Finite Group Databasebeta

← C3×#10591480 #13158493

Subgroups — C3×#10591480 #13158493

Order 1728.

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

normal subgroups only

This group has 702 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
1C3×#64149C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3C32×#832C212yesmaximal, complemented
4C3×#63564C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6C3×#4764—32nomaximal
7#10591480C311yesmaximal, characteristic, direct product factor
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2212yes—
14C3×Dic3⋊Dic3—22no—
15#64149C2×C311yescharacteristic
16C3×#940—32no—
17C3×#940—61no—
18C3×#890—32no—
19#63872C2×C311yescharacteristic, complemented
20C32×#102—32no—
21C3×#832—22no—
22C3×#832D312yescomplemented
23C3×#832C2×C312yescomplemented
24C3×#821—32no—
25C3×#815—32no—
26C3×#809—32no—
27#63564C2×C312yescomplemented
28#63555C2×C311yescharacteristic, complemented
29C23×C33—21no—
30C22×C3×C3⋊D3—21no—
31C33×D4—22no—
32C2×C33×C4C2311yescharacteristic
33C3×#638—44no—
34C2×C3×C3⋊Dic3D412yes—
35C3×C4⋊(C3⋊D3)—22no—
36C2×C32×Dic3—22no—
37C32×C3⋊C8—22no—
38C3×#490—91no—
39#4764—32no—
40C2×C32×D4—22no—
41C2×C32×D4—21no—
42C2×C32×D4—41no—
43C2×C32×D4C2×D312yes—
44C2×C32×D4C22×C311yescharacteristic
45C2×#638—22no—
46C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
47C2×C3×#67—62no—
48C2×C3×#67—62no—
49C2×C3×#67—64no—
50C2×C3×D12—32no—
51C2×C3×D12—61no—
52C2×C32×C8—32no—
53C32×C4⋊C4—32no—
54C4×C3⋊Dic3C22×C311yescharacteristic
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—62no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C22×C312yes—
59C3×Dic3⋊C4—62no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C22×C312yes—
65Dic3⋊Dic3—22no—
66C22×C33—41no—
67C22×C33—42no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—41no—
70C2×C3×C3⋊D3—42no—
71C33×C4#4112yes—
72C3×C3⋊Dic3—22no—
73C32×Dic3—42no—
74C3×#127—91no—
75C3×#125—91no—
76#940—32no—
77#940—61no—
78#890—32no—
79C3×#102—32no—
80C3×#102—32no—
81C3×#102—62no—
82C3×#102—92no—
83C3×#97—91no—
84#832C3×D312yescomplemented
85#821—32no—
86#815—32no—
87#809—32no—
88C23×C32—21no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—42no—
92C23×C32—42no—
93C22×C3⋊D3—21no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C32×D4—22no—
97C32×D4—24no—
98C32×D4—22no—
99C32×D4—22no—
100C32×D4—42no—
101C32×D4—42no—
102C32×D4—44no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—41no—
106C2×C32×C4—62no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C23×C311yescharacteristic
109#638—44no—
110C2×C3⋊Dic3C3×D412yes—
111C4⋊(C3⋊D3)—22no—
112C3×#67—128no—
113C3×#67—124no—
114C3×#67—124no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—62no—
121C3×D12—62no—
122C3×D12—62no—
123C3×D12—122no—
124C32×C8—62no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—42no—
128#490—91noSylow
129C2×C33—81no—
130C2×C33#12411yescharacteristic
131C2×C33AGL(1,ℤ/8)12yes—
132C3×C3⋊D3—81no—
133C2×C3×D4—22no—
134C2×C3×D4—21no—
135C2×C3×D4—41no—
136C2×C3×D4—91no—
137C2×C3×D4—182no—
138C2×C3×D4D3211yescharacteristic
139C2×C3×D4C2×C3×D312yes—
140C2×#67—62no—
141C2×#67—62no—
142C2×#67—64no—
143C2×D12—32no—
144C2×D12—61no—
145C2×C3×C8—32no—
146C2×C3×C8—32no—
147C2×C3×C8—62no—
148C3×C4⋊C4—32no—
149C3×C4⋊C4—32no—
150C3×C4⋊C4—62no—
151C3×C4⋊C4—182no—
152C3×C42—91no—
153C4⋊Dic3—62no—
154C4⋊Dic3C2×C3×D312yes—
155Dic3⋊C4—62no—
156C4×Dic3—32no—
157C4×Dic3—61no—
158C2×C3⋊C8C2×C3×D312yes—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—41no—
162C22×C32—44no—
163C22×C32—42no—
164C22×C32—44no—
165C22×C32—42no—
166C22×C32—41no—
167C22×C32—42no—
168C22×C32—44no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—82no—
