Finite Group Databasebeta

← C3×#10592867 #13163208

Subgroups — C3×#10592867 #13163208

Order 1728.

1052 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1052 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×#3756C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5C2×C3×#3753C211yesmaximal, characteristic
6C2×C3×#3744C211yesmaximal, characteristic, complemented
7C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592867C311yesmaximal, characteristic, direct product factor
9C3×#5168—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×SD16—22no—
15C2×C33×C8C2211yescharacteristic
16C3×#3757—22no—
17C3×#3756—22no—
18C3×#3753—22no—
19C3×#3752—22no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C3×#3744—22no—
23C2×C32×SD16—24no—
24C2×C32×SD16D314yescomplemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27C2×#3756C2×C311yescharacteristic, complemented
28#64145C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic
30C2×#3744C2×C311yescharacteristic, complemented
31C3×#947—34no—
32C2×C3×#216—34no—
33C3×#936—34no—
34C2×C3×#213—34no—
35C2×C3×#204—34no—
36C8×C3⋊Dic3C2×C311yescharacteristic, complemented
37C3×C8×Dic3—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C33×D4—21no—
43C33×D4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D412yes—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634—21no—
52C33×C8D412yes—
53C3×C3⋊(C3⋊C8)D412yes—
54C3×#495—91no—
55#5168—34no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C22×C311yescharacteristic
59C2×C32×D4—24no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6—34no—
66C32×SD16—216no—
67C32×SD16—28no—
68C32×SD16—22no—
69C2×C32×C8—24no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C22×C311yescharacteristic
72#3757—22no—
73#3756—22no—
74#3753—22no—
75#3752—22no—
76C4×C3⋊Dic3C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3744—22no—
79C3×#217—68no—
80C3×#216—68no—
81C3×#213—68no—
82C3×(C2×D3)⋊C4—68no—
83C3×C4×Dic3—34no—
84C2×C3×C3⋊C8—34no—
85C3×#204—68no—
86C22×C33—41no—
87C22×C33—42no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—42no—
91C33×C4—41no—
92C33×C4C2×D411yescharacteristic
93C33×C4C2×D411yescharacteristic
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—24no—
98C2×C3×SD16—91no—
99C2×C3×SD16C3×D314yescomplemented
100C2×C3×SD16C3⋊D311yescharacteristic, complemented
101C2×C3×D8—91no—
102C3×#124—91no—
103C3×#124—91no—
104#947—34no—
105C2×#216—34no—
106#936—34no—
107C2×#213—34no—
108C2×#204—34no—
109C3×C4×C8—91no—
110C8×Dic3—34no—
111C23×C32—21no—
112C23×C32—24no—
113C23×C32—44no—
114C22×C3⋊D3—21no—
115C22×C3×D3—64no—
116C32×Q8—28no—
117C32×Q8—24no—
118C32×Q8—21no—
119C32×Q8—21no—
120C32×Q8—24no—
121C32×Q8—44no—
122C32×D4—28no—
123C32×D4—24no—
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—24no—
127C32×D4—44no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—44no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D412yes—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—38no—
141C2×C3×Dic3—64no—
142C3×D12—64no—
143C3×D12—64no—
144C3×Dic6—64no—
145C3×Dic6—64no—
146C32×C8—28no—
147C32×C8#6718yes—
148C32×C8C3×D412yes—
149C3⋊(C3⋊C8)C3×D412yes—
150C3×C3⋊C8—38no—
151#495—91noSylow
152C2×C33—81no—
153C2×C33#12711yescharacteristic
154C2×C33#13512yes—
155C3×C3⋊D3—81no—
156C2×C3×Q8—24no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×C3×D314yes—
159C2×C3×Q8C2×C3⋊D311yescharacteristic
160C2×C3×D4—24no—
161C2×C3×D4—91no—
162C2×C3×D4C2×C3×D314yes—
163C2×C3×D4C2×C3⋊D311yescharacteristic
164C2×D12—34no—
165C2×Dic6—34no—
166C3×Q16—182no—
167C3×SD16—216no—
168C3×SD16—28no—
169C3×SD16—22no—
170C3×SD16—182no—
171C3×D8—182no—
172C2×C3×C8—24no—
173C2×C3×C8—91no—
174C2×C3×C8C2×C3×D314yes—
175C2×C3×C8C2×C3⋊D311yescharacteristic
176C3×C22⋊C4—182no—
177C3×C22⋊C4—182no—
178C3×C42—91no—
179#217—68no—
180#216—68no—
181#213—68no—
182(C2×D3)⋊C4—68no—
183C4×Dic3—34no—
184C2×C3⋊C8—34no—
185#204—68no—
186C22×C32—24no—
187C22×C32—48no—
188C22×C32—44no—
189C22×C32—41no—
190C22×C32—42no—
