Finite Group Databasebeta

← C3×#10592776 #13163115

Subgroups — C3×#10592776 #13163115

Order 1728.

1440 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 174 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
1C3×#64361C211yesmaximal, characteristic, complemented
2C33×#127C211yesmaximal, characteristic, complemented
3C3×#64141C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5C3×#63632C212yesmaximal, complemented
6#10592776C311yesmaximal, characteristic, direct product factor
7C3×#5077—34nomaximal
8C2×C33×D4—22no—
9C2×C33×D4C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic, complemented
11C2×C3×#638—22no—
12C3×#3830—22no—
13C3×D4×C3⋊D3—22no—
14C3×#3828C2212yes—
15C33×C42C2211yescharacteristic, complemented
16C3×#3754—22no—
17C3×#3753—22no—
18C3×#3748C2212yescomplemented
19#64361C2×C311yescharacteristic, complemented
20C3×#1004—34no—
21C32×#127—24no—
22C32×#127D314yescomplemented
23C32×#127C2×C311yescharacteristic, complemented
24#64141C2×C312yescomplemented
25C3×#932—38no—
26#63660C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#833—34no—
29C3×#805—38no—
30C23×C33—21no—
31C23×C33D412yes—
32C22×C3×C3⋊D3—22no—
33C33×D4—22no—
34C33×D4—22no—
35C33×D4—44no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C3×#638—24no—
39C3×#638—22no—
40C2×C3×C3⋊Dic3—22no—
41C3×C4⋊(C3⋊D3)D412yescomplemented
42C3×C4×C3⋊D3—22no—
43C3×#634D412yescomplemented
44C3×C3⋊(C3⋊C8)—22no—
45C3×#453—91no—
46#5077—34no—
47C2×C32×D4—216no—
48C2×C32×D4—24no—
49C2×C32×D4—24no—
50C2×C32×D4—28no—
51C2×C32×D4—22no—
52C2×C32×D4C2×D314yes—
53C2×C32×D4C2×D314yescomplemented
54C2×C32×D4C22×C311yescharacteristic
55C2×C32×D4C22×C311yescharacteristic, complemented
56C2×#638—22no—
57#3830—22no—
58D4×C3⋊D3—22no—
59#3828C22×C312yes—
60C2×C3×#67—68no—
61C3×#237—68no—
62C3×D3×D4—68no—
63C3×#235—38no—
64C32×C42—24no—
65C32×C42C2×D314yescomplemented
66C32×C42C22×C311yescharacteristic, complemented
67#3754—22no—
68#3753—22no—
69#3748C22×C312yescomplemented
70C3×#214—68no—
71C3×#213—68no—
72C3×#208—38no—
73C22×C33—22no—
74C22×C33—42no—
75C22×C33—42no—
76C22×C33—41no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—82no—
80C33×C4—22no—
81C33×C4C2×D412yes—
82C3×C3⋊Dic3—22no—
83C3×H(5,𝔽2)—91no—
84#1004—34no—
85C3×AGL(1,ℤ/8)—92no—
86C3×#127—24no—
87C3×#127C3×D314yescomplemented
88C3×#127C3⋊D311yescharacteristic, complemented
89#932—38no—
90C3×#104—92no—
91C3×#100—91no—
92#833—34no—
93#805—38no—
94C23×C32—28no—
95C23×C32—24no—
96C23×C32—21no—
97C23×C32—28no—
98C23×C32#6718yes—
99C23×C32C3×D412yes—
100C22×C3⋊D3—22no—
101C22×C3×D3—68no—
102C32×D4—22no—
103C32×D4—22no—
104C32×D4—28no—
105C32×D4—28no—
106C32×D4—48no—
107C32×D4—48no—
108C32×D4—416no—
109C32×D4—44no—
110C32×D4—432no—
111C2×C32×C4—28no—
112C2×C32×C4—24no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C23×C311yescharacteristic
117#638—24no—
118#638—22no—
119C2×C3⋊Dic3—22no—
120C4⋊(C3⋊D3)C3×D412yescomplemented
121C4×C3⋊D3—22no—
122#634C3×D412yescomplemented
123C3×#67—616no—
124C3×#67—68no—
125C2×C3×Dic3—68no—
126C3×D12—38no—
127C3×C4×D3—68no—
128C3×Dic6—38no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—68no—
131#453—91noSylow
132C2×C33—21no—
133C2×C33—42no—
134C2×C33—81no—
135C2×C33#12011yescharacteristic
136C3×C3⋊D3—42no—
137C3×#41—92no—
138C3×#41—182no—
139C2×C3×D4—216no—
140C2×C3×D4—28no—
141C2×C3×D4—22no—
142C2×C3×D4—24no—
143C2×C3×D4—24no—
144C2×C3×D4—182no—
145C2×C3×D4—182no—
146C2×C3×D4C2×C3×D314yes—
147C2×C3×D4C2×C3×D314yescomplemented
148C2×C3×D4C2×C3⋊D311yescharacteristic
149C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
