Finite Group Databasebeta

← C3×#10592869 #13163210

Subgroups — C3×#10592869 #13163210

Order 1728.

1220 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592869C311yesmaximal, characteristic, direct product factor
9C3×#5170—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×Q8×C3⋊D3—22no—
14C3×Q8×C3⋊D3—22no—
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C33×SD16—22no—
20C2×C33×C8C2211yescharacteristic
21C3×#3757—21no—
22C3×#3754—22no—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×Q8—34no—
27C2×C32×SD16—24no—
28C2×C32×SD16D314yescomplemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64147C2×C311yescharacteristic
31C2×#3754C2×C311yescharacteristic, complemented
32C3×#938—34no—
33C2×C3×#214—34no—
34#63662C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63643C2×C311yescharacteristic, complemented
37C3×#835—34no—
38C3×(C2×D3)⋊C8—34no—
39C3×#816—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yescomplemented
44C33×D4—21no—
45C33×D4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C3×#638—42no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—24no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#634—24no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#448—91no—
61#5170—34no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C22×C311yescharacteristic
65C2×C32×D4—24no—
66C2×C32×D4C2×D314yes—
67C2×C32×D4C22×C311yescharacteristic
68C2×#638—21no—
69Q8×C3⋊D3—22no—
70Q8×C3⋊D3—22no—
71C2×C4×C3⋊D3—21no—
72C2×C4×C3⋊D3C22×C311yescharacteristic
73C2×#634—21no—
74C2×#634C22×C311yescharacteristic
75C2×C3×#67—64no—
76C3×D3×Q8—68no—
77C3×D3×Q8—68no—
78C2×C3×C4×D3—34no—
79C2×C3×C4×D3—64no—
80C2×C3×Dic6—34no—
81C2×C3×Dic6—64no—
82C32×SD16—216no—
83C32×SD16—28no—
84C32×SD16—22no—
85C2×C32×C8—24no—
86C2×C32×C8C2×D314yes—
87C2×C32×C8C22×C311yescharacteristic
88#3757—21no—
89#3754—22no—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—64no—
93C3×#214—68no—
94C3×C4⋊Dic3—34no—
95C2×C3×C3⋊C8—34no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3SD1612yescomplemented
103C33×C4—22no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—22no—
108C3×C3⋊Dic3—41no—
109C22×C3×Q8—91no—
110C2×D3×Q8—34no—
111C2×C3×SD16—24no—
112C2×C3×SD16—91no—
113C2×C3×SD16C3×D314yescomplemented
114C2×C3×SD16C3⋊D311yescharacteristic, complemented
115C3×#121—91no—
116#938—34no—
117C2×#214—34no—
118C3×#103—91no—
119C3×#103—91no—
120C3×C22⋊C8—91no—
121#835—34no—
122(C2×D3)⋊C8—34no—
123#816—34no—
124C23×C32—21no—
125C23×C32—24no—
126C23×C32—44no—
127C22×C3⋊D3C3×D411yescharacteristic
128C22×C3×D3—34no—
129C32×Q8—21no—
130C32×Q8—28no—
131C32×Q8—24no—
132C32×Q8—44no—
133C32×Q8#6718yescomplemented
134C32×Q8C3×D412yescomplemented
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—24no—
140C32×D4—44no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—44no—
145C2×C32×C4C22×D314yes—
146C2×C32×C4C23×C311yescharacteristic
147#638—42no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3C3×D411yescharacteristic
151C4×C3⋊D3—24no—
152C4×C3⋊D3—21no—
153C4×C3⋊D3—21no—
154#634—24no—
155#634—21no—
156#634C3×D412yescomplemented