191C22×C32—48no—
192C22×C32—88no—
193C22×C32D3×D414yes—
194C22×C32C2×C3×D411yescharacteristic
195C2×C3⋊D3—41no—
196C2×C3⋊D3—42no—
197C2×C3×D3—128no—
198C2×C3×D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4D3×D414yes—
205C32×C4C2×#6714yes—
206C32×C4C2×C3×D411yescharacteristic
207C32×C4C2×C3×D411yescharacteristic
208C3⋊Dic3—22no—
209C3⋊Dic3—41no—
210C3×Dic3—68no—
211C3×Dic3—124no—
212C2×Q16—91no—
213C2×SD16—91no—
214C2×SD16C3×C3⋊D311yescharacteristic, complemented
215C2×D8—91no—
216#124—91no—
217#124—91no—
218C4×C8—91no—
219C33#49511yescharacteristic, complemented, Sylow
220C23×C3—24no—
221C23×C3—21no—
222C23×C3—44no—
223C23×C3—181no—
224C22×D3—64no—
225C3×Q8—28no—
226C3×Q8—24no—
227C3×Q8—24no—
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—181no—
232C3×Q8—181no—
233C3×D4—28no—
234C3×D4—24no—
235C3×D4—24no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—44no—
239C3×D4—181no—
240C3×D4—181no—
241C2×C3×C4—24no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—44no—
245C2×C3×C4—92no—
246C2×C3×C4—181no—
247C2×C3×C4C22×C3×D314yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249C2×Dic3—38no—
250C2×Dic3—64no—
251D12—64no—
252D12—64no—
253Dic6—64no—
254Dic6—64no—
255C3×C8—28no—
256C3×C8—92no—
257C3×C8C3×#6718yes—
258C3×C8#63812yes—
259C3⋊C8—38no—
260C2×C32—28no—
261C2×C32—24no—
262C2×C32—88no—
263C2×C32—84no—
264C2×C32—81no—
265C2×C32#91618yes—
266C2×C32#94014yes—
267C2×C32C3×#12711yescharacteristic
268C2×C32C3×#13512yes—
269C3⋊D3—81no—
270C3×D3—244no—
271C2×Q8—91no—
272C2×Q8C2×C3×C3⋊D311yescharacteristic
273C2×D4—91no—
274C2×D4C2×C3×C3⋊D311yescharacteristic
275Q16—182no—
276SD16—22no—
277SD16—182no—
278D8—182no—
279C2×C8—91no—
280C2×C8C2×C3×C3⋊D311yescharacteristic
281C22⋊C4—182no—
282C22⋊C4—182no—
283C42—91no—
284C22×C3—24no—
285C22×C3—48no—
286C22×C3—48no—
287C22×C3—44no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—88no—
291C22×C3—362no—
292C22×C3—361no—
293C22×C3C3×D3×D414yes—
294C22×C3D4×C3⋊D311yescharacteristic
295C2×D3—128no—
296C2×D3—124no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—48no—
300C3×C4—44no—
301C3×C4—41no—
302C3×C4—182no—
303C3×C4—361no—
304C3×C4C3×D3×D414yes—
305C3×C4C2×C3×#6714yes—
306C3×C4D4×C3⋊D311yescharacteristic
307C3×C4C2×#63811yescharacteristic
308Dic3—68no—
309Dic3—124no—
310C32—24no—
311C32#516814yescomplemented
312C32C3×#49511yescharacteristic, complemented
313C23—21no—
314C23—181no—
315Q8—21no—
316Q8—21no—
317Q8—181no—
318Q8—181no—
319D4—21no—
320D4—21no—
321D4—181no—
322D4—181no—
323C2×C4—21no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4C22×C3×C3⋊D311yescharacteristic
327C8—92no—
328C8C3×#63812yes—
329C2×C3—28no—
330C2×C3—24no—
331C2×C3—88no—
332C2×C3—84no—
333C2×C3—81no—
334C2×C3—721no—
335C2×C3C3×#91618yes—
336C2×C3C3×#94014yes—
337C2×C3#6412512yes—
338C2×C3#6414911yescharacteristic
339D3—244no—
340C22—42no—
341C22—41no—
342C22—362no—
343C22—361no—
344C22C3×D4×C3⋊D311yescharacteristic
345C4—41no—
346C4—182no—
347C4—361no—
348C4C3×D4×C3⋊D311yescharacteristic
349C4C2×C3×#63811yescharacteristic
350C3—24no—
351C3C3×#516814yescomplemented
352C3#1059286711yescharacteristic, direct product factor
353C2—81no—
354C2—721no—
355C2C3×#6412512yes—
356C2C3×#6414911yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#495
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×#3756C2semidirect
4. C3×#64145C2semidirect
6. C2×C3×#3744C2semidirect
7. C3×C8×C3⋊Dic3C2semidirect
8. #10592867C3direct (N × H)
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. C2×#3756C2×C3semidirect
28. #64145C2×C3semidirect
30. C2×#3744C2×C3semidirect
36. C8×C3⋊Dic3C2×C3semidirect
99. C2×C3×SD16C3×D3semidirect
100. C2×C3×SD16C3⋊D3semidirect
214. C2×SD16C3×C3⋊D3semidirect
219. C33#495semidirect
311. C32#5168semidirect
312. C32C3×#495semidirect
351. C3C3×#5168semidirect
352. C3#10592867direct (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×#37562
C3×#641452
C2×C3×#37532
C2×C3×#37442
C3×C8×C3⋊Dic32
#105928673
C3×#51683