150C2×#67—68no—
151#237—68no—
152D3×D4—68no—
153#235—38no—
154C3×SD16—182no—
155C3×D8—182no—
156C3×M16—92no—
157C3×C42—24no—
158C3×C42C2×C3×D314yescomplemented
159C3×C42C2×C3⋊D311yescharacteristic, complemented
160#214—68no—
161#213—68no—
162#208—38no—
163C22×C32—22no—
164C22×C32—24no—
165C22×C32—28no—
166C22×C32—48no—
167C22×C32—48no—
168C22×C32—42no—
169C22×C32—416no—
170C22×C32—416no—
171C22×C32—42no—
172C22×C32—48no—
173C22×C32—48no—
174C22×C32—44no—
175C22×C32—41no—
176C22×C32C2×#6714yes—
177C22×C32C2×C3×D411yescharacteristic
178C2×C3⋊D3—22no—
179C2×C3⋊D3—82no—
180C2×C3×D3—68no—
181C2×C3×D3—248no—
182C32×C4—216no—
183C32×C4—22no—
184C32×C4—28no—
185C32×C4—28no—
186C32×C4D3×D418yes—
187C32×C4C2×C3×D412yes—
188C3⋊Dic3—22no—
189C3×Dic3—68no—
190H(5,𝔽2)—91no—
191AGL(1,ℤ/8)—92no—
192#127C3×C3⋊D311yescharacteristic, complemented
193#104—92no—
194#100—91no—
195C33#45311yescharacteristic, complemented, Sylow
196C23×C3—28no—
197C23×C3—24no—
198C23×C3—21no—
199C23×C3—28no—
200C23×C3—182no—
201C23×C3C3×#6718yes—
202C23×C3#63812yes—
203C22×D3—68no—
204C3×Q8—92no—
205C3×D4—28no—
206C3×D4—28no—
207C3×D4—22no—
208C3×D4—22no—
209C3×D4—48no—
210C3×D4—48no—
211C3×D4—416no—
212C3×D4—44no—
213C3×D4—432no—
214C3×D4—92no—
215C3×D4—184no—
216C3×D4—182no—
217C2×C3×C4—28no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—182no—
222C2×C3×C4—182no—
223C2×C3×C4C22×C3×D314yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225#67—616no—
226#67—68no—
227C2×Dic3—68no—
228D12—38no—
229C4×D3—68no—
230Dic6—38no—
231C3×C8—182no—
232C3⋊C8—68no—
233C2×C32—28no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—42no—
238C2×C32—48no—
239C2×C32—88no—
240C2×C32—88no—
241C2×C32—81no—
242C2×C32—84no—
243C2×C32#93714yes—
244C2×C32C3×#12011yescharacteristic
245C3⋊D3—42no—
246C3×D3—128no—
247#41—92no—
248#41—182no—
249C2×D4—22no—
250C2×D4—182no—
251C2×D4—182no—
252C2×D4C2×C3×C3⋊D311yescharacteristic
253C2×D4C2×C3×C3⋊D311yescharacteristic, complemented
254SD16—182no—
255D8—182no—
256M16—92no—
257C42C2×C3×C3⋊D311yescharacteristic, complemented
258C22×C3—28no—
259C22×C3—22no—
260C22×C3—24no—
261C22×C3—48no—
262C22×C3—48no—
263C22×C3—42no—
264C22×C3—416no—
265C22×C3—416no—
266C22×C3—42no—
267C22×C3—48no—
268C22×C3—48no—
269C22×C3—44no—
270C22×C3—41no—
271C22×C3—182no—
272C22×C3—722no—
273C22×C3C2×C3×#6714yes—
274C22×C3C2×#63811yescharacteristic
275C2×D3—68no—
276C2×D3—248no—
277C3×C4—216no—
278C3×C4—22no—
279C3×C4—28no—
280C3×C4—28no—
281C3×C4—182no—
282C3×C4C3×D3×D418yes—
283C3×C4D4×C3⋊D312yes—
284Dic3—68no—
285C32—24no—
286C32#507714yescomplemented
287C32C3×#45311yescharacteristic, complemented
288C23—21no—
289C23—182no—
290C23C3×#63812yes—
291Q8—92no—
292D4—22no—
293D4—22no—
294D4—44no—
295D4—92no—
296D4—184no—
297D4—182no—
298C2×C4—21no—
299C2×C4—182no—
300C2×C4—182no—
301C2×C4C22×C3×C3⋊D311yescharacteristic
302C8—182no—
303C2×C3—28no—
304C2×C3—21no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—42no—
308C2×C3—48no—
309C2×C3—88no—
310C2×C3—88no—
311C2×C3—81no—
312C2×C3—84no—
313C2×C3—362no—
314C2×C3C3×#93714yes—
315C2×C3#6414611yescharacteristic
316D3—128no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—182no—
322C22—722no—
323C22C2×C3×#63811yescharacteristic
324C4—22no—
325C4—182no—
326C4C3×D4×C3⋊D312yes—
327C3—24no—
328C3C3×#507714yescomplemented
329C3#1059277611yescharacteristic, direct product factor
330C2—21no—
331C2—42no—
332C2—81no—
333C2—362no—
334C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#127C2
socleC2×C33#120
hypercenterC3×#127C3⋊D3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#127

