Finite Group Databasebeta

← C3×#10592885 #13163226

Subgroups — C3×#10592885 #13163226

Order 1728.

1128 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×C33×Q16C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic
4C2×C3×#3756C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592885C311yesmaximal, characteristic, direct product factor
9C3×#5186—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×Q8C2211yescharacteristic
12C3×Q8×C3⋊D3—22no—
13C3×Q8×C3⋊D3—22no—
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634—21no—
17C2×C3×#634C2211yescharacteristic
18C33×Q16—22no—
19C2×C33×C8C2211yescharacteristic
20C3×#3756—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750—21no—
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×Q8×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×Q8—34no—
27C2×C32×Q16—24no—
28C2×C32×Q16D314yescomplemented
29C2×C32×Q16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic
31C2×#3756C2×C311yescharacteristic, complemented
32C3×#946—34no—
33C2×C3×#216—34no—
34#63662C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic
36#63643C2×C311yescharacteristic, complemented
37C3×#835—34no—
38C3×(C2×D3)⋊C8—34no—
39C3×#816—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C33×Q8—21no—
42C33×Q8—21no—
43C33×Q8—21no—
44C33×Q8D412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—24no—
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×#634—24no—
55C3×#634—21no—
56C3×#634D412yes—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#451—91no—
60#5186—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C2×D314yes—
65C2×C32×Q8C22×C311yescharacteristic
66C2×C32×Q8C22×C311yescharacteristic
67Q8×C3⋊D3—22no—
68Q8×C3⋊D3—22no—
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×#634—21no—
72C2×#634C22×C311yescharacteristic
73C3×D3×Q8—68no—
74C3×D3×Q8—68no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—64no—
77C2×C3×Dic6—34no—
78C2×C3×Dic6—64no—
79C32×Q16—216no—
80C32×Q16—28no—
81C32×Q16—22no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C22×C311yescharacteristic
85#3756—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)C22×C311yescharacteristic
88#3750—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#216—68no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—64no—
94C2×C3×C3⋊C8—34no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3Q1612yescomplemented
99C33×C4—22no—
100C33×C4—41no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3×C3⋊Dic3—21no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—41no—
106C22×C3×Q8—91no—
107C2×D3×Q8—34no—
108C2×C3×Q16—24no—
109C2×C3×Q16—91no—
110C2×C3×Q16C3×D314yescomplemented
111C2×C3×Q16C3⋊D311yescharacteristic, complemented
112C3×C22⋊Q8—91no—
113#946—34no—
114C2×#216—34no—
115C3×#103—91no—
116C3×#103—91no—
117C3×C22⋊C8—91no—
118#835—34no—
119(C2×D3)⋊C8—34no—
120#816—34no—
121C22×C3⋊D3C3×D411yescharacteristic
122C22×C3×D3—34no—
123C32×Q8—21no—
124C32×Q8—28no—
125C32×Q8—24no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—24no—
131C32×Q8—44no—
132C32×Q8—44no—
133C32×Q8#6718yes—
134C32×Q8C3×D412yes—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—44no—
141C2×C32×C4—44no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C23×C311yescharacteristic
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3C3×D411yescharacteristic
147C4×C3⋊D3—24no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—24no—
151#634—21no—
152#634C3×D412yes—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—64no—
156C3×C4×D3—616no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×Dic6—38no—
160C3×Dic6—616no—
161C3×Dic6—64no—
162C32×C8—28no—
163C32×C8—24no—
164C32×C8—21no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—64no—
167#451—91noSylow
168C2×C33#12011yescharacteristic
169C2×C33C2×Q1611yescharacteristic
170C2×C33#13711yescharacteristic
171C3×C3⋊D3—22no—
172C2×C3×Q8—24no—
173C2×C3×Q8—24no—
174C2×C3×Q8—91no—
175C2×C3×Q8—181no—
176C2×C3×Q8—182no—
177C2×C3×Q8—182no—
178C2×C3×Q8C2×C3×D314yes—
179C2×C3×Q8C2×C3×D314yes—
180C2×C3×Q8C2×C3⋊D311yescharacteristic
181C2×C3×Q8C2×C3⋊D311yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184D3×Q8—68no—
185D3×Q8—68no—
186C2×C4×D3—34no—
187C2×C4×D3—64no—
188C2×Dic6—34no—
189C2×Dic6—64no—
190C3×Q16—216no—
191C3×Q16—28no—
192C3×Q16—22no—
193C3×Q16—182no—
194C2×C3×C8—24no—
195C2×C3×C8—91no—
196C2×C3×C8C2×C3×D314yes—
197C2×C3×C8C2×C3⋊D311yescharacteristic
198C3×C4⋊C4—91no—
199C3×C4⋊C4—181no—
200C3×C22⋊C4—181no—
201#216—68no—
