Finite Group Databasebeta

← C3×#10592481 #13162819

Subgroups — C3×#10592481 #13162819

Order 1728.

980 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C211yesmaximal, characteristic, complemented
2C2×C3×#3753C211yesmaximal, characteristic, complemented
3C2×C3×#3743C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592481C311yesmaximal, characteristic, direct product factor
9C3×#4782—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4⋊(C3⋊D3)—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C33×C8C2211yescharacteristic
16C33×C4⋊C4C2211yescharacteristic
17C3×#3757—21no—
18C3×#3753—22no—
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3743—22no—
23#64147C2×C311yescharacteristic, complemented
24C2×#3753C2×C311yescharacteristic, complemented
25C2×#3743C2×C311yescharacteristic, complemented
26#64104C2×C311yescharacteristic, complemented
27C3×#938—34no—
28C2×C3×#213—34no—
29C2×C3×ΓL(1,𝔽25)—34no—
30C3×#895—34no—
31C32×#102—24no—
32C32×#102D314yescomplemented
33C32×#102C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63635C2×C311yescharacteristic, complemented
36C3×#821—34no—
37C3×#808—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4—21no—
44C2×C33×C4C2311yescharacteristic
45C3×#638—42no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—42no—
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C33×C8—21no—
54C3×C3⋊(C3⋊C8)D412yescomplemented
55C3×#469—91no—
56#4782—34no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C22×C311yescharacteristic
60C2×#638—21no—
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×C4×C3⋊D3C22×C311yescharacteristic
64C2×C3×#67—64no—
65C2×C3×D12—34no—
66C2×C3×D12—64no—
67C2×C3×C4×D3—34no—
68C2×C32×C8—24no—
69C2×C32×C8C2×D314yes—
70C2×C32×C8C22×C311yescharacteristic
71C32×C4⋊C4—24no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C22×C311yescharacteristic
74#3757—21no—
75#3753—22no—
76#3752—21no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79#3743—22no—
80C3×#217—64no—
81C3×#213—68no—
82C3×(C2×D3)⋊C4—64no—
83C3×C4⋊Dic3—34no—
84C2×C3×C3⋊C8—34no—
85C3×ΓL(1,𝔽25)—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—21no—
100C3×C3⋊Dic3—41no—
101C2×C3×D8—91no—
102C2×C3×M16—91no—
103C3×#121—91no—
104C3×#121—91no—
105#938—34no—
106C2×#213—34no—
107C2×ΓL(1,𝔽25)—34no—
108#895—34no—
109C3×#106—91no—
110C3×#102—24no—
111C3×#102—91no—
112C3×#102C3×D314yescomplemented
113C3×#102C3⋊D311yescharacteristic, complemented
114#821—34no—
115#808—34no—
116C23×C32—21no—
117C23×C32—24no—
118C23×C32—44no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3C3×D411yescharacteristic
121C22×C3×D3—34no—
122C22×C3×D3—64no—
123C32×D4—28no—
124C32×D4—24no—
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—24no—
128C32×D4—44no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—44no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C23×C311yescharacteristic
135#638—42no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3C3×D411yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—42no—
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143C3×#67—128no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C3×D12—64no—
147C3×D12—64no—
148C3×D12—128no—
149C3×C4×D3—64no—
150C3×C4×D3—64no—
151C32×C8—21no—
152C32×C8—24no—
153C32×C8—44no—
154C3⋊(C3⋊C8)C3×D412yescomplemented
155C3×C3⋊C8—38no—
156#469—91noSylow
157C2×C33—81no—
158C2×C33#12111yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—41no—
162C3×C3⋊D3—81no—
163C2×C3×D4—24no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C2×C3×D4—181no—
167C2×C3×D4C2×C3×D314yes—
168C2×C3×D4C2×C3⋊D311yescharacteristic
169C22×C3×C4—91no—
170C2×#67—64no—
171C2×D12—34no—
172C2×D12—64no—
173C2×C4×D3—34no—
174C3×D8—182no—
175C3×M16—182no—
176C2×C3×C8—24no—
177C2×C3×C8—91no—
178C2×C3×C8C2×C3×D314yes—
179C2×C3×C8C2×C3⋊D311yescharacteristic
180C3×C4⋊C4—24no—
181C3×C4⋊C4—91no—
182C3×C4⋊C4C2×C3×D314yes—
183C3×C4⋊C4C2×C3⋊D311yescharacteristic
184C3×C22⋊C4—181no—
185C3×C22⋊C4—181no—
186#217—64no—
187#213—68no—
188(C2×D3)⋊C4—64no—
189C4⋊Dic3—34no—
190C2×C3⋊C8—34no—
191ΓL(1,𝔽25)—68no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—44no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—84no—
201C22×C32—84no—
202C22×C32D3×D414yes—
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—21no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—41no—
219C32×C4—44no—