157C3×#67—128no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3—64no—
161C3×C4×D3—616no—
162C3×C4×D3—64no—
163C3×C4×D3—64no—
164C3×Dic6—38no—
165C3×Dic6—616no—
166C3×Dic6—64no—
167C32×C8—28no—
168C32×C8—24no—
169C32×C8—21no—
170C3⋊(C3⋊C8)—21no—
171C3×C3⋊C8—64no—
172#448—91noSylow
173C2×C33—81no—
174C2×C33#12011yescharacteristic
175C2×C33C2×SD1611yescharacteristic
176C2×C33#13711yescharacteristic
177C3×C3⋊D3—22no—
178C2×C3×Q8—24no—
179C2×C3×Q8—91no—
180C2×C3×Q8—181no—
181C2×C3×Q8—182no—
182C2×C3×Q8—182no—
183C2×C3×Q8C2×C3×D314yes—
184C2×C3×Q8C2×C3⋊D311yescharacteristic
185C2×C3×D4—24no—
186C2×C3×D4—181no—
187C2×C3×D4C2×C3×D314yes—
188C2×C3×D4C2×C3⋊D311yescharacteristic
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191C2×#67—64no—
192D3×Q8—68no—
193D3×Q8—68no—
194C2×C4×D3—34no—
195C2×C4×D3—64no—
196C2×Dic6—34no—
197C2×Dic6—64no—
198C3×SD16—216no—
199C3×SD16—28no—
200C3×SD16—22no—
201C3×SD16—182no—
202C2×C3×C8—24no—
203C2×C3×C8—91no—
204C2×C3×C8C2×C3×D314yes—
205C2×C3×C8C2×C3⋊D311yescharacteristic
206C3×C4⋊C4—91no—
207C3×C22⋊C4—181no—
208#217—64no—
209#214—68no—
210C4⋊Dic3—34no—
211C2×C3⋊C8—34no—
212C22×C32—24no—
213C22×C32—48no—
214C22×C32—44no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—44no—
220C22×C32—84no—
221C22×C32—84no—
222C22×C32D3×D414yes—
223C22×C32C2×C3×D411yescharacteristic
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3C3×SD1612yescomplemented
227C2×C3×D3—38no—
228C2×C3×D3—64no—
229C2×C3×D3—64no—
230C32×C4—22no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—28no—
234C32×C4—48no—
235C32×C4D3×D414yes—
236C32×C4C2×#6714yes—
237C32×C4C2×C3×D411yescharacteristic
238C32×C4C2×C3×D411yescharacteristic
239C3⋊Dic3—21no—
240C3⋊Dic3—22no—
241C3⋊Dic3—41no—
242C3×Dic3—68no—
243C3×Dic3—64no—
244C3×Dic3—124no—
245C22×Q8—91no—
246C2×SD16—91no—
247C2×SD16C3×C3⋊D311yescharacteristic, complemented
248#121—91no—
249#103—91no—
250#103—91no—
251C22⋊C8—91no—
252C33#44811yescharacteristic, complemented, Sylow
253C23×C3—24no—
254C23×C3—21no—
255C23×C3—44no—
256C23×C3—91no—
257C22×D3—34no—
258C3×Q8—24no—
259C3×Q8—21no—
260C3×Q8—28no—
261C3×Q8—44no—
262C3×Q8—92no—
263C3×Q8—184no—
264C3×Q8—181no—
265C3×Q8C3×#6718yescomplemented
266C3×Q8#63812yescomplemented
267C3×D4—28no—
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—44no—
273C3×D4—362no—
274C2×C3×C4—24no—
275C2×C3×C4—21no—
276C2×C3×C4—24no—
277C2×C3×C4—44no—
278C2×C3×C4—91no—
279C2×C3×C4—184no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C22×C3×D314yes—
285C2×C3×C4C22×C3⋊D311yescharacteristic
286#67—128no—
287C2×Dic3—34no—
288C2×Dic3—64no—
289C2×Dic3—64no—
290C4×D3—616no—
291C4×D3—64no—
292C4×D3—64no—
293Dic6—38no—
294Dic6—616no—
295Dic6—64no—
296C3×C8—28no—
297C3×C8—24no—
298C3×C8—21no—
299C3×C8—181no—
300C3⋊C8—64no—
301C2×C32—24no—
302C2×C32—24no—
303C2×C32—24no—
304C2×C32—88no—
305C2×C32—84no—
306C2×C32—81no—
307C2×C32D3×SD1614yes—
308C2×C32#91514yes—
309C2×C32#93714yes—
310C2×C32C3×#12011yescharacteristic
311C2×C32C2×C3×SD1611yescharacteristic
312C2×C32C3×#13711yescharacteristic
313C3⋊D3—22no—
314C3×D3—68no—
315C2×Q8—91no—
316C2×Q8—181no—
