Finite Group Databasebeta

← #10589340⋊C3 #13135343

Subgroups — #10589340⋊C3 #13135343

Order 1728.

353 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 50 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#64149⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#99C211yesmaximal, characteristic, complemented
3#63661⋊C3C211yesmaximal, characteristic, complemented
4#10589340C311yesmaximal, characteristic, complemented
5C3×#4474—31nomaximal
6C2×D4×H(3,𝔽3)C2211yescharacteristic
7C2×#638⋊C3—21no—
8C2×#638⋊C3—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
10C22⋊C4×H(3,𝔽3)—21no—
11#3757⋊C3—21no—
12C4×(C3⋊Dic3)⋊C3C411yescharacteristic, complemented
13#64149C2×C311yescharacteristic
14C3×#940—31no—
15C32×#99—21no—
16C32×#99D311yescharacteristic, complemented
17C32×#99C2×C311yescharacteristic, complemented
18#63661C2×C311yescharacteristic, complemented
19C3×#834—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C23×H(3,𝔽3)D411yescharacteristic
22C22×(C3⋊D3)⋊C3—21no—
23D4×H(3,𝔽3)—41no—
24C2×C4×H(3,𝔽3)—41no—
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26#638⋊C3—42no—
27#638⋊C3—42no—
28C2×(C3⋊Dic3)⋊C3—21no—
29C2×(C3⋊Dic3)⋊C3—41no—
30C4⋊((C3⋊D3)⋊C3)—41no—
31C3×#353—91no—
32#4474—31no—
33#4474—91no—
34C2×C32×D4—21no—
35C2×C32×D4C2×D311yescharacteristic
36C2×C32×D4C22×C311yescharacteristic
37C2×#638—21no—
38C2×#638—21no—
39C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
40C2×C3×#67—61no—
41C2×C3×#67—61no—
42C2×C3×D12—31no—
43C32×C22⋊C4—22no—
44C32×C22⋊C4—21no—
45C32×C22⋊C4—21no—
46#3757—21no—
47C4×C3⋊Dic3C3×C411yescharacteristic, complemented
48C3×#217—61no—
49C3×C4×Dic3—31no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—41no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)C22⋊C411yescharacteristic
55C2×(C3⋊D3)⋊C3—41no—
56C2×(C3⋊D3)⋊C3—41no—
57C2×(C3⋊D3)⋊C3—41no—
58C4×H(3,𝔽3)—21no—
59C4×H(3,𝔽3)—81no—
60(C3⋊Dic3)⋊C3—22no—
61(C3⋊Dic3)⋊C3—81no—
62C3×#127—91no—
63#940—31no—
64#940—91no—
65C3×#99—31no—
66C3×#99—31no—
67C3×#99—61no—
68C3×#99—91no—
69C3×#99C3×D311yescharacteristic
70#834—31no—
71#834—91no—
72C23×C32—21no—
73C23×C32—21no—
74C23×C32D1211yescharacteristic
75C23×C32#6711yescharacteristic
76C23×C32C3×D411yescharacteristic
77C23×C32C3×D411yescharacteristic
78C22×C3⋊D3—21no—
79C22×C3×D3—61no—
80C32×D4—41no—
81C32×D4—41no—
82C32×D4—42no—
83C2×C32×C4—21no—
84C2×C32×C4—42no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4C4×D311yescharacteristic
88C2×C32×C4C2×C3×C411yescharacteristic
89#638—42no—
90#638—42no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—41no—
93C4⋊(C3⋊D3)—41no—
94C3×#67—122no—
95C3×#67—122no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—121no—
98C3×D12—121no—
99#353—91noSylow
100C2×H(3,𝔽3)—21no—
101C2×H(3,𝔽3)—41no—
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)#9911yescharacteristic
104(C3⋊D3)⋊C3—81no—
105C2×C3×D4—31no—
106C2×C3×D4—31no—
107C2×C3×D4—61no—
108C2×C3×D4—91no—
109C2×C3×D4—181no—
110C2×C3×D4—181no—
111C2×C3×D4C2×C3×D311yescharacteristic
112C2×#67—61no—
113C2×#67—61no—
114C2×#67—181no—
115C2×#67—181no—
116C2×D12—31no—
117C2×D12—91no—
118C3×C22⋊C4—21no—
119C3×C22⋊C4—61no—
120C3×C22⋊C4—61no—
121C3×C22⋊C4—62no—
122C3×C22⋊C4—181no—
123C3×C42—91no—
124#217—61no—
125#217—181no—
126C4×Dic3—31no—
127C4×Dic3—91no—
128C22×C32—22no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—81no—
140C22×C32—81no—
141C22×C32(C2×D3)⋊C411yescharacteristic
142C22×C32C3×C22⋊C411yescharacteristic
143C2×C3⋊D3—41no—
144C2×C3⋊D3—41no—
145C2×C3⋊D3—41no—
146C2×C3×D3—121no—
147C2×C3×D3—121no—
148C2×C3×D3—121no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—41no—
152C32×C4—82no—
153C32×C4—81no—
154C32×C4—81no—
155C3⋊Dic3—22no—
156C3⋊Dic3—81no—
157C3×Dic3—62no—
