Finite Group Databasebeta

← C3×#10592864 #13163204

Subgroups — C3×#10592864 #13163204

Order 1728.

1016 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×SD16C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63641C211yesmaximal, characteristic, complemented
8#10592864C311yesmaximal, characteristic, direct product factor
9C3×#5165—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C33×SD16—22no—
15C2×C33×C8C2211yescharacteristic
16C3×#3757—21no—
17C3×#3753—22no—
18C3×#3752—21no—
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C2×C32×SD16—24no—
24C2×C32×SD16D314yescomplemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×#3753C2×C311yescharacteristic
29C3×#947—34no—
30C3×D4×Dic3—34no—
31C2×C3×#213—34no—
32#63662C2×C311yescharacteristic, complemented
33#63648C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#835—34no—
36C3×#821—34no—
37C3×Dic3⋊C8—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C33×D4—21no—
43C33×D4D412yes—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—42no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C33×C8—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#463—91no—
55#5165—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C22×C3⋊Dic3—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C22×C3×Dic3—64no—
65C2×C3×D12—34no—
66C32×SD16—216no—
67C32×SD16—28no—
68C32×SD16—22no—
69C2×C32×C8—24no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C22×C311yescharacteristic
72#3757—21no—
73#3753—22no—
74#3752—21no—
75#3752—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C4×C3⋊Dic3C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#217—64no—
80C3×#213—68no—
81C3×(C2×D3)⋊C4—64no—
82C3×(C2×D3)⋊C4—64no—
83C3×C4⋊Dic3—34no—
84C3×C4×Dic3—34no—
85C2×C3×C3⋊C8—34no—
86C22×C33—22no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C3×C3⋊Dic3—41no—
99C2×C3×SD16—24no—
100C2×C3×SD16C3×D314yescomplemented
101C2×C3×SD16C3⋊D311yescharacteristic, complemented
102C2×C3×D8—91no—
103C3×#124—91no—
104C3×C4×D4—91no—
105#947—34no—
106D4×Dic3—34no—
107C2×#213—34no—
108C3×C4⋊C8—91no—
109C3×#103—91no—
110C3×#102—91no—
111#835—34no—
112#821—34no—
113Dic3⋊C8—34no—
114C23×C32—21no—
115C23×C32—24no—
116C23×C32—44no—
117C22×C3⋊D3—21no—
118C22×C3×D3—64no—
119C32×Q8—28no—
120C32×Q8—24no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×D4—21no—
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4#6718yes—
130C32×D4C3×D412yes—
131C2×C32×C4—21no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—44no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—42no—
139C2×C3⋊Dic3C3×D411yescharacteristic
140C2×C3⋊Dic3C3×D411yescharacteristic
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—128no—
147C3×D12—64no—
148C3×D12—64no—
149C32×C8—28no—
150C32×C8—24no—
151C32×C8—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—64no—
154#463—91noSylow
155C2×C33—42no—
156C2×C33#12111yescharacteristic
157C2×C33#13511yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C3×C3⋊D3—81no—
160C2×C3×Q8—24no—
161C2×C3×Q8C2×C3×D314yes—
162C2×C3×Q8C2×C3⋊D311yescharacteristic
163C2×C3×D4—24no—
164C2×C3×D4—91no—
165C2×C3×D4C2×C3×D314yes—
166C2×C3×D4C2×C3⋊D311yescharacteristic
167C22×C3×C4—181no—
168C22×Dic3—64no—
169C2×D12—34no—
170C3×SD16—216no—
171C3×SD16—28no—
172C3×SD16—22no—
173C3×D8—182no—
174C2×C3×C8—24no—
175C2×C3×C8—91no—
176C2×C3×C8C2×C3×D314yes—
177C2×C3×C8C2×C3⋊D311yescharacteristic
178C3×C4⋊C4—91no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C22⋊C4—181no—
182C3×C42—91no—
183#217—64no—
184#213—68no—
185(C2×D3)⋊C4—64no—
186(C2×D3)⋊C4—64no—
187C4⋊Dic3—34no—
188C4×Dic3—34no—
189C2×C3⋊C8—34no—
190C22×C32—22no—
191C22×C32—24no—
192C22×C32—28no—
193C22×C32—48no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—44no—
197C22×C32—44no—
198C22×C32—84no—
199C22×C32—84no—
200C22×C32D3×D414yes—
201C22×C32C2×C3×D411yescharacteristic
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4#23714yes—
214C32×C4C2×#6714yes—
215C32×C4C2×C3×D411yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—21no—
