Finite Group Databasebeta

← #10592537⋊C3 #13142059

Subgroups — #10592537⋊C3 #13142059

Order 1728.

465 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3743⋊C3C212yesmaximal, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
4#63647⋊C3C212yesmaximal, complemented
5C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
6#10592537C311yesmaximal, characteristic, complemented
7C3×#4838—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#3752⋊C3C412yes—
13C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16#3743⋊C3—24no—
17C2×#3743C2×C312yescomplemented
18C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C2×C3×ΓL(1,𝔽25)—32no—
20C3×C4×D12—31no—
21C32×C4⋊C8—21no—
22C32×C4⋊C8D311yescharacteristic, complemented
23C32×C4⋊C8C2×C311yescharacteristic, complemented
24#63647C2×C312yescomplemented
25C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
26C3×(C2×D3)⋊C8—32no—
27C3×C4×C3⋊C8—31no—
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3C2×C412yes—
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4×(C3⋊D3)⋊C3—22no—
35C4×(C3⋊D3)⋊C3—22no—
36C8×H(3,𝔽3)—22no—
37(C3⋊(C3⋊C8))⋊C3—21no—
38(C3⋊(C3⋊C8))⋊C3D412yescomplemented
39C3×#436—91no—
40#4838—31no—
41#4838—91no—
42C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
43C2×C4×C3⋊D3C22×C312yes—
44C2×C3×D12—31no—
45C2×C3×C4×D3—32no—
46C2×C32×C8—22no—
47C2×C32×C8C2×D312yes—
48C2×C32×C8C22×C312yes—
49C32×C42—21no—
50C32×C42C2×D311yescharacteristic
51C32×C42C22×C311yescharacteristic
52#3752C3×C412yes—
53C4⋊(C3⋊Dic3)C3×C411yescharacteristic
54C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
55C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
56#3743—24no—
57C3×(C2×D3)⋊C4—32no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C3×ΓL(1,𝔽25)—64no—
62C22×H(3,𝔽3)C22×C411yescharacteristic
63C2×(C3⋊D3)⋊C3—22no—
64C2×(C3⋊D3)⋊C3—22no—
65C2×(C3⋊D3)⋊C3—22no—
66C4×H(3,𝔽3)—21no—
67C4×H(3,𝔽3)M1612yescomplemented
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70(C3⋊Dic3)⋊C3—22no—
71C2×C3×M16—92no—
72C3×C4×D4—91no—
73C2×ΓL(1,𝔽25)—32no—
74C2×ΓL(1,𝔽25)—92no—
75C4×D12—31no—
76C4×D12—91no—
77C3×C4⋊C8—31no—
78C3×C4⋊C8—31no—
79C3×C4⋊C8—61no—
80C3×C4⋊C8C3×D311yescharacteristic
81C3×C22⋊C8—92no—
82C3×C4×C8—91no—
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—92no—
85C4×C3⋊C8—31no—
86C4×C3⋊C8—91no—
87C22×C3⋊D3C2×C3×C412yes—
88C22×C3×D3—32no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4C4×D311yescharacteristic
93C2×C32×C4C4×D311yescharacteristic
94C2×C32×C4C2×C3×C411yescharacteristic
95C2×C32×C4C2×C3×C411yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97C2×C32×C4C23×C311yescharacteristic
98C2×C3⋊Dic3C2×C3×C412yes—
99C4⋊(C3⋊D3)—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3—22no—
102C2×C3×Dic3—32no—
103C3×D12—62no—
104C3×C4×D3—62no—
105C3×C4×D3—62no—
106C32×C8—22no—
107C32×C8—22no—
108C32×C8—42no—
109C3⋊(C3⋊C8)—21no—
110C3⋊(C3⋊C8)C3×D412yescomplemented
111C3×C3⋊C8—32no—
112C3×C3⋊C8—61no—
113#436—91noSylow
114C2×H(3,𝔽3)C4×D411yescharacteristic
115C2×H(3,𝔽3)C2×M1611yescharacteristic
116C2×H(3,𝔽3)#13111yescharacteristic
117(C3⋊D3)⋊C3—42no—
118C2×C3×D4—91no—
119C22×C3×C4—92no—
120C2×D12—31no—
121C2×D12—91no—
122C2×C4×D3—32no—
123C2×C4×D3—92no—
124C3×M16—184no—
125C2×C3×C8—32no—
126C2×C3×C8—32no—
127C2×C3×C8—62no—
128C2×C3×C8—91no—
129C2×C3×C8—91no—
130C2×C3×C8C2×C3×D312yes—
131C3×C4⋊C4—91no—
132C3×C22⋊C4—92no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—61no—
136C3×C42C2×C3×D311yescharacteristic
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—92no—
139C4⋊Dic3—31no—
140C4⋊Dic3—91no—
141C2×C3⋊C8—31no—
142C2×C3⋊C8—31no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145ΓL(1,𝔽25)—64no—
146ΓL(1,𝔽25)—184no—
147C22×C32—21no—
148C22×C32C2×C4×D311yescharacteristic
149C22×C32C22×C3×C411yescharacteristic
150C2×C3⋊D3—22no—
151C2×C3⋊D3—22no—
152C2×C3⋊D3—22no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C32×C4—21no—
157C32×C4—21no—
