no—
249C32#487211yescharacteristic, complemented
250C32C3×C4×D811yescharacteristic, complemented
251C23—182no—
252D4—94no—
253D4—182no—
254C2×C4—22no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—364no—
258C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
259C8—181no—
260C8C4×(C3⋊D3)⋊C312yescomplemented
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—61no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—364no—
271C2×C3C3×#89311yescharacteristic
272C2×C3C3×D3×D811yescharacteristic
273C2×C3C3×#91911yescharacteristic
274D3—124no—
275D3—364no—
276C22—184no—
277C22—362no—
278C22—362no—
279C22D4×(C3⋊D3)⋊C311yescharacteristic
280C4—42no—
281C4—92no—
282C4—181no—
283C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
284C4#8872211yescharacteristic
285C3—31no—
286C3—31no—
287C3—61no—
288C3C3×#487211yescharacteristic
289C2—364no—
290C2#64102⋊C311yescharacteristic
291C2D8×(C3⋊D3)⋊C311yescharacteristic
292C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC2×#3745⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #64102⋊C3C2semidirect
3. H(3,𝔽3)×#107C2semidirect
4. C8×(C3⋊Dic3)⋊C3C2semidirect
5. #63636⋊C3C2semidirect
6. #10592571C3semidirect
15. #3745⋊C3C4semidirect
17. #64102C2×C3semidirect
21. C32×#107D3semidirect
22. C32×#107C2×C3semidirect
23. C8×C3⋊Dic3C2×C3semidirect
24. #63636C2×C3semidirect
33. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
35. C8×H(3,𝔽3)C2×C4semidirect
53. #3745C3×C4semidirect
66. (C3⋊Dic3)⋊C3D8semidirect
94. C4⋊(C3⋊D3)C2×C3×C4semidirect
102. C32×C8C4×D3semidirect
103. C32×C8C2×C3×C4semidirect
156. C3⋊Dic3C3×D8semidirect
161. #107(C3⋊D3)⋊C3semidirect
164. H(3,𝔽3)C4×D8semidirect
249. C32#4872semidirect
250. C32C3×C4×D8semidirect
260. C8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#64102⋊C32
H(3,𝔽3)×#1072
C8×(C3⋊Dic3)⋊C32
#63636⋊C32
#105925713
C3×#48723