172C22×C32—82no—
173C22×C32—84no—
174C22×C32D3×D412yes—
175C22×C32C2×C3×D411yescharacteristic
176C2×C3⋊D3—41no—
177C2×C3⋊D3—42no—
178C2×C3×D3—122no—
179C2×C3×D3—124no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—122no—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C32×C4C3×#4112yes—
191C3⋊Dic3—22no—
192C3×Dic3—42no—
193C3×Dic3—42no—
194C3×Dic3—64no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C3×Dic3—82no—
198#127—91no—
199#125—91no—
200#102—32no—
201#102—92no—
202#97—91no—
203C33#49011yescharacteristic, complemented, Sylow
204C23×C3—22no—
205C23×C3—21no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—181no—
210C22×D3—62no—
211C22×D3—62no—
212C3×D4—24no—
213C3×D4—22no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—44no—
219C3×D4—182no—
220C3×D4—364no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—92no—
227C2×C3×C4—122no—
228C2×C3×C4C2×D3211yescharacteristic
229C2×C3×C4C22×C3×D312yes—
230#67—128no—
231#67—124no—
232#67—124no—
233C2×Dic3—22no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—62no—
237D12—62no—
238D12—62no—
239D12—122no—
240C3×C8—62no—
241C3×C8—62no—
242C3×C8—122no—
243C3⋊C8—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32—42no—
251C2×C32—82no—
252C2×C32—84no—
253C2×C32—84no—
254C2×C32—82no—
255C2×C32—81no—
256C2×C32#88512yes—
257C2×C32#91112yes—
258C2×C32#91412yes—
259C2×C32C3×#12411yescharacteristic
260C2×C32C3×AGL(1,ℤ/8)12yes—
261C3⋊D3—81no—
262C3×D3—242no—
263C3×D3—242no—
264C2×D4—91no—
265C2×D4—182no—
266C2×D4C3×D3211yescharacteristic
267C2×C8—32no—
268C4⋊C4—32no—
269C4⋊C4—182no—
270C42—91no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—42no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—84no—
286C22×C3—362no—
287C22×C3—361no—
288C22×C3#381111yescharacteristic
289C22×C3C3×D3×D412yes—
290C2×D3—122no—
291C2×D3—124no—
292C2×D3—122no—
293C2×D3—122no—
294C2×D3—122no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4—122no—
300C3×C4—122no—
301C3×C4—182no—
302C3×C4—242no—
303C3×C4#380412yes—
304C3×C4C3×#23512yes—
305C3×C4C3×#23712yes—
306Dic3—42no—
307Dic3—64no—
308Dic3—62no—
309Dic3—62no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32#476412yescomplemented
314C32C3×#49011yescharacteristic, complemented
315C23—21no—
316C23—181no—
317D4—22no—
318D4—182no—
319D4—364no—
320C2×C4—62no—
321C2×C4—92no—
322C2×C4C2×C3×D3211yescharacteristic
323C8—62no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—22no—
328C2×C3—22no—
329C2×C3—41no—
330C2×C3—42no—
331C2×C3—82no—
332C2×C3—84no—
333C2×C3—84no—
334C2×C3—82no—
335C2×C3—81no—
336C2×C3—721no—
337C2×C3#6393712yes—
338C2×C3#6395311yescharacteristic
339C2×C3C3×#88512yes—
340C2×C3C3×#91112yes—
341C2×C3C3×#91412yes—
342D3—242no—
343D3—242no—
344C22—41no—
345C22—42no—
346C22—361no—
347C22—362no—
348C22C3×#381111yescharacteristic
349C4—122no—
350C4—182no—
351C4C3×#380412yes—
352C3—21no—
353C3—22no—
354C3—41no—
355C3#1059148011yescharacteristic, direct product factor
356C3C3×#476412yescomplemented
357C2—81no—
358C2—721no—
359C2C3×#6393712yes—
360C2C3×#6395311yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#490
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63872C2semidirect
3. C32×#832C2semidirect
4. C3×#63564C2semidirect
5. C3×#63555C2semidirect
7. #10591480C3direct (N × H)
19. #63872C2×C3semidirect
22. C3×#832D3semidirect
23. C3×#832C2×C3semidirect
27. #63564C2×C3semidirect
28. #63555C2×C3semidirect
84. #832C3×D3semidirect
203. C33#490semidirect
313. C32#4764semidirect
314. C32C3×#490semidirect
355. C3#10591480direct (N × H)
356. C3C3×#4764semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C3×#638722
C32×#8322
C3×#635642
C3×#635552
C3×#47643
#105914803