158C3×Dic3—241no—
159#127—91no—
160#99—91no—
161#99(C3⋊D3)⋊C311yescharacteristic, complemented
162H(3,𝔽3)#35311yescharacteristic, complemented, Sylow
163C23×C3—31no—
164C23×C3—31no—
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—61no—
168C23×C3—61no—
169C23×C3—181no—
170C23×C3C3×D1211yescharacteristic
171C23×C3C3×#6711yescharacteristic
172C22×D3—61no—
173C22×D3—181no—
174C3×D4—41no—
175C3×D4—121no—
176C3×D4—121no—
177C3×D4—122no—
178C3×D4—361no—
179C3×D4—362no—
180C3×D4—362no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—41no—
184C2×C3×C4—61no—
185C2×C3×C4—121no—
186C2×C3×C4—121no—
187C2×C3×C4—122no—
188C2×C3×C4—181no—
189C2×C3×C4—361no—
190C2×C3×C4C3×C4×D311yescharacteristic
191#67—122no—
192#67—122no—
193#67—362no—
194#67—362no—
195C2×Dic3—61no—
196C2×Dic3—121no—
197C2×Dic3—181no—
198C2×Dic3—361no—
199D12—121no—
200D12—361no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—41no—
205C2×C32—42no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—81no—
211C2×C32#80611yescharacteristic
212C2×C32C3×#9911yescharacteristic
213C3⋊D3—81no—
214C3×D3—241no—
215C2×D4—91no—
216C2×D4—181no—
217C2×D4—181no—
218C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
219C22⋊C4—21no—
220C22⋊C4—181no—
221C42—91no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—61no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—62no—
233C22×C3—61no—
234C22×C3—121no—
235C22×C3—121no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3—121no—
239C22×C3—241no—
240C22×C3—241no—
241C22×C3—361no—
242C22×C3—361no—
243C22×C3—361no—
244C22×C3C3×(C2×D3)⋊C411yescharacteristic
245C2×D3—121no—
246C2×D3—121no—
247C2×D3—121no—
248C2×D3—361no—
249C2×D3—361no—
250C2×D3—361no—
251C3×C4—21no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—81no—
255C3×C4—121no—
256C3×C4—182no—
257C3×C4—241no—
258C3×C4—241no—
259C3×C4—242no—
260C3×C4—721no—
261Dic3—62no—
262Dic3—182no—
263Dic3—241no—
264Dic3—721no—
265C32—21no—
266C32#447411yescharacteristic, complemented
267C32C3×#35311yescharacteristic, complemented
268C23—181no—
269C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
270C23#638⋊C311yescharacteristic
271D4—41no—
272D4—361no—
273D4—362no—
274D4—362no—
275C2×C4—41no—
276C2×C4—181no—
277C2×C4—361no—
278C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
279C2×C3—21no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—61no—
287C2×C3—121no—
288C2×C3—121no—
289C2×C3—121no—
290C2×C3—121no—
291C2×C3—121no—
292C2×C3—122no—
293C2×C3—241no—
294C2×C3—721no—
295C2×C3C3×#80611yescharacteristic
296D3—241no—
297D3—721no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—361no—
303C22—361no—
304C22—361no—
305C22#3752⋊C311yescharacteristic
306C4—21no—
307C4—81no—
308C4—182no—
309C4—721no—
310C3—31no—
311C3—31no—
312C3—61no—
313C3C3×#447411yescharacteristic
314C2—21no—
315C2—41no—
316C2—41no—
317C2—721no—
318C2#63633⋊C311yescharacteristic
subgroupG/H
centerC2#63633⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
FittingH(3,𝔽3)×#99C2
socleC2×C3C3×#806
prime radical#64149⋊C3C2
cosocleC2×C32×D4C22×C3
hypercenter#99(C3⋊D3)⋊C3
hypocenterC32C3×#353
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #99

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#99C2semidirect
3. #63661⋊C3C2semidirect
4. #10589340C3semidirect
9. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
12. C4×(C3⋊Dic3)⋊C3C4semidirect
16. C32×#99D3semidirect
17. C32×#99C2×C3semidirect
18. #63661C2×C3semidirect
39. C2×C4⋊(C3⋊D3)C3×C4semidirect
47. C4×C3⋊Dic3C3×C4semidirect
161. #99(C3⋊D3)⋊C3semidirect
162. H(3,𝔽3)#353semidirect
266. C32#4474semidirect
267. C32C3×#353semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
H(3,𝔽3)×#992
#63661⋊C32
#105893403
C3×#44743