Finite Group Databasebeta

← #10592842⋊C3 #13143257

Subgroups — #10592842⋊C3 #13143257

Order 1728.

412 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#131C211yesmaximal, characteristic
2#64160⋊C3C212yesmaximal
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#63651⋊C3C212yesmaximal, complemented
5#63646⋊C3C211yesmaximal, characteristic, complemented
6#10592842C311yesmaximal, characteristic, complemented
7C3×#5143—31nomaximal
8#41×H(3,𝔽3)C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10M16×H(3,𝔽3)—21no—
11M16×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#3756⋊C3—22no—
15#3754⋊C3—22no—
16#3748⋊C3C2212yes—
17#3746⋊C3—21no—
18#3746⋊C3—21no—
19C32×#131—21no—
20C32×#131D311yescharacteristic
21C32×#131C2×C311yescharacteristic
22#64160C2×C312yes—
23C2×#3746C2×C311yescharacteristic, complemented
24C3×#951—32no—
25C2×C3×Dic12—31no—
26#63651C2×C312yescomplemented
27#63646C2×C311yescharacteristic, complemented
28C3×#824—32no—
29C3×#819—31no—
30Q8×H(3,𝔽3)D411yescharacteristic
31D4×H(3,𝔽3)—21no—
32D4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—22no—
37#634⋊C3—22no—
38C8×H(3,𝔽3)—21no—
39C8×H(3,𝔽3)D411yescharacteristic
40C8×H(3,𝔽3)D411yescharacteristic
41(C3⋊(C3⋊C8))⋊C3—22no—
42C3×#473—91no—
43#5143—31no—
44#5143—91no—
45C32×#41—21no—
46C32×#41C2×D311yescharacteristic
47C32×#41C22×C311yescharacteristic
48C2×#634C22×C312yes—
49C2×C3×Dic6—32no—
50C32×M16—22no—
51C32×M16—21no—
52C32×M16—21no—
53C32×M16—21no—
54C32×M16C2×D311yescharacteristic
55C32×M16C22×C311yescharacteristic
56C2×C32×C8—22no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62#3756—22no—
63#3754—22no—
64#3748C22×C312yes—
65#3746—21no—
66#3746—21no—
67C3×#216—62no—
68C3×#214—62no—
69C3×#208—32no—
70C3×Dic12—61no—
71C3×Dic12—61no—
72C22×H(3,𝔽3)—21no—
73C22×H(3,𝔽3)#4111yescharacteristic
74C4×H(3,𝔽3)—21no—
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)C2×D411yescharacteristic
77(C3⋊Dic3)⋊C3—42no—
78C3×#137—92no—
79C2×C3×Q16—91no—
80C3×#131—31no—
81C3×#131—31no—
82C3×#131—61no—
83C3×#131C3×D311yescharacteristic
84#951—32no—
85#951—92no—
86C2×Dic12—31no—
87C2×Dic12—91no—
88C3×#108—91no—
89C3×#101—92no—
90#824—32no—
91#824—92no—
92#819—31no—
93#819—91no—
94C32×Q8—21no—
95C32×Q8D1211yescharacteristic
96C32×Q8C3×D411yescharacteristic
97C32×D4—22no—
98C32×D4—21no—
99C32×D4—21no—
100C32×D4—21no—
101C32×D4D1211yescharacteristic
102C32×D4C3×D411yescharacteristic
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109C2×C3⋊Dic3—22no—
110#634—22no—
111#634—22no—
112C2×C3×Dic3—62no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C32×C8—22no—
116C32×C8—21no—
117C32×C8—21no—
118C32×C8—21no—
119C32×C8—21no—
120C32×C8#6711yescharacteristic
121C32×C8#6711yescharacteristic
122C32×C8C3×D411yescharacteristic
123C32×C8C3×D411yescharacteristic
124C3⋊(C3⋊C8)—22no—
125C3×C3⋊C8—62no—
126#473—91noSylow
127C2×H(3,𝔽3)—21no—
128C2×H(3,𝔽3)—41no—
129C2×H(3,𝔽3)#12111yescharacteristic
130C3×#41—31no—
131C3×#41—31no—
132C3×#41—61no—
133C3×#41C2×C3×D311yescharacteristic
134C2×C3×Q8—92no—
135C2×Dic6—32no—
136C2×Dic6—92no—
137C3×Q16—181no—
138C3×Q16—181no—
139C3×Q16—182no—
140C3×SD16—182no—
141C3×M16—21no—
142C3×M16—31no—
143C3×M16—31no—
144C3×M16—61no—
145C3×M16—61no—
146C3×M16—62no—
147C3×M16—61no—
148C3×M16—92no—
149C3×M16C2×C3×D311yescharacteristic
150C2×C3×C8—21no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—61no—
155C2×C3×C8—62no—
156C2×C3×C8—61no—
157C2×C3×C8C2×C3×D311yescharacteristic
158#216—62no—
159#216—182no—
160#214—62no—
161#214—182no—
162#208—32no—
163#208—92no—
164Dic12—61no—
165Dic12—61no—