220C32×C4—84no—
221C32×C4#23514yes—
222C32×C4D3×D414yes—
223C32×C4C2×C3×D411yescharacteristic
224C32×C4C3×#4111yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—41no—
227C3×Dic3—64no—
228C3×Dic3—124no—
229C2×D8—91no—
230C2×M16—91no—
231#121—91no—
232#121—91no—
233#106—91no—
234#102—91no—
235#102C3×C3⋊D311yescharacteristic, complemented
236C33#46911yescharacteristic, complemented, Sylow
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—44no—
240C23×C3—91no—
241C23×C3—181no—
242C22×D3—34no—
243C22×D3—64no—
244C3×D4—28no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—44no—
250C3×D4—181no—
251C3×D4—181no—
252C3×D4—362no—
253C3×D4—362no—
254C2×C3×C4—24no—
255C2×C3×C4—21no—
256C2×C3×C4—24no—
257C2×C3×C4—44no—
258C2×C3×C4—91no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C22×C3×D314yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264#67—128no—
265C2×Dic3—34no—
266C2×Dic3—64no—
267D12—64no—
268D12—64no—
269D12—128no—
270C4×D3—64no—
271C4×D3—64no—
272C3×C8—24no—
273C3×C8—21no—
274C3×C8—44no—
275C3×C8—92no—
276C3⋊C8—38no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—88no—
281C2×C32—84no—
282C2×C32—81no—
283C2×C32#88414yes—
284C2×C32#91114yes—
285C2×C32#91414yes—
286C2×C32C3×#12111yescharacteristic
287C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
288C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
289C3⋊D3—41no—
290C3⋊D3—81no—
291C3×D3—124no—
292C3×D3—244no—
293C2×D4—91no—
294C2×D4—181no—
295C2×D4—181no—
296C2×D4C2×C3×C3⋊D311yescharacteristic
297C22×C4—91no—
298D8—182no—
299M16—182no—
300C2×C8—91no—
301C2×C8C2×C3×C3⋊D311yescharacteristic
302C4⋊C4—91no—
303C4⋊C4C2×C3×C3⋊D311yescharacteristic
304C22⋊C4—181no—
305C22⋊C4—181no—
306C22×C3—24no—
307C22×C3—48no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—44no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—84no—
315C22×C3—84no—
316C22×C3—181no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3C3×D3×D414yes—
323C22×C3D4×C3⋊D311yescharacteristic
324C2×D3—64no—
325C2×D3—64no—
326C2×D3—64no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—44no—
333C3×C4—41no—
334C3×C4—84no—
335C3×C4—181no—
336C3×C4—361no—
337C3×C4C3×#23514yes—
338C3×C4C3×D3×D414yes—
339C3×C4#382811yescharacteristic
340C3×C4D4×C3⋊D311yescharacteristic
341Dic3—64no—
342Dic3—124no—
343C32—24no—
344C32#478214yescomplemented
345C32C3×#46911yescharacteristic, complemented
346C23—21no—
347C23—91no—
348C23—181no—
349D4—21no—
350D4—21no—
351D4—181no—
352D4—181no—
353D4—362no—
354D4—362no—
355C2×C4—21no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4C22×C3×C3⋊D311yescharacteristic
361C8—21no—
362C8—92no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—88no—
367C2×C3—84no—
368C2×C3—81no—
369C2×C3—361no—
370C2×C3—721no—
371C2×C3C3×#88414yes—
372C2×C3C3×#91114yes—
373C2×C3C3×#91414yes—
374C2×C3#6409311yescharacteristic
375C2×C3#6412011yescharacteristic
376C2×C3#6412311yescharacteristic
377D3—124no—
378D3—244no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—181no—
383C22—181no—
384C22—181no—
385C22—361no—
386C22—361no—
387C22—361no—
388C22C3×D4×C3⋊D311yescharacteristic
389C4—41no—
390C4—181no—
391C4—361no—
392C4C3×#382811yescharacteristic
393C4C3×D4×C3⋊D311yescharacteristic
394C3—24no—
395C3C3×#478214yescomplemented
396C3#1059248111yescharacteristic, direct product factor
397C2—81no—
398C2—361no—
399C2—721no—
400C2C3×#6409311yescharacteristic
401C2C3×#6412011yescharacteristic
402C2C3×#6412311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. C3×#64147C2semidirect
2. C2×C3×#3753C2semidirect
3. C2×C3×#3743C2semidirect
4. C3×#64104C2semidirect
5. C33×#102C2semidirect
6. C3×#63648C2semidirect
7. C3×#63635C2semidirect
8. #10592481C3direct (N × H)
23. #64147C2×C3semidirect
24. C2×#3753C2×C3semidirect
25. C2×#3743C2×C3semidirect
26. #64104C2×C3semidirect
32. C32×#102D3semidirect
33. C32×#102C2×C3semidirect
34. #63648C2×C3semidirect
35. #63635C2×C3semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
112. C3×#102C3×D3semidirect
113. C3×#102C3⋊D3semidirect
154. C3⋊(C3⋊C8)C3×D4semidirect
235. #102C3×C3⋊D3semidirect
236. C33#469semidirect
344. C32#4782semidirect
345. C32C3×#469semidirect
395. C3C3×#4782semidirect
396. C3#10592481direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641472
C2×C3×#37532
C2×C3×#37432
C3×#641042
C33×#1022
C3×#636482
C3×#636352
#105924813
C3×#47823