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. C33×#127C2semidirect
3. C3×#64141C2semidirect
4. C3×#63660C2semidirect
5. C3×#63632C2semidirect
6. #10592776C3direct (N × H)
10. C2×C33×D4C22semidirect
15. C33×C42C22semidirect
18. C3×#3748C22semidirect
19. #64361C2×C3semidirect
22. C32×#127D3semidirect
23. C32×#127C2×C3semidirect
24. #64141C2×C3semidirect
26. #63660C2×C3semidirect
27. #63632C2×C3semidirect
41. C3×C4⋊(C3⋊D3)D4semidirect
43. C3×#634D4semidirect
53. C2×C32×D4C2×D3semidirect
55. C2×C32×D4C22×C3semidirect
65. C32×C42C2×D3semidirect
66. C32×C42C22×C3semidirect
69. #3748C22×C3semidirect
87. C3×#127C3×D3semidirect
88. C3×#127C3⋊D3semidirect
120. C4⋊(C3⋊D3)C3×D4semidirect
122. #634C3×D4semidirect
147. C2×C3×D4C2×C3×D3semidirect
149. C2×C3×D4C2×C3⋊D3semidirect
158. C3×C42C2×C3×D3semidirect
159. C3×C42C2×C3⋊D3semidirect
192. #127C3×C3⋊D3semidirect
195. C33#453semidirect
253. C2×D4C2×C3×C3⋊D3semidirect
257. C42C2×C3×C3⋊D3semidirect
286. C32#5077semidirect
287. C32C3×#453semidirect
328. C3C3×#5077semidirect
329. C3#10592776direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C33×#1272
C3×#641412
C3×#636602
C3×#636322
#105927763
C3×#50773