166Dic12—181no—
167Dic12—181no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32#23511yescharacteristic
173C22×C32C3×#4111yescharacteristic
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4C2×D1211yescharacteristic
180C32×C4C2×#6711yescharacteristic
181C32×C4C2×C3×D411yescharacteristic
182C32×C4C2×C3×D411yescharacteristic
183C3⋊Dic3—42no—
184C3×Dic3—122no—
185#137—92no—
186C2×Q16—91no—
187#131(C3⋊D3)⋊C311yescharacteristic
188#108—91no—
189#101—92no—
190H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
191C3×Q8—31no—
192C3×Q8—31no—
193C3×Q8—61no—
194C3×Q8—182no—
195C3×Q8—182no—
196C3×Q8C3×D1211yescharacteristic
197C3×D4—21no—
198C3×D4—31no—
199C3×D4—31no—
200C3×D4—61no—
201C3×D4—61no—
202C3×D4—62no—
203C3×D4—61no—
204C3×D4C3×D1211yescharacteristic
205C2×C3×C4—21no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—61no—
212C2×C3×C4—182no—
213C2×C3×C4C22×C3×D311yescharacteristic
214C2×Dic3—62no—
215C2×Dic3—182no—
216Dic6—62no—
217Dic6—62no—
218Dic6—182no—
219Dic6—182no—
220C3×C8—21no—
221C3×C8—31no—
222C3×C8—31no—
223C3×C8—31no—
224C3×C8—31no—
225C3×C8—61no—
226C3×C8—61no—
227C3×C8—62no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—182no—
231C3×C8C3×#6711yescharacteristic
232C3×C8C3×#6711yescharacteristic
233C3⋊C8—62no—
234C3⋊C8—182no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—41no—
239C2×C32—42no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32#93011yescharacteristic
243C2×C32C3×#12111yescharacteristic
244#41C2×(C3⋊D3)⋊C311yescharacteristic
245C2×Q8—92no—
246Q16—182no—
247Q16—181no—
248Q16—181no—
249SD16—182no—
250M16—21no—
251M16—92no—
252M16C2×(C3⋊D3)⋊C311yescharacteristic
253C2×C8—21no—
254C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
255C22×C3—21no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—62no—
261C22×C3—61no—
262C22×C3C3×#23511yescharacteristic
263C3×C4—21no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—62no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—362no—
274C3×C4C2×C3×D1211yescharacteristic
275C3×C4C2×C3×#6711yescharacteristic
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#514311yescharacteristic, complemented
280C32C3×#47311yescharacteristic, complemented
281Q8—182no—
282Q8—182no—
283Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
284D4—21no—
285D4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
286C2×C4—21no—
287C2×C4—182no—
288C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
289C8—21no—
290C8—182no—
291C8#638⋊C311yescharacteristic
292C8#638⋊C311yescharacteristic
293C2×C3—21no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—41no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—121no—
301C2×C3—121no—
302C2×C3—121no—
303C2×C3—122no—
304C2×C3C3×#93011yescharacteristic
305C22—21no—
306C22#3828⋊C311yescharacteristic
307C4—21no—
308C4—362no—
309C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
310C4C2×#638⋊C311yescharacteristic
311C3—31no—
312C3—31no—
313C3—61no—
314C3C3×#514311yescharacteristic
315C2—21no—
316C2—41no—
317C2#64139⋊C311yescharacteristic
subgroupG/H
centerC2#64139⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C3×#930
prime radical#41×H(3,𝔽3)C22
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
4. #63651⋊C3C2semidirect
5. #63646⋊C3C2semidirect
6. #10592842C3semidirect
23. C2×#3746C2×C3semidirect
26. #63651C2×C3semidirect
27. #63646C2×C3semidirect
190. H(3,𝔽3)#473semidirect
279. C32#5143semidirect
280. C32C3×#473semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1312
#64160⋊C32
C2×#3746⋊C32
#63651⋊C32
#63646⋊C32
#105928423
C3×#51433