202(C2×D3)⋊C4—64no—
203C4⋊Dic3—34no—
204Dic3⋊C4—64no—
205C2×C3⋊C8—34no—
206C22×C32—24no—
207C22×C32D3×D414yes—
208C22×C32C2×C3×D411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3C3×Q1612yescomplemented
212C2×C3×D3—38no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C32×C4—22no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—28no—
219C32×C4—48no—
220C32×C4—48no—
221C32×C4—44no—
222C32×C4—41no—
223C32×C4D3×D414yes—
224C32×C4C2×#6714yes—
225C32×C4C2×C3×D411yescharacteristic
226C32×C4C2×C3×D411yescharacteristic
227C3⋊Dic3—21no—
228C3⋊Dic3—22no—
229C3⋊Dic3—41no—
230C3×Dic3—68no—
231C3×Dic3—64no—
232C3×Dic3—124no—
233C22×Q8—91no—
234C2×Q16—91no—
235C2×Q16C3×C3⋊D311yescharacteristic, complemented
236C22⋊Q8—91no—
237#103—91no—
238#103—91no—
239C22⋊C8—91no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—34no—
243C3×Q8—24no—
244C3×Q8—21no—
245C3×Q8—28no—
246C3×Q8—24no—
247C3×Q8—24no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×Q8—28no—
251C3×Q8—44no—
252C3×Q8—44no—
253C3×Q8—92no—
254C3×Q8—184no—
255C3×Q8—181no—
256C3×Q8C3×#6718yes—
257C3×Q8#63812yes—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—91no—
266C2×C3×C4—184no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C22×C3×D314yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273C2×Dic3—34no—
274C2×Dic3—64no—
275C2×Dic3—64no—
276C4×D3—616no—
277C4×D3—64no—
278C4×D3—64no—
279Dic6—38no—
280Dic6—616no—
281Dic6—64no—
282C3×C8—28no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—181no—
286C3⋊C8—64no—
287C2×C32—24no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32D3×Q1614yes—
291C2×C32#91814yes—
292C2×C32#93714yes—
293C2×C32C3×#12011yescharacteristic
294C2×C32C2×C3×Q1611yescharacteristic
295C2×C32C3×#13711yescharacteristic
296C3⋊D3—22no—
297C3×D3—68no—
298C2×Q8—91no—
299C2×Q8—181no—
300C2×Q8—182no—
301C2×Q8—182no—
302C2×Q8C2×C3×C3⋊D311yescharacteristic
303C2×Q8C2×C3×C3⋊D311yescharacteristic
304C22×C4—91no—
305C22×C4—181no—
306Q16—22no—
307Q16—182no—
308C2×C8—91no—
309C2×C8C2×C3×C3⋊D311yescharacteristic
310C4⋊C4—91no—
311C4⋊C4—181no—
312C22⋊C4—181no—
313C22×C3—24no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3C3×D3×D414yes—
318C22×C3D4×C3⋊D311yescharacteristic
319C2×D3—38no—
320C2×D3—64no—
321C2×D3—64no—
322C3×C4—28no—
323C3×C4—22no—
324C3×C4—24no—
325C3×C4—24no—
326C3×C4—48no—
327C3×C4—48no—
328C3×C4—44no—
329C3×C4—41no—
330C3×C4—181no—
331C3×C4—182no—
332C3×C4—361no—
333C3×C4C3×D3×D414yes—
334C3×C4C2×C3×#6714yes—
335C3×C4D4×C3⋊D311yescharacteristic
336C3×C4C2×#63811yescharacteristic
337Dic3—68no—
338Dic3—64no—
339Dic3—124no—
340C32—24no—
341C32#518614yescomplemented
342C32C3×#45111yescharacteristic, complemented
343C23—91no—
344Q8—21no—
345Q8—21no—
346Q8—21no—
347Q8—92no—
348Q8—184no—
349Q8—181no—
350Q8C3×#63812yes—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—91no—
354C2×C4—184no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4C22×C3×C3⋊D311yescharacteristic
360C8—21no—
361C8—181no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—182no—
366C2×C3C3×D3×Q1614yes—
367C2×C3C3×#91814yes—
368C2×C3C3×#93714yes—
369C2×C3Q16×C3⋊D311yescharacteristic
370C2×C3#6412711yescharacteristic
371C2×C3#6414611yescharacteristic
372D3—68no—
373C22—92no—
374C22—181no—
375C22—181no—
376C22C3×D4×C3⋊D311yescharacteristic
377C4—22no—
378C4—41no—
379C4—181no—
380C4—182no—
381C4—361no—
382C4C3×D4×C3⋊D311yescharacteristic
383C4C2×C3×#63811yescharacteristic
384C3—24no—
385C3C3×#518614yescomplemented
386C3#1059288511yescharacteristic, direct product factor
387C2—182no—
388C2C3×Q16×C3⋊D311yescharacteristic
389C2C3×#6412711yescharacteristic
390C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×Q16C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×Q16C3⋊D3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C33×Q16C2semidirect
4. C2×C3×#3756C2semidirect
5. C3×#63662C2semidirect
7. C3×#63643C2semidirect
8. #10592885C3direct (N × H)
28. C2×C32×Q16D3semidirect
29. C2×C32×Q16C2×C3semidirect
31. C2×#3756C2×C3semidirect
34. #63662C2×C3semidirect
36. #63643C2×C3semidirect
98. C2×C3×C3⋊D3Q16semidirect
110. C2×C3×Q16C3×D3semidirect
111. C2×C3×Q16C3⋊D3semidirect
211. C2×C3⋊D3C3×Q16semidirect
235. C2×Q16C3×C3⋊D3semidirect
240. C33#451semidirect
341. C32#5186semidirect
342. C32C3×#451semidirect
385. C3C3×#5186semidirect
386. C3#10592885direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C33×Q162
C3×#641552
C2×C3×#37562
C3×#636622
C3×#636472
C3×#636432
#105928853
C3×#51863