Finite Group Databasebeta

← C3×#10592503 #13162841

Subgroups — C3×#10592503 #13162841

Order 1728.

1116 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×C3×#3832C211yesmaximal, characteristic
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#64101C211yesmaximal, characteristic, complemented
4C33×#103C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592503C311yesmaximal, characteristic, direct product factor
9C3×#4804—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C33×C8C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752—21no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4×C3⋊Dic3—21no—
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×C8×C3⋊D3—22no—
24C2×#3832C2×C311yescharacteristic
25C2×C3×#239—34no—
26C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
27#64101C2×C311yescharacteristic, complemented
28C2×C3×D3×C8—34no—
29C3×#892—34no—
30C32×#103—24no—
31C32×#103D314yescomplemented
32C32×#103C2×C311yescharacteristic, complemented
33#63662C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#835—34no—
37C3×#821—34no—
38C3×#809—34no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8C2×C412yes—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—24no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)C2×C412yes—
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3C2×C412yes—
53C3×C4×C3⋊D3D412yes—
54C33×C8—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×#416—91no—
57#4804—34no—
58C2×C32×Q8—24no—
59C2×C32×Q8C2×D314yes—
60C2×C32×Q8C22×C311yescharacteristic
61#3832—22no—
62#3832C3×C414yescomplemented
63C2×C4⋊(C3⋊D3)—21no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C4×C3⋊D3—21no—
66C2×C4×C3⋊D3C22×C311yescharacteristic
67C3×#239—316no—
68C3×#239—68no—
69C2×C3×D12—34no—
70C2×C3×D12—64no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—64no—
73C2×C32×C8—24no—
74C2×C32×C8C2×D314yes—
75C2×C32×C8C22×C311yescharacteristic
76C32×C4⋊C4—24no—
77C32×C4⋊C4C2×D314yes—
78C32×C4⋊C4C22×C311yescharacteristic
79#3752—21no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4×C3⋊Dic3—21no—
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C8×C3⋊D3—22no—
84C3×(C2×D3)⋊C4—64no—
85C3×C4⋊Dic3—34no—
86C3×C4×Dic3—64no—
87C2×C3×C3⋊C8—34no—
88C3×D3×C8—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3C22⋊C412yes—
96C33×C4—22no—
97C33×C4—41no—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×D411yescharacteristic
100C3×C3⋊Dic3—41no—
101C3×C3⋊Dic3C22⋊C412yes—
102C2×C3×#41—91no—
103C2×#239—34no—
104C22×C3×C8—91no—
105C3×#117—91no—
106C2×D3×C8—34no—
107#892—34no—
108C3×#103—24no—
109C3×#103—91no—
110C3×#103C3×D314yescomplemented
111C3×#103C3⋊D311yescharacteristic, complemented
112C3×#102—91no—
113C3×#102—91no—
114#835—34no—
115#821—34no—
116#809—34no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3C3×D411yescharacteristic
119C22×C3×D3—34no—
120C22×C3×D3—64no—
121C32×Q8—21no—
122C32×Q8—28no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8C4×D318yes—
126C32×Q8C2×C3×C412yes—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—44no—
133C2×C32×C4—44no—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C23×C311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3C3×D411yescharacteristic
138C4⋊(C3⋊D3)—24no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C2×C3×C412yes—
141C4×C3⋊D3—24no—
142C4×C3⋊D3C2×C3×C412yes—
143C4×C3⋊D3C3×D412yes—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—64no—
146C3×D12—38no—
147C3×D12—616no—
148C3×D12—64no—
149C3×C4×D3—38no—
150C3×C4×D3—38no—
151C3×C4×D3—616no—
152C32×C8—21no—
153C32×C8—24no—
154C32×C8—44no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—64no—
157#416—91noSylow
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#13511yescharacteristic
160C2×C33#13511yescharacteristic
161C3×C3⋊D3—22no—
162C3×C3⋊D3—42no—
163C3×#41—94no—
164C3×#41—182no—
165C2×C3×Q8—24no—
166C2×C3×Q8C2×C3×D314yes—
167C2×C3×Q8C2×C3⋊D311yescharacteristic
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—91no—
171C22×C3×C4—181no—
172#239—316no—
173#239—68no—
174C2×D12—34no—
175C2×D12—64no—
176C2×C4×D3—34no—
177C2×C4×D3—64no—
178C2×C3×C8—24no—
179C2×C3×C8—91no—
180C2×C3×C8—182no—
181C2×C3×C8C2×C3×D314yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C4⋊C4—24no—
184C3×C4⋊C4—91no—
185C3×C4⋊C4C2×C3×D314yes—
186C3×C4⋊C4C2×C3⋊D311yescharacteristic
187C3×C22⋊C4—181no—
188C3×C42—181no—
189(C2×D3)⋊C4—64no—
190C4⋊Dic3—34no—
191C4×Dic3—64no—
192C2×C3⋊C8—34no—
193D3×C8—68no—
194C22×C32—24no—
195C22×C32D3×D414yes—
