Finite Group Databasebeta

← #10592404⋊C3 #13141529

Subgroups — #10592404⋊C3 #13141529

Order 1728.

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

normal subgroups only

This group has 425 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
1C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic, complemented
3#63647⋊C3C212yesmaximal
4#63641⋊C3C212yesmaximal, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10592404C311yesmaximal, characteristic, complemented
7C3×#4705—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C411yescharacteristic
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13C4×(C3⋊Dic3)⋊C3C411yescharacteristic
14C4×(C3⋊Dic3)⋊C3C411yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C42×C3⋊D3C2×C311yescharacteristic
18C3×C42×D3—31no—
19C32×#97—21no—
20C32×#97D311yescharacteristic, complemented
21C32×#97C2×C311yescharacteristic, complemented
22#63647C2×C312yes—
23#63641C2×C312yescomplemented
24#63630C2×C311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—32no—
26C3×Dic3⋊C8—32no—
27C3×#803—31no—
28C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
33C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
34C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
35C4×(C3⋊D3)⋊C3—22no—
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3—22no—
38C8×H(3,𝔽3)—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40C3×#432—91no—
41#4705—31no—
42#4705—91no—
43C2×C4×C3⋊D3C3×C411yescharacteristic
44C2×C4×C3⋊D3C3×C411yescharacteristic
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×C4×D3—31no—
47C2×C3×C4×D3—31no—
48C2×C3×C4×D3—31no—
49C2×C32×C8—22no—
50C2×C32×C8C2×D312yes—
51C2×C32×C8C22×C312yes—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C22×C311yescharacteristic
55C4×C3⋊Dic3C3×C411yescharacteristic
56C4×C3⋊Dic3C3×C411yescharacteristic
57C4×C3⋊Dic3C22×C311yescharacteristic
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×C4×Dic3—31no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3—21no—
66C2×(C3⋊D3)⋊C3M1612yes—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)#4112yes—
70(C3⋊Dic3)⋊C3—21no—
71(C3⋊Dic3)⋊C3—21no—
72(C3⋊Dic3)⋊C3M1612yes—
73C2×C3×C42—91no—
74C42×D3—31no—
75C42×D3—91no—
76C3×C4⋊C8—92no—
77C3×C22⋊C8—92no—
78C3×#97—31no—
79C3×#97—31no—
80C3×#97—61no—
81C3×#97—91no—
82C3×#97C3×D311yescharacteristic
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—92no—
85Dic3⋊C8—32no—
86Dic3⋊C8—92no—
87#803—31no—
88#803—91no—
89C22×C3⋊D3C2×C3×C411yescharacteristic
90C22×C3×D3—31no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4C4×D311yescharacteristic
95C2×C32×C4C4×D311yescharacteristic
96C2×C32×C4C2×C3×C411yescharacteristic
97C2×C32×C4C2×C3×C411yescharacteristic
98C2×C32×C4C22×D311yescharacteristic
99C2×C32×C4C23×C311yescharacteristic
100C2×C3⋊Dic3C2×C3×C411yescharacteristic
101C2×C3⋊Dic3C2×C3×C411yescharacteristic
102C2×C3⋊Dic3C2×C3×C411yescharacteristic
103C4×C3⋊D3—22no—
104C4×C3⋊D3—22no—
105C4×C3⋊D3—22no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C3×C4×D3—62no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C32×C8—24no—
113C32×C8—22no—
114C32×C8—22no—
115C3⋊(C3⋊C8)—22no—
116C3×C3⋊C8—62no—
117#432—91noSylow
118C2×H(3,𝔽3)#11711yescharacteristic
119C2×H(3,𝔽3)C2×M1611yescharacteristic
120C2×H(3,𝔽3)C2×M1611yescharacteristic
121(C3⋊D3)⋊C3—22no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C2×C4×D3—31no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—91no—
129C2×C4×D3—91no—
130C2×C4×D3—91no—
131C2×C3×C8—32no—
132C2×C3×C8—32no—
133C2×C3×C8—62no—
134C2×C3×C8—92no—
135C2×C3×C8C2×C3×D312yes—
136C3×C42—31no—
137C3×C42—31no—
138C3×C42—61no—
139C3×C42—91no—
140C3×C42—91no—
141C3×C42—91no—
142C3×C42C2×C3×D311yescharacteristic
143C4×Dic3—31no—
144C4×Dic3—31no—
145C4×Dic3—31no—
146C4×Dic3—91no—
147C4×Dic3—91no—
