Finite Group Databasebeta

← C3×#10592900 #13163239

Subgroups — C3×#10592900 #13163239

Order 1728.

1148 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2C33×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3C3×#64145C211yesmaximal, characteristic, complemented
4C3×#64141C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63651C211yesmaximal, characteristic, complemented
8#10592900C311yesmaximal, characteristic, direct product factor
9C3×#5201—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C3×Q8×C3⋊D3—21no—
13C3×#3830—21no—
14C3×#3830—22no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C33×SD16—21no—
20C33×D8—21no—
21C33×M16C2211yescharacteristic
22C3×#3757—22no—
23C3×#3754—21no—
24C3×#3753—21no—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×#3748C2211yescharacteristic
27#64367C2×C311yescharacteristic, complemented
28C3×#1010—34no—
29C32×AGL(1,ℤ/8)—24no—
30C32×AGL(1,ℤ/8)D314yescomplemented
31C32×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
32#64145C2×C311yescharacteristic, complemented
33#64141C2×C311yescharacteristic, complemented
34C3×#936—34no—
35C3×#932—34no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39C3×#837—34no—
40C3×#825—34no—
41C3×#824—34no—
42C23×C33—21no—
43C33×Q8D411yescharacteristic
44C33×D4—21no—
45C33×D4—21no—
46C33×D4—21no—
47C33×D4D411yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—22no—
51C3×#638—21no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D412yes—
55C3×C4⋊(C3⋊D3)D411yescharacteristic
56C3×C4×C3⋊D3—22no—
57C3×C4×C3⋊D3—21no—
58C3×#634—22no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#454—91no—
65#5201—34no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C22×C311yescharacteristic
69C2×C32×D4—24no—
70C2×C32×D4C2×D314yes—
71C2×C32×D4C22×C311yescharacteristic
72Q8×C3⋊D3—21no—
73#3830—21no—
74#3830—22no—
75#3828—21no—
76#3828C22×C311yescharacteristic
77C2×#634—21no—
78C2×#634C22×C311yescharacteristic
79C3×D3×Q8—64no—
80C3×#237—68no—
81C3×#237—64no—
82C3×#235—34no—
83C3×#235—64no—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—64no—
86C32×SD16—28no—
87C32×SD16—24no—
88C32×SD16—21no—
89C32×D8—28no—
90C32×D8—24no—
91C32×D8—21no—
92C32×M16—24no—
93C32×M16C2×D314yes—
94C32×M16C22×C311yescharacteristic
95#3757—22no—
96#3754—21no—
97#3753—21no—
98C4×C3⋊Dic3C22×C311yescharacteristic
99#3748C22×C311yescharacteristic
100C3×#217—68no—
101C3×#214—64no—
102C3×#213—64no—
103C3×C4×Dic3—34no—
104C3×#208—34no—
105C22×C33—21no—
106C22×C33—42no—
107C22×C33—41no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—21no—
110C33×C4—21no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C3×2−1+4—91no—
117#1010—34no—
118C3×AGL(1,ℤ/8)—24no—
119C3×AGL(1,ℤ/8)—91no—
120C3×AGL(1,ℤ/8)C3×D314yescomplemented
121C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic, complemented
122C3×#124—91no—
123#936—34no—
124#932—34no—
125C3×#104—91no—
126C3×#104—91no—
127C3×#101—91no—
128#837—34no—
129#825—34no—
130#824—34no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32—44no—
134C32×Q8—24no—
135C32×Q8#6714yes—
136C32×Q8C3×D411yescharacteristic
137C32×D4—21no—
138C32×D4—28no—
139C32×D4—24no—
140C32×D4—21no—
141C32×D4—21no—
142C32×D4—24no—
143C32×D4—24no—