196C22×C32C2×C3×D411yescharacteristic
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3C3×C22⋊C412yes—
203C2×C3×D3—38no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—68no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C32×C4—22no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—28no—
213C32×C4—48no—
214C32×C4—41no—
215C32×C4—44no—
216C32×C4—84no—
217C32×C4C2×C4×D314yes—
218C32×C4D3×D414yes—
219C32×C4C22×C3×C411yescharacteristic
220C32×C4C2×C3×D411yescharacteristic
221C3⋊Dic3—41no—
222C3⋊Dic3C3×C22⋊C412yes—
223C3×Dic3—38no—
224C3×Dic3—124no—
225C2×#41—91no—
226C22×C8—91no—
227#117—91no—
228#103—91no—
229#103C3×C3⋊D311yescharacteristic, complemented
230#102—91no—
231#102—91no—
232C33#41611yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C23×C3—181no—
235C22×D3—34no—
236C22×D3—64no—
237C3×Q8—24no—
238C3×Q8—21no—
239C3×Q8—28no—
240C3×Q8—44no—
241C3×Q8C3×C4×D318yes—
242C3×Q8C4×C3⋊D312yes—
243C3×D4—92no—
244C3×D4—184no—
245C3×D4—181no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4—91no—
254C2×C3×C4—92no—
255C2×C3×C4—92no—
256C2×C3×C4—184no—
257C2×C3×C4—181no—
258C2×C3×C4C22×C3×D314yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260C2×Dic3—34no—
261C2×Dic3—64no—
262D12—38no—
263D12—616no—
264D12—64no—
265C4×D3—38no—
266C4×D3—38no—
267C4×D3—616no—
268C3×C8—24no—
269C3×C8—21no—
270C3×C8—44no—
271C3×C8—181no—
272C3⋊C8—64no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32D3×C22⋊C414yes—
277C2×C32#91614yes—
278C2×C32#91914yes—
279C2×C32C2×C3×C22⋊C411yescharacteristic
280C2×C32C3×#13511yescharacteristic
281C2×C32C3×#13511yescharacteristic
282C3⋊D3—22no—
283C3⋊D3—42no—
284C3×D3—68no—
285C3×D3—128no—
286#41—94no—
287#41—182no—
288C2×Q8C2×C3×C3⋊D311yescharacteristic
289C2×D4—91no—
290C2×D4—181no—
291C22×C4—91no—
292C22×C4—181no—
293C2×C8—91no—
294C2×C8—182no—
295C2×C8C2×C3×C3⋊D311yescharacteristic
296C4⋊C4—91no—
297C4⋊C4C2×C3×C3⋊D311yescharacteristic
298C22⋊C4—181no—
299C42—181no—
300C22×C3—24no—
301C22×C3—92no—
302C22×C3—181no—
303C22×C3—181no—
304C22×C3—182no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3C3×D3×D414yes—
308C22×C3D4×C3⋊D311yescharacteristic
309C2×D3—38no—
310C2×D3—64no—
311C2×D3—64no—
312C2×D3—68no—
313C2×D3—124no—
314C2×D3—124no—
315C3×C4—28no—
316C3×C4—22no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—48no—
320C3×C4—44no—
321C3×C4—41no—
322C3×C4—84no—
323C3×C4—92no—
324C3×C4—361no—
325C3×C4C2×C3×C4×D314yes—
326C3×C4C3×D3×D414yes—
327C3×C4C2×C4×C3⋊D311yescharacteristic
328C3×C4D4×C3⋊D311yescharacteristic
329Dic3—38no—
330Dic3—124no—
331C32—24no—
332C32#480414yescomplemented
333C32C3×#41611yescharacteristic, complemented
334C23—91no—
335C23—181no—
336Q8—21no—
337Q8C3×C4×C3⋊D312yes—
338D4—92no—
339D4—184no—
340D4—181no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—91no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—184no—
347C2×C4—181no—
348C2×C4C22×C3×C3⋊D311yescharacteristic
349C8—21no—
350C8—181no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—182no—
355C2×C3—362no—
356C2×C3C3×D3×C22⋊C414yes—
357C2×C3C3×#91614yes—
358C2×C3C3×#91914yes—
359C2×C3C22⋊C4×C3⋊D311yescharacteristic
360C2×C3#6412511yescharacteristic
361C2×C3#6412811yescharacteristic
362D3—68no—
363D3—128no—
364C22—92no—
365C22—181no—
366C22—181no—
367C22—182no—
368C22—361no—
369C22—361no—
370C22C3×D4×C3⋊D311yescharacteristic
371C4—22no—
372C4—41no—
373C4—92no—
374C4—361no—
375C4C2×C3×C4×C3⋊D311yescharacteristic
376C4C3×D4×C3⋊D311yescharacteristic
377C3—24no—
378C3C3×#480414yescomplemented
379C3#1059250311yescharacteristic, direct product factor
380C2—182no—
381C2—362no—
382C2C3×C22⋊C4×C3⋊D311yescharacteristic
383C2C3×#6412511yescharacteristic
384C2C3×#6412811yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×C3⋊D3C2semidirect
3. C3×#64101C2semidirect
4. C33×#103C2semidirect
5. C3×#63662C2semidirect
6. C3×#63648C2semidirect
7. C3×#63636C2semidirect
8. #10592503C3direct (N × H)
12. C3×#3832C4semidirect
26. C2×C8×C3⋊D3C2×C3semidirect
27. #64101C2×C3semidirect
31. C32×#103D3semidirect
32. C32×#103C2×C3semidirect
33. #63662C2×C3semidirect
34. #63648C2×C3semidirect
35. #63636C2×C3semidirect
62. #3832C3×C4semidirect
110. C3×#103C3×D3semidirect
111. C3×#103C3⋊D3semidirect
229. #103C3×C3⋊D3semidirect
232. C33#416semidirect
332. C32#4804semidirect
333. C32C3×#416semidirect
378. C3C3×#4804semidirect
379. C3#10592503direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×C8×C3⋊D32
C3×#641012
C33×#1032
C3×#636622
C3×#636482
C3×#636362
#105925033
C3×#48043