Finite Group Databasebeta

← C3×#10592831 #13163169

Subgroups — C3×#10592831 #13163169

Order 1728.

1172 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C33×M16C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4C3×#63651C212yesmaximal, complemented
5C3×#63650C212yesmaximal, complemented
6#10592831C311yesmaximal, characteristic, direct product factor
7C3×#5132—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×C3×#638C412yes—
10C3×#3828—21no—
11C3×#3828—21no—
12C3×#3828—21no—
13C3×#3828—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C412yes—
16C2×C3×#634C2211yescharacteristic
17C33×M16—21no—
18C33×M16C2212yes—
19C2×C33×C8—21no—
20C3×#3748—21no—
21C3×#3748C2212yes—
22C2×C3×C3⋊(C3⋊C8)—21no—
23C2×#3828C2×C311yescharacteristic
24C2×C3×#235—34no—
25C2×C32×M16—24no—
26C2×C32×M16D314yescomplemented
27C2×C32×M16C2×C311yescharacteristic, complemented
28C2×#3748C2×C311yescharacteristic, complemented
29C2×C3×#208—34no—
30#63651C2×C312yescomplemented
31#63650C2×C312yescomplemented
32C3×#824—38no—
33C3×#823—38no—
34C23×C33C2×C411yescharacteristic
35C22×C3×C3⋊D3C2×C412yes—
36C2×C33×C4C2×C411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C3×#638—22no—
43C3×#638—22no—
44C2×C3×C3⋊Dic3C2×C412yes—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4×C3⋊D3—22no—
48C3×C4×C3⋊D3—22no—
49C3×#634—21no—
50C3×#634—21no—
51C33×C8—22no—
52C3×C3⋊(C3⋊C8)—22no—
53C3×#413—91no—
54#5132—34no—
55C22×C32×C4—24no—
56C22×C32×C4C2×D314yes—
57C22×C32×C4C22×C311yescharacteristic
58C2×#638C3×C412yes—
59#3828—21no—
60#3828—21no—
61#3828—21no—
62#3828—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C4×C3⋊D3C3×C412yes—
65C2×#634C22×C311yescharacteristic
66C2×C3×#67—38no—
67C3×#235—64no—
68C3×#235—64no—
69C3×#235—64no—
70C3×#235—64no—
71C2×C3×D12—34no—
72C2×C3×C4×D3—38no—
73C2×C3×Dic6—34no—
74C32×M16—28no—
75C32×M16—28no—
76C32×M16—24no—
77C32×M16—21no—
78C32×M16C2×D318yes—
79C32×M16C22×C312yes—
80C2×C32×C8—28no—
81C2×C32×C8—24no—
82C2×C32×C8—21no—
83#3748—21no—
84#3748C22×C312yes—
85C2×C3⋊(C3⋊C8)—21no—
86C3×#208—38no—
87C3×#208—64no—
88C2×C3×C3⋊C8—64no—
89C22×C33—41no—
90C22×C33C22⋊C411yescharacteristic
91C22×C33C22⋊C411yescharacteristic
92C22×C33C22×C411yescharacteristic
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—42no—
95C33×C4C22⋊C411yescharacteristic
96C33×C4C22⋊C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3×C3⋊Dic3—22no—
100C2×C3×#41—91no—
101C2×#235—34no—
102C2×C3×M16—24no—
103C2×C3×M16—91no—
104C2×C3×M16C3×D314yescomplemented
105C2×C3×M16C3⋊D311yescharacteristic, complemented
106C2×#208—34no—
107C3×#101—92no—
108C3×#100—92no—
109#824—38no—
110#823—38no—
111C23×C32—24no—
112C23×C32C4×D314yes—
113C23×C32C2×C3×C411yescharacteristic
114C22×C3⋊D3C2×C3×C412yes—
115C22×C3×D3—38no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4C4×D314yes—
123C2×C32×C4D1214yes—
124C2×C32×C4D1214yes—
125C2×C32×C4#6714yes—
126C2×C32×C4#6714yes—
127C2×C32×C4C2×C3×C411yescharacteristic
128C2×C32×C4C3×D411yescharacteristic
129C2×C32×C4C3×D411yescharacteristic
130C2×C32×C4C3×D411yescharacteristic
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C22×D314yes—
