Finite Group Databasebeta

← C3×#10592875 #13163205

Subgroups — C3×#10592875 #13163205

Order 1728.

1136 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×SD16C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4C3×#64149C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic, complemented
6C2×C3×#3745C211yesmaximal, characteristic, complemented
7C3×#63642C211yesmaximal, characteristic, complemented
8#10592875C311yesmaximal, characteristic, direct product factor
9C3×#5176—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—22no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C33×SD16—22no—
16C2×C33×C8C2211yescharacteristic
17C3×#3755—22no—
18C3×#3753—22no—
19C3×#3752—22no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3745—22no—
23C2×C32×SD16—24no—
24C2×C32×SD16D314yescomplemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28#64149C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic, complemented
30C2×#3745C2×C311yescharacteristic, complemented
31C3×#947—34no—
32C2×C3×#215—34no—
33C3×#940—34no—
34C2×C3×#213—34no—
35C2×C3×D24—34no—
36#63642C2×C311yescharacteristic, complemented
37C3×#815—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C33×Q8—21no—
42C33×Q8—21no—
43C33×D4—21no—
44C33×D4—21no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C3×#638—44no—
48C2×C3×C3⋊Dic3D412yes—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C33×C8D412yescomplemented
54C3×C3⋊(C3⋊C8)D412yescomplemented
55C3×#496—91no—
56#5176—34no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yes—
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—24no—
61C2×C32×D4C2×D314yes—
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—22no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×#67—68no—
67C2×C3×D12—34no—
68C2×C3×D12—34no—
69C32×SD16—216no—
70C32×SD16—28no—
71C32×SD16—22no—
72C2×C32×C8—24no—
73C2×C32×C8C2×D314yes—
74C2×C32×C8C22×C311yescharacteristic
75#3755—22no—
76#3753—22no—
77#3752—22no—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80#3745—22no—
81C3×#215—68no—
82C3×#213—68no—
83C3×(C2×D3)⋊C4—68no—
84C3×C4×Dic3—34no—
85C2×C3×C3⋊C8—34no—
86C3×D24—68no—
87C22×C33—41no—
88C22×C33—42no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—42no—
93C2×C3×C3⋊D3—42no—
94C33×C4—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—22no—
98C2×C3×SD16—24no—
99C2×C3×SD16—91no—
100C2×C3×SD16C3×D314yescomplemented
101C2×C3×SD16C3⋊D311yescharacteristic, complemented
102C2×C3×D8—91no—
103C2×C3×D8—91no—
104C3×#127—91no—
105C3×#124—91no—
106#947—34no—
107C2×#215—34no—
108#940—34no—
109C2×#213—34no—
110C2×D24—34no—
111C3×#97—91no—
112#815—34no—
113C23×C32—21no—
114C23×C32—24no—
115C23×C32—44no—
116C22×C3⋊D3—21no—
117C22×C3⋊D3—21no—
118C22×C3×D3—64no—
119C22×C3×D3—64no—
120C32×Q8—28no—
121C32×Q8—24no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—24no—
125C32×Q8—44no—
126C32×D4—28no—
127C32×D4—24no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—24no—
131C32×D4—44no—
132C2×C32×C4—21no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—44no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C23×C311yescharacteristic
138#638—44no—
139C2×C3⋊Dic3C3×D412yes—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C3×#67—1216no—
145C2×C3×Dic3—38no—
146C3×D12—64no—
147C3×D12—64no—
148C3×D12—64no—
149C3×D12—64no—
150C32×C8—28no—
151C32×C8#6718yescomplemented
152C32×C8C3×D412yescomplemented
153C3⋊(C3⋊C8)C3×D412yescomplemented
154C3×C3⋊C8—38no—
155#496—91noSylow
156C2×C33—81no—
157C2×C33#12711yescharacteristic
158C2×C33AGL(1,ℤ/8)12yes—
159C3×C3⋊D3—81no—
160C3×C3⋊D3—81no—
161C2×C3×Q8—24no—
162C2×C3×Q8C2×C3×D314yes—
163C2×C3×Q8C2×C3⋊D311yescharacteristic
164C2×C3×D4—24no—
165C2×C3×D4—91no—
166C2×C3×D4—91no—
167C2×C3×D4—182no—
168C2×C3×D4C2×C3×D314yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C2×#67—68no—
171C2×D12—34no—
172C2×D12—34no—
173C3×SD16—216no—
174C3×SD16—28no—
175C3×SD16—22no—
176C3×SD16—182no—
177C3×D8—182no—
178C3×D8—182no—
179C2×C3×C8—24no—
180C2×C3×C8—91no—
181C2×C3×C8C2×C3×D314yes—
182C2×C3×C8C2×C3⋊D311yescharacteristic
183C3×C22⋊C4—182no—
184C3×C42—91no—
185#215—68no—
186#213—68no—
187(C2×D3)⋊C4—68no—
188C4×Dic3—34no—
189C2×C3⋊C8—34no—
190D24—68no—
191C22×C32—24no—
192C22×C32—48no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32—42no—
196C22×C32—48no—
197C22×C32—88no—