144C32×D4—24no—
145C32×D4—44no—
146C32×D4—44no—
147C32×D4#6714yes—
148C32×D4C3×D411yescharacteristic
149C2×C32×C4—21no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—44no—
153C2×C32×C4C22×D314yes—
154C2×C32×C4C23×C311yescharacteristic
155#638—22no—
156#638—21no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3C3×D412yes—
160C4⋊(C3⋊D3)C3×D411yescharacteristic
161C4×C3⋊D3—22no—
162C4×C3⋊D3—21no—
163#634—22no—
164#634—21no—
165#634—21no—
166#634C3×D411yescharacteristic
167C3×#67—68no—
168C3×#67—64no—
169C2×C3×Dic3—38no—
170C2×C3×Dic3—68no—
171C2×C3×Dic3—64no—
172C3×D12—34no—
173C3×C4×D3—68no—
174C3×C4×D3—64no—
175C3×Dic6—34no—
176C3×Dic6—68no—
177C3×Dic6—64no—
178C3×Dic6—64no—
179C32×C8—28no—
180C32×C8—24no—
181C32×C8—21no—
182C3⋊(C3⋊C8)—21no—
183C3×C3⋊C8—64no—
184#454—91noSylow
185C2×C33—21no—
186C2×C33—41no—
187C2×C33—81no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—41no—
190C3×#41—24no—
191C3×#41—91no—
192C3×#41—181no—
193C3×#41—182no—
194C3×#41—181no—
195C3×#41C2×C3×D314yes—
196C3×#41C2×C3⋊D311yescharacteristic
197C2×C3×Q8—91no—
198C2×C3×Q8—181no—
199C2×C3×Q8—181no—
200C2×C3×D4—24no—
201C2×C3×D4C2×C3×D314yes—
202C2×C3×D4C2×C3⋊D311yescharacteristic
203D3×Q8—64no—
204#237—68no—
205#237—64no—
206#235—34no—
207#235—64no—
208C2×Dic6—34no—
209C2×Dic6—64no—
210C3×SD16—28no—
211C3×SD16—24no—
212C3×SD16—21no—
213C3×SD16—181no—
214C3×D8—28no—
215C3×D8—24no—
216C3×D8—21no—
217C3×D8—181no—
218C3×M16—24no—
219C3×M16—91no—
220C3×M16C2×C3×D314yes—
221C3×M16C2×C3⋊D311yescharacteristic
222C3×C22⋊C4—182no—
223C3×C42—91no—
224#217—68no—
225#214—64no—
226#213—64no—
227C4×Dic3—34no—
228#208—34no—
229C22×C32—21no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—44no—
233C22×C32—48no—
234C22×C32—42no—
235C22×C32—44no—
236C22×C32—41no—
237C22×C32—48no—
238C22×C32—88no—
239C22×C32D3×D414yes—
240C22×C32C2×C3×D411yescharacteristic
241C2×C3⋊D3—21no—
242C2×C3×D3—64no—
243C32×C4—21no—
244C32×C4—24no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—44no—
248C32×C4D3×D414yes—
249C32×C4C2×#6714yes—
250C32×C4C2×C3×D411yescharacteristic
251C32×C4C2×C3×D411yescharacteristic
252C3⋊Dic3—22no—
253C3⋊Dic3—21no—
254C3⋊Dic3—41no—
255C3×Dic3—64no—
256C3×Dic3—68no—
257C3×Dic3—124no—
2582−1+4—91no—
259AGL(1,ℤ/8)—91no—
260AGL(1,ℤ/8)C3×C3⋊D311yescharacteristic, complemented
261#124—91no—
262#104—91no—
263#104—91no—
264#101—91no—
265C33#45411yescharacteristic, complemented, Sylow
266C23×C3—24no—
267C23×C3—21no—
268C23×C3—44no—
269C3×Q8—24no—
270C3×Q8—91no—
271C3×Q8—182no—
272C3×Q8—181no—
273C3×Q8—181no—
274C3×Q8C3×#6714yes—
275C3×Q8#63811yescharacteristic
276C3×D4—24no—
277C3×D4—21no—
278C3×D4—28no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—21no—
282C3×D4—21no—
283C3×D4—24no—
284C3×D4—44no—
285C3×D4—44no—
286C3×D4—91no—
287C3×D4—182no—
288C3×D4—181no—
289C3×D4C3×#6714yes—
290C3×D4#63811yescharacteristic
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—24no—
294C2×C3×C4—44no—
295C2×C3×C4—92no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×C3×C4—181no—
299C2×C3×C4—181no—
300C2×C3×C4C22×C3×D314yes—
301C2×C3×C4C22×C3⋊D311yescharacteristic
302#67—68no—
303#67—64no—
304C2×Dic3—38no—
305C2×Dic3—68no—
306C2×Dic3—64no—
307D12—34no—