133C2×C32×C4C23×C311yescharacteristic
134#638—22no—
135#638—22no—
136C2×C3⋊Dic3C2×C3×C412yes—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4×C3⋊D3—22no—
140C4×C3⋊D3—22no—
141#634—21no—
142#634—21no—
143C3×#67—68no—
144C3×#67—68no—
145C2×C3×Dic3—38no—
146C3×D12—64no—
147C3×D12—64no—
148C3×C4×D3—68no—
149C3×C4×D3—68no—
150C3×Dic6—64no—
151C3×Dic6—64no—
152C32×C8—216no—
153C32×C8—28no—
154C32×C8—22no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—68no—
157#413—91noSylow
158C2×C33—21no—
159C2×C33—21no—
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—42no—
163C3×#41—181no—
164C3×#41—181no—
165C3×#41—181no—
166C3×#41—181no—
167C2×C3×Q8—91no—
168C2×C3×D4—92no—
169C2×C3×D4—91no—
170C22×C3×C4—24no—
171C22×C3×C4—92no—
172C22×C3×C4C2×C3×D314yes—
173C22×C3×C4C2×C3⋊D311yescharacteristic
174C2×#67—38no—
175#235—64no—
176#235—64no—
177#235—64no—
178#235—64no—
179C2×D12—34no—
180C2×C4×D3—38no—
181C2×Dic6—34no—
182C3×M16—28no—
183C3×M16—24no—
184C3×M16—21no—
185C3×M16—28no—
186C3×M16—92no—
187C3×M16—181no—
188C3×M16C2×C3×D318yes—
189C3×M16C2×C3⋊D312yes—
190C2×C3×C8—28no—
191C2×C3×C8—24no—
192C2×C3×C8—21no—
193C2×C3×C8—181no—
194#208—38no—
195#208—64no—
196C2×C3⋊C8—64no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32—41no—
202C22×C32—48no—
203C22×C32(C2×D3)⋊C414yes—
204C22×C32(C2×D3)⋊C414yes—
205C22×C32C3×C22⋊C411yescharacteristic
206C22×C32C3×C22⋊C411yescharacteristic
207C22×C32C2×C4×D314yes—
208C22×C32C22×C3×C411yescharacteristic
209C2×C3⋊D3—22no—
210C2×C3⋊D3—42no—
211C2×C3×D3—68no—
212C2×C3×D3—128no—
213C32×C4—24no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4(C2×D3)⋊C414yes—
218C32×C4(C2×D3)⋊C414yes—
219C32×C4C3×C22⋊C411yescharacteristic
220C32×C4C3×C22⋊C411yescharacteristic
221C32×C4C2×D1214yes—
222C32×C4C2×#6714yes—
223C32×C4C2×C3×D411yescharacteristic
224C32×C4C2×C3×D411yescharacteristic
225C3⋊Dic3—22no—
226C3×Dic3—68no—
227C2×#41—91no—
228C2×M16—91no—
229C2×M16C3×C3⋊D311yescharacteristic, complemented
230#101—92no—
231#100—92no—
232C33#41311yescharacteristic, complemented, Sylow
233C23×C3—24no—
234C23×C3—92no—
235C23×C3C3×C4×D314yes—
236C23×C3C4×C3⋊D311yescharacteristic
237C22×D3—38no—
238C3×Q8—181no—
239C3×Q8—181no—
240C3×D4—182no—
241C3×D4—182no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—92no—
251C2×C3×C4—182no—
252C2×C3×C4—182no—
253C2×C3×C4C3×C4×D314yes—
254C2×C3×C4C3×D1214yes—
255C2×C3×C4C3×D1214yes—
256C2×C3×C4C3×#6714yes—
257C2×C3×C4C3×#6714yes—
258C2×C3×C4C4×C3⋊D311yescharacteristic
259C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
260C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
261C2×C3×C4#63811yescharacteristic
262C2×C3×C4#63811yescharacteristic
263C2×C3×C4C22×C3×D314yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265#67—68no—
266#67—68no—
267C2×Dic3—38no—
268D12—64no—
269D12—64no—
270C4×D3—68no—
271C4×D3—68no—
272Dic6—64no—
273Dic6—64no—
274C3×C8—216no—
275C3×C8—28no—
276C3×C8—22no—
277C3×C8—182no—
278C3⋊C8—68no—
279C2×C32—28no—
280C2×C32—24no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—44no—
288C2×C32—44no—
289C2×C32C2×(C2×D3)⋊C414yes—