219C3⋊Dic3—41no—
220C3×Dic3—64no—
221C3×Dic3—64no—
222C3×Dic3—124no—
223C2×SD16C3×C3⋊D311yescharacteristic, complemented
224C2×D8—91no—
225#124—91no—
226C4×D4—91no—
227C4⋊C8—91no—
228#103—91no—
229#102—91no—
230C33#46311yescharacteristic, complemented, Sylow
231C23×C3—24no—
232C23×C3—21no—
233C23×C3—44no—
234C23×C3—181no—
235C22×D3—64no—
236C3×Q8—28no—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—44no—
242C3×D4—24no—
243C3×D4—21no—
244C3×D4—28no—
245C3×D4—44no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4C3×#6718yes—
249C3×D4#63812yes—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—44no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—181no—
257C2×C3×C4—362no—
258C2×C3×C4C22×C3×D314yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260C2×Dic3—34no—
261C2×Dic3—34no—
262C2×Dic3—64no—
263C2×Dic3—128no—
264D12—64no—
265D12—64no—
266C3×C8—28no—
267C3×C8—24no—
268C3×C8—21no—
269C3×C8—181no—
270C3⋊C8—64no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—42no—
275C2×C32—48no—
276C2×C32—88no—
277C2×C32#91414yes—
278C2×C32#91614yes—
279C2×C32#93914yes—
280C2×C32C3×#12111yescharacteristic
281C2×C32C3×#13511yescharacteristic
282C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
283C3⋊D3—81no—
284C3×D3—244no—
285C2×Q8C2×C3×C3⋊D311yescharacteristic
286C2×D4—91no—
287C2×D4C2×C3×C3⋊D311yescharacteristic
288C22×C4—181no—
289SD16—22no—
290D8—182no—
291C2×C8—91no—
292C2×C8C2×C3×C3⋊D311yescharacteristic
293C4⋊C4—91no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22⋊C4—181no—
297C42—91no—
298C22×C3—28no—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—48no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—84no—
307C22×C3—84no—
308C22×C3—361no—
309C22×C3—361no—
310C22×C3—361no—
311C22×C3C3×D3×D414yes—
312C22×C3D4×C3⋊D311yescharacteristic
313C2×D3—124no—
314C2×D3—124no—
315C2×D3—124no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—48no—
319C3×C4—44no—
320C3×C4—41no—
321C3×C4—181no—
322C3×C4—181no—
323C3×C4—361no—
324C3×C4C3×#23714yes—
325C3×C4C2×C3×#6714yes—
326C3×C4#383011yescharacteristic
327C3×C4C2×#63811yescharacteristic
328Dic3—64no—
329Dic3—64no—
330Dic3—124no—
331C32—24no—
332C32#516514yescomplemented
333C32C3×#46311yescharacteristic, complemented
334C23—21no—
335C23—181no—
336Q8—21no—
337Q8—21no—
338D4—21no—
339D4—181no—
340D4—181no—
341D4C3×#63812yes—
342C2×C4—21no—
343C2×C4—91no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—362no—
347C2×C4C22×C3×C3⋊D311yescharacteristic
348C8—21no—
349C8—181no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—48no—
354C2×C3—42no—
355C2×C3—88no—
356C2×C3—721no—
357C2×C3C3×#91414yes—
358C2×C3C3×#91614yes—
359C2×C3C3×#93914yes—
360C2×C3#6412311yescharacteristic
361C2×C3#6412511yescharacteristic
362C2×C3#6414811yescharacteristic
363D3—244no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—361no—
368C22—361no—
369C22—361no—
370C22C3×D4×C3⋊D311yescharacteristic
371C4—41no—
372C4—181no—
373C4—181no—
374C4—361no—
375C4C3×#383011yescharacteristic
376C4C2×C3×#63811yescharacteristic
377C3—24no—
378C3C3×#516514yescomplemented
379C3#1059286411yescharacteristic, direct product factor
380C2—42no—
381C2—721no—
382C2C3×#6412311yescharacteristic
383C2C3×#6412511yescharacteristic
384C2C3×#6414811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C33×SD16C2semidirect
2. C3×#64156C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
5. C3×#63662C2semidirect
6. C3×#63648C2semidirect
7. C3×#63641C2semidirect
8. #10592864C3direct (N × H)
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. D4×C3⋊Dic3C2×C3semidirect
32. #63662C2×C3semidirect
33. #63648C2×C3semidirect
34. #63641C2×C3semidirect
100. C2×C3×SD16C3×D3semidirect
101. C2×C3×SD16C3⋊D3semidirect
223. C2×SD16C3×C3⋊D3semidirect
230. C33#463semidirect
332. C32#5165semidirect
333. C32C3×#463semidirect
378. C3C3×#5165semidirect
379. C3#10592864direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×SD162
C3×#641562
C3×D4×C3⋊Dic32
C2×C3×#37532
C3×#636622
C3×#636482
C3×#636412
#105928643
C3×#51653