198C22×C32D3×D414yes—
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3—42no—
204C2×C3×D3—128no—
205C2×C3×D3—128no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4D3×D414yes—
214C32×C4C2×#6714yes—
215C32×C4C2×C3×D411yescharacteristic
216C32×C4C2×C3×D411yescharacteristic
217C3⋊Dic3—22no—
218C3×Dic3—68no—
219C2×SD16—91no—
220C2×SD16C3×C3⋊D311yescharacteristic, complemented
221C2×D8—91no—
222C2×D8—91no—
223#127—91no—
224#124—91no—
225#97—91no—
226C33#49611yescharacteristic, complemented, Sylow
227C23×C3—24no—
228C23×C3—21no—
229C23×C3—44no—
230C23×C3—181no—
231C23×C3—181no—
232C22×D3—64no—
233C22×D3—64no—
234C3×Q8—28no—
235C3×Q8—24no—
236C3×Q8—24no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—44no—
240C3×D4—28no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—44no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4—181no—
249C3×D4—181no—
250C3×D4—364no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—44no—
255C2×C3×C4—92no—
256C2×C3×C4C22×C3×D314yes—
257C2×C3×C4C22×C3⋊D311yescharacteristic
258#67—1216no—
259C2×Dic3—38no—
260D12—64no—
261D12—64no—
262D12—64no—
263D12—64no—
264C3×C8—28no—
265C3×C8—92no—
266C3×C8C3×#6718yescomplemented
267C3×C8#63812yescomplemented
268C3⋊C8—38no—
269C2×C32—28no—
270C2×C32—24no—
271C2×C32—88no—
272C2×C32—84no—
273C2×C32—81no—
274C2×C32#91418yes—
275C2×C32#94014yes—
276C2×C32C3×#12711yescharacteristic
277C2×C32C3×AGL(1,ℤ/8)12yes—
278C3⋊D3—81no—
279C3⋊D3—81no—
280C3×D3—244no—
281C3×D3—244no—
282C2×Q8C2×C3×C3⋊D311yescharacteristic
283C2×D4—91no—
284C2×D4—91no—
285C2×D4—182no—
286C2×D4C2×C3×C3⋊D311yescharacteristic
287SD16—22no—
288SD16—182no—
289D8—182no—
290D8—182no—
291C2×C8—91no—
292C2×C8C2×C3×C3⋊D311yescharacteristic
293C22⋊C4—182no—
294C42—91no—
295C22×C3—24no—
296C22×C3—48no—
297C22×C3—48no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—88no—
302C22×C3—362no—
303C22×C3—362no—
304C22×C3—361no—
305C22×C3—361no—
306C22×C3C3×D3×D414yes—
307C22×C3D4×C3⋊D311yescharacteristic
308C2×D3—128no—
309C2×D3—128no—
310C2×D3—124no—
311C2×D3—124no—
312C3×C4—24no—
313C3×C4—24no—
314C3×C4—48no—
315C3×C4—44no—
316C3×C4—41no—
317C3×C4—182no—
318C3×C4C3×D3×D414yes—
319C3×C4C2×C3×#6714yes—
320C3×C4D4×C3⋊D311yescharacteristic
321C3×C4C2×#63811yescharacteristic
322Dic3—68no—
323C32—24no—
324C32#517614yescomplemented
325C32C3×#49611yescharacteristic, complemented
326C23—21no—
327C23—181no—
328C23—181no—
329Q8—21no—
330Q8—21no—
331D4—21no—
332D4—21no—
333D4—181no—
334D4—181no—
335D4—181no—
336D4—181no—
337D4—364no—
338C2×C4—21no—
339C2×C4—92no—
340C2×C4C22×C3×C3⋊D311yescharacteristic
341C8—92no—
342C8C3×#63812yescomplemented
343C2×C3—28no—
344C2×C3—24no—
345C2×C3—88no—
346C2×C3—84no—
347C2×C3—81no—
348C2×C3—721no—
349C2×C3—721no—
350C2×C3C3×#91418yes—
351C2×C3C3×#94014yes—
352C2×C3#6412312yes—
353C2×C3#6414911yescharacteristic
354D3—244no—
355D3—244no—
356C22—42no—
357C22—41no—
358C22—362no—
359C22—362no—
360C22—361no—
361C22—361no—
362C22C3×D4×C3⋊D311yescharacteristic
363C4—41no—
364C4—182no—
365C4C3×D4×C3⋊D311yescharacteristic
366C4C2×C3×#63811yescharacteristic
367C3—24no—
368C3C3×#517614yescomplemented
369C3#1059287511yescharacteristic, direct product factor
370C2—81no—
371C2—721no—
372C2—721no—
373C2C3×#6412312yes—
374C2C3×#6414911yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#496
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×C33×SD16C2semidirect
2. C3×#64156C2semidirect
3. C2×C3×#3755C2semidirect
4. C3×#64149C2semidirect
5. C2×C3×#3753C2semidirect
6. C2×C3×#3745C2semidirect
7. C3×#63642C2semidirect
8. #10592875C3direct (N × H)
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. #64149C2×C3semidirect
29. C2×#3753C2×C3semidirect
30. C2×#3745C2×C3semidirect
36. #63642C2×C3semidirect
53. C33×C8D4semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
100. C2×C3×SD16C3×D3semidirect
101. C2×C3×SD16C3⋊D3semidirect
151. C32×C8#67semidirect
152. C32×C8C3×D4semidirect
153. C3⋊(C3⋊C8)C3×D4semidirect
220. C2×SD16C3×C3⋊D3semidirect
226. C33#496semidirect
266. C3×C8C3×#67semidirect
267. C3×C8#638semidirect
324. C32#5176semidirect
325. C32C3×#496semidirect
342. C8C3×#638semidirect
368. C3C3×#5176semidirect
369. C3#10592875direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×SD162
C3×#641562
C2×C3×#37552
C3×#641492
C2×C3×#37532
C2×C3×#37452
C3×#636422
#105928753
C3×#51763