317C2×Q8—182no—
318C2×Q8—182no—
319C2×Q8C2×C3×C3⋊D311yescharacteristic
320C2×D4—181no—
321C2×D4C2×C3×C3⋊D311yescharacteristic
322C22×C4—91no—
323C22×C4—181no—
324SD16—22no—
325SD16—182no—
326C2×C8—91no—
327C2×C8C2×C3×C3⋊D311yescharacteristic
328C4⋊C4—91no—
329C22⋊C4—181no—
330C22×C3—24no—
331C22×C3—48no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—44no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—84no—
339C22×C3—84no—
340C22×C3—92no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3C3×D3×D414yes—
344C22×C3D4×C3⋊D311yescharacteristic
345C2×D3—38no—
346C2×D3—64no—
347C2×D3—64no—
348C3×C4—28no—
349C3×C4—22no—
350C3×C4—24no—
351C3×C4—24no—
352C3×C4—48no—
353C3×C4—181no—
354C3×C4—182no—
355C3×C4—361no—
356C3×C4C3×D3×D414yes—
357C3×C4C2×C3×#6714yes—
358C3×C4D4×C3⋊D311yescharacteristic
359C3×C4C2×#63811yescharacteristic
360Dic3—68no—
361Dic3—64no—
362Dic3—124no—
363C32—24no—
364C32#517014yescomplemented
365C32C3×#44811yescharacteristic, complemented
366C23—21no—
367C23—91no—
368Q8—21no—
369Q8—92no—
370Q8—184no—
371Q8—181no—
372Q8C3×#63812yescomplemented
373D4—21no—
374D4—21no—
375D4—362no—
376C2×C4—21no—
377C2×C4—91no—
378C2×C4—184no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4C22×C3×C3⋊D311yescharacteristic
384C8—21no—
385C8—181no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—24no—
389C2×C3—88no—
390C2×C3—84no—
391C2×C3—81no—
392C2×C3—182no—
393C2×C3C3×D3×SD1614yes—
394C2×C3C3×#91514yes—
395C2×C3C3×#93714yes—
396C2×C3SD16×C3⋊D311yescharacteristic
397C2×C3#6412411yescharacteristic
398C2×C3#6414611yescharacteristic
399D3—68no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22C3×D4×C3⋊D311yescharacteristic
407C4—22no—
408C4—181no—
409C4—182no—
410C4—361no—
411C4C3×D4×C3⋊D311yescharacteristic
412C4C2×C3×#63811yescharacteristic
413C3—24no—
414C3C3×#517014yescomplemented
415C3#1059286911yescharacteristic, direct product factor
416C2—81no—
417C2—182no—
418C2C3×SD16×C3⋊D311yescharacteristic
419C2C3×#6412411yescharacteristic
420C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#448
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×C3×Q8×C3⋊D3C2semidirect
2. C2×C33×SD16C2semidirect
4. C2×C3×#3754C2semidirect
5. C3×#63662C2semidirect
6. C3×#63647C2semidirect
7. C3×#63643C2semidirect
8. #10592869C3direct (N × H)
25. C2×Q8×C3⋊D3C2×C3semidirect
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
31. C2×#3754C2×C3semidirect
34. #63662C2×C3semidirect
35. #63647C2×C3semidirect
36. #63643C2×C3semidirect
43. C33×Q8D4semidirect
57. C3×#634D4semidirect
102. C2×C3×C3⋊D3SD16semidirect
113. C2×C3×SD16C3×D3semidirect
114. C2×C3×SD16C3⋊D3semidirect
133. C32×Q8#67semidirect
134. C32×Q8C3×D4semidirect
156. #634C3×D4semidirect
226. C2×C3⋊D3C3×SD16semidirect
247. C2×SD16C3×C3⋊D3semidirect
252. C33#448semidirect
265. C3×Q8C3×#67semidirect
266. C3×Q8#638semidirect
364. C32#5170semidirect
365. C32C3×#448semidirect
372. Q8C3×#638semidirect
414. C3C3×#5170semidirect
415. C3#10592869direct (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×SD162
C3×#641472
C2×C3×#37542
C3×#636622
C3×#636472
C3×#636432
#105928693
C3×#51703