290C2×C32C2×C3×C22⋊C411yescharacteristic
291C3⋊D3—42no—
292C3×D3—128no—
293#41—181no—
294#41—181no—
295#41—181no—
296#41—181no—
297C2×Q8—91no—
298C2×D4—92no—
299C2×D4—91no—
300C22×C4—92no—
301C22×C4C2×C3×C3⋊D311yescharacteristic
302M16—21no—
303M16—92no—
304M16—181no—
305M16C2×C3×C3⋊D312yes—
306C2×C8—21no—
307C2×C8—181no—
308C22×C3—24no—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—44no—
312C22×C3—41no—
313C22×C3—48no—
314C22×C3—182no—
315C22×C3—362no—
316C22×C3C3×(C2×D3)⋊C414yes—
317C22×C3C3×(C2×D3)⋊C414yes—
318C22×C3#375211yescharacteristic
319C22×C3#375211yescharacteristic
320C22×C3C2×C3×C4×D314yes—
321C22×C3C2×C4×C3⋊D311yescharacteristic
322C2×D3—68no—
323C2×D3—128no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—182no—
329C3×C4C3×(C2×D3)⋊C414yes—
330C3×C4C3×(C2×D3)⋊C414yes—
331C3×C4#375211yescharacteristic
332C3×C4#375211yescharacteristic
333C3×C4C2×C3×D1214yes—
334C3×C4C2×C3×#6714yes—
335C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
336C3×C4C2×#63811yescharacteristic
337Dic3—68no—
338C32—24no—
339C32#513214yescomplemented
340C32C3×#41311yescharacteristic, complemented
341C23—92no—
342C23C3×C4×C3⋊D311yescharacteristic
343Q8—181no—
344Q8—181no—
345D4—182no—
346D4—182no—
347D4—181no—
348D4—181no—
349C2×C4—92no—
350C2×C4—182no—
351C2×C4—182no—
352C2×C4C3×C4×C3⋊D311yescharacteristic
353C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
354C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
355C2×C4C3×#63811yescharacteristic
356C2×C4C3×#63811yescharacteristic
357C2×C4C22×C3×C3⋊D311yescharacteristic
358C8—22no—
359C8—182no—
360C2×C3—28no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3—44no—
369C2×C3—44no—
370C2×C3—362no—
371C2×C3C2×C3×(C2×D3)⋊C414yes—
372C2×C3C2×#375211yescharacteristic
373D3—128no—
374C22—41no—
375C22—182no—
376C22—362no—
377C22C3×#375211yescharacteristic
378C22C3×#375211yescharacteristic
379C22C2×C3×C4×C3⋊D311yescharacteristic
380C4—182no—
381C4C3×#375211yescharacteristic
382C4C3×#375211yescharacteristic
383C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
384C4C2×C3×#63811yescharacteristic
385C3—24no—
386C3C3×#513214yescomplemented
387C3#1059283111yescharacteristic, direct product factor
388C2—21no—
389C2—21no—
390C2—21no—
391C2—362no—
392C2C2×C3×#375211yescharacteristic
subgroupG/H
centerC3×C4#3752
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×M16C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×M16C3⋊D3
hypocenterC32C3×#413
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
2. C2×C33×M16C2semidirect
3. C2×C3×#3748C2semidirect
4. C3×#63651C2semidirect
5. C3×#63650C2semidirect
6. #10592831C3direct (N × H)
26. C2×C32×M16D3semidirect
27. C2×C32×M16C2×C3semidirect
28. C2×#3748C2×C3semidirect
30. #63651C2×C3semidirect
31. #63650C2×C3semidirect
104. C2×C3×M16C3×D3semidirect
105. C2×C3×M16C3⋊D3semidirect
229. C2×M16C3×C3⋊D3semidirect
232. C33#413semidirect
339. C32#5132semidirect
340. C32C3×#413semidirect
386. C3C3×#5132semidirect
387. C3#10592831direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C33×M162
C2×C3×#37482
C3×#636512
C3×#636502
#105928313
C3×#51323