158C32×C4—22no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—41no—
162C32×C4ΓL(1,𝔽25)12yescomplemented
163C32×C4C3×M1612yescomplemented
164C32×C4D3×D411yescharacteristic
165C32×C4#23911yescharacteristic
166C32×C4C2×C3×D411yescharacteristic
167C32×C4C3×#4111yescharacteristic
168C3⋊Dic3—22no—
169C3×Dic3—62no—
170C2×M16—92no—
171C4×D4—91no—
172C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
173C22⋊C8—92no—
174C4×C8—91no—
175H(3,𝔽3)#43611yescharacteristic, complemented, Sylow
176C23×C3—92no—
177C22×D3—32no—
178C22×D3—92no—
179C3×D4—182no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—92no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4C3×C4×D311yescharacteristic
193C2×C3×C4C3×C4×D311yescharacteristic
194C2×C3×C4C22×C3×D311yescharacteristic
195C2×Dic3—32no—
196C2×Dic3—92no—
197D12—62no—
198D12—182no—
199C4×D3—62no—
200C4×D3—62no—
201C4×D3—182no—
202C4×D3—182no—
203C3×C8—22no—
204C3×C8—62no—
205C3×C8—62no—
206C3×C8—92no—
207C3×C8—122no—
208C3×C8—181no—
209C3⋊C8—32no—
210C3⋊C8—61no—
211C3⋊C8—92no—
212C3⋊C8—181no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32#89311yescharacteristic
217C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
218C2×C32#90711yescharacteristic
219C2×C32C3×C4×D411yescharacteristic
220C2×C32C2×C3×M1611yescharacteristic
221C2×C32C3×#13111yescharacteristic
222C3⋊D3—42no—
223C3×D3—122no—
224C2×D4—91no—
225C22×C4—92no—
226M16—184no—
227C2×C8—91no—
228C2×C8—91no—
229C2×C8C2×(C3⋊D3)⋊C312yes—
230C4⋊C4—91no—
231C22⋊C4—92no—
232C42C2×(C3⋊D3)⋊C311yescharacteristic
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—182no—
237C22×C3—182no—
238C22×C3—182no—
239C22×C3C2×C3×C4×D311yescharacteristic
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—182no—
244C2×D3—182no—
245C2×D3—182no—
246C3×C4—21no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—32no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—62no—
258C3×C4—121no—
259C3×C4—182no—
260C3×C4C3×ΓL(1,𝔽25)12yes—
261C3×C4C3×D3×D411yescharacteristic
262C3×C4C3×#23911yescharacteristic
263Dic3—62no—
264Dic3—182no—
265C32—21no—
266C32#483811yescharacteristic, complemented
267C32C3×#43611yescharacteristic, complemented
268C23—92no—
269D4—182no—
270C2×C4—92no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
274C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
275C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
276C8—22no—
277C8—92no—
278C8—181no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—61no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—362no—
289C2×C3C3×#89311yescharacteristic
290C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
291C2×C3C3×#90711yescharacteristic
292D3—122no—
293D3—362no—
294C22—182no—
295C22—182no—
296C22—182no—
297C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
298C4—21no—
299C4—182no—
300C4#3743⋊C312yescomplemented
301C4D4×(C3⋊D3)⋊C311yescharacteristic
302C4#8872211yescharacteristic
303C3—31no—
304C3—31no—
305C3—61no—
306C3C3×#483811yescharacteristic
307C2—362no—
308C2#64102⋊C311yescharacteristic
309C2C2×#3743⋊C311yescharacteristic
310C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#436
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#3743⋊C3C2semidirect
3. H(3,𝔽3)×C4⋊C8C2semidirect
4. #63647⋊C3C2semidirect
5. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
6. #10592537C3semidirect
17. C2×#3743C2×C3semidirect
22. C32×C4⋊C8D3semidirect
23. C32×C4⋊C8C2×C3semidirect
24. #63647C2×C3semidirect
25. C4×C3⋊(C3⋊C8)C2×C3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
67. C4×H(3,𝔽3)M16semidirect
110. C3⋊(C3⋊C8)C3×D4semidirect
162. C32×C4ΓL(1,𝔽25)semidirect
163. C32×C4C3×M16semidirect
172. C4⋊C8(C3⋊D3)⋊C3semidirect
175. H(3,𝔽3)#436semidirect
266. C32#4838semidirect
267. C32C3×#436semidirect
300. C4#3743⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×C4⋊C82
#63647⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105925373
C3×#48383