148C4×Dic3—91no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—92no—
151C22×C32—21no—
152C22×C32C2×C4×D311yescharacteristic
153C22×C32C22×C3×C411yescharacteristic
154C2×C3⋊D3—21no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3C3×M1612yes—
157C2×C3×D3—32no—
158C2×C3×D3—61no—
159C2×C3×D3—61no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—22no—
165C32×C4—41no—
166C32×C4—41no—
167C32×C4#23512yes—
168C32×C4C3×#4112yes—
169C3⋊Dic3—21no—
170C3⋊Dic3—21no—
171C3⋊Dic3C3×M1612yes—
172C3×Dic3—32no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C2×C42—91no—
176C4⋊C8—92no—
177C22⋊C8—92no—
178#97—91no—
179#97(C3⋊D3)⋊C311yescharacteristic, complemented
180H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
181C23×C3—91no—
182C22×D3—31no—
183C22×D3—91no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4—182no—
199C2×C3×C4C3×C4×D311yescharacteristic
200C2×C3×C4C3×C4×D311yescharacteristic
201C2×C3×C4C22×C3×D311yescharacteristic
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—91no—
206C2×Dic3—91no—
207C2×Dic3—91no—
208C4×D3—62no—
209C4×D3—62no—
210C4×D3—62no—
211C4×D3—182no—
212C4×D3—182no—
213C4×D3—182no—
214C3×C8—22no—
215C3×C8—62no—
216C3×C8—62no—
217C3×C8—64no—
218C3×C8—182no—
219C3⋊C8—62no—
220C3⋊C8—182no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32#87211yescharacteristic
225C2×C32D3×M1611yescharacteristic
226C2×C32D3×M1611yescharacteristic
227C2×C32C3×#11711yescharacteristic
228C2×C32C2×C3×M1611yescharacteristic
229C2×C32C2×C3×M1611yescharacteristic
230C3⋊D3—22no—
231C3×D3—62no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C2×C8—92no—
236C2×C8C2×(C3⋊D3)⋊C312yes—
237C42—91no—
238C42—91no—
239C42—91no—
240C42C2×(C3⋊D3)⋊C311yescharacteristic
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—61no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3C2×C3×C4×D311yescharacteristic
248C2×D3—32no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—92no—
252C2×D3—181no—
253C2×D3—181no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—32no—
257C3×C4—32no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—62no—
263C3×C4—92no—
264C3×C4—121no—
265C3×C4—121no—
266C3×C4—181no—
267C3×C4—181no—
268C3×C4C3×#23512yes—
269Dic3—32no—
270Dic3—61no—
271Dic3—61no—
272Dic3—92no—
273Dic3—181no—
274Dic3—181no—
275C32—21no—
276C32#470511yescharacteristic, complemented
277C32C3×#43211yescharacteristic, complemented
278C23—91no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
286C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
287C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
288C8—22no—
289C8—182no—
290C2×C3—31no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—61no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—182no—
300C2×C3C3×#87211yescharacteristic
301C2×C3C3×D3×M1611yescharacteristic
302C2×C3C3×D3×M1611yescharacteristic
303D3—62no—
304D3—182no—
305C22—92no—
306C22—181no—
307C22—181no—
308C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
309C4—21no—
310C4—21no—
311C4—92no—
312C4—181no—
313C4—181no—
314C4#3828⋊C312yes—
315C3—31no—
316C3—31no—
317C3—61no—
318C3C3×#470511yescharacteristic
319C2—182no—
320C2#64081⋊C311yescharacteristic
321C2M16×(C3⋊D3)⋊C311yescharacteristic
322C2M16×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#432
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#97C2semidirect
4. #63641⋊C3C2semidirect
5. #63630⋊C3C2semidirect
6. #10592404C3semidirect
20. C32×#97D3semidirect
21. C32×#97C2×C3semidirect
23. #63641C2×C3semidirect
24. #63630C2×C3semidirect
179. #97(C3⋊D3)⋊C3semidirect
180. H(3,𝔽3)#432semidirect
276. C32#4705semidirect
277. C32C3×#432semidirect

Maximal subgroups

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

subgroupindex
C42×(C3⋊D3)⋊C32
H(3,𝔽3)×#972
#63647⋊C32
#63641⋊C32
#63630⋊C32
#105924043
C3×#47053