308C4×D3—68no—
309C4×D3—64no—
310Dic6—34no—
311Dic6—68no—
312Dic6—64no—
313Dic6—64no—
314C3×C8—28no—
315C3×C8—24no—
316C3×C8—21no—
317C3×C8—181no—
318C3⋊C8—64no—
319C2×C32—21no—
320C2×C32—24no—
321C2×C32—24no—
322C2×C32—44no—
323C2×C32—41no—
324C2×C32—44no—
325C2×C32—84no—
326C2×C32—88no—
327C2×C32—84no—
328C2×C32—81no—
329C2×C32#93714yes—
330C2×C32C3×#12011yescharacteristic
331C3⋊D3—41no—
332C3×D3—124no—
333#41—91no—
334#41—181no—
335#41—182no—
336#41—181no—
337#41C2×C3×C3⋊D311yescharacteristic
338C2×Q8—91no—
339C2×Q8—181no—
340C2×Q8—181no—
341C2×D4C2×C3×C3⋊D311yescharacteristic
342SD16—21no—
343SD16—181no—
344D8—21no—
345D8—181no—
346M16—91no—
347M16C2×C3×C3⋊D311yescharacteristic
348C22⋊C4—182no—
349C42—91no—
350C22×C3—24no—
351C22×C3—21no—
352C22×C3—24no—
353C22×C3—44no—
354C22×C3—48no—
355C22×C3—42no—
356C22×C3—48no—
357C22×C3—44no—
358C22×C3—41no—
359C22×C3—88no—
360C22×C3—181no—
361C22×C3C3×D3×D414yes—
362C22×C3D4×C3⋊D311yescharacteristic
363C2×D3—64no—
364C3×C4—24no—
365C3×C4—21no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—44no—
369C3×C4—182no—
370C3×C4—181no—
371C3×C4—361no—
372C3×C4C3×D3×D414yes—
373C3×C4C2×C3×#6714yes—
374C3×C4D4×C3⋊D311yescharacteristic
375C3×C4C2×#63811yescharacteristic
376Dic3—64no—
377Dic3—68no—
378Dic3—124no—
379C32—24no—
380C32#520114yescomplemented
381C32C3×#45411yescharacteristic, complemented
382C23—21no—
383Q8—91no—
384Q8—182no—
385Q8—181no—
386Q8—181no—
387Q8C3×#63811yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—91no—
392D4—182no—
393D4—181no—
394D4C3×#63811yescharacteristic
395C2×C4—21no—
396C2×C4—92no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4—181no—
400C2×C4—181no—
401C2×C4C22×C3×C3⋊D311yescharacteristic
402C8—21no—
403C8—181no—
404C2×C3—24no—
405C2×C3—21no—
406C2×C3—24no—
407C2×C3—44no—
408C2×C3—44no—
409C2×C3—41no—
410C2×C3—84no—
411C2×C3—88no—
412C2×C3—84no—
413C2×C3—81no—
414C2×C3—361no—
415C2×C3C3×#93714yes—
416C2×C3#6414611yescharacteristic
417D3—124no—
418C22—21no—
419C22—42no—
420C22—41no—
421C22—181no—
422C22C3×D4×C3⋊D311yescharacteristic
423C4—21no—
424C4—182no—
425C4—181no—
426C4—361no—
427C4C3×D4×C3⋊D311yescharacteristic
428C4C2×C3×#63811yescharacteristic
429C3—24no—
430C3C3×#520114yescomplemented
431C3#1059290011yescharacteristic, direct product factor
432C2—21no—
433C2—41no—
434C2—81no—
435C2—361no—
436C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33#120
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. C33×AGL(1,ℤ/8)C2semidirect
3. C3×#64145C2semidirect
4. C3×#64141C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63651C2semidirect
8. #10592900C3direct (N × H)
27. #64367C2×C3semidirect
30. C32×AGL(1,ℤ/8)D3semidirect
31. C32×AGL(1,ℤ/8)C2×C3semidirect
32. #64145C2×C3semidirect
33. #64141C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63651C2×C3semidirect
120. C3×AGL(1,ℤ/8)C3×D3semidirect
121. C3×AGL(1,ℤ/8)C3⋊D3semidirect
260. AGL(1,ℤ/8)C3×C3⋊D3semidirect
265. C33#454semidirect
380. C32#5201semidirect
381. C32C3×#454semidirect
430. C3C3×#5201semidirect
431. C3#10592900direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643672
C33×AGL(1,ℤ/8)2
C3×#641452
C3×#641412
C3×#636642
C3×#636522
C3×#636512
#105929003
C3×#52013